ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Review Article
Volume 5 Issue 1 - 2017
Predictive Influence of Variables on the Odds Ratio and in the Logistic Model
S K Bhattacharjee1, Atanu Biswas2, Ganesh Dutta3, S Rao Jammalamadaka4* and M Masoom Ali5
1Indian Statistical Institute, North-East Centre, Tezpur, Assam-784028, India
2Indian Statistical Institute, India
3Basanti Devi College, India
4Department of Statistics and Applied Probability, University of California, USA
5Department of Mathematical Sciences, Ball State University, USA
Received: October 01, 2016 | Published: February 01, 2017
*Corresponding author: S Rao Jammalamadaka, Department of Statistics and Applied Probability, University of California, USA, Email:
Citation: Bhattacharjee SK, Biswas A, Dutta G, Jammalamadaka SR, Ali MM (2017) Predictive Influence of Variables on the Odds Ratio and in the Logistic Model. Biom Biostat Int J 5(1): 00125. DOI: 10.15406/bbij.2017.05.00125

Abstract

We study the influence of explanatory variables in prediction by looking at the distribution of the log-odds ratio. We also consider the predictive influence of a subset of unobserved future variables on the distribution of log-odds ratio as well as in a logistic model, via the Bayesian predictive density of a future observation. This problem is considered for dichotomous, as well as continuous explanatory variables.

AMS Subject Classification: Primary 62J12, Secondary 62B10, 62F15

Keywords: Predictive density/probability; Log-odds ratio; Logistic model; Predictive in u-ence; Missing/unobserved variable; Kullback-Leibler divergence

Introduction

Odds ratio (OR) is perhaps the most popular measure of treatment difference for binary outcomes and is extensively used in dealing with 2 2 tables in biomedical studies and clinical trials. The distribution of the log of sample OR is often approximated by a normal distribution with true log OR as the mean and with variance estimated by the sum of the reciprocal of the four cell frequencies in the 2 2 table Breslow [1]. Bohning et al. [2] provide detailed book-length discussion on the OR. For logistic regression, ORs enable one to examine the e ect of explanatory variables in that relationship.

Logistic link is perhaps the most popular way to model the success probabilities of a binary variable. Pregibon [3], Cook and Weisberg [4] and Johnson [5] have considered the problem of the influence of observations for logistic regression models. Several measures have been suggested to identify observations in the data set which are influential relative to the estimation of the vector of regression coefficients, the deviance, and the determination of predictive probabilities and the classification of future observations.

Bhattacharjee & Dunsmore [6] considered the effect on the predictive probability of a future observation of the omission of subsets of the explanatory variables. Mercier et al. [7] used logistic regression to determine whether age and/or gender were a factor influencing severity of injuries suffered in head-on automobile collisions on rural highways. Zellner et al. [8] considered the problem of variable selection in logistic regression to compare the performance of stepwise selection procedures with a bagging method.

In the present paper, our aim is to measure the predictive influence of a subset of explanatory variables in log-odds ratio of a logistic model using a Bayesian approach. We are also interested in studying the effect of missing future explanatory variables on Bayes prediction, on a logistic model as well as on the log-odds ratio.

In Section 2, we derive the predictive densities of a future log-odds ratio for both the full model and a subset deleted model. We derive the predictive density of log-odds ratio in Section 3, when a subset of future explanatory variables is missing. To derive the predictive densities we assume that the future explanatory variables xf are distributed as multivariate normal, both when these xf's are independent or dependent. In Section 4, we discuss the influence of future missing explanatory variables by considering the predictive probability of a future response in a logistic model. This is done by assuming that the future explanatory variables xf are multivariate normal for the continuous case. Also considered is the dichotomous case. Since the predictive probabilities are not mathematically tractable for the logistic model, we use several approximations.

In Section 2 and 3 we employ Kullback-Leibler [9] directed measure of divergence DKL to assess the influence of variables and also the influence of future missing variables on the log-odds ratio. The form of the Kullback-Leibler [9] measure used here is given by
D KL =f( a' W f |. )log( f( a' W f |. ) f ( r+s ) ( a'| W f |. ) )d( a' W f ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadseada WgaaqcfasaaiaadUeacaWGmbaajuaGbeaacqGH9aqpcqGHRiI8caWG MbWaaeWaaeaacaGGHbGaai4jaiaadEfadaahaaqabKqbGeaacaWGMb aaaKqbakaacYhacaGGUaaacaGLOaGaayzkaaGaciiBaiaac+gacaGG NbWaaeWaaeaadaWcaaqaaiaadAgadaqadaqaaiaacggacaGGNaGaam 4vamaaCaaabeqcfasaaiaadAgaaaqcfaOaaiiFaiaac6caaiaawIca caGLPaaaaeaacaWGMbWaaSbaaeaadaWgaaqcfasaaKqbaoaabmaaju aibaGaamOCaiabgUcaRiaadohaaiaawIcacaGLPaaaaKqbagqaamaa bmaabaGaamyyaiaacEcacaGG8bGaam4vamaaCaaabeqcfasaaiaadA gaaaqcfaOaaiiFaiaac6caaiaawIcacaGLPaaaaeqaaaaaaiaawIca caGLPaaacaWGKbWaaeWaaeaacaWGHbGaai4jaiaadEfadaahaaqabK qbGeaacaWGMbaaaaqcfaOaayjkaiaawMcaaiaac6caaaa@687A@

To assess the influence of missing future variables or to measure the predictive probability in a logistic model we use the absolute difference of the two predictive probabilities.

Influence of variables in Log-odds Ratio

Consider a phase III clinical trial with two competing treatments, say A and B, having binary responses. Suppose n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gaaa a@376C@ patients are randomly allocated with n A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada WgaaqcfasaaiaadgeaaKqbagqaaaaa@390F@ and n B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada WgaaqcfasaaiaadkeaaKqbagqaaaaa@3910@  patients to treatments A and B respectively. The patient responses are influenced by a covariate vector x p×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada ahaaqabKqbGeaacaWGWbGaey41aqRaaGymaaaaaaa@3B8D@ where one component of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhaaa a@3776@ may be 1 (which covers the constant term). Let ( Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMfada WgaaqcfasaaiaadMgaaKqbagqaaaaa@3922@ ; Z i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaadMgaaKqbagqaaaaa@3923@ ; x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadMgaaKqbagqaaaaa@3941@ ) be the data corresponding to its patient, where Yi is the indicator of response ( Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMfada WgaaqcfasaaiaadMgaaKqbagqaaaaa@3922@ =1 or 0 for a success or failure), Z i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaadMgaaKqbagqaaaaa@3923@  is the indicator of the treatment assignment ( Z i =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaadMgaaKqbagqaaiabg2da9iaaigdaaaa@3AE4@ or 0 according as treatment A or B is applied to the its patient), and x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhaaa a@3776@ is the covariate vector. We assume a logit model for the responses:

Pr( Y i =1| Z i , x i )= exp( Δ Z i + x i β ) 1+exp( Δ Z i + x i β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWGzbWaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH 9aqpcaaIXaGaaiiFaiaadQfadaWgaaqcfasaaiaadMgaaKqbagqaai aacYcacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIcacaGL PaaacqGH9aqpdaWcaaqaaiGacwgacaGG4bGaaiiCamaabmaabaGaey iLdqKaamOwamaaBaaajuaibaGaamyAaaqcfayabaGaey4kaSIaamiE amaaBaaajuaibaGaamyAaaqcfayabaGaeqOSdigacaGLOaGaayzkaa aabaGaaGymaiabgUcaRiGacwgacaGG4bGaaiiCamaabmaabaGaeyiL dqKaamOwamaaBaaajuaibaGaamyAaaqcfayabaGaey4kaSIaamiEam aaBaaajuaibaGaamyAaaqcfayabaGaeqOSdigacaGLOaGaayzkaaaa aaaa@638D@ i=1,2,....,n. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6cacaGGUaGa aiOlaiaacYcacaWGUbGaaiOlaaaa@4061@ A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaaa a@373F@ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkeaaa a@3740@    (i)

Then the odds for treatments A and B with covariate vector xi are respectively
O A = Pr( Y i =1| Z i =1, x i ) Pr( Y i =0| Z i =1, x i ) =exp( Δ+ x i β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eada WgaaqcfasaaiaadgeaaKqbagqaaiabg2da9maalaaabaGaciiuaiaa ckhadaqadaqaaiaadMfadaWgaaqcfasaaiaadMgaaKqbagqaaiabg2 da9iaaigdacaGG8bGaamOwamaaBaaajuaibaGaamyAaaqcfayabaGa eyypa0JaaGymaiaacYcacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbe aaaiaawIcacaGLPaaaaeaaciGGqbGaaiOCamaabmaabaGaamywamaa BaaajuaibaGaamyAaaqcfayabaGaeyypa0JaaGimaiaacYhacaWGAb WaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH9aqpcaaIXaGaaiilaiaa dIhadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaaaaacq GH9aqpciGGLbGaaiiEaiaacchadaqadaqaaiabgs5aejabgUcaRiaa dIhadaWgaaqcfasaaiaadMgaaKqbagqaaiabek7aIbGaayjkaiaawM caaaaa@6765@ , O B = Pr( Y i =1| Z i =0, x i ) Pr( Y i =0| Z i =0, x i ) =exp( x i β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eada WgaaqcfasaaiaadkeaaKqbagqaaiabg2da9maalaaabaGaciiuaiaa ckhadaqadaqaaiaadMfadaWgaaqcfasaaiaadMgaaKqbagqaaiabg2 da9iaaigdacaGG8bGaamOwamaaBaaajuaibaGaamyAaaqcfayabaGa eyypa0JaaGimaiaacYcacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbe aaaiaawIcacaGLPaaaaeaaciGGqbGaaiOCamaabmaabaGaamywamaa BaaajuaibaGaamyAaaqcfayabaGaeyypa0JaaGimaiaacYhacaWGAb WaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH9aqpcaaIWaGaaiilaiaa dIhadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaaaaacq GH9aqpciGGLbGaaiiEaiaacchadaqadaqaaiaadIhadaWgaaqcfasa aiaadMgaaKqbagqaaiabek7aIbGaayjkaiaawMcaaaaa@651B@
and hence the log-odds ratio is
logOR= log O A log O B =Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGacYgaca GGVbGaai4zaiaad+eacaWGsbGaeyypa0ZaaSaaaeaaciGGSbGaai4B aiaacEgacaWGpbWaaSbaaKqbGeaacaWGbbaajuaGbeaaaeaaciGGSb Gaai4BaiaacEgacaWGpbWaaSbaaKqbGeaacaWGcbaajuaGbeaaaaGa eyypa0JaeyiLdqeaaa@4906@

Let us partition
xβ= x A β A + x B β B + x AB β AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhacq aHYoGycqGH9aqpcaWG4bWaaSbaaKqbGeaacaWGbbaajuaGbeaacqaH YoGydaWgaaqcfasaaiaadgeaaKqbagqaaiabgUcaRiaadIhadaWgaa qcfasaaiaadkeaaKqbagqaaiabek7aInaaBaaajuaibaGaamOqaaqc fayabaGaey4kaSIaamiEamaaBaaajuaibaGaamyqaiaadkeaaKqbag qaaiabek7aInaaBaaajuaibaGaamyqaiaadkeaaKqbagqaaaaa@4F1D@

Where x A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadgeaaKqbagqaaaaa@3919@  indicates the variables used in treatment A only, x B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadkeaaKqbagqaaaaa@391A@  is for treatment B only, and x AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavababe qaaKqzadGaamyqaiaadkeaaKqbagqabaqcLbsacaWG4baaaaaa@3B7C@ is for both treatments A and B. Then the model can be partitioned for treatments A and B as:
log O A =u=Δ+ x A x B + x AB β AB = x ( A ) β ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGacYgaca GGVbGaai4zaiaad+eadaWgaaqcfasaaiaadgeaaKqbagqaaiabg2da 9iaadwhacqGH9aqpcqGHuoarcqGHRaWkcaWG4bWaaSbaaKqbGeaaca WGbbaajuaGbeaacaWG4bWaaSbaaKqbGeaacaWGcbaajuaGbeaacqGH RaWkcaWG4bWaaSbaaKqbGeaacaWGbbGaamOqaaqcfayabaGaeqOSdi 2aaSbaaKqbGeaacaWGbbGaamOqaaqcfayabaGaeyypa0JaamiEamaa Baaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaaju aGbeaacqaHYoGydaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGa ayjkaiaawMcaaaqcfayabaaaaa@5A18@ (ii)
log O B =v= x A x B + x AB β AB = x ( B ) β ( B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGacYgaca GGVbGaai4zaiaad+eadaWgaaqcfasaaiaadkeaaKqbagqaaiabg2da 9iaadAhacqGH9aqpcaWG4bWaaSbaaKqbGeaacaWGbbaajuaGbeaaca WG4bWaaSbaaKqbGeaacaWGcbaajuaGbeaacqGHRaWkcaWG4bWaaSba aKqbGeaacaWGbbGaamOqaaqcfayabaGaeqOSdi2aaSbaaKqbGeaaca WGbbGaamOqaaqcfayabaGaeyypa0JaamiEamaaBaaajuaibaqcfa4a aeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaaajuaGbeaacqaHYoGyda WgaaqcfasaaKqbaoaabmaajuaibaGaamOqaaGaayjkaiaawMcaaaqc fayabaaaaa@57D3@ (iii)

The predictive density of future log-odds for A, u f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwhada ahaaqabKqbGeaacaWGMbaaaaaa@38AE@  , for non-informative prior (vague prior) with normal or any spherical symmetric errors is of Student form Jammalamadaka et al. [10] and is given by
f( u f | x ( A ) f ,data )St( nk, x ( A ) f β ^ ( A ) , s ( A ) 2 ( 1+ x ( A ) f' ( x ' ( A ) ) 1 x ( A ) f ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadwhadaahaaqabKqbGeaacaWGMbaaaKqbakaacYhacaWG 4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPa aaaeaacaWGMbaaaKqbakaacYcacaWGKbGaamyyaiaadshacaWGHbaa caGLOaGaayzkaaGaeyyyIORaam4uaiaadshadaqadaqaaiaad6gacq GHsislcaWGRbGaaiilaiaadIhadaqhaaqcfasaaKqbaoaabmaajuai baGaamyqaaGaayjkaiaawMcaaaqaaiaadAgaaaqcfaOafqOSdiMbaK aadaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMca aaqcfayabaGaaiilaiaadohadaqhaaqcfasaaKqbaoaabmaajuaiba GaamyqaaGaayjkaiaawMcaaaqaaiaaikdaaaqcfa4aaeWaaeaacaaI XaGaey4kaSIaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbb aacaGLOaGaayzkaaaabaGaamOzaiaacEcaaaqcfa4aaeWaaeaacaWG 4bGaai4jamaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOa GaayzkaaaajuaGbeaaaiaawIcacaGLPaaadaahaaqabKqbGeaacqGH sislcaaIXaaaaKqbakaadIhadaqhaaqcfasaaKqbaoaabmaajuaiba GaamyqaaGaayjkaiaawMcaaaqaaiaadAgaaaaajuaGcaGLOaGaayzk aaaacaGLOaGaayzkaaaaaa@7A4B@

where β ^ ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaWaaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGL PaaaaKqbagqaaaaa@3C12@ is the MLE of β ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa juaGbeaaaaa@3C02@ , s ( A ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohada qhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqa aiaaikdaaaaaaa@3B88@  is the MLE of A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqaS WaaWbaaeqabaqcLbmacaaIYaaaaaaa@3957@ and k is the number of parameters in the model (ii). See Bhattacharjee et al. [11] in this context. If the sample size is large then this predictive density can be well approximated by its asymptotic normal form
N( x ( A ) f β ^ ( A ), s ( A ) 2 ( 1+ x ( A ) f' ( x ( A ) ' x ( A ) ) 1 x ( A ) f )( nk )/( nk2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6eada qadaqaaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGa ayjkaiaawMcaaaqaaiaadAgaaaqcfaOafqOSdiMbaKaadaqadaqaai aadgeaaiaawIcacaGLPaaacaGGSaGaam4CamaaDaaajuaibaqcfa4a aeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaabaGaaGOmaaaajuaGda qadaqaaiaaigdacqGHRaWkcaWG4bWaa0baaKqbGeaajuaGdaqadaqc fasaaiaadgeaaiaawIcacaGLPaaaaeaacaWGMbGaai4jaaaajuaGda qadaqaaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGa ayjkaiaawMcaaaqcfayaaiaacEcaaaGaamiEamaaBaaajuaibaqcfa 4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaajuaGbeaaaiaawIca caGLPaaadaahaaqabKqbGeaacqGHsislcaaIXaaaaKqbakaadIhada qhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqa aiaadAgaaaaajuaGcaGLOaGaayzkaaWaaeWaaeaacaWGUbGaeyOeI0 Iaam4AaaGaayjkaiaawMcaaiaac+cadaqadaqaaiaad6gacqGHsisl caWGRbGaeyOeI0IaaGOmaaGaayjkaiaawMcaaaGaayjkaiaawMcaaa aa@71F9@

