ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 4 - 2016
Modified Data Analysis in Two Period Cross-Over Design
Oyeka ICA1 and Okeh UM2*
1Department of Statistics, Nnamdi Azikiwe University Awka, Nigeria
2Department of Industrial Mathematics and Applied Statistics, Ebonyi State University Abakaliki, Nigeria
Received: March 05, 2016 | Published: March 30, 2016
*Corresponding author: Okeh UM, Department of Industrial Mathematics and Applied Statistics, Ebonyi State University Abakaliki, Nigeria, Email:
Citation: Oyeka ICA, Okeh UM (2016) Modified Data Analysis in Two Period Cross-Over Design. Biom Biostat Int J 3(4): 00071. DOI: 10.15406/bbij.2016.03.00071

Abstract

This paper proposes and presents a chi-square statistical method for the analysis of response from one period cross over design for two sample data in which the sampled populations may be measurements that are numeric (assuming real values) and non-numeric assuming only values on the nominal scale. Test statistics are developed for testing the null hypothesis that subjects who receive each of the treatments first do not differ in their response as well as the null hypothesis that subjects exposed to one of the treatment or experimental conditions first do not on the average differ in their responses with those exposed to the other treatment or experimental condition first. Estimates of the proportions responding positive; experiencing no change in response or responding negative are provided for subjects exposed to each treatment first as well as for the two treatments together. The proposed method which is illustrated with some sample data can be used with either numeric or non-numeric data and is shown to be at least as powerful as the traditional two sample small ’t’ test.

Keywords: Cross over; Treatment; Chi-square; Design; Patients

Introduction

Suppose subjects for a clinical trial are first matched on characteristics associated with the outcome understudy such as a disease and randomly assigned the treatments T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38EA@ and. In particular, suppose as in a cross over design each subject serves as his own control, that is, each patient receives each treatment. One half of the sample of 2n patients or subjects is randomly selected to be given the two treatments in one order and the other half to be given the treatments in the reversed order. That is n of the random sample of the 2n patients or subjects is given treatment T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38EA@ first and treatment T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38EB@ later and the remaining n subjects is given treatment T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38EB@ first and treatment T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38EA@  later.

A number of factors must be guarded against in analyzing the data from such studies. However, the order in which the treatments are given may affect the response [1]. A test that is valid when order effects are present has been described by Gart [2]. Another factor to be guarded against is the possibility that a treatment’s effectiveness may be long lasting and hence may affect the response to the treatment given after it. When this so-called carry over effect operates and when it is unequal for the two treatments, then for comparing their effectiveness, only the data from the first period may be used [3]. Specifically, the responses by the subjects given one of the treatments first must be compared with the responses by the subjects given the other treatment first. In this paper we present a method for analyzing data from a crossover design in which each subjects serves as his own control and analysis is based on responses by patients given one of the treatments first and responses by patients given the other treatment first. Here allowance is made for the possibility that patients or subjects may die or drop out of the study.

The Proposed Method

In the regular two crossover design were subjects served as own control in controlled clinical trials or diagnosis screening test to study the differential effects of two procedures such as drugs or treatments. Random samples of matched pairs might in terms of some demographic characteristics such as age, gender or body mass index are used. A randomly selected subject from each of the matched pairs of subjects is given or administered one of the 2 treatments or drugs first, while the remaining subjects in the matched pair of subjects is given or administered the remaining test drug or treatment first. This procedure is later repeated in the reverse order. That is the randomly selected subject in each matched pair of subjects given one of the two days first is now given the other drug or treatments while the remaining subject in the pair earlier given the 2nd treatment first is now given the first treatment or drug. Because of some of the problems that may often arise in these type of clinical trials in which the effects of the drugs may be long lasting, each having carry-over effects with long dry out periods, the usual practice is often to base statistical analysis and comparison of subject responses to the two treatments on only subject responses to treatments, tests or drug administered first, while treating responses obtained during the second administration of the drugs perhaps only to gauge the pattern of responses.

We here however propose a modification of this approach. Here only those subjects in each matched pairs of subjects who failed to respond positive when administered one of the treatments or tests will be administered a second treatment or test later. Similarly only those subjects in each matched pair of subjects who respond negative when administered the second drug or treatment first will later be administered the other treatment. This approach would enable the researcher not only compare the differential effects of the 2 drugs or treatment when they are administered to subjects in the matched pairs of subjects with one of the treatments given one of the subjects first and the other treatments given to the remaining subjects in the pair first. The procedure will also enable the researcher determine whether on the average the proportion of matched pairs of subjects who fail to respond positive when administered one of the 2 treatment first but respond positive when administered the other treatment later are equal to a proportion of subjects in a matched pairs of subjects who respond negative when administered the second treatment first but respond positive when administered the first treatment later.

To develop a statistical method to compare the proportion of subjects in the matched pairs of subjects who respond positive when administered the test, drug or treatment T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38EA@ say first with the proportion of subjects in the matched pairs of subjects who test or respond positive when administered test, drug, or treatment T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38EB@ first we may proceed as follows:

Suppose n is a number of randomly selected matched pairs of subjects to be used in a screening test or clinical trials. Suppose further one subject in a randomly selected matched pairs of subjects is administered treatment T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38EA@ say and the remaining subjects in the matched pair of subject is administered treatment T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38EB@ say first.