Similarly one can find the same for treatment B, v f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAhada ahaaqabKqbGeaacaWGMbaaaaaa@38AF@ .
Let us de ne w f = ( u f , v f ) ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEhada ahaaqabKqbGeaacaWGMbaaaKqbakabg2da9maabmaabaGaamyDamaa CaaabeqcfasaaiaadAgaaaqcfaOaaiilaiaadAhadaahaaqabKqbGe aacaWGMbaaaaqcfaOaayjkaiaawMcaamaaCaaabeqaaiaacEcaaaaa aa@42D1@ and a= ( 1,1 ) ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggacq GH9aqpdaqadaqaaiaaigdacaGGSaGaeyOeI0IaaGymaaGaayjkaiaa wMcaamaaCaaabeqaaiaacEcaaaaaaa@3DCE@ : Then the predictive density of future log odds ratio a ' w f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada ahaaqabeaacaGGNaaaaiaadEhadaahaaqabKqbGeaacaWGMbaaaaaa @3A63@ is given by
f( a ' w f | x ( A ) f , x ( B ) f ,data )N( θ, δ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadggadaahaaqabeaacaGGNaaaaiaadEhadaahaaqabKqb GeaacaWGMbaaaKqbakaacYhacaWG4bWaa0baaKqbGeaajuaGdaqada qcfasaaiaadgeaaiaawIcacaGLPaaaaeaacaWGMbaaaKqbakaacYca caWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIcaca GLPaaaaeaacaWGMbaaaKqbakaacYcacaWGKbGaamyyaiaadshacaWG HbaacaGLOaGaayzkaaGaeyisISRaamOtamaabmaabaGaeqiUdeNaai ilaiabes7aKnaaCaaabeqcfasaaiaaikdaaaaajuaGcaGLOaGaayzk aaaaaa@58C8@   (iv)
Where
θ= x ( A ) f β ^ ( A ) x ( B ) f β ^ ( B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXj abg2da9iaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGa ayjkaiaawMcaaaqaaiaadAgaaaqcfaOafqOSdiMbaKaadaWgaaqcfa saaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqabaqcfaOa eyOeI0IaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGcbaaca GLOaGaayzkaaaabaGaamOzaaaajuaGcuaHYoGygaqcamaaBaaajuai baqcfa4aaeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaaajuaGbeaaaa a@50F8@
and

δ 2 = s ( A ) 2 ( 1+ x ( A ) f' ( x ( A ) ' x ( A ) ) 1 x ( A ) f )( nk )/( nk2 )+ s ( B ) 2 ( ( 1+ x ( B ) f' ( x ( B ) ' x ( B ) ) 1 x ( B ) f )( nq )/( nq2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes7aKn aaCaaabeqcfasaaiaaikdaaaqcfaOaeyypa0Jaam4CamaaDaaajuai baqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaabaGaaGOmaa aajuaGdaqadaqaaiaaigdacqGHRaWkcaWG4bWaa0baaKqbGeaajuaG daqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeaacaWGMbGaai4jaa aajuaGdaqadaqaaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGa amyqaaGaayjkaiaawMcaaaqaaiaacEcaaaqcfaOaamiEamaaBaaaju aibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaajuaGbeaa aiaawIcacaGLPaaadaahaaqabKqbGeaacqGHsislcaaIXaaaaKqbak aadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaa wMcaaaqaaiaadAgaaaaajuaGcaGLOaGaayzkaaWaaeWaaeaacaWGUb GaeyOeI0Iaam4AaaGaayjkaiaawMcaaiaac+cadaqadaqaaiaad6ga cqGHsislcaWGRbGaeyOeI0IaaGOmaaGaayjkaiaawMcaaiabgUcaRi aadohadaqhaaqcfasaaKqbaoaabmaajuaibaGaamOqaaGaayjkaiaa wMcaaaqaaiaaikdaaaqcfa4aaeWaaeaadaqadaqaaiaaigdacqGHRa WkcaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIca caGLPaaaaeaacaWGMbGaai4jaaaajuaGdaqadaqaaiaadIhadaqhaa qcfasaaKqbaoaabmaajuaibaGaamOqaaGaayjkaiaawMcaaaqcfaya aiaacEcaaaGaamiEamaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGcb aacaGLOaGaayzkaaaajuaGbeaaaiaawIcacaGLPaaadaahaaqabeaa cqGHsislcaaIXaaaaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaiba GaamOqaaGaayjkaiaawMcaaaqaaiaadAgaaaaajuaGcaGLOaGaayzk aaWaaeWaaeaacaWGUbGaeyOeI0IaamyCaaGaayjkaiaawMcaaiaac+ cadaqadaqaaiaad6gacqGHsislcaWGXbGaeyOeI0IaaGOmaaGaayjk aiaawMcaaaGaayjkaiaawMcaaaaa@99C4@

Our interest is to measure the influence of explanatory variables in the predictive density (iv) for the following cases:
Case 1: Influence of r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhaaa a@3770@ explanatory variables x A r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeaaeaacaWGYbaaaaaa@3983@  of x A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadgeaaKqbagqaaaaa@3919@  in treatment A.
Case 2: Influence of r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhaaa a@3770@  explanatory variables x B r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadkeaaeaacaWGYbaaaaaa@3984@  of x B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadkeaaKqbagqaaaaa@391A@  in treatment B.
Case 3: Influence of s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohaaa a@3771@  explanatory variables x AB s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeacaWGcbaabaGaam4Caaaaaaa@3A4B@  of x AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavababe qaaKqzadGaamyqaiaadkeaaKqbagqabaqcLbsacaWG4baaaaaa@3B7C@ in treatment A.
Case 2: Influence of r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhaaa a@3770@  explanatory variables x B r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadkeaaeaacaWGYbaaaaaa@3984@ of x B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadkeaaKqbagqaaaaa@391A@  in treatment B.
Case 3: Influence of s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohaaa a@3771@  explanatory variables x AB s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeacaWGcbaabaGaam4Caaaaaaa@3A4B@  of x AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavababe qaaKqzadGaamyqaiaadkeaaKqbagqabaqcLbsacaWG4baaaaaa@3B7C@ in treatment A.
Case 4: Influence of s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohaaa a@3771@  explanatory variables x AB s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeacaWGcbaabaGaam4Caaaaaaa@3A4B@ of x AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavababe qaaKqzadGaamyqaiaadkeaaKqbagqabaqcLbsacaWG4baaaaaa@3B7C@ in treatment B.
Case 5: Joint influence of r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhaaa a@3770@  explanatory variables x A r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeaaeaacaWGYbaaaaaa@3983@ of x A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadgeaaKqbagqaaaaa@3919@  and s explanatory variables x AB s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeacaWGcbaabaGaam4Caaaaaaa@3A4B@ of x AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavababe qaaKqzadGaamyqaiaadkeaaKqbagqabaqcLbsacaWG4baaaaaa@3B7C@ in treatment A.
Case 6: Joint influence of r explanatory variables x B r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadkeaaeaacaWGYbaaaaaa@3984@  of x B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadkeaaKqbagqaaaaa@391A@  and s explanatory variables x AB s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeacaWGcbaabaGaam4Caaaaaaa@3A4B@ of x AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavababe qaaKqzadGaamyqaiaadkeaaKqbagqabaqcLbsacaWG4baaaaaa@3B7C@ in treatment B.

To see the influence of explanatory variables in log-odds ratio, we construct a reduced log-odds model deleting a subset of explanatory variables. Then we derive the predictive density of future log-odds ratio for reduced model and compare it with the predictive density (iv) for full model. It is enough to consider Case 5 for illustration. We construct the reduced model by deleting variables x A r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeaaeaacaWGYbaaaaaa@3983@ of x A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadgeaaKqbagqaaaaa@3919@  and x AB s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadgeacaWGcbaabaGaam4Caaaaaaa@3A4B@  of x AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavababe qaaKqzadGaamyqaiaadkeaaKqbagqabaqcLbsacaWG4baaaaaa@3B7C@ in (ii) as
u=Δ+ x A * β A * + x A * | B β AB * = x ( A ) * β ( A ) * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwhacq GH9aqpcqGHuoarcqGHRaWkcaWG4bWaa0baaKqbGeaacaWGbbaabaGa aiOkaaaajuaGcqaHYoGydaqhaaqcfasaaiaadgeaaeaacaGGQaaaaK qbakabgUcaRiaadIhadaqhaaqcfasaaiaadgeaaeaacaGGQaaaaKqb akaacYhadaWgaaqcfasaaiaadkeaaKqbagqaaiabek7aInaaDaaaju aibaGaamyqaiaadkeaaeaacaGGQaaaaKqbakabg2da9iaadIhadaqh aaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqaai aacQcaaaqcfaOaeqOSdi2aa0baaKqbGeaajuaGdaqadaqcfasaaiaa dgeaaiaawIcacaGLPaaaaeaacaGGQaaaaaaa@59D7@
Then the predictive density of u f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaCaaabeqcfasaaiaadAgaaaaaaa@38B9@ is given by
f( u f | x ( A ) *f ,data )=St( nk+r+s, x ( A ) *f β ^ ( A ) * , S ( A ) *2 ( 1+ x ( A ) *f' ( x ( A ) *' x ( A ) * ) 1 x ( A ) *f ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadwhadaahaaqcfasabeaacaWGMbaaaKqbakaacYhacaWG 4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPa aaaeaacaGGQaGaamOzaaaacaGGSaqcfaOaamizaiaadggacaWG0bGa amyyaaGaayjkaiaawMcaaiabg2da9iaadofacaWG0bWaaeWaaeaaca WGUbGaeyOeI0Iaam4AaiabgUcaRiaadkhacqGHRaWkcaWGZbGaaiil aiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkai aawMcaaaqaaiaacQcacaWGMbaaaKqbakqbek7aIzaajaWaa0baaKqb GeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeaacaGGQa aaaKqbakaacYcacaWGtbWaa0baaKqbGeaajuaGdaqadaqcfasaaiaa dgeaaiaawIcacaGLPaaaaeaacaGGQaGaaGOmaaaajuaGdaqadaqaai aaigdacqGHRaWkcaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaa dgeaaiaawIcacaGLPaaaaeaacaGGQaGaamOzaiaacEcaaaqcfa4aae WaaeaacaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaa wIcacaGLPaaaaeaacaGGQaGaai4jaaaajuaGcaWG4bWaa0baaKqbGe aajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeaacaGGQaaa aaqcfaOaayjkaiaawMcaamaaCaaabeqaaiabgkHiTiaaigdaaaGaam iEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzk aaaabaGaaiOkaiaadAgaaaaajuaGcaGLOaGaayzkaaaacaGLOaGaay zkaaaaaa@86B7@
The normal approximation of the predictive density is
N( x ( A ) *f β ^ ( A ) * , s ( A ) *2 |( 1+ x ( A ) *f' ( x ( A ) *' x ( A ) * ) 1 x ( A ) *f ( nk+r+s )/( nk+r+s2 ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6eada qadaqaaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGa ayjkaiaawMcaaaqaaiaacQcacaWGMbaaaKqbakqbek7aIzaajaWaa0 baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeaa caGGQaaaaKqbakaacYcacaWGZbWaa0baaKqbGeaajuaGdaqadaqcfa saaiaadgeaaiaawIcacaGLPaaaaeaacaGGQaGaaGOmaaaajuaGcaGG 8bWaaeWaaeaacaaIXaGaey4kaSIaamiEamaaDaaajuaibaqcfa4aae WaaKqbGeaacaWGbbaacaGLOaGaayzkaaaabaGaaiOkaiaadAgacaGG NaaaaKqbaoaabmaabaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGe aacaWGbbaacaGLOaGaayzkaaaabaGaaiOkaiaacEcaaaqcfaOaamiE amaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaa aabaGaaiOkaaaaaKqbakaawIcacaGLPaaadaahaaqabKqbGeaacqGH sislcaaIXaaaaKqbakaadIhadaqhaaqcfasaaKqbaoaabmaajuaiba GaamyqaaGaayjkaiaawMcaaaqaaiaacQcacaWGMbaaaKqbaoaabmaa baGaamOBaiabgkHiTiaadUgacqGHRaWkcaWGYbGaey4kaSIaam4Caa GaayjkaiaawMcaaiaac+cadaqadaqaaiaad6gacqGHsislcaWGRbGa ey4kaSIaamOCaiabgUcaRiaadohacqGHsislcaaIYaaacaGLOaGaay zkaaaacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@80BC@

Since no variable is missing in υ=log O B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu Naeyypa0JaciiBaiaac+gacaGGNbGaam4tamaaBaaajuaibaGaamOq aaqcfayabaaaaa@3E99@ , the predictive density of υ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaWbaaeqajuaibaGaamOzaaaaaaa@3986@ is unaltered along with its normal approximation. Hence the predictive density of log-odds ratio a ' w f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyam aaCaaabeqaaiaacEcaaaGaam4DamaaCaaabeqcfasaaiaadAgaaaaa aa@3A6E@  under Case 5 is given by
f ( r+s ) ( a ' ω f | x ( A ) *f , x ( B ) f ,data )N( θ * , δ *2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGa ayjkaiaawMcaaaqcfayabaWaaeWaaeaacaWGHbWaaWbaaeqabaGaai 4jaaaacqaHjpWDdaahaaqabKqbGeaacaWGMbaaaKqbakaacYhacaWG 4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPa aaaeaacaGGQaGaamOzaaaajuaGcaGGSaGaamiEamaaDaaajuaibaqc fa4aaeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaaabaGaamOzaaaaju aGcaGGSaGaamizaiaadggacaWG0bGaamyyaaGaayjkaiaawMcaaiab gIKi7kaad6eadaqadaqaaiabeI7aXnaaCaaabeqaaiaacQcaaaGaai ilaiabes7aKnaaCaaabeqcfasaaiaacQcacaaIYaaaaaqcfaOaayjk aiaawMcaaaaa@61C3@   (v)
Where
θ * = x ( A ) *f β ^ * ( A ) x ( B ) f β ^ ( B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaWbaaeqabaGaaiOkaaaacqGH9aqpcaWG4bWaa0baaKqbGeaajuaG daqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeaacaGGQaGaamOzaa aajuaGcuaHYoGygaqcamaaCaaabeqaaiaacQcaaaWaaSbaaKqbGeaa juaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeqaaKqbakabgk HiTiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamOqaaGaayjk aiaawMcaaaqaaiaadAgaaaqcfaOafqOSdiMbaKaadaWgaaqcfasaaK qbaoaabmaajuaibaGaamOqaaGaayjkaiaawMcaaaqcfayabaaaaa@5351@
and

δ *2 = s ( A ) *2 ( 1+ x ( A ) *f' ( x ( A ) *' ) 1 x ( A ) *f )( nk+r+s )/( nk+r+s2 )+ s ( B ) 2 ( 1+ x ( B ) f' ( x ( B ) ' x ( B ) ) 1 x ( B ) f )( nq )/( nq2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaWbaaeqajuaibaGaaiOkaiaaikdaaaqcfaOaeyypa0Jaam4Camaa Daaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaaba GaaiOkaiaaikdaaaqcfa4aaeWaaeaacaaIXaGaey4kaSIaamiEamaa Daaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaaba GaaiOkaiaadAgacaGGNaaaaKqbaoaabmaabaGaamiEamaaDaaajuai baqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaabaGaaiOkai aacEcaaaaajuaGcaGLOaGaayzkaaWaaWbaaeqajuaibaGaeyOeI0Ia aGymaaaajuaGcaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadg eaaiaawIcacaGLPaaaaeaacaGGQaGaamOzaaaaaKqbakaawIcacaGL Paaadaqadaqaaiaad6gacqGHsislcaWGRbGaey4kaSIaamOCaiabgU caRiaadohaaiaawIcacaGLPaaacaGGVaWaaeWaaeaacaWGUbGaeyOe I0Iaam4AaiabgUcaRiaadkhacqGHRaWkcaWGZbGaeyOeI0IaaGOmaa GaayjkaiaawMcaaiabgUcaRiaadohadaqhaaqcfasaaKqbaoaabmaa juaibaGaamOqaaGaayjkaiaawMcaaaqaaiaaikdaaaqcfa4aaeWaae aacaaIXaGaey4kaSIaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaa caWGcbaacaGLOaGaayzkaaaabaGaamOzaiaacEcaaaqcfa4aaeWaae aacaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIca caGLPaaaaeaacaGGNaaaaKqbakaadIhadaWgaaqcfasaaKqbaoaabm aajuaibaGaamOqaaGaayjkaiaawMcaaaqabaaajuaGcaGLOaGaayzk aaWaaWbaaeqajuaibaGaeyOeI0IaaGymaaaajuaGcaWG4bWaa0baaK qbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIcacaGLPaaaaeaacaWG MbaaaaqcfaOaayjkaiaawMcaamaabmaabaGaamOBaiabgkHiTiaadg haaiaawIcacaGLPaaacaGGVaWaaeWaaeaacaWGUbGaeyOeI0IaamyC aiabgkHiTiaaikdaaiaawIcacaGLPaaaaaa@9EE9@

To access the influence of the deleted variables we employ the Kullback-Leibler [9] directed measure of divergence D KL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaaaa@39CB@ between the predictive densities of a ' w f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyam aaCaaabeqaaiaacEcaaaGaam4DamaaCaaabeqcfasaaiaadAgaaaaa aa@3A6E@ for full model (iv) and reduced model (v). The form of K-L measure used here is given by
D KL = f ( r+s ) ( a' ω f |. )log( f ( r+s ) ( a' ω f |. ) f( a' ω f |. ) )d a ' ω f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaiabg2da9maapeaabaGa amOzamaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam 4CaaGaayjkaiaawMcaaaqabaaajuaGbeqabiabgUIiYdWaaeWaaeaa caWGHbGaai4jaiabeM8a3naaCaaabeqcfasaaiaadAgaaaqcfaOaai iFaiaac6caaiaawIcacaGLPaaaciGGSbGaai4BaiaacEgadaqadaqa amaalaaabaGaamOzamaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGYb Gaey4kaSIaam4CaaGaayjkaiaawMcaaaqabaqcfa4aaeWaaeaacaWG HbGaai4jaiabeM8a3naaCaaabeqcfasaaiaadAgaaaqcfaOaaiiFai aac6caaiaawIcacaGLPaaaaeaacaWGMbWaaeWaaeaacaWGHbGaai4j aiabeM8a3naaCaaabeqcfasaaiaadAgaaaqcfaOaaiiFaiaac6caai aawIcacaGLPaaaaaaacaGLOaGaayzkaaGaamizaiaadggadaahaaqa beaacaGGNaaaaiabeM8a3naaCaaabeqcfasaaiaadAgaaaaaaa@6E8E@
The discrepancy measure D KL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaaaa@39CB@ between the predictive densities (iv) and (v) reduces to
D KL = ( θθ* ) 2 2 δ 2 + 1 2 ( δ *2 δ 2 log( δ *2 δ 2 )1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaiabg2da9maalaaabaWa aeWaaeaacqaH4oqCcqGHsislcqaH4oqCcaGGQaaacaGLOaGaayzkaa WaaWbaaeqajuaibaGaaGOmaaaaaKqbagaacaaIYaGaeqiTdq2aaWba aeqajuaibaGaaGOmaaaaaaqcfaOaey4kaSYaaSaaaeaacaaIXaaaba GaaGOmaaaadaqadaqaamaalaaabaGaeqiTdq2aaWbaaeqajuaibaGa aiOkaiaaikdaaaaajuaGbaGaeqiTdq2aaWbaaeqajuaibaGaaGOmaa aaaaqcfaOaeyOeI0IaciiBaiaac+gacaGGNbWaaeWaaeaadaWcaaqa aiabes7aKnaaCaaabeqcfasaaiaacQcacaaIYaaaaaqcfayaaiabes 7aKnaaCaaabeqcfasaaiaaikdaaaaaaaqcfaOaayjkaiaawMcaaiab gkHiTiaaigdaaiaawIcacaGLPaaaaaa@605E@
Here L= ( θ θ * ) 2 2 δ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maalaaabaWaaeWaaeaacqaH4oqCcqGHsislcqaH4oqCdaah aaqabeaacaGGQaaaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaik daaaaajuaGbaGaaGOmaiabes7aKnaaCaaabeqcfasaaiaaikdaaaaa aaaa@4424@ is due to difference of location parameters and S= 1 2 ( δ *2 δ 2 log( δ *2 δ 2 )1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai abg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaadaWcaaqa aiabes7aKnaaCaaabeqcfasaaiaacQcacaaIYaaaaaqcfayaaiabes 7aKnaaCaaabeqcfasaaiaaikdaaaaaaKqbakabgkHiTiGacYgacaGG VbGaai4zamaabmaabaWaaSaaaeaacqaH0oazdaahaaqabKqbGeaaca GGQaGaaGOmaaaaaKqbagaacqaH0oazdaahaaqabKqbGeaacaaIYaaa aaaaaKqbakaawIcacaGLPaaacqGHsislcaaIXaaacaGLOaGaayzkaa aaaa@50D8@ due to difference of scale parameters of the two predictive densities (iv) and (v).

Example 1: Here we have considered a u shot Data Pregibon [3]. A local health clinic sent fliers to its clients to encourage everyone, but especially older persons at high risk of complications, to get a u shot for protection against an expected u epidemic. In a pilot follow-up study, 159 clients were randomly selected and asked whether they actually received a u shot. A client who received a u shot was coded Y=1; and a client who did not receive a u shot was coded Y=0. In addition, data were collected on their age ( x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIXaaabeaaaKqbakaawIcacaGLPaaa aaa@3AA2@ and their health awareness ( x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIYaaabeaaaKqbakaawIcacaGLPaaa aaa@3AA3@ . Also included in the data were client gender ( x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIZaaabeaaaKqbakaawIcacaGLPaaa aaa@3AA4@ , with males coded x 3 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaG4maaqcfayabaGaeyypa0JaaGymaaaa@3ADC@ and females coded x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaG4maaqcfayabaGaeyypa0JaaGimaaaa@3ADB@ . Here we have divided whole data set into two groups A and B on the basis of gender that is group A corresponds to the male and group B corresponds to the female. We have computed D KL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaaaa@39CB@ to measure the influence of the deleted variable x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3919@  in group A and B separately and the discrepancies are drawn in Figure

  1. Similar gure can be obtained by deleting x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@391A@ . From this gure the discrepancy is less around the mean of the deleted variable.

Example 2: This is a simulation exercise. Here we have drawn sample of size 159 from bivariate normal distribution and we have used means, variances and correlation coefficient of x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3919@ and x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@391A@ of the above u shot data of size 159 for generating the sample. Now using these x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3919@ and x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@391A@ , we got response that is Y values and thereafter using this whole generated data set we have computed D KL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaaaa@39CB@ . Now we have repeated whole process 1000 times and computed means of D KL s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaiaadohaaaa@3AC3@ . The mean discrepancies are shown in Figure 2. Here we get the same conclusion as in the data example.

Influence of Missing Future Explanatory Variables in Log-Odds Ratio

Here the aim is to detect the predictive influence of a set of missing future explanatory variables in log-odds ratio of logistic model (i). Our interest is to detect the influence of missing future explanatory variables in the six cases pointed out in Section 2. Let in treatment A, r future variables missing from x A f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamyqaaqaaiaadAgaaaaaaa@3982@ and s future variables missing from x AB f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamyqaiaadkeaaeaacaWGMbaaaaaa@3A49@ be denoted by x ( A ) ( r+|s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaaiiFaiaadohaaiaawI cacaGLPaaacaWGMbaaaaaa@41DD@ . Similarly in treatment B, r future missing variables from x B f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamOqaaqaaiaadAgaaaaaaa@3983@ and s future variables missing from x AB f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamyqaiaadkeaaeaacaWGMbaaaaaa@3A49@ be denoted by x ( B ) ( r+|s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaaiiFaiaadohaaiaawI cacaGLPaaacaWGMbaaaaaa@41DE@ . We assume that the errors of models (ii) and (iii) are normally distributed with zero means and variances τ ( A ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3aa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aeaacqGHsislcaaIXaaaaaaa@3D4C@ and τ ( B ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3aa0baaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIcacaGLPaaa aeaacqGHsislcaaIXaaaaaaa@3D4D@ , respectively. We also assume that the conditional density of x ( A ) ( r+|s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaaiiFaiaadohaaiaawI cacaGLPaaacaWGMbaaaaaa@41DD@ given x ( A ) *f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baGaaiOkaiaadAgaaaaaaa@3C75@ is independent of β ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aKqbagqaaaaa@3C0D@ and τ ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aKqbagqaaaaa@3C31@ and x ( B ) ( r+|s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaaiiFaiaadohaaiaawI cacaGLPaaacaWGMbaaaaaa@41DE@  given x ( B ) *f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaaa baGaaiOkaiaadAgaaaaaaa@3C76@ is independent of β ( B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIcacaGLPaaa aKqbagqaaaaa@3C0E@ and τ ( B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIcacaGLPaaa aKqbagqaaaaa@3C32@ , i.e.,
f( x ( . ) ( r+s )f | x ( . ) *f , β ( . ) , τ ( . ) )=f( x ( . ) ( r+s )f | x ( . ) *f ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaGGUaaa caGLOaGaayzkaaaabaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam 4CaaGaayjkaiaawMcaaiaadAgaaaqcfaOaaiiFaiaadIhadaqhaaqc fasaaKqbaoaabmaajuaibaGaaiOlaaGaayjkaiaawMcaaaqaaiaacQ cacaWGMbaaaKqbakaacYcacqaHYoGydaWgaaqcfasaaKqbaoaabmaa juaibaGaaiOlaaGaayjkaiaawMcaaaqcfayabaGaaiilaiabes8a0n aaBaaajuaibaqcfa4aaeWaaKqbGeaacaGGUaaacaGLOaGaayzkaaaa juaGbeaaaiaawIcacaGLPaaacqGH9aqpcaWGMbWaaeWaaeaacaWG4b Waa0baaKqbGeaajuaGdaqadaqcfasaaiaac6caaiaawIcacaGLPaaa aeaajuaGdaqadaqcfasaaiaadkhacqGHRaWkcaWGZbaacaGLOaGaay zkaaGaamOzaaaajuaGcaGG8bGaamiEamaaDaaajuaibaqcfa4aaeWa aKqbGeaacaGGUaaacaGLOaGaayzkaaaabaGaaiOkaiaadAgaaaaaju aGcaGLOaGaayzkaaaaaa@6D5C@
where x ( . ) *f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaGGUaaacaGLOaGaayzkaaaa baGaaiOkaiaadAgaaaaaaa@3C61@ denotes the future explanatory variables x ( . ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaGGUaaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BB3@ without x ( . ) ( r+s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaGGUaaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGaayjkaiaawM caaiaadAgaaaaaaa@40C9@ .

Explanatory variables are continuous

We assume that x i f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamyAaaqaaiaadAgaaaaaaa@39AA@ 's are dependent and the distribution of x ( A ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BC7@ is ( k1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGRbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaa@3AA5@ -dimensional multivariate normal, i.e. f( x ( A ) f ) N k1 ( η,ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaa caGLOaGaayzkaaaabaGaamOzaaaaaKqbakaawIcacaGLPaaacqGHHj IUcaWGobWaaSbaaKqbGeaacaWGRbGaeyOeI0IaaGymaaqcfayabaWa aeWaaeaacqaH3oaAcaGGSaGaeqiYdKhacaGLOaGaayzkaaaaaa@4A8D@ .
The conditional density of x ( A ) ( r+|s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaaiiFaiaadohaaiaawI cacaGLPaaacaWGMbaaaaaa@41DD@  given x ( A ) *f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baGaaiOkaiaadAgaaaaaaa@3C75@ is given by
f( x ( A ) ( r+s )f | x ( A ) *f ) N r+s ( η ( r+s ) * , ψ ( r+s ) * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaa caGLOaGaayzkaaaabaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam 4CaaGaayjkaiaawMcaaiaadAgaaaqcfaOaaiiFaiaadIhadaqhaaqc fasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqaaiaacQ cacaWGMbaaaaqcfaOaayjkaiaawMcaaiabggMi6kaad6eadaWgaaqc fasaaiaadkhacqGHRaWkcaWGZbaajuaGbeaadaqadaqaaiabeE7aOn aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGa ayjkaiaawMcaaaqcfayaaiaacQcaaaGaaiilaiabeI8a5naaDaaaju aibaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGaayjkaiaa wMcaaaqcfayaaiaacQcaaaaacaGLOaGaayzkaaaaaa@649F@ ,

Where
η=( η * , η r+s ), x ( A ) f =( x ( A ) *f , x ( A ) ( r+s )f ),ψ=( ψ 11      ψ 12 ψ 21      ψ 22 ),   η r+s * = η r+s + ψ 21 ψ 11 1 ( x ( A ) *f η * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG Maeyypa0ZaaeWaaeaacqaH3oaAdaahaaqabeaacaGGQaaaaiaacYca cqaH3oaAdaWgaaqcfasaaiaadkhacqGHRaWkcaWGZbaajuaGbeaaai aawIcacaGLPaaacaGGSaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqb GeaacaWGbbaacaGLOaGaayzkaaaabaGaamOzaaaajuaGcqGH9aqpda qadaqaaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGa ayjkaiaawMcaaaqaaiaacQcacaWGMbaaaKqbakaacYcacaWG4bWaa0 baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeaa juaGdaqadaqcfasaaiaadkhacqGHRaWkcaWGZbaacaGLOaGaayzkaa GaamOzaaaaaKqbakaawIcacaGLPaaacaGGSaGaeqiYdKNaeyypa0Za aeWaaqaabeqaaiabeI8a5naaBaaajuaibaGaaGymaiaaigdaaKqbag qaaabaaaaaaaaapeGaaiiOaiaacckacaGGGcGaaiiOaiabeI8a5naa BaaajuaibaGaaGymaiaaikdaaKqbagqaaaqaaiabeI8a5naaBaaaju aibaGaaGOmaiaaigdaaKqbagqaaiaacckacaGGGcGaaiiOaiaaccka cqaHipqEdaWgaaqcfasaaiaaikdacaaIYaaajuaGbeaaaaWdaiaawI cacaGLPaaacaGGSaWdbiaacckacaGGGcGaeq4TdG2aa0baaKqbGeaa caWGYbGaey4kaSIaam4CaaqcfayaaiaacQcaaaGaeyypa0Jaeq4TdG 2aaSbaaKqbGeaacaWGYbGaey4kaSIaam4CaaqcfayabaGaey4kaSIa eqiYdK3aaSbaaKqbGeaacaaIYaGaaGymaaqcfayabaGaeqiYdK3aa0 baaKqbGeaacaaIXaGaaGymaaqaaiabgkHiTiaaigdaaaqcfa4aaeWa aeaacaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawI cacaGLPaaaaeaacaGGQaGaamOzaaaajuaGcqGHsislcqaH3oaAdaah aaqabeaacaGGQaaaaaGaayjkaiaawMcaaaaa@A306@  
and
ψ ( r+s ) * = ψ 22 ψ 21 ψ 11 1 ψ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aa0baaKqbGeaajuaGdaqadaqcfasaaiaadkhacqGHRaWkcaWGZbaa caGLOaGaayzkaaaajuaGbaGaaiOkaaaacqGH9aqpcqaHipqEdaWgaa qcfasaaiaaikdacaaIYaaajuaGbeaacqGHsislcqaHipqEdaWgaaqc fasaaiaaikdacaaIXaaajuaGbeaacqaHipqEdaqhaaqcfasaaiaaig dacaaIXaaabaGaeyOeI0IaaGymaaaajuaGcqaHipqEdaWgaaqcfasa aiaaigdacaaIYaaajuaGbeaaaaa@5318@
As earlier it is enough to consider Case 5 to see the joint influence of r missing future ex-planatory variables x A rf MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamyqaaqaaiaadkhacaWGMbaaaaaa@3A79@ of x A f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamyqaaqaaiaadAgaaaaaaa@3982@ and s missing future explanatory variables x AB sf MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamyqaiaadkeaaeaacaWGZbGaamOzaaaaaaa@3B41@ of x AB f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaGaamyqaiaadkeaaeaacaWGMbaaaaaa@3A49@  in treatment A. The density of u f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaCaaabeqcfasaaiaadAgaaaaaaa@38B9@ when x ( A ) ( r+s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGaayjkaiaawM caaiaadAgaaaaaaa@40DD@ is missing is given by