Let

u il1 ={ 1,ifintheithpairofmatchsubjects,arandomlyselected subjectisadministeredtestdrugtreatment T l first 0,otherwise fori=1,2,....,nthpairs;l=1,2,....,treatments. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WG1bWaaSbaaKqbGeaacaWGPbGaamiBaiaaigdaaKqbagqaaiabg2da 9maaceaabaqbaeqabiqaaaabaeqabaGaaGymaiaaysW7caGGSaGaam yAaiaadAgacaaMe8UaamyAaiaad6gacaaMe8UaamiDaiaadIgacaWG LbGaaGzaVlaaysW7caWGPbGaamiDaiaadIgacaaMe8UaamiCaiaadg gacaWGPbGaamOCaiaaysW7caWGVbGaamOzaiaaysW7caWGTbGaamyy aiaadshacaWGJbGaamiAaiaaysW7caWGZbGaamyDaiaadkgacaWGQb GaamyzaiaadogacaWG0bGaam4CaiaacYcacaaMe8UaamyyaiaaykW7 caWGYbGaamyyaiaad6gacaWGKbGaam4Baiaad2gacaWGSbGaamyEai aaysW7caWGZbGaamyzaiaadYgacaWGLbGaam4yaiaadshacaWGLbGa amizaaqaaiaadohacaWG1bGaamOyaiaadQgacaWGLbGaam4yaiaads hacaaMe8UaamyAaiaadohacaaMe8UaamyyaiaadsgaciGGTbGaaiyA aiaac6gacaWGPbGaam4CaiaadshacaWGLbGaamOCaiaadwgacaWGKb GaaGjbVlaadshacaWGLbGaam4CaiaadshacaaMe8Uaamizaiaadkha caWG1bGaam4zaiaaysW7caWG0bGaamOCaiaadwgacaWGHbGaamiDai aad2gacaWGLbGaamOBaiaadshacaaMe8UaamivamaaBaaabaGaamiB aaqabaGaaGjbVlaadAgacaWGPbGaamOCaiaadohacaWG0bGaaGjbVd aabaGaaGimaiaacYcacaaMe8Uaam4BaiaadshacaWGObGaamyzaiaa dkhacaWG3bGaamyAaiaadohacaWGLbGaaGjbVlaaysW7caaMe8UaaG jbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaM e8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaays W7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjb VlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8 UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7 caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVl aaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8Ua aGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8oaaaGaay5Eaaaake aajuaGcaaMb8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8Ua aGjbVlaaysW7caWGMbGaam4BaiaadkhacaaMe8UaamyAaiabg2da9i aaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGa aiilaiaad6gacaWG0bGaamiAaiaaysW7caWGWbGaamyyaiaadMgaca WGYbGaam4CaiaacUdacaWGSbGaeyypa0JaaGymaiaacYcacaaIYaGa aiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGSaGaamiDaiaadkhaca WGLbGaamyyaiaadshacaWGTbGaamyzaiaad6gacaWG0bGaam4Caiaa c6caaaaa@5D27@  (1)

Let

π l1 + =P( u il1 )=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda 3aa0baaKqbGeaacaWGSbGaaGymaaqaaiabgUcaRaaajuaGcqGH9aqp caWGqbWaaeWaaeaacaWG1bWaaSbaaKqbGeaacaWGPbGaamiBaiaaig daaKqbagqaaaGaayjkaiaawMcaaiabg2da9iaaigdaaaa@4543@  (2)

And

W l1 = i=1 n u il1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4vam aaBaaajuaibaGaamiBaiaaigdaaKqbagqaaiabg2da9maaqahabaGa amyDamaaBaaajuaibaGaamyAaiaadYgacaaIXaaajuaGbeaaaKqbGe aacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaaaa@45E8@  (3)

Now the expected value and variance of u il1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaamyAaiaadYgacaaIXaaajuaGbeaaaaa@3AF5@ are respectively

E( u il1 )= π l1 + ;Var( u il1 )= π l1 + ( 1 π l1 + ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaamyDamaaBaaajuaibaGaamyAaiaadYgacaaIXaaajuaG beaaaiaawIcacaGLPaaacqGH9aqpcqaHapaCdaqhaaqcfasaaiaadY gacaaIXaaabaGaey4kaScaaKqbakaacUdacaWGwbGaamyyaiaadkha daqadaqaaiaadwhadaWgaaqcfasaaiaadMgacaWGSbGaaGymaaqcfa yabaaacaGLOaGaayzkaaGaeyypa0JaeqiWda3aa0baaKqbGeaacaWG SbGaaGymaaqaaiabgUcaRaaajuaGdaqadaqaaiaaigdacqGHsislcq aHapaCdaqhaaqcfasaaiaadYgacaaIXaaabaGaey4kaScaaaqcfaOa ayjkaiaawMcaaaaa@5B71@ (4)

Similarly the expected value and variance W l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4vam aaBaaajuaibaGaamiBaiaaigdaaKqbagqaaaaa@39E9@ are respectively