f( u f | x ( A ) *f ,β | ( A ) , τ ( A ) )= f( u f | x ( A ) f , β ( A ) , τ ( A ) ) f( x ( A ) ( r+s )f | x ( A ) *f )d x ( A ) ( r+s )f N( i=0 i=0 krs1 x ( A )i f β ( A )i + i=krs k1 η i * β ( A )i , i=krs k1 β ( A )i β ( A )j ψ ij * + τ ( A ) 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamyDamaaCaaabeqcfasaaiaadAgaaaqcfaOaaiiFaiaa dIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawM caaaqaaiaacQcacaWGMbaaaKqbakaacYcacqaHYoGycaGG8bWaaSba aKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeqaaK qbakaacYcacqaHepaDdaWgaaqcfasaaKqbaoaabmaajuaibaGaamyq aaGaayjkaiaawMcaaaqabaaajuaGcaGLOaGaayzkaaGaeyypa0Zaa8 qaaeaacaWGMbWaaeWaaeaacaWG1bWaaWbaaeqajuaibaGaamOzaaaa juaGcaGG8bGaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbb aacaGLOaGaayzkaaaabaGaamOzaaaajuaGcaGGSaGaeqOSdi2aaSba aKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaKqbag qaaiaacYcacqaHepaDdaWgaaqcfasaaKqbaoaabmaajuaibaGaamyq aaGaayjkaiaawMcaaaqcfayabaaacaGLOaGaayzkaaaabeqabiabgU IiYdGaamOzamaabmaabaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqb GeaacaWGbbaacaGLOaGaayzkaaaabaqcfa4aaeWaaKqbGeaacaWGYb Gaey4kaSIaam4CaaGaayjkaiaawMcaaiaadAgaaaqcfaOaaiiFaiaa dIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawM caaaqaaiaacQcacaWGMbaaaaqcfaOaayjkaiaawMcaaiaadsgacaWG 4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPa aaaeaajuaGdaqadaqcfasaaiaadkhacqGHRaWkcaWGZbaacaGLOaGa ayzkaaGaamOzaaaajuaGcqGHHjIUcaWGobWaaeWaaeaacaWGPbGaey ypa0JaaGimamaaqahabaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqb GeaacaWGbbaacaGLOaGaayzkaaGaamyAaaqaaiaadAgaaaqcfaOaeq OSdi2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGL PaaacaWGPbaabeaajuaGcqGHRaWkdaaeWbqaaiabeE7aOnaaDaaaju aibaGaamyAaaqaaiaacQcaaaqcfaOaeqOSdi2aaSbaaKqbGeaajuaG daqadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaabeaajuaGca GGSaWaaabCaeaacqaHYoGydaWgaaqcfasaaKqbaoaabmaajuaibaGa amyqaaGaayjkaiaawMcaaiaadMgaaKqbagqaaaqcfasaaiaadMgacq GH9aqpcaWGRbGaeyOeI0IaamOCaiabgkHiTiaadohaaeaacaWGRbGa eyOeI0IaaGymaaqcfaOaeyyeIuoacqaHYoGydaWgaaqcfasaaKqbao aabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadQgaaKqbagqaaiab eI8a5naaDaaajuaibaGaamyAaiaadQgaaeaacaGGQaaaaKqbakabgU caRiabes8a0naaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGL OaGaayzkaaaabaGaeyOeI0IaaGymaaaaaeaacaWGPbGaeyypa0Jaam 4AaiabgkHiTiaadkhacqGHsislcaWGZbaabaGaam4AaiabgkHiTiaa igdaaKqbakabggHiLdaajuaibaGaamyAaiabg2da9iaaicdaaeaaca WGRbGaeyOeI0IaamOCaiabgkHiTiaadohacqGHsislcaaIXaaajuaG cqGHris5aaGaayjkaiaawMcaaaaa@EB26@

Where η i * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG 2aa0baaKqbGeaacaWGPbaabaGaaiOkaaaaaaa@3A1C@ is the η i * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG 2aa0baaKqbGeaacaWGPbaabaGaaiOkaaaaaaa@3A1C@ th component of η ( r+s ) * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG 2aa0baaKqbGeaajuaGdaqadaqcfasaaiaadkhacqGHRaWkcaWGZbaa caGLOaGaayzkaaaajuaGbaGaaiOkaaaaaaa@3ED2@ and ψ ij * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aa0baaKqbGeaacaWGPbGaamOAaaqcfayaaiaacQcaaaaaaa@3BBB@  is the ( i,j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGPbGaaiilaiaadQgaaiaawIcacaGLPaaaaaa@3A9A@ th component of ψ ( r+s ) * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aa0baaKqbGeaajuaGdaqadaqcfasaaiaadkhacqGHRaWkcaWGZbaa caGLOaGaayzkaaaajuaGbaGaaiOkaaaaaaa@3EF4@ .
See Bhattacharjee et al [11] in this context. Using Taylor's expansion and improper prior density for both β ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aeqaaaaa@3B7F@  and τ ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aeqaaaaa@3BA3@ , the approximate predictive density of u f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaCaaabeqcfasaaiaadAgaaaaaaa@38B9@ when x ( A ) ( r+s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGaayjkaiaawM caaiaadAgaaaaaaa@40DD@ is missing is given by

f ( r+s ) ( u f | x ( A ) *f ,data )N( i=0 krs1 x ( A )i f β ^ ( A )i + i=krs k1 η i * β ^ ( A )i , i,j=krs k1 β ^ ( A )i β ^ ( A )j ψ ij * + s ( A ) 2 γ * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGa ayjkaiaawMcaaaqcfayabaWaaeWaaeaacaWG1bWaaWbaaeqajuaiba GaamOzaaaajuaGcaGG8bGaamiEamaaDaaajuaibaqcfa4aaeWaaKqb GeaacaWGbbaacaGLOaGaayzkaaaabaGaaiOkaiaadAgaaaqcfaOaai ilaiaadsgacaWGHbGaamiDaiaadggaaiaawIcacaGLPaaacqGHHjIU caWGobWaaeWaaeaadaaeWbqaaiaadIhadaqhaaqcfasaaKqbaoaabm aajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaaeaacaWGMbaaaKqb akqbek7aIzaajaWaaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaai aawIcacaGLPaaacaWGPbaajuaGbeaacqGHRaWkdaaeWbqaaiabeE7a OnaaDaaajuaibaGaamyAaaqcfayaaiaacQcaaaGafqOSdiMbaKaada WgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaa dMgaaeqaaKqbakaacYcadaaeWbqaaiqbek7aIzaajaWaaSbaaKqbGe aajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaabeaa juaGcuaHYoGygaqcamaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGbb aacaGLOaGaayzkaaGaamOAaaqcfayabaaajuaibaGaamyAaiaacYca caWGQbGaeyypa0Jaam4AaiabgkHiTiaadkhacqGHsislcaWGZbaaba Gaam4AaiabgkHiTiaaigdaaKqbakabggHiLdGaeqiYdK3aa0baaKqb GeaacaWGPbGaamOAaaqcfayaaiaacQcaaaGaey4kaSIaam4CamaaDa aajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaabaGa aGOmaaaajuaGcqaHZoWzdaahaaqabeaacaGGQaaaaaqcfasaaiaadM gacqGH9aqpcaWGRbGaeyOeI0IaamOCaiabgkHiTiaadohaaeaacaWG RbGaeyOeI0IaaGymaaqcfaOaeyyeIuoaaKqbGeaacaWGPbGaeyypa0 JaaGimaaqaaiaadUgacqGHsislcaWGYbGaeyOeI0Iaam4CaiabgkHi TiaaigdaaKqbakabggHiLdaacaGLOaGaayzkaaaaaa@AC5B@
evaluated at  β ^ ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaa wMcaaaqcfayabaaaaa@3C1D@ and s ( A ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Cam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baGaaGOmaaaaaaa@3B93@ where
γ * =( 1+ 1 2 0 k1 Q ij * ( β ( A ) , τ ( A ) )Cov( β ( A )i , β ( A )j )+ 1 2 Q τ ( A ) 2 ( β ( A ) , τ ( A ) )Var( τ ( A ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC 2aaWbaaeqabaGaaiOkaaaacqGH9aqpdaqadaqaaiaaigdacqGHRaWk daWcaaqaaiaaigdaaeaacaaIYaaaamaaqahabaGaamyuamaaDaaaju aibaGaamyAaiaadQgaaKqbagaacaGGQaaaamaabmaabaGaeqOSdi2a aSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaK qbagqaaiaacYcacqaHepaDdaWgaaqcfasaaKqbaoaabmaajuaibaGa amyqaaGaayjkaiaawMcaaaqcfayabaaacaGLOaGaayzkaaGaam4qai aad+gacaWG2bWaaeWaaeaacqaHYoGydaWgaaqcfasaaKqbaoaabmaa juaibaGaamyqaaGaayjkaiaawMcaaiaadMgaaKqbagqaaiaacYcacq aHYoGydaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaa wMcaaiaadQgaaKqbagqaaaGaayjkaiaawMcaaiabgUcaRmaalaaaba GaaGymaaqaaiaaikdaaaaajuaibaGaaGimaaqaaiaadUgacqGHsisl caaIXaaajuaGcqGHris5aiaadgfadaqhaaqcfasaaiabes8a0Lqbao aaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa beaaaeaacaaIYaaaaKqbaoaabmaabaGaeqOSdi2aaSbaaKqbGeaaju aGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeqaaKqbakaacYca cqaHepaDdaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkai aawMcaaaqcfayabaaacaGLOaGaayzkaaGaamOvaiaadggacaWGYbWa aeWaaeaacqaHepaDdaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaa GaayjkaiaawMcaaaqabaaajuaGcaGLOaGaayzkaaaacaGLOaGaayzk aaaaaa@8A56@
is the multiplicative factor for the second order Taylor's approximation. If x ( A ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BC7@ 's are independent the corresponding approximate predictive density of u f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaCaaabeqcfasaaiaadAgaaaaaaa@38B9@ is
f ( r+s ) ( u f | x ( A ) *f ,data )N( i=0 krs1 x ( A )i f β ^ ( A )i + i=krs k1 η i * β ^ ( A )i , i,j=krs k1 β ^ ( A )i β ^ ( A )j ψ ij * + s ( A ) 2 γ * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGa ayjkaiaawMcaaaqcfayabaWaaeWaaeaacaWG1bWaaWbaaeqajuaiba GaamOzaaaajuaGcaGG8bGaamiEamaaDaaajuaibaqcfa4aaeWaaKqb GeaacaWGbbaacaGLOaGaayzkaaaabaGaaiOkaiaadAgaaaqcfaOaai ilaiaadsgacaWGHbGaamiDaiaadggaaiaawIcacaGLPaaacqGHHjIU caWGobWaaeWaaeaadaaeWbqaaiaadIhadaqhaaqcfasaaKqbaoaabm aajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaaeaacaWGMbaaaKqb akqbek7aIzaajaWaaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaai aawIcacaGLPaaacaWGPbaajuaGbeaacqGHRaWkdaaeWbqaaiabeE7a OnaaDaaajuaibaGaamyAaaqcfayaaiaacQcaaaGafqOSdiMbaKaada WgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaa dMgaaeqaaKqbakaacYcadaaeWbqaaiqbek7aIzaajaWaaSbaaKqbGe aajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaabeaa juaGcuaHYoGygaqcamaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGbb aacaGLOaGaayzkaaGaamOAaaqcfayabaaajuaibaGaamyAaiaacYca caWGQbGaeyypa0Jaam4AaiabgkHiTiaadkhacqGHsislcaWGZbaaba Gaam4AaiabgkHiTiaaigdaaKqbakabggHiLdGaeqiYdK3aa0baaKqb GeaacaWGPbGaamOAaaqcfayaaiaacQcaaaGaey4kaSIaam4CamaaDa aajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaabaGa aGOmaaaajuaGcqaHZoWzdaahaaqabeaacaGGQaaaaaqcfasaaiaadM gacqGH9aqpcaWGRbGaeyOeI0IaamOCaiabgkHiTiaadohaaeaacaWG RbGaeyOeI0IaaGymaaqcfaOaeyyeIuoaaKqbGeaacaWGPbGaeyypa0 JaaGimaaqaaiaadUgacqGHsislcaWGYbGaeyOeI0Iaam4CaiabgkHi TiaaigdaaKqbakabggHiLdaacaGLOaGaayzkaaaaaa@AC5B@
evaluated at β ^ ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaa wMcaaaqcfayabaaaaa@3C1D@ and s ( A ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Cam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baGaaGOmaaaaaaa@3B93@ , where η i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG 2aaSbaaKqbGeaacaWGPbaajuaGbeaaaaa@39FB@ and ψ i 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aa0baaKqbGeaacaWGPbaabaGaaGOmaaaaaaa@3A4C@  are mean and variance of the ith missing variable and γ=( 1+ 1 2 0 k1 Q ij ( β ( A ) , τ ( A ) )Cov( β ( A )i , β ( A )j )+ 1 2 Q τ ( A ) 2 ( β ( A ) , τ ( A ) )Var( τ ( A ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaeWaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGa aGOmaaaadaaeWbqaaiaadgfadaWgaaqcfasaaiaadMgacaWGQbaaju aGbeaadaqadaqaaiabek7aInaaBaaajuaibaqcfa4aaeWaaKqbGeaa caWGbbaacaGLOaGaayzkaaaajuaGbeaacaGGSaGaeqiXdq3aaSbaaK qbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaKqbagqa aaGaayjkaiaawMcaaiaadoeacaWGVbGaamODamaabmaabaGaeqOSdi 2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa caWGPbaajuaGbeaacaGGSaGaeqOSdi2aaSbaaKqbGeaajuaGdaqada qcfasaaiaadgeaaiaawIcacaGLPaaacaWGQbaajuaGbeaaaiaawIca caGLPaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaaqcfasaai aaicdaaeaacaWGRbGaeyOeI0IaaGymaaqcfaOaeyyeIuoacaWGrbWa a0baaKqbGeaacqaHepaDjuaGdaWgaaqcfasaaKqbaoaabmaajuaiba GaamyqaaGaayjkaiaawMcaaaqabaaabaGaaGOmaaaajuaGdaqadaqa aiabek7aInaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOa GaayzkaaaajuaGbeaacaGGSaGaeqiXdq3aaSbaaKqbGeaajuaGdaqa daqcfasaaiaadgeaaiaawIcacaGLPaaaaKqbagqaaaGaayjkaiaawM caaiaadAfacaWGHbGaamOCamaabmaabaGaeqiXdq3aaSbaaKqbGeaa juaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeqaaaqcfaOaay jkaiaawMcaaaGaayjkaiaawMcaaaaa@88D7@ . Since no future variable is missing in υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu haaa@384B@ , the approximate predictive density of υ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaWbaaeqajuaibaGaamOzaaaaaaa@3986@ is same as obtained in Section 2. Thus when x ( A ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BC7@ s are dependent the approximate predictive density of log-odds ratio a ' w f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyam aaCaaabeqaaiaacEcaaaGaam4DamaaCaaabeqcfasaaiaadAgaaaaa aa@3A6E@ for x ( A ) ( r+s )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGaayjkaiaawM caaiaadAgaaaaaaa@40DD@ missing is given by
f ( r+s ) ( a ' w f | x ( A ) *f , x ( B ) f ;data ) γ * N( ξ, ω 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbGaey4kaSIaam4CaaGa ayjkaiaawMcaaaqabaqcfa4aaeWaaeaacaWGHbWaaWbaaeqabaGaai 4jaaaacaWG3bWaaWbaaeqajuaibaGaamOzaaaajuaGcaGG8bGaamiE amaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaa aabaGaaiOkaiaadAgaaaqcfaOaaiilaiaadIhadaqhaaqcfasaaKqb aoaabmaajuaibaGaamOqaaGaayjkaiaawMcaaaqaaiaadAgaaaqcfa Oaai4oaiaadsgacaWGHbGaamiDaiaadggaaiaawIcacaGLPaaacqGH HjIUcqaHZoWzdaahaaqabeaacaGGQaaaaiaad6eadaqadaqaaiabe6 7a4jaacYcacqaHjpWDdaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjk aiaawMcaaaaa@6247@ ,  (vi)
Where
ξ= i=0 krs1 x ( A )i f β ^ ( A )i + i=krs k1 η i * β ^ ( A )i x ( B ) f β ^ ( B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOVdG Naeyypa0ZaaabCaeaacaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasa aiaadgeaaiaawIcacaGLPaaacaWGPbaabaGaamOzaaaaaeaacaWGPb Gaeyypa0JaaGimaaqaaiaadUgacqGHsislcaWGYbGaeyOeI0Iaam4C aiabgkHiTiaaigdaaKqbakabggHiLdGafqOSdiMbaKaadaWgaaqcfa saaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaaKqb agqaaiabgUcaRmaaqahabaGaeq4TdG2aa0baaKqbGeaacaWGPbaaju aGbaGaaiOkaaaaaKqbGeaacaWGPbGaeyypa0Jaam4AaiabgkHiTiaa dkhacqGHsislcaWGZbaabaGaam4AaiabgkHiTiaaigdaaKqbakabgg HiLdGafqOSdiMbaKaadaWgaaqcfasaaKqbaoaabmaajuaibaGaamyq aaGaayjkaiaawMcaaiaadMgaaKqbagqaaiabgkHiTiaadIhadaqhaa qcfasaaKqbaoaabmaajuaibaGaamOqaaGaayjkaiaawMcaaaqaaiaa dAgaaaqcfaOafqOSdiMbaKaadaWgaaqcfasaaKqbaoaabmaajuaiba GaamOqaaGaayjkaiaawMcaaaqabaaaaa@7551@
and
ω 2 =( i,j=krs k1 β ^ ( A )i β ( A )j ψ ij * + s ( A ) 2 )+ s ( B ) 2 ( 1+ x ( B ) f ( X ( B ) ' X ( B ) ) 1 x ( B ) f' ) nq nq+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyYdC 3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaqadaqaamaaqaha baGafqOSdiMbaKaadaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaa GaayjkaiaawMcaaiaadMgaaKqbagqaaaqcfasaaiaadMgacaGGSaGa amOAaiabg2da9iaadUgacqGHsislcaWGYbGaeyOeI0Iaam4Caaqaai aadUgacqGHsislcaaIXaaajuaGcqGHris5aiabek7aInaaBaaajuai baqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGaamOAaaqaba qcfaOaeqiYdK3aa0baaKqbGeaacaWGPbGaamOAaaqcfayaaiaacQca aaGaey4kaSIaam4CamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbb aacaGLOaGaayzkaaaabaGaaGOmaaaaaKqbakaawIcacaGLPaaacqGH RaWkcaWGZbWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawI cacaGLPaaaaeaacaaIYaaaaKqbaoaabmaabaGaaGymaiabgUcaRiaa dIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamOqaaGaayjkaiaawM caaaqaaiaadAgaaaqcfa4aaeWaaeaacaWGybWaa0baaKqbGeaajuaG daqadaqcfasaaiaadkeaaiaawIcacaGLPaaaaKqbagaacaGGNaaaai aadIfadaWgaaqcfasaaKqbaoaabmaajuaibaGaamOqaaGaayjkaiaa wMcaaaqcfayabaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaeyOeI0 IaaGymaaaajuaGcaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaa dkeaaiaawIcacaGLPaaaaeaacaWGMbGaai4jaaaaaKqbakaawIcaca GLPaaadaWcaaqaaiaad6gacqGHsislcaWGXbaabaGaamOBaiabgkHi TiaadghacqGHRaWkcaaIYaaaaaaa@8E8F@

The K-L directed measure of divergence between the predictive densities (iv) when no variable is missing and the predictive density (3.1) when r+s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai abgUcaRiaadohaaaa@3955@ future variables are missing is given by
D KL = f( a ' w f | x ( A ) f , x ( B ) f ,data ) log( f( a ' w f | x ( A ) f , x ( B ) f ,data ) f ( r+s ) ( a ' w f | x ( A ) *f , x ( B ) f ,data ) )d a ' w f = 1 2 ω 2 ( θξ ) 2 + 1 2 ( δ 2 ω 2 log( δ 2 ω 2 )1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaiabg2da9maapeaabaGa amOzamaabmaabaGaamyyamaaCaaabeqaaiaacEcaaaGaam4DamaaCa aabeqcfasaaiaadAgaaaqcfaOaaiiFaiaadIhadaqhaaqcfasaaKqb aoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqaaiaadAgaaaqcfa OaaiilaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamOqaaGa ayjkaiaawMcaaaqaaiaadAgaaaqcfaOaaiilaiaadsgacaWGHbGaam iDaiaadggaaiaawIcacaGLPaaaaeqabeGaey4kIipaciGGSbGaai4B aiaacEgadaqadaqaamaalaaabaGaamOzamaabmaabaGaamyyamaaCa aabeqaaiaacEcaaaGaam4DamaaCaaabeqcfasaaiaadAgaaaqcfaOa aiiFaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaay jkaiaawMcaaaqaaiaadAgaaaqcfaOaaiilaiaadIhadaqhaaqcfasa aKqbaoaabmaajuaibaGaamOqaaGaayjkaiaawMcaaaqaaiaadAgaaa qcfaOaaiilaiaadsgacaWGHbGaamiDaiaadggaaiaawIcacaGLPaaa aeaacaWGMbWaaSbaaKqbGeaajuaGdaqadaqcfasaaiaadkhacqGHRa WkcaWGZbaacaGLOaGaayzkaaaajuaGbeaadaqadaqaaiaadggadaah aaqabeaacaGGNaaaaiaadEhadaahaaqabKqbGeaacaWGMbaaaKqbak aacYhacaWG4bWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaa wIcacaGLPaaaaeaacaGGQaGaamOzaaaajuaGcaGGSaGaamiEamaaDa aajuaibaqcfa4aaeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaaabaGa amOzaaaajuaGcaGGSaGaamizaiaadggacaWG0bGaamyyaaGaayjkai aawMcaaaaaaiaawIcacaGLPaaacaWGKbGaamyyamaaCaaabeqaaiaa cEcaaaGaam4DamaaCaaabeqcfasaaiaadAgaaaqcfaOaeyypa0ZaaS aaaeaacaaIXaaabaGaaGOmaiabeM8a3naaCaaabeqcfasaaiaaikda aaaaaKqbaoaabmaabaGaeqiUdeNaeyOeI0IaeqOVdGhacaGLOaGaay zkaaWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaWcaaqaaiaa igdaaeaacaaIYaaaamaabmaabaWaaSaaaeaacqaH0oazdaahaaqabK qbGeaacaaIYaaaaaqcfayaaiabeM8a3naaCaaabeqcfasaaiaaikda aaaaaKqbakabgkHiTiGacYgacaGGVbGaai4zamaabmaabaWaaSaaae aacqaH0oazdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeM8a3naa CaaabeqcfasaaiaaikdaaaaaaaqcfaOaayjkaiaawMcaaiabgkHiTi aaigdaaiaawIcacaGLPaaaaaa@BCBF@
= 1 2 i,j=0 k1 E( Q ij * ( β ( A ) , τ ( A ) )Cov( τ ( A )i , τ ( A )j ) ) 1 2 E( Q τ( A ) 2 ( β ( A ) , τ ( A ) )var( τ ( A ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 YaaSaaaeaacaaIXaaabaGaaGOmaaaadaaeWbqaaiaadweadaqadaqa aiaadgfadaqhaaqcfasaaiaadMgacaWGQbaabaGaaiOkaaaajuaGda qadaqaaiabek7aInaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaa caGLOaGaayzkaaaajuaGbeaacaGGSaGaeqiXdq3aaSbaaKqbGeaaju aGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaKqbagqaaaGaayjk aiaawMcaaiaadoeacaWGVbGaamODamaabmaabaGaeqiXdq3aaSbaaK qbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaa juaGbeaacaGGSaGaeqiXdq3aaSbaaKqbGeaajuaGdaqadaqcfasaai aadgeaaiaawIcacaGLPaaacaWGQbaajuaGbeaaaiaawIcacaGLPaaa aiaawIcacaGLPaaaaKqbGeaacaWGPbGaaiilaiaadQgacqGH9aqpca aIWaaabaGaam4AaiabgkHiTiaaigdaaKqbakabggHiLdGaeyOeI0Ya aSaaaeaacaaIXaaabaGaaGOmaaaacaWGfbWaaeWaaeaacaWGrbWaa0 baaKqbGeaacqaHepaDjuaGdaqadaqcfasaaiaadgeaaiaawIcacaGL PaaaaeaacaaIYaaaaKqbaoaabmaabaGaeqOSdi2aaSbaaKqbGeaaju aGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeqaaKqbakaacYca cqaHepaDdaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkai aawMcaaaqcfayabaaacaGLOaGaayzkaaGaciODaiaacggacaGGYbWa aeWaaeaacqaHepaDdaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaa GaayjkaiaawMcaaaqcfayabaaacaGLOaGaayzkaaaacaGLOaGaayzk aaaaaa@8C6E@ (vii)
If x ( A ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BC7@ ‘s are independent the predictive density of a ' w f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyam aaCaaabeqaaiaacEcaaaGaam4DamaaCaaabeqcfasaaiaadAgaaaaa aa@3A6E@ when ( r+s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGYbGaey4kaSIaam4CaaGaayjkaiaawMcaaaaa@3ADE@ future variables are missing is same as (vi) and the corresponding K-L [9] measure D KL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaaaa@39CB@ is same as (vii) but replacing η i * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG 2aa0baaKqbGeaacaWGPbaabaGaaiOkaaaaaaa@3A1C@  by η i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG 2aaSbaaKqbGeaacaWGPbaajuaGbeaaaaa@39FB@ ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOVdG haaa@3847@ , β ^ ( A )i β ^ ( A )j ψ ij * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaa wMcaaiaadMgaaKqbagqaaiqbek7aIzaajaWaaSbaaKqbGeaajuaGda qadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWGQbaajuaGbeaacqaH ipqEdaqhaaqcfasaaiaadMgacaWGQbaabaGaaiOkaaaaaaa@483C@  by β ^ ( A )i 2 ψ i 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaa wMcaaiaadMgaaeaacaaIYaaaaKqbakabeI8a5naaDaaajuaibaGaam yAaaqaaiaaikdaaaaaaa@4190@ in ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyYdC 3aaWbaaeqajuaibaGaaGOmaaaaaaa@395D@ and Q ij * ( β ( A ) , τ ( A ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyuam aaDaaajuaibaGaamyAaiaadQgaaeaacaGGQaaaaKqbaoaabmaabaGa eqOSdi2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcaca GLPaaaaKqbagqaaiaacYcacqaHepaDdaWgaaqcfasaaKqbaoaabmaa juaibaGaamyqaaGaayjkaiaawMcaaaqcfayabaaacaGLOaGaayzkaa aaaa@4832@  by  Q ij ( β ( A ) , τ ( A ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyuam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaamaabmaabaGaeqOSdi2a aSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaK qbagqaaiaacYcacqaHepaDdaWgaaqcfasaaKqbaoaabmaajuaibaGa amyqaaGaayjkaiaawMcaaaqabaaajuaGcaGLOaGaayzkaaaaaa@4783@  in γ * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC 2aaWbaaeqabaGaaiOkaaaaaaa@38FB@ , where η i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG 2aaSbaaKqbGeaacaWGPbaajuaGbeaaaaa@39FB@ and ψ i 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aa0baaKqbGeaacaWGPbaabaGaaGOmaaaaaaa@3A4C@ are mean and variance of the i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgaaaa@3793@ th missing variable.

Explanatory variables are dichotomous

Here we assume that all the explanatory variables are dichotomous and independent. We assume that the errors of models (ii) and (iii) are normally distributed with means zero and variances τ ( A ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3aa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aeaacqGHsislcaaIXaaaaaaa@3D4C@ and τ ( B ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3aa0baaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIcacaGLPaaa aeaacqGHsislcaaIXaaaaaaa@3D4D@ respectively. To assess the influence of the missing variables in treatment A, we consider that x ( A )i f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGa amyAaaqaaiaadAgaaaaaaa@3CB5@ is distributed as
Pr( X ( A )i f = x ( A )i f )= θ ( A )i x ( A )i f ( 1 θ ( A )i ) 1 x ( A )i f , x ( A )i f =0,1,    i=1,2,...,k1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiuai aackhadaqadaqaaiaadIfadaqhaaqcfasaaKqbaoaabmaajuaibaGa amyqaaGaayjkaiaawMcaaiaadMgaaeaacaWGMbaaaKqbakabg2da9i aadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaa wMcaaiaadMgaaeaacaWGMbaaaaqcfaOaayjkaiaawMcaaiabg2da9i abeI7aXnaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGa ayzkaaGaamyAaaqaaiaadIhajuaGdaqhaaqcfasaaKqbaoaabmaaju aibaGaamyqaaGaayjkaiaawMcaaiaadMgaaeaacaWGMbaaaaaajuaG daqadaqaaiaaigdacqGHsislcqaH4oqCdaWgaaqcfasaaKqbaoaabm aajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaaKqbagqaaaGaayjk aiaawMcaamaaCaaabeqaaiaaigdacqGHsislcaWG4bWaa0baaKqbGe aajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaabaGa amOzaaaaaaqcfaOaaiilaiaadIhadaqhaaqcfasaaKqbaoaabmaaju aibaGaamyqaaGaayjkaiaawMcaaiaadMgaaeaacaWGMbaaaKqbakab g2da9iaaicdacaGGSaGaaGymaiaacYcaqaaaaaaaaaWdbiaacckaca GGGcGaaiiOaiaacckacaWGPbGaeyypa0JaaGymaiaacYcacaaIYaGa aiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGRbGaeyOeI0IaaGymaa aa@82F8@
The density of a future u f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaCaaabeqcfasaaiaadAgaaaaaaa@38B9@ is
f( u f | x ( A ) f , β ( A ) , τ ( A ) )N( i=0 k1 x ( A )i f β ( A )i , τ ( A ) 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamyDamaaCaaabeqcfasaaiaadAgaaaqcfaOaaiiFaiaa dIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawM caaaqaaiaadAgaaaqcfaOaaiilaiabek7aInaaBaaajuaibaqcfa4a aeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaabeaajuaGcaGGSaGaeq iXdq3aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGL PaaaaKqbagqaaaGaayjkaiaawMcaaiabggMi6kaad6eadaqadaqaam aaqahabaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaa caGLOaGaayzkaaGaamyAaaqaaiaadAgaaaqcfaOaeqOSdi2aaSbaaK qbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaa juaGbeaacaGGSaGaeqiXdq3aa0baaKqbGeaajuaGdaqadaqcfasaai aadgeaaiaawIcacaGLPaaaaeaacqGHsislcaaIXaaaaaqaaiaadMga cqGH9aqpcaaIWaaabaGaam4AaiabgkHiTiaaigdaaKqbakabggHiLd aacaGLOaGaayzkaaaaaa@6FF3@

If x ( A ) ( r )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbaacaGLOaGaayzkaaGaamOzaaaaaa a@3F03@ future variables are missing in treatment A, then the density of a future u f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaCaaabeqcfasaaiaadAgaaaaaaa@38B9@ is given

f( u f | x ( A ) *f , β ( A ) , τ ( A ) 1 )= x ( A )k1 f =0 1 N( i=0 k1 x ( A )i f β ( A )i , τ ( A ) 1 ) i=kr k1 θ ( A )i x ( A )i f ( 1 θ ( A )i ) 1 x ( A )i f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamyDamaaCaaabeqcfasaaiaadAgaaaqcfaOaaiiFaiaa dIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawM caaaqaaiaacQcacaWGMbaaaKqbakaacYcacqaHYoGydaWgaaqcfasa aKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqcfayabaGaai ilaiabes8a0naaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGL OaGaayzkaaaabaGaeyOeI0IaaGymaaaaaKqbakaawIcacaGLPaaacq GH9aqpdaaeWbqaaiaad6eadaqadaqaamaaqahabaGaamiEamaaDaaa juaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGaamyAaa qaaiaadAgaaaqcfaOaeqOSdi2aaSbaaKqbGeaajuaGdaqadaqcfasa aiaadgeaaiaawIcacaGLPaaacaWGPbaajuaGbeaacaGGSaGaeqiXdq 3aa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aeaacqGHsislcaaIXaaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaam 4AaiabgkHiTiaaigdaaKqbakabggHiLdaacaGLOaGaayzkaaWaaebC aeaacqaH4oqCdaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaay jkaiaawMcaaiaadMgaaeaacaWG4bqcfa4aa0baaKqbGeaajuaGdaqa daqcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaabaGaamOzaaaaaa aabaGaamyAaiabg2da9iaadUgacqGHsislcaWGYbaabaGaam4Aaiab gkHiTiaaigdaaKqbakabg+GivdaabaGaamiEamaaDaaajuaibaqcfa 4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGaam4AaiabgkHiTiaa igdaaeaacaWGMbaaaKqbakabg2da9iaaicdaaeaacaaIXaaacqGHri s5amaabmaabaGaaGymaiabgkHiTiabeI7aXnaaBaaajuaibaqcfa4a aeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGaamyAaaqabaaajuaGca GLOaGaayzkaaWaaWbaaeqabaGaaGymaiabgkHiTiaadIhadaqhaaqc fasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaae aacaWGMbaaaaaaaaa@A712@