E( W l1 )= i=1 n E( u il1 ) =n. π l1 + ;Var( W l1 )= i=1 n Var( u il1 ) =n. π l1 + ( 1 π l1 + ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaam4vamaaBaaajuaibaGaamiBaiaaigdaaKqbagqaaaGa ayjkaiaawMcaaiabg2da9maaqahabaGaamyramaabmaabaGaamyDam aaBaaajuaibaGaamyAaiaadYgacaaIXaaajuaGbeaaaiaawIcacaGL PaaaaKqbGeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabgg HiLdGaeyypa0JaamOBaiaac6cacqaHapaCdaqhaaqcfasaaiaadYga caaIXaaabaGaey4kaScaaKqbakaacUdacaWGwbGaamyyaiaadkhada qadaqaaiaadEfadaWgaaqcfasaaiaadYgacaaIXaaajuaGbeaaaiaa wIcacaGLPaaacqGH9aqpdaaeWbqaaiaadAfacaWGHbGaamOCamaabm aabaGaamyDamaaBaaajuaibaGaamyAaiaadYgacaaIXaaajuaGbeaa aiaawIcacaGLPaaaaKqbGeaacaWGPbGaeyypa0JaaGymaaqaaiaad6 gaaKqbakabggHiLdGaeyypa0JaamOBaiaac6cacqaHapaCdaqhaaqc fasaaiaadYgacaaIXaaabaGaey4kaScaaKqbaoaabmaabaGaaGymai abgkHiTiabec8aWnaaDaaajuaibaGaamiBaiaaigdaaeaacqGHRaWk aaaajuaGcaGLOaGaayzkaaaaaa@7B35@ (5)

Now π l1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda 3aa0baaKqbGeaacaWGSbGaaGymaaqaaiabgUcaRaaaaaa@3B1F@  is the proportion of the probability that a subject in randomly selected matched pair of subjects test or responds positive when administered test, or treatment T l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGSbaabeaaaaa@37EC@ first in a two period controlled trial or diagnostic screening test, for l=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBai abg2da9iaaikdaaaa@3937@ its sample estimate is

π ^ l1 + = P l 1 = f l 1 n = W l 1 n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaadYgacaaIXaaabaGaey4kaScaaKqbakab g2da9iaadcfadaWgaaqcfasaaiaadYgajuaGdaWgaaqcfasaaiaaig daaeqaaaqcfayabaGaeyypa0ZaaSaaaeaacaWGMbWaaSbaaKqbGeaa caWGSbqcfa4aaSbaaKqbGeaacaaIXaaabeaaaKqbagqaaaqaaiaad6 gaaaGaeyypa0ZaaSaaaeaacaWGxbWaaSbaaKqbGeaacaWGSbqcfa4a aSbaaKqbGeaacaaIXaaabeaaaKqbagqaaaqaaiaad6gaaaaaaa@4DA3@  (6)

Where f l 1 + = W l 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGymaaqabaaabaGa ey4kaScaaKqbakabg2da9iaadEfadaWgaaqcfasaaiaadYgajuaGda Wgaaqcfasaaiaaigdaaeqaaaqcfayabaaaaa@4100@ is the total number of subjects in the matched pairs of subjects who test or respond positive when administered treatment T l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamiBaaqcfayabaaaaa@392B@ first in a diagnostic screening test or controlled clinical trial. In other words, f l 1 + = W l 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGymaaqabaaabaGa ey4kaScaaKqbakabg2da9iaadEfadaWgaaqcfasaaiaadYgajuaGda Wgaaqcfasaaiaaigdaaeqaaaqcfayabaaaaa@4100@ is the total number of 1’s in the frequency distribution of the n values of 0s and 1s in u i l 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGPbGaamiBamaaBaaameaacaaIXaaabeaaaSqabaaaaa@39EE@ , for l=1,2,...,n;l=1,2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6ca caGGSaGaamOBaiaacUdacaWGSbGaeyypa0JaaGymaiaacYcacaaIYa aaaa@43E8@ . The corresponding sample estimate of the variance of π ^ l 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aadaqhaaWcbaGaamiBamaaBaaameaacaaIXaaabeaaaSqaaiabgUca Raaaaaa@3AB6@  is

Var( π ^ l 1 + )=Var ( W l 1 ) n 2 = π ^ l 1 + ( 1 π ^ l 1 + ) n = P l 1 ( 1 P l 1 ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aadggacaWGYbWaaeWaaeaacuaHapaCgaqcamaaDaaajuaibaGaamiB aKqbaoaaBaaajuaibaGaaGymaaqabaaabaGaey4kaScaaaqcfaOaay jkaiaawMcaaiabg2da9iaadAfacaWGHbGaamOCamaalaaabaWaaeWa aeaacaWGxbWaaSbaaKqbGeaacaWGSbqcfa4aaSbaaKqbGeaacaaIXa aabeaaaKqbagqaaaGaayjkaiaawMcaaaqaaiaad6gadaahaaqabKqb GeaacaaIYaaaaaaajuaGcqGH9aqpdaWcaaqaaiqbec8aWzaajaWaa0 baaKqbGeaacaWGSbqcfa4aaSbaaKqbGeaacaaIXaaabeaaaeaacqGH RaWkaaqcfa4aaeWaaeaacaaIXaGaeyOeI0IafqiWdaNbaKaadaqhaa qcfasaaiaadYgajuaGdaWgaaqcfasaaiaaigdaaeqaaaqaaiabgUca RaaaaKqbakaawIcacaGLPaaaaeaacaWGUbaaaiabg2da9maalaaaba GaamiuamaaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGymaaqa baaajuaGbeaadaqadaqaaiaaigdacqGHsislcaWGqbWaaSbaaKqbGe aacaWGSbqcfa4aaSbaaKqbGeaacaaIXaaabeaaaKqbagqaaaGaayjk aiaawMcaaaqaaiaad6gaaaaaaa@6C17@    (7)

A null hypothesis that is usually of interest in two period cross over design is that the proportion of subjects in the period populations of subjects administered test, drug, or treatment T l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamiBaaqcfayabaaaaa@392B@  first is the same as the proportion of subjects in the paired populations of subjects administered test, drug, or treatment T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@38F6@  first in a control clinical trial, or the null hypothesis