The predictive density of u f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaCaaabeqcfasaaiaadAgaaaaaaa@38B9@ when x ( A ) ( r )f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaa baqcfa4aaeWaaKqbGeaacaWGYbaacaGLOaGaayzkaaGaamOzaaaaaa a@3F03@ is missing is given by
f( u f | x ( A ) *f , β ( A ) , τ ( A ) 1 )f( β ( A ) |data )d β ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamyDamaaCaaabeqcfasaaiaadAgaaaqcfaOaaiiFaiaa dIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawM caaaqaaiaacQcacaWGMbaaaKqbakaacYcacqaHYoGydaWgaaqcfasa aKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqcfayabaGaai ilaiabes8a0naaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGL OaGaayzkaaaabaGaeyOeI0IaaGymaaaaaKqbakaawIcacaGLPaaaca WGMbWaaeWaaeaacqaHYoGydaWgaaqcfasaaKqbaoaabmaajuaibaGa amyqaaGaayjkaiaawMcaaaqabaqcfaOaaiiFaiaadsgacaWGHbGaam iDaiaadggaaiaawIcacaGLPaaacaWGKbGaeqOSdi2aaSbaaKqbGeaa juaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaKqbagqaaaaa@6496@

which is not mathematically tractable. For vague prior densities for β ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aeqaaaaa@3B7F@ and τ ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aeqaaaaa@3BA3@ Taylor's expansion, and using the approximate predictive density of (viii) is

f( u f |x * ( A ) f ,data )= x ( A )fr f =0 1 ... x ( A )f1 f =0 1 N( i=0 k1 x ( A )i f β ^ ( A )i s ( A ) 2 ) i=kr k1 θ ( A )i x ( A )i f ( 1 θ ( A ) i ) 1 x ( A )i f ( 1+ i,j=0 k1 Q ij ( β ^ , s ( A ) 2 ) cov( β ( A )i , β ( A )j ) 2 + Q T ( A ) 2 ( β ^ ( A ) , s ( A ) 2 ) var( T ( A ) ) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadwhadaahaaqabKqbGeaacaWGMbaaaKqbakaacYhacaWG 4bGaaiOkamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOa GaayzkaaaabaGaamOzaaaajuaGcaGGSaGaamizaiaadggacaWG0bGa amyyaaGaayjkaiaawMcaaiabg2da9maaqahabaGaaiOlaiaac6caca GGUaWaaabCaeaacaWGobWaaeWaaeaadaaeWbqaaiaadIhadaqhaaqc fasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaae aacaWGMbaaaKqbakqbek7aIzaajaWaaSbaaKqbGeaajuaGdaqadaqc fasaaiaadgeaaiaawIcacaGLPaaacaWGPbaajuaGbeaaaKqbGeaaca WGPbGaeyypa0JaaGimaaqaaiaadUgacqGHsislcaaIXaaajuaGcqGH ris5aiaacohadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaay jkaiaawMcaaaqaaiaaikdaaaaajuaGcaGLOaGaayzkaaaajuaibaGa amiEaKqbaoaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOa GaayzkaaGaamOzaiabgkHiTiaaigdaaeaacaWGMbaaaiabg2da9iaa icdaaeaacaaIXaaajuaGcqGHris5aaqcfasaaiaadIhajuaGdaqhaa qcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadAga cqGHsislcaWGYbaabaGaamOzaaaacqGH9aqpcaaIWaaabaGaaGymaa qcfaOaeyyeIuoadaqeWbqaaiabeI7aXnaaDaaajuaibaqcfa4aaeWa aKqbGeaacaWGbbaacaGLOaGaayzkaaGaamyAaaqaaiaadIhajuaGda qhaaqcfasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaa dMgaaeaacaWGMbaaaaaaaeaacaWGPbGaeyypa0Jaam4AaiabgkHiTi aadkhaaeaacaWGRbGaeyOeI0IaaGymaaqcfaOaey4dIunadaqadaqa aiaaigdacqGHsislcqaH4oqCdaWgaaqcfasaaKqbaoaabmaajuaiba GaamyqaaGaayjkaiaawMcaaaqcfayabaqcfaIaamyAaaqcfaOaayjk aiaawMcaamaaCaaabeqcfasaaiaaigdacqGHsislcaWG4bqcfa4aa0 baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWG PbaabaGaamOzaaaaaaqcfa4aaeWaaeaacaaIXaGaey4kaSYaaabCae aacaWGrbWaaSbaaKqbGeaacaWGPbGaamOAaaqcfayabaaajuaibaGa amyAaiaacYcacaWGQbGaeyypa0JaaGimaaqaaiaadUgacqGHsislca aIXaaajuaGcqGHris5amaabmaabaGafqOSdiMbaKaacaGGSaGaam4C amaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaa aabaGaeyOeI0IaaGOmaaaaaKqbakaawIcacaGLPaaadaWcaaqaaiGa cogacaGGVbGaaiODamaabmaabaGaeqOSdi2aaSbaaKqbGeaajuaGda qadaqcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaajuaGbeaacaGG SaGaeqOSdi2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawI cacaGLPaaacaWGQbaajuaGbeaaaiaawIcacaGLPaaaaeaacaaIYaaa aiabgUcaRiaadgfadaWgaaqcfasaaiaadsfajuaGdaqhaaqcfasaaK qbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqaaiaaikdaaaaa juaGbeaadaqadaqaaiqbek7aIzaajaWaaSbaaKqbGeaajuaGdaqada qcfasaaiaadgeaaiaawIcacaGLPaaaaKqbagqaaiaacYcacaWGZbWa a0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaae aacqGHsislcaaIYaaaaaqcfaOaayjkaiaawMcaamaalaaabaGaciOD aiaacggacaGGYbWaaeWaaeaacaWGubWaaSbaaKqbGeaajuaGdaqada qcfasaaiaadgeaaiaawIcacaGLPaaaaKqbagqaaaGaayjkaiaawMca aaqaaiaaikdaaaaacaGLOaGaayzkaaaaaa@FC77@ (viii)

Since there are no missing variables in υ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaWbaaeqajuaibaGaamOzaaaaaaa@3986@ , the density of υ f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyXdu 3aaWbaaeqajuaibaGaamOzaaaaaaa@3986@ is same as that can be obtained in Section 2. Then the predictive density of a ' w f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyam aaCaaabeqaaiaacEcaaaGaam4DamaaCaaabeqcfasaaiaadAgaaaaa aa@3A6E@ is given by
f( a' w f |x * ( A ) f , x ( B ) f ,data )= x ( A )fr f =0 1 ... x ( A )f1 f =0 1 N( i=0 k1 ( x ( A )i f β ^ ( A )i x ( B )i f β ^ ( B )i ) , s ( A ) 2 + s ( B ) 2 ( 1+ x ( B ) f ( X ' ( B ) X ( B ) ) 1 x ' ( B ) ) ) i=kr k1 θ ( A )i x ( A )i f ( 1 θ ( A ) i ) 1 x ( A )i f ( 1+ i,j=0 k1 Q ij ( β ^ ( A ) , s ( A ) 2 ) cov( β ( A )i , β ( A )j ) 2 + Q T ( A ) 2 ( β ^ ( A ) , s ( A ) 2 ) var( T ( A ) ) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam OzamaabmaabaGaamyyaiaacEcacaWG3bWaaWbaaeqabaGaamOzaaaa caGG8bGaamiEaiaacQcadaqhaaqcfasaaKqbaoaabmaajuaibaGaam yqaaGaayjkaiaawMcaaaqaaiaadAgaaaqcfaOaaiilaiaadIhadaqh aaqcfasaaKqbaoaabmaajuaibaGaamOqaaGaayjkaiaawMcaaaqaai aadAgaaaqcfaOaaiilaiaadsgacaWGHbGaamiDaiaadggaaiaawIca caGLPaaacqGH9aqpdaaeWbqaaiaac6cacaGGUaGaaiOlamaaqahaba GaamOtamaabmaabaWaaabCaeaadaqadaqaaiaadIhadaqhaaqcfasa aKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaaeaaca WGMbaaaKqbakqbek7aIzaajaWaaSbaaKqbGeaajuaGdaqadaqcfasa aiaadgeaaiaawIcacaGLPaaacaWGPbaajuaGbeaacqGHsislcaWG4b Waa0baaKqbGeaajuaGdaqadaqcfasaaiaadkeaaiaawIcacaGLPaaa caWGPbaabaGaamOzaaaajuaGcuaHYoGygaqcamaaBaaajuaibaqcfa 4aaeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaGaamyAaaqcfayabaaa caGLOaGaayzkaaaajuaibaGaamyAaiabg2da9iaaicdaaeaacaWGRb GaeyOeI0IaaGymaaqcfaOaeyyeIuoacaGGSaGaai4CamaaDaaajuai baqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaabaGaaGOmaa aajuaGcqGHRaWkcaGGZbWaa0baaKqbGeaajuaGdaqadaqcfasaaiaa dkeaaiaawIcacaGLPaaaaeaacaaIYaaaaKqbaoaabmaabaGaaGymai abgUcaRiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamOqaaGa ayjkaiaawMcaaaqaaiaadAgaaaqcfa4aaeWaaeaacaWGybGaai4jam aaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGcbaacaGLOaGaayzkaaaa juaGbeaacaWGybWaaSbaaKqbGeaajuaGdaqadaqcfasaaiaadkeaai aawIcacaGLPaaaaKqbagqaaaGaayjkaiaawMcaamaaCaaabeqcfasa aiabgkHiTiaaigdaaaqcfaOaamiEaiaacEcadaWgaaqcfasaaKqbao aabmaajuaibaGaamOqaaGaayjkaiaawMcaaaqcfayabaaacaGLOaGa ayzkaaaacaGLOaGaayzkaaaajuaibaGaamiEaKqbaoaaDaaajuaiba qcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGaamOzaiabgkHi TiaaigdaaeaacaWGMbaaaiabg2da9iaaicdaaeaacaaIXaaajuaGcq GHris5aaqcfasaaiaadIhajuaGdaqhaaqcfasaaKqbaoaabmaajuai baGaamyqaaGaayjkaiaawMcaaiaadAgacqGHsislcaWGYbaabaGaam OzaaaacqGH9aqpcaaIWaaabaGaaGymaaqcfaOaeyyeIuoaaOqaaKqb aoaarahabaGaeqiUde3aa0baaKqbGeaajuaGdaqadaqcfasaaiaadg eaaiaawIcacaGLPaaacaWGPbaabaGaamiEaKqbaoaaDaaajuaibaqc fa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGaamyAaaqaaiaadA gaaaaaaaqaaiaadMgacqGH9aqpcaWGRbGaeyOeI0IaamOCaaqaaiaa dUgacqGHsislcaaIXaaajuaGcqGHpis1amaabmaabaGaaGymaiabgk HiTiabeI7aXnaaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGL OaGaayzkaaaajuaGbeaajuaicaWGPbaajuaGcaGLOaGaayzkaaWaaW baaeqajuaibaGaaGymaiabgkHiTiaadIhajuaGdaqhaaqcfasaaKqb aoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaaeaacaWGMb aaaaaajuaGdaqadaqaaiaaigdacqGHRaWkdaaeWbqaaiaadgfadaWg aaqcfasaaiaadMgacaWGQbaajuaGbeaaaKqbGeaacaWGPbGaaiilai aadQgacqGH9aqpcaaIWaaabaGaam4AaiabgkHiTiaaigdaaKqbakab ggHiLdWaaeWaaeaacuaHYoGygaqcamaaBaaajuaibaqcfa4aaeWaaK qbGeaacaWGbbaacaGLOaGaayzkaaaajuaGbeaacaGGSaGaam4Camaa Daaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaaba GaeyOeI0IaaGOmaaaaaKqbakaawIcacaGLPaaadaWcaaqaaiGacoga caGGVbGaaiODamaabmaabaGaeqOSdi2aaSbaaKqbGeaajuaGdaqada qcfasaaiaadgeaaiaawIcacaGLPaaacaWGPbaajuaGbeaacaGGSaGa eqOSdi2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcaca GLPaaacaWGQbaajuaGbeaaaiaawIcacaGLPaaaaeaacaaIYaaaaiab gUcaRiaadgfadaWgaaqcfasaaiaadsfajuaGdaqhaaqcfasaaKqbao aabmaajuaibaGaamyqaaGaayjkaiaawMcaaaqaaiaaikdaaaaajuaG beaadaqadaqaaiqbek7aIzaajaWaaSbaaKqbGeaajuaGdaqadaqcfa saaiaadgeaaiaawIcacaGLPaaaaKqbagqaaiaacYcacaWGZbWaa0ba aKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaeaacq GHsislcaaIYaaaaaqcfaOaayjkaiaawMcaamaalaaabaGaciODaiaa cggacaGGYbWaaeWaaeaacaWGubWaaSbaaKqbGeaajuaGdaqadaqcfa saaiaadgeaaiaawIcacaGLPaaaaKqbagqaaaGaayjkaiaawMcaaaqa aiaaikdaaaaacaGLOaGaayzkaaaaaaa@3C13@ (ix)
Analytical solution of D KL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaaaa@39CB@ between the predictive densities (iv) and (ix) is very difficult to obtain but numerical solution can be obtained. In Some situations it is seen that among the explanatory variables, some of the variables are dichotomous and some of the variables are continuous. Among the k1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb GaeyOeI0IaaGymaaaa@391D@ -explanatory variables, without loss of generality we assume that the first  are dichotomous and the remaining last kl1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb GaeyOeI0IaamiBaiabgkHiTiaaigdaaaa@3AFB@ are continuous variables. We also assume that out of l dichotomous future variables last d variables are missing and out of (kl1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa Gaam4AaiabgkHiTiaadYgacqGHsislcaaIXaGaaiykaaaa@3C54@ continuous future variables last g variables are missing. Then the predictive density of future log-odds ratio a ' w f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyam aaCaaabeqaaiaacEcaaaGaam4DamaaCaaabeqcfasaaiaadAgaaaaa aa@3A6E@ when d dichotomous and g continuous variables are missing is given by
f( u f |x * ( A ) f ,data )= x ( A )fr f =0 1 ... x ( A )f1 f =0 1 N( i=0 k1 x ( A )i f β ^ ( A )i s ( A ) 2 ) i=kr k1 θ ( A )i x ( A )i f ( 1 θ ( A ) i ) 1 x ( A )i f ( 1+ i,j=0 k1 Q ij ( β ^ , s ( A ) 2 ) cov( β ( A )i , β ( A )j ) 2 + Q T ( A ) 2 ( β ^ ( A ) , s ( A ) 2 ) var( T ( A ) ) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam OzamaabmaabaGaamyDamaaCaaabeqcfasaaiaadAgaaaqcfaOaaiiF aiaadIhacaGGQaWaa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaai aawIcacaGLPaaaaeaacaWGMbaaaKqbakaacYcacaWGKbGaamyyaiaa dshacaWGHbaacaGLOaGaayzkaaGaeyypa0ZaaabCaeaacaGGUaGaai Olaiaac6cadaaeWbqaaiaad6eadaqadaqaamaaqahabaGaamiEamaa Daaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGaam yAaaqaaiaadAgaaaqcfaOafqOSdiMbaKaadaWgaaqcfasaaKqbaoaa bmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaaKqbagqaaaqcfa saaiaadMgacqGH9aqpcaaIWaaabaGaam4AaiabgkHiTiaaigdaaKqb akabggHiLdGaai4CamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbb aacaGLOaGaayzkaaaabaGaaGOmaaaaaKqbakaawIcacaGLPaaaaKqb GeaacaWG4bqcfa4aa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaai aawIcacaGLPaaacaWGMbGaeyOeI0IaaGymaaqaaiaadAgaaaGaeyyp a0JaaGimaaqaaiaaigdaaKqbakabggHiLdaajuaibaGaamiEaKqbao aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGa amOzaiabgkHiTiaadkhaaeaacaWGMbaaaiabg2da9iaaicdaaeaaca aIXaaajuaGcqGHris5aaGcbaqcfa4aaebCaeaacqaH4oqCdaqhaaqc fasaaKqbaoaabmaajuaibaGaamyqaaGaayjkaiaawMcaaiaadMgaae aacaWG4bqcfa4aa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaa wIcacaGLPaaacaWGPbaabaGaamOzaaaaaaaabaGaamyAaiabg2da9i aadUgacqGHsislcaWGYbaabaGaam4AaiabgkHiTiaaigdaaKqbakab g+GivdWaaeWaaeaacaaIXaGaeyOeI0IaeqiUde3aaSbaaKqbGeaaju aGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaaaKqbagqaaKqbGiaa dMgaaKqbakaawIcacaGLPaaadaahaaqabKqbGeaacaaIXaGaeyOeI0 IaamiEaKqbaoaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGL OaGaayzkaaGaamyAaaqaaiaadAgaaaaaaKqbaoaabmaabaGaaGymai abgUcaRmaaqahabaGaamyuamaaBaaajuaibaGaamyAaiaadQgaaKqb agqaaaqcfasaaiaadMgacaGGSaGaamOAaiabg2da9iaaicdaaeaaca WGRbGaeyOeI0IaaGymaaqcfaOaeyyeIuoadaqadaqaaiqbek7aIzaa jaGaaiilaiaadohadaqhaaqcfasaaKqbaoaabmaajuaibaGaamyqaa GaayjkaiaawMcaaaqaaiabgkHiTiaaikdaaaaajuaGcaGLOaGaayzk aaWaaSaaaeaaciGGJbGaai4BaiaacAhadaqadaqaaiabek7aInaaBa aajuaibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaGaamyA aaqcfayabaGaaiilaiabek7aInaaBaaajuaibaqcfa4aaeWaaKqbGe aacaWGbbaacaGLOaGaayzkaaGaamOAaaqcfayabaaacaGLOaGaayzk aaaabaGaaGOmaaaacqGHRaWkcaWGrbWaaSbaaKqbGeaacaWGubqcfa 4aa0baaKqbGeaajuaGdaqadaqcfasaaiaadgeaaiaawIcacaGLPaaa aeaacaaIYaaaaaqcfayabaWaaeWaaeaacuaHYoGygaqcamaaBaaaju aibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaajuaGbeaa caGGSaGaam4CamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGbbaaca GLOaGaayzkaaaabaGaeyOeI0IaaGOmaaaaaKqbakaawIcacaGLPaaa daWcaaqaaiGacAhacaGGHbGaaiOCamaabmaabaGaamivamaaBaaaju aibaqcfa4aaeWaaKqbGeaacaWGbbaacaGLOaGaayzkaaaajuaGbeaa aiaawIcacaGLPaaaaeaacaaIYaaaaaGaayjkaiaawMcaaaaaaa@FD16@ (x)
Again, analytical solution of D KL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaaaa@39CB@ between the predictive densities (iv) and (x) is very difficult but we can obtain its numerical solution. In similar way we can derive the predictive density of future log-odds ratio when some future variables are missing in treatment B.

Example 1 revisited: This example is based on the u shot data of Example 1. From Figure 3 we have observed same as Examples 1 and 2 that the discrepancies are less around the mean of the missing variables. Moreover we have observed from Figures 1 and 3 that the discrepancies of the missing variables are less as compared to the discrepancies of the deleted variables.

Example 2 revisited: This example is based on the simulation data of Example 2 and here we have also got same conclusion as Example 1 revisited (Figures 2 & 4).

Group A Group B

Figure 1: Three dimensional scatter plots based on real data for DKL
x1 is deleted

Group A Group B

Figure 2: Three dimensional scatter plots based on simulated data for DKL
x1 is deleted

Group A Group B

Figure 3: Three dimensional scatter plots based on real data for DKL
xf1 is missing

Group A Group B

Figure 4: Three dimensional scatter plots based on simulated data for DKL
xf1 is missing

Examples 1 and 2 revisited: In this example, we have used D KL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaam4saiaadYeaaKqbagqaaaaa@39CB@  values for real data for drawing box plots for each cases (deleted and missing). From Figure 5, we have observed that x2 is more in uential than x1. Moreover the discrepancies are much less in missing case than deleted case. We have got same result in simulation study and are illustrated in Figure 6.

Treatment A Treatment B

Figure 5: Box plot for DKL based on real data

Treatment A Treatment B

Figure 6: Box plot for DKL based on simulated data

Evaluation of Predictive Probability of a Logistic Model

We consider the logistic model as
Pr( y=1|x,β )=exp( xβ )/( 1+exp( xβ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bGaeyypa0JaaGymaiaacYhacaWG4bGaaiil aiabek7aIbGaayjkaiaawMcaaiabg2da9iGacwgacaGG4bGaaiiCam aabmaabaGaamiEaiabek7aIbGaayjkaiaawMcaaiaac+cadaqadaqa aiaaigdacqGHRaWkciGGLbGaaiiEaiaacchadaqadaqaaiaadIhacq aHYoGyaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@53BE@
The probability that a future response yf will be a success is given by
Pr( y f =1| x f ,β )=exp( x f β )/( 1+exp( x f β ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhadaahaaqabKqbGeaacaWGMbaaaKqbak aacYcacqaHYoGyaiaawIcacaGLPaaacqGH9aqpciGGLbGaaiiEaiaa cchadaqadaqaaiaadIhadaahaaqabKqbGeaacaWGMbaaaKqbakabek 7aIbGaayjkaiaawMcaaiaac+cadaqadaqaaiaaigdacqGHRaWkciGG LbGaaiiEaiaacchadaqadaqaaiaadIhadaahaaqabKqbGeaacaWGMb aaaKqbakabek7aIbGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@5AE2@
We assume that the conditional density of xf(r) given xf is independent of, where xf denotes the future explanatory variables without variables xf(r). Then predictive probabilities of yf will be a success for models are given by
Pr( y f =1| x f ,data )= Pr( y f =1| x f ,β ) f( β|data )dβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhadaahaaqabKqbGeaacaWGMbaaaKqbak aacYcacaWGKbGaamyyaiaadshacaWGHbaacaGLOaGaayzkaaGaeyyp a0Zaa8qaaeaaciGGqbGaaiOCamaabmaabaGaamyEamaaCaaabeqcfa saaiaadAgaaaqcfaOaeyypa0JaaGymaiaacYhacaWG4bWaaWbaaeqa juaibaGaamOzaaaajuaGcaGGSaGaeqOSdigacaGLOaGaayzkaaaabe qabiabgUIiYdGaamOzamaabmaabaGaeqOSdiMaaiiFaiaadsgacaWG HbGaamiDaiaadggaaiaawIcacaGLPaaacaWGKbGaeqOSdigaaa@62A4@
and
Pr( y f =1|x * f ,data )= Pr( y f =1|x * f ,β ) f( β|data )dβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhacaGGQaWaaWbaaeqajuaibaGaamOzaa aajuaGcaGGSaGaamizaiaadggacaWG0bGaamyyaaGaayjkaiaawMca aiabg2da9maapeaabaGaciiuaiaackhadaqadaqaaiaadMhadaahaa qabKqbGeaacaWGMbaaaKqbakabg2da9iaaigdacaGG8bGaamiEaiaa cQcadaahaaqabKqbGeaacaWGMbaaaKqbakaacYcacqaHYoGyaiaawI cacaGLPaaaaeqabeGaey4kIipacaWGMbWaaeWaaeaacqaHYoGycaGG 8bGaamizaiaadggacaWG0bGaamyyaaGaayjkaiaawMcaaiaadsgacq aHYoGyaaa@6400@
respectively. Simple analytically tractable priors are not available here. Numerical integration techniques might be used for some specified priors to approximate Pr( y f =1|x * f ,data ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhacaGGQaWaaWbaaeqajuaibaGaamOzaa aajuaGcaGGSaGaamizaiaadggacaWG0bGaamyyaaGaayjkaiaawMca aaaa@4728@  and Pr( y f =1| x f ,data ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhadaahaaqabKqbGeaacaWGMbaaaKqbak aacYcacaWGKbGaamyyaiaadshacaWGHbaacaGLOaGaayzkaaaaaa@467A@ , respectively.

Normal approximation for the posterior density

Let us suppose that the sample size is large. Lindley [12] stated that the posterior density f( β|data ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiabek7aIjaacYhacaWGKbGaamyyaiaadshacaWGHbaacaGL OaGaayzkaaaaaa@3F3C@ may then be well approximated by its asymptotic normal form as
f( β|data ) N p ( β ^ , ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiabek7aIjaacYhacaWGKbGaamyyaiaadshacaWGHbaacaGL OaGaayzkaaGaeyisISRaamOtamaaBaaajuaibaGaamiCaaqcfayaba WaaeWaaeaacuaHYoGygaqcaiaacYcacqGHris5aiaawIcacaGLPaaa aaa@4920@
where ^ _ is the maximum likelihood estimate of _, _ = ( H) 1 and H is the Hessian of
log L(_) evaluated at b_.

h jl ( β ^ )= i=1 n x ij x il exp( x i β ^ ) ( 1+exp( x i β ^ ) ) 2 ,j,l=0,1,...,k, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIgada WgaaqcfasaaiaadQgacaWGSbaajuaGbeaadaqadaqaaiqbek7aIzaa jaaacaGLOaGaayzkaaGaeyypa0JaeyOeI0YaaabCaeaadaWcaaqaai aadIhadaWgaaqcfasaaiaadMgacaWGQbaajuaGbeaacaWG4bWaaSba aKqbGeaacaWGPbGaamiBaaqcfayabaGaciyzaiaacIhacaGGWbWaae WaaeaacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaacuaHYoGygaqc aaGaayjkaiaawMcaaaqaamaabmaabaGaaGymaiabgUcaRiGacwgaca GG4bGaaiiCamaabmaabaGaamiEamaaBaaajuaibaGaamyAaaqcfaya baGafqOSdiMbaKaaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaahaa qabKqbGeaacaaIYaaaaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaa d6gaaKqbakabggHiLdGaaiilaiaacQgacaGGSaGaaiiBaiabg2da9i aaicdacaGGSaGaaGymaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGa ai4AaiaacYcaaaa@6E94@
Where xij is the jth component of xi with xi0 = 1: For given xf , z = xf will have approximately a posteriori a normal distribution with mean bxf = xf b and variance d2xf = xf xf0, and with probability density function (zjbxf ; d2xf ). Using the transformation we can approximate f( jxf ; data) by
Pr( y f =1| x f ,data ) exp( z ) 1+exp( z ) ϕ( z| b x f , d x f 2 )dz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhadaahaaqabKqbGeaacaWGMbaaaKqbak aacYcacaWGKbGaamyyaiaadshacaWGHbaacaGLOaGaayzkaaGaeyis IS7aa8qaaeaadaWcaaqaaiGacwgacaGG4bGaaiiCamaabmaabaGaam OEaaGaayjkaiaawMcaaaqaaiaaigdacqGHRaWkciGGLbGaaiiEaiaa cchadaqadaqaaiaadQhaaiaawIcacaGLPaaaaaaabeqabiabgUIiYd Gaeqy1dy2aaeWaaeaacaWG6bGaaiiFaiaadkgadaWgaaqaamaaBaaa juaibaGaamiEaKqbaoaaCaaajuaibeqaaiaadAgaaaaajuaGbeaaca GGSaGaamizamaaDaaajuaibaGaamiEaKqbaoaaCaaajuaibeqaaiaa dAgaaaaabaGaaGOmaaaaaKqbagqaaaGaayjkaiaawMcaaiaadsgaca WG6baaaa@685D@
Analytical evaluation of (4.1) is very di cult. We can however evaluate then by numerical integration techniques viz Gauss-Hermite Quadrature Abramowitz and Stegun [13], Normal approximation Cox [14], Laplace's approximation de Bruijn [15].

If the sample size is small, the posterior normality assumption may not be accurate. Therefore, we consider Flat prior approximation Tierney and Kadane [16] as an alternative approach using the Laplace's method for integrals

Effect of the variables x f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaWbaaSqabeaajugWaiaadAgaaaaaaa@3A56@

Here we assume that the future variables x f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaWbaaSqabeaajugWaiaadAgaaaaaaa@3A56@ are dependent and the density of x f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaWbaaSqabeaajugWaiaadAgaaaaaaa@3A56@ is p-dimensional multivariate normal i.e.
f( x f ) N p ( n,ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadIhadaahaaqabKqbGeaacaWGMbaaaaqcfaOaayjkaiaa wMcaaiabggMi6kaad6eadaWgaaqcfasaaiaadchaaKqbagqaamaabm aabaGaamOBaiaacYcacqaHipqEaiaawIcacaGLPaaaaaa@451B@
The conditional density of x ( r ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BF8@  for given x *f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b WcdaahaaqabeaajugWaiaacQcacaWGMbaaaaaa@3A76@ is  
f( x ( r ) f |x * f ) N r ( n * ( r ) ,ψ * ( r ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadIhadaqhaaqcfasaaKqbaoaabmaajuaibaGaamOCaaGa ayjkaiaawMcaaaqaaiaadAgaaaqcfaOaaiiFaiaadIhacaGGQaWaaW baaeqajuaibaGaamOzaaaaaKqbakaawIcacaGLPaaacqGHHjIUcaWG obWaaSbaaKqbGeaacaWGYbaajuaGbeaadaqadaqaaiaad6gacaGGQa WaaSbaaKqbGeaajuaGdaqadaqcfasaaiaadkhaaiaawIcacaGLPaaa aKqbagqaaiaacYcacqaHipqEcaGGQaWaaSbaaKqbGeaajuaGdaqada qcfasaaiaadkhaaiaawIcacaGLPaaaaKqbagqaaaGaayjkaiaawMca aaaa@565B@
The probability of y f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b WcdaahaaqabeaajugWaiaadAgaaaaaaa@39C9@ as a success when x ( r ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BF8@ is missing given by

Pr( y f =1| x f ,data )=| g( β )f( β|data ) dβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhadaahaaqabKqbGeaacaWGMbaaaKqbak aacYcacaGGKbGaaiyyaiaacshacaGGHbaacaGLOaGaayzkaaGaeyyp a0JaaiiFamaapeaabaGaam4zamaabmaabaGaeqOSdigacaGLOaGaay zkaaGaamOzamaabmaabaGaeqOSdiMaaiiFaiaacsgacaGGHbGaaiiD aiaacggaaiaawIcacaGLPaaaaeqabeGaey4kIipacaWGKbGaeqOSdi gaaa@59BE@
ϕ( ( i=0 kr x i f β i + i=kr+1 k n i β i )/ ( k 2 + i=kr+1 k β i 2 Ψ i 2 ) 1/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgIKi7k abew9aMnaabmaabaWaaeWaaeaadaaeWbqaaiaadIhadaqhaaqcfasa aiaadMgaaeaacaWGMbaaaKqbakabek7aInaaBaaajuaibaGaamyAaa qcfayabaGaey4kaSYaaabCaeaacaWGUbWaaSbaaKqbGeaacaWGPbaa juaGbeaacqaHYoGydaWgaaqcfasaaiaadMgaaKqbagqaaaqcfasaai aadMgacqGH9aqpcaWGRbGaeyOeI0IaamOCaiabgUcaRiaaigdaaeaa caWGRbaajuaGcqGHris5aaqcfasaaiaadMgacqGH9aqpcaaIWaaaba Gaam4AaiabgkHiTiaadkhaaKqbakabggHiLdaacaGLOaGaayzkaaGa ai4lamaabmaabaGaam4AamaaCaaabeqcfasaaiaaikdaaaqcfaOaey 4kaSYaaabCaeaacqaHYoGydaqhaaqcfasaaiaadMgaaeaacaaIYaaa aKqbakabfI6aznaaDaaajuaibaGaamyAaaqaaiaaikdaaaaabaGaam yAaiabg2da9iaadUgacqGHsislcaWGYbGaey4kaSIaaGymaaqaaiaa dUgaaKqbakabggHiLdaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG ymaiaac+cacaaIYaaaaaqcfaOaayjkaiaawMcaaaaa@7842@

The integral in (ii) can be evaluated as the integral in (i) using Taylor's and Laplace's approximations.