H 0 : π l1 + = π l2 + versus H 1 : π l1 + π l2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaGaaiOoaiabec8aWnaaDaaajuai baGaamiBaiaaigdaaeaacqGHRaWkaaqcfaOaeyypa0JaeqiWda3aa0 baaKqbGeaacaWGSbGaaGOmaaqaaiabgUcaRaaajuaGcaaMe8UaamOD aiaadwgacaWGYbGaam4CaiaadwhacaWGZbGaaGjbVlaadIeadaWgaa qcfasaaiaaigdaaKqbagqaaiaacQdacqaHapaCdaqhaaqcfasaaiaa dYgacaaIXaaabaGaey4kaScaaKqbakabgcMi5kabec8aWnaaDaaaju aibaGaamiBaiaaikdaaeaacqGHRaWkaaaaaa@5C8E@      (8)

Now the sample estimate of the difference in proportion, π l1 + π l2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda 3aa0baaKqbGeaacaWGSbGaaGymaaqaaiabgUcaRaaajuaGcqGHsisl cqaHapaCdaqhaaqcfasaaiaadYgacaaIYaaabaGaey4kaScaaaaa@4136@  is

π ^ 1l + π ^ 2l + = p 1l p 2l = f 1l + f 2l + n = W 1l W 2l n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaaigdacaWGSbaabaGaey4kaScaaKqbakab gkHiTiqbec8aWzaajaWaa0baaKqbGeaacaaIYaGaamiBaaqaaiabgU caRaaajuaGcqGH9aqpcaWGWbWaaSbaaKqbGeaacaaIXaGaamiBaaqc fayabaGaeyOeI0IaamiCamaaBaaajuaibaGaaGOmaiaadYgaaKqbag qaaiabg2da9maalaaabaGaamOzamaaDaaajuaibaGaaGymaiaadYga aeaacqGHRaWkaaqcfaOaeyOeI0IaamOzamaaDaaajuaibaGaaGOmai aadYgaaeaacqGHRaWkaaaajuaGbaGaamOBaaaacqGH9aqpdaWcaaqa aiaadEfadaWgaaqcfasaaiaaigdacaWGSbaajuaGbeaacqGHsislca WGxbWaaSbaaKqbGeaacaaIYaGaamiBaaqcfayabaaabaGaamOBaaaa aaa@603A@     (9)

Whose estimated variance is

Var( π ^ 1l + π ^ 1l + )=Var( p 1l p 2l )=Var ( W 1l W 2l ) n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aadggacaWGYbWaaeWaaeaacuaHapaCgaqcamaaDaaajuaibaGaaGym aiaadYgaaeaacqGHRaWkaaqcfaOaeyOeI0IafqiWdaNbaKaadaqhaa qcfasaaiaaigdacaWGSbaabaGaey4kaScaaaqcfaOaayjkaiaawMca aiabg2da9iaadAfacaWGHbGaamOCamaabmaabaGaamiCamaaBaaaju aibaGaaGymaiaadYgaaKqbagqaaiabgkHiTiaadchadaWgaaqcfasa aiaaikdacaWGSbaajuaGbeaaaiaawIcacaGLPaaacqGH9aqpcaWGwb GaamyyaiaadkhadaWcaaqaamaabmaabaGaam4vamaaBaaajuaibaGa aGymaiaadYgaaKqbagqaaiabgkHiTiaadEfadaWgaaqcfasaaiaaik dacaWGSbaajuaGbeaaaiaawIcacaGLPaaaaeaacaWGUbWaaWbaaeqa juaibaGaaGOmaaaaaaaaaa@6263@

Now it is easily shown using the specifications of equations 1-3 that Cov( W 1l ; W 2l )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aad+gacaWG2bWaaeWaaeaacaWGxbWaaSbaaKqbGeaacaaIXaGaamiB aaqcfayabaGaai4oaiaadEfadaWgaaqcfasaaiaaikdacaWGSbaaju aGbeaaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@440E@

Hence

Var( π ^ 1l + π ^ 2l + )=Var( p 1l p 2l )= Var( W 1l )+Var( W 2l ) n 2 = π ^ 1l + ( 1 π ^ 1l + )+ π ^ 2l + ( 1 π ^ 2l + ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aadggacaWGYbWaaeWaaeaacuaHapaCgaqcamaaDaaajuaibaGaaGym aiaadYgaaeaacqGHRaWkaaqcfaOaeyOeI0IafqiWdaNbaKaadaqhaa qcfasaaiaaikdacaWGSbaabaGaey4kaScaaaqcfaOaayjkaiaawMca aiabg2da9iaadAfacaWGHbGaamOCamaabmaabaGaamiCamaaBaaaju aibaGaaGymaiaadYgaaKqbagqaaiabgkHiTiaadchadaWgaaqcfasa aiaaikdacaWGSbaajuaGbeaaaiaawIcacaGLPaaacqGH9aqpdaWcaa qaaiaadAfacaWGHbGaamOCamaabmaabaGaam4vamaaBaaajuaibaGa aGymaiaadYgaaKqbagqaaaGaayjkaiaawMcaaiabgUcaRiaadAfaca WGHbGaamOCamaabmaabaGaam4vamaaBaaajuaibaGaaGOmaiaadYga aKqbagqaaaGaayjkaiaawMcaaaqaaiaad6gadaahaaqabKqbGeaaca aIYaaaaaaajuaGcqGH9aqpdaWcaaqaaiqbec8aWzaajaWaa0baaKqb GeaacaaIXaGaamiBaaqaaiabgUcaRaaajuaGdaqadaqaaiaaigdacq GHsislcuaHapaCgaqcamaaDaaajuaibaGaaGymaiaadYgaaeaacqGH RaWkaaaajuaGcaGLOaGaayzkaaGaey4kaSIafqiWdaNbaKaadaqhaa qcfasaaiaaikdacaWGSbaabaGaey4kaScaaKqbaoaabmaabaGaaGym aiabgkHiTiqbec8aWzaajaWaa0baaKqbGeaacaaIYaGaamiBaaqaai abgUcaRaaaaKqbakaawIcacaGLPaaaaeaacaWGUbaaaaaa@855B@     (10)