If, instead, the future variables x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@ ,…, x k f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadUgaaeaacaWGMbaaaaaa@39A1@ are independently and normally distributed with mean and variance ψ i 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aa0baaKqbGeaacaWGPbaabaGaaGOmaaaaaaa@3A4C@  (i = 1, 2, … , k), then the conditional density of x ( r ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BF8@ is
f( x ( r ) f | x *f )f( x ( r ) f ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEamaaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbaa caGLOaGaayzkaaaabaGaamOzaaaajuaGcaGG8bqcLbsacaWG4bWcda ahaaqcfayabeaajugWaiaacQcacaWGMbaaaaqcfaOaayjkaiaawMca aiabggMi6kaadAgadaqadaqaaiaadIhadaqhaaqcfasaaKqbaoaabm aajuaibaGaamOCaaGaayjkaiaawMcaaaqaaiaadAgaaaaajuaGcaGL OaGaayzkaaaaaa@4FD5@
Consequently, we get
Pr( y f =1|x * f ,data )= h( β )f( β|data ) dβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhacaGGQaWaaWbaaeqajuaibaGaamOzaa aajuaGcaGGSaGaaiizaiaacggacaGG0bGaaiyyaaGaayjkaiaawMca aiabg2da9maapeaabaGaamiAamaabmaabaGaeqOSdigacaGLOaGaay zkaaGaamOzamaabmaabaGaeqOSdiMaaiiFaiaacsgacaGGHbGaaiiD aiaacggaaiaawIcacaGLPaaaaeqabeGaey4kIipacaWGKbGaeqOSdi gaaa@596D@
See Aitchison and Begg (1976) in this context. Again,
Pr( y f =1|x * f ,data )= h( β )f( β|data ) dβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhacaGGQaWaaWbaaeqajuaibaGaamOzaa aajuaGcaGGSaGaaiizaiaacggacaGG0bGaaiyyaaGaayjkaiaawMca aiabg2da9maapeaabaGaamiAamaabmaabaGaeqOSdigacaGLOaGaay zkaaGaamOzamaabmaabaGaeqOSdiMaaiiFaiaacsgacaGGHbGaaiiD aiaacggaaiaawIcacaGLPaaaaeqabeGaey4kIipacaWGKbGaeqOSdi gaaa@596D@

Variables xf are dichotomous

Here we assume that the variables x f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b WcdaWgaaqaaKqzadGaamOzaaWcbeaaaaa@39D2@ are independent and they can take only two values 0 or 1. We also assume that x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  is distributed as
Pr( X i f = X i f )= θ i x i f ( 1 θ i ) 1 x i f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWGybWaa0baaKqbGeaacaWGPbaabaGaamOzaaaa juaGcqGH9aqpcaWGybWaa0baaKqbGeaacaWGPbaabaGaamOzaaaaaK qbakaawIcacaGLPaaacqGH9aqpcqaH4oqCdaqhaaqcfasaaiaadMga aeaacaWG4bqcfa4aa0baaKqbGeaacaWGPbaabaGaamOzaaaaaaqcfa 4aaeWaaeaacaaIXaGaeyOeI0IaeqiUde3aaSbaaKqbGeaacaWGPbaa juaGbeaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIXaGaeyOeI0 IaamiEaKqbaoaaDaaajuaibaGaamyAaaqaaiaadAgaaaaaaaaa@5696@ x i f =0,1, i=1,2,....,k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaadMgaaeaacaWGMbaaaKqbakabg2da9iaaicdacaGG SaGaaGymaiaacYcaqaaaaaaaaaWdbiaacckapaGaamyAaiabg2da9i aaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGa aiilaiaadUgaaaa@488E@
If x ( r ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaKqbaoaabmaajuaibaGaamOCaaGaayjkaiaawMcaaaqa aiaadAgaaaaaaa@3BED@ is missing the probability of y f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMhada ahaaqabKqbGeaacaWGMbaaaaaa@38B2@ as a success is given by
Pr( y f =1|x * f ,β )= x kr+1 f =0 1 ... x k f 1 exp( x f β ) 1+exp( x f β ) i=kr+1 k θ i x i f ( 1 θ i ) 1 x i f =h( β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhacaGGQaWaaWbaaeqajuaibaGaamOzaa aajuaGcaGGSaGaeqOSdigacaGLOaGaayzkaaGaeyypa0ZaaabCaeaa caGGUaGaaiOlaiaac6caaeaajuaicaWG4bqcfa4aa0baaKqbGeaaca WGRbGaeyOeI0IaamOCaiabgUcaRiaaigdaaeaacaWGMbaaaiabg2da 9iaaicdaaeaacaaIXaaajuaGcqGHris5amaaqahabaWaaSaaaeaaci GGLbGaaiiEaiaacchadaqadaqaaiaadIhadaahaaqabKqbGeaacaWG MbaaaKqbakabek7aIbGaayjkaiaawMcaaaqaaiaaigdacqGHRaWkci GGLbGaaiiEaiaacchadaqadaqaaiaadIhadaahaaqabKqbGeaacaWG MbaaaKqbakabek7aIbGaayjkaiaawMcaaaaaaKqbGeaacaWG4bqcfa 4aa0baaKqbGeaacaWGRbaabaGaamOzaaaaaeaacaaIXaaajuaGcqGH ris5amaarahabaGaeqiUde3aa0baaKqbGeaacaWGPbaabaGaamiEaK qbaoaaDaaajuaibaGaamyAaaqaaiaadAgaaaaaaaqaaiaadMgacqGH 9aqpcaWGRbGaeyOeI0IaamOCaiabgUcaRiaaigdaaeaacaWGRbaaju aGcqGHpis1amaabmaabaGaaGymaiabgkHiTiabeI7aXnaaBaaajuai baGaamyAaaqcfayabaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG ymaiabgkHiTiaadIhajuaGdaqhaaqcfasaaiaadMgaaeaacaWGMbaa aaaajuaGcqGH9aqpcaWGObWaaeWaaeaacqaHYoGyaiaawIcacaGLPa aaaaa@9216@
If   x ( r ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaDaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbaacaGLOaGaayzkaaaa baGaamOzaaaaaaa@3BF8@ is missing the probability of y f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b qcfa4aaWbaaSqabeaajugWaiaadAgaaaaaaa@3A57@ as a success is given by
Pr( y f =1|x * f ,β )= x kr+1 f =0 1 ... x k f 1 exp( x f β ) 1+exp( x f β ) i=kr+1 k θ i x i f ( 1 θ i ) 1 x i f =h( β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhacaGGQaWaaWbaaeqajuaibaGaamOzaa aajuaGcaGGSaGaeqOSdigacaGLOaGaayzkaaGaeyypa0ZaaabCaeaa caGGUaGaaiOlaiaac6caaeaajuaicaWG4bqcfa4aa0baaKqbGeaaca WGRbGaeyOeI0IaamOCaiabgUcaRiaaigdaaeaacaWGMbaaaiabg2da 9iaaicdaaeaacaaIXaaajuaGcqGHris5amaaqahabaWaaSaaaeaaci GGLbGaaiiEaiaacchadaqadaqaaiaadIhadaahaaqabKqbGeaacaWG MbaaaKqbakabek7aIbGaayjkaiaawMcaaaqaaiaaigdacqGHRaWkci GGLbGaaiiEaiaacchadaqadaqaaiaadIhadaahaaqabKqbGeaacaWG MbaaaKqbakabek7aIbGaayjkaiaawMcaaaaaaKqbGeaacaWG4bqcfa 4aa0baaKqbGeaacaWGRbaabaGaamOzaaaaaeaacaaIXaaajuaGcqGH ris5amaarahabaGaeqiUde3aa0baaKqbGeaacaWGPbaabaGaamiEaK qbaoaaDaaajuaibaGaamyAaaqaaiaadAgaaaaaaaqaaiaadMgacqGH 9aqpcaWGRbGaeyOeI0IaamOCaiabgUcaRiaaigdaaeaacaWGRbaaju aGcqGHpis1amaabmaabaGaaGymaiabgkHiTiabeI7aXnaaBaaajuai baGaamyAaaqcfayabaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG ymaiabgkHiTiaadIhajuaGdaqhaaqcfasaaiaadMgaaeaacaWGMbaa aaaajuaGcqGH9aqpcaWGObWaaeWaaeaacqaHYoGyaiaawIcacaGLPa aaaaa@9216@
The predictive probability of y f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMhada ahaaqabKqbGeaacaWGMbaaaaaa@38B2@ as a success when x ( r ) f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaKqbaoaabmaajuaibaGaamOCaaGaayjkaiaawMcaaaqa aiaadAgaaaaaaa@3BED@ is missing is given by

Pr( y f =1|x * f ,data )= h( β )f( β|data ) dβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaWG5bWaaWbaaeqajuaibaGaamOzaaaajuaGcqGH 9aqpcaaIXaGaaiiFaiaadIhacaGGQaWaaWbaaeqajuaibaGaamOzaa aajuaGcaGGSaGaaiizaiaacggacaGG0bGaaiyyaaGaayjkaiaawMca aiabg2da9maapeaabaGaamiAamaabmaabaGaeqOSdigacaGLOaGaay zkaaGaamOzamaabmaabaGaeqOSdiMaaiiFaiaacsgacaGGHbGaaiiD aiaacggaaiaawIcacaGLPaaaaeqabeGaey4kIipacaWGKbGaeqOSdi gaaa@596D@

If the sample size is large, assuming the normality assumption for the posterior density we can approximate (iii) using Taylor's theorem, Laplace's method and normal approximation.

Example: one variable case

Here we consider two different logistic models based on any single variable either x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaigdaaKqbagqaaaaa@390E@  or x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaikdaaKqbagqaaaaa@390F@ . We want to measure the discrepancies between the predictive probability p ^ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaamyAaaqcfayabaaaaa@3949@ , based on a single variable x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaS WaaSbaaeaajugWaiaadMgaaSqabaaaaa@39CA@ when x i f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaS Waa0baaeaajugWaiaadMgaaSqaaKqzadGaamOzaaaaaaa@3BE4@ is known, and the predictive probability p ^ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGimaaqcfayabaaaaa@3915@ , based on xi alone when x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@ is missing, to assess the influence of the missing variable x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  , i = 1; 2. The predictive probability p ^ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaamyAaaqcfayabaaaaa@3949@ is determined using quadrature approximation and the predictive probability p ^ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGimaaqcfayabaaaaa@3915@ is determined using second order Taylor's approximation.

We assume that the marginal densities of the future variables x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@ and x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@  are normal with means 33.35, 78.24 and variances 65.39, 1827.0 respectively, where means and variances are the estimated sample means and sample variances from the observed data. We employ the absolute difference of probabilities and Kullback-Leibler divergence measure to assess the influence of the missing variable. The discrepancies are drawn in Figure 7. Here we see that the discrepancies due to missing x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  in the predictive probability based on x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaigdaaKqbagqaaaaa@390E@ are very large compared to the discrepancies due to missing x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@  in the predictive probability based on x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaikdaaKqbagqaaaaa@390F@ . The discrepancies are less around the mean of the missing variable.

xf1 is missing xf2 is missing
Kullback-Leibler directed divergence DKL

Figure 7: Box plot for DKL based on simulated data

Example: two-variable Case

Now we consider that the predictive probability based on two variables x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  and x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@  when both x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  and x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@  are known is denoted by p ^ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGymaiaaikdaaKqbagqaaaaa@39D2@ and the predictive probability p ^ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaaa@3A38@ , i=0,1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaIWaGaaiilaiaaigdaaaa@3A92@ , j=0,2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpcaaIWaGaaiilaiaaikdaaaa@3A94@ and ( i,j )( 1,2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamyAaiaacYcacaWGQbaacaGLOaGaayzkaaGaeyiyIK7aaeWaaeaa caaIXaGaaiilaiaaikdaaiaawIcacaGLPaaaaaa@4006@  based on x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaigdaaKqbagqaaaaa@390E@  and x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaikdaaKqbagqaaaaa@390F@  when any future variable is missing. " indicates missing variable. Here also the predictive probability p ^ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGymaiaaikdaaKqbagqaaaaa@39D2@ is determined using quadrature approximation and predictive probabilities p ^ 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGymaiaaicdaaKqbagqaaaaa@39D0@ , p ^ 02 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGimaiaaikdaaKqbagqaaaaa@39D1@ and p ^ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGimaaqcfayabaaaaa@3915@ are determined using second order Taylor's approximation. Here we assume that the joint density of x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  and x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@ is bivariate normal with correlation coefficient 0:33 which is the estimated sample correlation coefficient from the observed data. The absolute differences of the two predictive probabilities p ^ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGymaiaaikdaaKqbagqaaaaa@39D2@ and p ^ 02 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGimaiaaikdaaKqbagqaaaaa@39D1@ when x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@ is missing and the absolute differences of the two predictive probabilities p ^ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGymaiaaikdaaKqbagqaaaaa@39D2@ and p ^ 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaaGymaiaaicdaaKqbagqaaaaa@39D0@ when x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@  is missing are drawn in Figure 8. Kullback-Leibler directed divergence DKL are drawn in Figure 9. The discrepancies when x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  is missing and for different given values of the other variable for both the cases are close together since the correlation between x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@ and x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@  are very small. The discrepancies due to missing x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  are very large compared to missing x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@  except near the mean of the missing variable. If both x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  and x 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaikdaaeaacaWGMbaaaaaa@396D@  are missing the discrepancies are drawn in Figure 10. These discrepancies are very similar to the discrepancies due to missing x 1 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada qhaaqcfasaaiaaigdaaeaacaWGMbaaaaaa@396C@  alone in the predictive probability based on x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaigdaaKqbagqaaaaa@390E@  and x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaikdaaKqbagqaaaaa@390F@  since the contribution of x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaikdaaKqbagqaaaaa@390F@ is negligible.

xf1 is missing
Absolute difference jp^12 p^10j

xf2 is missing

Figure 8: Absolute difference jp^12p^02j

xf1 is missing
Kullback-Leibler directed divergence DKL

xf2 is missing

Figure 9: Box plot for DKL based on simulated data

Kullback-Leibler directed divergence DKL

xf1 and xf2 are both missing

Figure 10: Absolute difference jp^12p^00j.

Concluding Remarks

In our present study we have observed that the discrepancies are minimum around the mean of the deleted variables as well as the mean of the missing future variables in both the logistic model and the log-odds ratio; the discrepancies are larger if the deleted or missing variables are more influential; the discrepancies in the deleted case are higher than the missing case.

In this present paper we studied the important problem of predictive influence of variables on the log odds ratio under a Bayesian set up. The treatment difference
Pr( Y i =1| Z i =1, x i )Pr( Y i =1| Z i =0, x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaGGzbWaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH 9aqpcaaIXaGaaiiFaiaadQfadaWgaaqcfasaaiaadMgaaKqbagqaai abg2da9iaaigdacaGGSaGaaiiEamaaBaaajuaibaGaamyAaaqcfaya baaacaGLOaGaayzkaaGaeyOeI0IaciiuaiaackhadaqadaqaaiaacM fadaWgaaqcfasaaiaadMgaaKqbagqaaiabg2da9iaaigdacaGG8bGa amOwamaaBaaajuaibaGaamyAaaqcfayabaGaeyypa0JaaGimaiaacY cacaGG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIcacaGLPaaa aaa@58A5@
Or the risk of ratio
Pr( Y i =1| Z i =1, x i )/Pr( Y i =1| Z i =1, x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGaccfaca GGYbWaaeWaaeaacaGGzbWaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH 9aqpcaaIXaGaaiiFaiaadQfadaWgaaqcfasaaiaadMgaaKqbagqaai abg2da9iaaigdacaGGSaGaaiiEamaaBaaajuaibaGaamyAaaqcfaya baaacaGLOaGaayzkaaGaai4laiGaccfacaGGYbWaaeWaaeaacaGGzb WaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH9aqpcaaIXaGaaiiFaiaa dQfadaWgaaqcfasaaiaadMgaaKqbagqaaiabg2da9iaaigdacaGGSa GaaiiEamaaBaaajuaibaGaamyAaaqcfayabaaacaGLOaGaayzkaaaa aa@586C@
can also be studied along the same lines.

We have also considered the influence of missing future explanatory variables in a logistic model. Influence of missing future explanatory variables in a Probit and complementary log-log models can also be studied in similar fashion.

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