Hence the chi-square test statistic for the null hypothesis H0 of equation 8 is

χ 2 = ( π ^ 1l + π ^ 2l + ) 2 Var( π ^ 1l + π ^ 1l + ) = ( W 1l W 2l ) 2 Var( W 1l )+Var( W 2l ) = n ( π ^ 1l + π ^ 2l + ) 2 π ^ 1l + ( 1 π ^ 1l + )+ π ^ 2l + ( 1 π ^ 2l + ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaamaabmaa baGafqiWdaNbaKaadaqhaaqcfasaaiaaigdacaWGSbaabaGaey4kaS caaKqbakabgkHiTiqbec8aWzaajaWaa0baaKqbGeaacaaIYaGaamiB aaqaaiabgUcaRaaaaKqbakaawIcacaGLPaaadaahaaqabKqbGeaaca aIYaaaaaqcfayaaiaadAfacaWGHbGaamOCamaabmaabaGafqiWdaNb aKaadaqhaaqcfasaaiaaigdacaWGSbaabaGaey4kaScaaKqbakabgk HiTiqbec8aWzaajaWaa0baaKqbGeaacaaIXaGaamiBaaqaaiabgUca RaaaaKqbakaawIcacaGLPaaaaaGaeyypa0ZaaSaaaeaadaqadaqaai aadEfadaWgaaqcfasaaiaaigdacaWGSbaajuaGbeaacqGHsislcaWG xbWaaSbaaKqbGeaacaaIYaGaamiBaaqcfayabaaacaGLOaGaayzkaa WaaWbaaeqajuaibaGaaGOmaaaaaKqbagaacaWGwbGaamyyaiaadkha daqadaqaaiaadEfadaWgaaqcfasaaiaaigdacaWGSbaajuaGbeaaai aawIcacaGLPaaacqGHRaWkcaWGwbGaamyyaiaadkhadaqadaqaaiaa dEfadaWgaaqcfasaaiaaikdacaWGSbaajuaGbeaaaiaawIcacaGLPa aaaaGaeyypa0ZaaSaaaeaacaWGUbWaaeWaaeaacuaHapaCgaqcamaa DaaajuaibaGaaGymaiaadYgaaeaacqGHRaWkaaqcfaOaeyOeI0Iafq iWdaNbaKaadaqhaaqcfasaaiaaikdacaWGSbaabaGaey4kaScaaaqc faOaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaaajuaGbaGafq iWdaNbaKaadaqhaaqcfasaaiaaigdacaWGSbaabaGaey4kaScaaKqb aoaabmaabaGaaGymaiabgkHiTiqbec8aWzaajaWaa0baaKqbGeaaca aIXaGaamiBaaqaaiabgUcaRaaaaKqbakaawIcacaGLPaaacqGHRaWk cuaHapaCgaqcamaaDaaajuaibaGaaGOmaiaadYgaaeaacqGHRaWkaa qcfa4aaeWaaeaacaaIXaGaeyOeI0IafqiWdaNbaKaadaqhaaqcfasa aiaaikdacaWGSbaabaGaey4kaScaaaqcfaOaayjkaiaawMcaaaaaaa a@A1E4@    (11)

Which under the null hypothesis of equation 8 has approximately the chi-square distribution with 1 degree of freedom for sufficiently large n?

Where π ^ l1 + = p l1 ,forl=1,2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaadYgacaaIXaaabaGaey4kaScaaKqbakab g2da9iaadchadaWgaaqcfasaaiaadYgacaaIXaaajuaGbeaacaGGSa GaamOzaiaad+gacaWGYbGaaGjbVlaadYgacqGH9aqpcaaIXaGaaiil aiaaikdaaaa@4972@

The null hypothesis H0 of equation 8 is rejected at the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ level of significant if χ 2 χ 1α;1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHLjYScqaHhpWydaqhaaqc fasaaiaaigdacqGHsislcqaHXoqycaGG7aGaaGymaaqaaiaaikdaaa aaaa@431F@ , otherwise the null hypothesis H0 is accepted. As earlier noted above an additional and modified method of or approach to the analysis of data obtained in a two period cross over design is to also compare the responses of those subjects in the matched paired populations of subjects who failed to test or respond positive to one of the two treatment when administered first but respond positive when the other treatment is administered to them later with the responses of the remaining subjects who failed to respond positive when administered the second test or treatment first but respond positive when administered the first test or treatment later that is at the second trial. In these cases interest is then only in the n l 1 =n f j 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGymaaqabaaajuaG beaacqGH9aqpcaWGUbGaeyOeI0IaamOzamaaDaaajuaibaGaamOAaK qbaoaaBaaajuaibaGaaGymaaqabaaabaGaey4kaScaaaaa@4267@ subjects who failed to respond positive when administered test or treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamOAaaqcfayabaaaaa@3929@ first but respond positive when administered test or treatment T l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamiBaaqcfayabaaaaa@392B@ later, that is at the second clinical trial or diagnostic screening test, for l,j=1,2;lj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBai aacYcacaWGQbGaeyypa0JaaGymaiaacYcacaaIYaGaai4oaiaadYga cqGHGjsUcaWGQbaaaa@40A7@ . To conduct this additional and modified analysis of response data, we may let

u i l 2 ;j ={ 1,iffortheithnightpairofsubjects,thesubject administeredtreatment T j firstfailstorespond positivebutrespondpositivewhenthesamesubsetisadministered treatment T l laterthatisatthesecondtrial 0,otherwise fori=1,2,....,n l 2 =n f j + ;l,j=1,2;lj. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WG1bWaaSbaaKqbGeaacaWGPbGaamiBaKqbaoaaBaaajuaibaGaaGOm aaqabaGaai4oaiaadQgaaKqbagqaaiabg2da9maaceaabaqbaeqabi qaaaabaeqabaGaaGymaiaaysW7caGGSaGaamyAaiaadAgacaaMe8Ua amOzaiaad+gacaWGYbGaaGjbVlaadshacaWGObGaamyzaiaaygW7ca aMe8UaamyAaiaadshacaWGObGaaGjbVlaad6gacaWGPbGaam4zaiaa dIgacaWG0bGaaGjbVlaadchacaWGHbGaamyAaiaadkhacaaMe8Uaam 4BaiaadAgacaaMe8Uaam4CaiaadwhacaWGIbGaamOAaiaadwgacaWG JbGaamiDaiaadohacaGGSaGaaGjbVlaadshacaWGObGaamyzaiaays W7caWGZbGaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0bGaaGjb VdqaaiaadggacaWGKbGaciyBaiaacMgacaGGUbGaamyAaiaadohaca WG0bGaamyzaiaadkhacaWGLbGaamizaiaaysW7caWG0bGaamOCaiaa dwgacaWGHbGaamiDaiaad2gacaWGLbGaamOBaiaadshacaaMe8Uaam ivamaaBaaabaGaamOAaaqabaGaaGjbVlaadAgacaWGPbGaamOCaiaa dohacaWG0bGaaGjbVlaadAgacaWGHbGaamyAaiaadYgacaWGZbGaaG jbVlaadshacaWGVbGaaGjbVlaadkhacaWGLbGaam4CaiaadchacaWG VbGaamOBaiaadsgacaaMe8oabaGaamiCaiaad+gacaWGZbGaamyAai aadshacaWGPbGaamODaiaadwgacaaMe8UaamOyaiaadwhacaWG0bGa aGjbVlaadkhacaWGLbGaam4CaiaadchacaWGVbGaamOBaiaadsgaca aMe8UaamiCaiaad+gacaWGZbGaamyAaiaadshacaWGPbGaamODaiaa dwgacaaMe8Uaam4DaiaadIgacaWGLbGaamOBaiaaysW7caWG0bGaam iAaiaadwgacaaMe8Uaam4CaiaadggacaWGTbGaamyzaiaaysW7caWG ZbGaamyDaiaadkgacaWGZbGaamyzaiaadshacaaMe8UaamyAaiaado hacaaMe8UaamyyaiaadsgaciGGTbGaaiyAaiaac6gacaWGPbGaam4C aiaadshacaWGLbGaamOCaiaadwgacaWGKbGaaGjbVdqaaiaadshaca WGYbGaamyzaiaadggacaWG0bGaamyBaiaadwgacaWGUbGaamiDaiaa ysW7caWGubWaaSbaaeaacaWGSbGaaGjbVdqabaGaamiBaiaadggaca WG0bGaamyzaiaadkhacaaMe8UaamiDaiaadIgacaWGHbGaamiDaiaa ysW7caWGPbGaam4CaiaaysW7caWGHbGaamiDaiaaysW7caWG0bGaam iAaiaadwgacaaMe8Uaci4CaiaacwgacaGGJbGaam4Baiaad6gacaWG KbGaaGjbVlaadshacaWGYbGaamyAaiaadggacaWGSbGaaGjbVdaaba GaaGimaiaacYcacaaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7 caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8Uaam4Bai aadshacaWGObGaamyzaiaadkhacaWG3bGaamyAaiaadohacaWGLbaa aaGaay5EaaaakeaajuaGcaaMb8UaaGjbVlaaysW7caaMe8UaaGjbVl aaysW7caaMe8UaaGjbVlaaysW7caWGMbGaam4BaiaadkhacaaMe8Ua amyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlai aac6cacaGGUaGaaiilaiaad6gacaWGSbWaaSbaaKqbGeaacaaIYaaa juaGbeaacqGH9aqpcaWGUbGaeyOeI0IaamOzamaaDaaajuaibaGaam OAaaqaaiabgUcaRaaajuaGcaGG7aGaamiBaiaacYcacaWGQbGaeyyp a0JaaGymaiaacYcacaaIYaGaai4oaiaadYgacqGHGjsUcaWGQbGaai Olaaaaaa@7255@  (13)

Let

π l2 + =P( u i l 2 ;j =1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaa8ajuaGcq aHapaCdaqhaaqcfasaaiaadYgacaaIYaaabaGaey4kaScaaKqbakab g2da9iaadcfadaqadaqaaiaadwhadaWgaaqcfasaaiaadMgacaWGSb qcfa4aaSbaaKqbGeaacaaIYaaabeaacaGG7aGaamOAaaqcfayabaGa eyypa0JaaGymaaGaayjkaiaawMcaaaaa@4896@    (14)

And

W l2 = i=1 n l 2 u i l 2 ;j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaGRajuaGca WGxbWaaSbaaKqbGeaacaWGSbGaaGOmaaqcfayabaGaeyypa0ZaaabC aeaacaWG1bWaaSbaaKqbGeaacaWGPbGaamiBaKqbaoaaBaaajuaiba GaaGOmaaqabaGaai4oaiaadQgaaKqbagqaaaqcfasaaiaadMgacqGH 9aqpcaaIXaaabaGaamOBaKqbaoaaBaaajuaibaGaamiBaKqbaoaaBa aajuaibaGaaGOmaaqabaaabeaaaKqbakabggHiLdaaaa@4C99@     (15)

Now the expected value and variance of u i l 2 ;j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGPbGaamiBamaaBaaameaacaaIYaaabeaaliaacUdacaWG Qbaabeaaaaa@3B9D@  are respectively

E( u i l 2 ;j )= π l2 + ;Var( u i l 2 ;j )= π l2 + ( 1 π l2 + ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaamyDamaaBaaajuaibaGaamyAaiaadYgajuaGdaWgaaqc fasaaiaaikdaaeqaaiaacUdacaWGQbaajuaGbeaaaiaawIcacaGLPa aacqGH9aqpcqaHapaCdaqhaaqcfasaaiaadYgacaaIYaaabaGaey4k aScaaKqbakaacUdacaWGwbGaamyyaiaadkhadaqadaqaaiaadwhada WgaaqcfasaaiaadMgacaWGSbqcfa4aaSbaaKqbGeaacaaIYaaabeaa caGG7aGaamOAaaqcfayabaaacaGLOaGaayzkaaGaeyypa0JaeqiWda 3aa0baaKqbGeaacaWGSbGaaGOmaaqaaiabgUcaRaaajuaGdaqadaqa aiaaigdacqGHsislcqaHapaCdaqhaaqcfasaaiaadYgacaaIYaaaba Gaey4kaScaaaqcfaOaayjkaiaawMcaaaaa@608C@     (16)

Similarly the expected value and variance of W l 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaWGSbWaaSbaaWqaaiaaikdaaeqaaaWcbeaaaaa@38E3@  are respectively

E( W l 2 )= l=1 n l 2 E( u i l 2 ;j )=n l 2 . π l2 + ;Var( W l 2 )= l=1 n l 2 Var( u i l 2 ;j )=n l 2 . π l2 + ( 1 π l2 + ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaam4vamaaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGa aGOmaaqabaaajuaGbeaaaiaawIcacaGLPaaacqGH9aqpdaaeWbqaai aadweadaqadaqaaiaadwhadaWgaaqcfasaaiaadMgacaWGSbqcfa4a aSbaaKqbGeaacaaIYaaabeaacaGG7aGaamOAaaqcfayabaaacaGLOa GaayzkaaGaeyypa0JaamOBaiaadYgadaWgaaqcfasaaiaaikdaaKqb agqaaiaac6caaKqbGeaacaWGSbGaeyypa0JaaGymaaqaaiaad6gaju aGdaWgaaqcfasaaiaadYgajuaGdaWgaaqcfasaaiaaikdaaeqaaaqa baaajuaGcqGHris5aiabec8aWnaaDaaajuaibaGaamiBaiaaikdaae aacqGHRaWkaaqcfaOaai4oaiaadAfacaWGHbGaamOCamaabmaabaGa am4vamaaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGOmaaqaba aajuaGbeaaaiaawIcacaGLPaaacqGH9aqpdaaeWbqaaiaadAfacaWG HbGaamOCamaabmaabaGaamyDamaaBaaajuaibaGaamyAaiaadYgaju aGdaWgaaqcfasaaiaaikdaaeqaaiaacUdacaWGQbaajuaGbeaaaiaa wIcacaGLPaaacqGH9aqpcaWGUbGaamiBamaaBaaajuaibaGaaGOmaa qcfayabaGaaiOlaaqcfasaaiaadYgacqGH9aqpcaaIXaaabaGaamOB aKqbaoaaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGOmaaqaba aabeaaaKqbakabggHiLdGaeqiWda3aa0baaKqbGeaacaWGSbGaaGOm aaqaaiabgUcaRaaajuaGdaqadaqaaiaaigdacqGHsislcqaHapaCda qhaaqcfasaaiaadYgacaaIYaaabaGaey4kaScaaaqcfaOaayjkaiaa wMcaaaaa@8DF4@     (17)

Now π l2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aa0 baaSqaaiaadYgacaaIYaaabaGaey4kaScaaaaa@3A6F@  is the proportion or the probability that a randomly selected subject in the matched pairs of subjects administered test or treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqEcjuaGca WGubWaaSbaaKqbGeaacaWGQbaajuaGbeaaaaa@3A19@ first fail to respond positive but this same subject respond positive when administered test or treatment T l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamiBaaqcfayabaaaaa@392B@ later, that is at the second trial. Its sample estimate is

π ^ l2 + = P l2 = f l2 + n l2 = W l2 n l2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaadYgacaaIYaaabaGaey4kaScaaKqbakab g2da9iaadcfadaWgaaqcfasaaiaadYgacaaIYaaajuaGbeaacqGH9a qpdaWcaaqaaiaadAgadaqhaaqcfasaaiaadYgacaaIYaaabaGaey4k aScaaaqcfayaaiaad6gadaWgaaqcfasaaiaadYgacaaIYaaajuaGbe aaaaGaeyypa0ZaaSaaaeaacaWGxbWaaSbaaKqbGeaacaWGSbGaaGOm aaqcfayabaaabaGaamOBamaaBaaajuaibaGaamiBaiaaikdaaKqbag qaaaaaaaa@5107@    (18)

Where f l 2 + = W l 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGOmaaqabaaabaGa ey4kaScaaKqbakabg2da9iaadEfadaWgaaqcfasaaiaadYgajuaGda Wgaaqcfasaaiaaikdaaeqaaaqcfayabaaaaa@4102@ are the total number of subjects in the matched pairs of subjects who failed to respond positive when administered test for treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqEcjuaGca WGubWaaSbaaKqbGeaacaWGQbaajuaGbeaaaaa@3A19@ first but respond positive when administered test or treatment f l 2 + = W l 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGOmaaqabaaabaGa ey4kaScaaKqbakabg2da9iaadEfadaWgaaqcfasaaiaadYgajuaGda Wgaaqcfasaaiaaikdaaeqaaaqcfayabaaaaa@4102@ later, that at the second trial. In other words, f l 2 + = W l 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGOmaaqabaaabaGa ey4kaScaaKqbakabg2da9iaadEfadaWgaaqcfasaaiaadYgajuaGda Wgaaqcfasaaiaaikdaaeqaaaqcfayabaaaaa@4102@ is the total number of 1s in the frequency distribution of the n l 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGSbWaaSbaaWqaaiaaikdaaeqaaaWcbeaaaaa@38FA@ values of 0s and 1s in u i l 2 ;j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaamyAaiaadYgajuaGdaWgaaqcfasaaiaaikdaaeqa aiaacUdacaWGQbaajuaGbeaaaaa@3D81@ , for i=1,2,..., n l 2 =n f l1 + ,( l,j=1,2;lj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6ca caGGSaGaamOBamaaBaaabaGaamiBamaaBaaajuaibaGaaGOmaaqcfa yabaaabeaacqGH9aqpcaWGUbGaeyOeI0IaamOzamaaDaaajuaibaGa amiBaiaaigdaaeaacqGHRaWkaaqcfaOaaiilamaabmaabaGaamiBai aacYcacaWGQbGaeyypa0JaaGymaiaacYcacaaIYaGaai4oaiaadYga cqGHGjsUcaWGQbaacaGLOaGaayzkaaaaaa@554C@ .

The sample estimate of the variance of π l 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aa0 baaSqaaiaadYgadaWgaaadbaGaaGOmaaqabaaaleaacqGHRaWkaaaa aa@3AA7@ is

Var( π ^ l2 + )= Var( p l2 ) n l 2 =Var( W l2 )= π ^ l2 + ( 1 π ^ l2 + ) n l 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aadggacaWGYbWaaeWaaeaacuaHapaCgaqcamaaDaaajuaibaGaamiB aiaaikdaaeaacqGHRaWkaaaajuaGcaGLOaGaayzkaaGaeyypa0ZaaS aaaeaacaWGwbGaamyyaiaadkhadaqadaqaaiaadchadaWgaaqcfasa aiaadYgacaaIYaaajuaGbeaaaiaawIcacaGLPaaaaeaacaWGUbWaaS baaKqbGeaacaWGSbqcfa4aaSbaaKqbGeaacaaIYaaabeaaaKqbagqa aaaacqGH9aqpcaWGwbGaamyyaiaadkhadaqadaqaaiaadEfadaWgaa qcfasaaiaadYgacaaIYaaajuaGbeaaaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiqbec8aWzaajaWaa0baaKqbGeaacaWGSbGaaGOmaaqaai abgUcaRaaajuaGdaqadaqaaiaaigdacqGHsislcuaHapaCgaqcamaa DaaajuaibaGaamiBaiaaikdaaeaacqGHRaWkaaaajuaGcaGLOaGaay zkaaaabaGaamOBamaaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGa aGOmaaqabaaajuaGbeaaaaaaaa@68F1@     (19)

As noted above, an additional null hypothesis that may be of further research interest when expressed in terms of the difference between population proportions is

H 0 : π 12 + = π 22 + versus H 1 : π 12 + π 22 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaGaaiOoaiabec8aWnaaDaaajuai baGaaGymaiaaikdaaeaacqGHRaWkaaqcfaOaeyypa0JaeqiWda3aa0 baaKqbGeaacaaIYaGaaGOmaaqaaiabgUcaRaaajuaGcaaMe8UaamOD aiaadwgacaWGYbGaam4CaiaadwhacaWGZbGaaGjbVlaadIeadaWgaa qcfasaaiaaigdaaKqbagqaaiaacQdacqaHapaCdaqhaaqcfasaaiaa igdacaaIYaaabaGaey4kaScaaKqbakabgcMi5kabec8aWnaaDaaaju aibaGaaGOmaiaaikdaaeaacqGHRaWkaaaaaa@5BBA@     (20)

Now the sample estimate of the difference in population proportion is

π ^ 12 + π ^ 22 + = P 12 P 22 = f 12 + n 12 f 22 + n 22 = W 12 n 12 W 22 n 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaaigdacaaIYaaabaGaey4kaScaaKqbakab gkHiTiqbec8aWzaajaWaa0baaKqbGeaacaaIYaGaaGOmaaqaaiabgU caRaaajuaGcqGH9aqpcaWGqbWaaSbaaKqbGeaacaaIXaGaaGOmaaqc fayabaGaeyOeI0IaamiuamaaBaaajuaibaGaaGOmaiaaikdaaKqbag qaaiabg2da9maalaaabaGaamOzamaaDaaajuaibaGaaGymaiaaikda aeaacqGHRaWkaaaajuaGbaGaamOBamaaBaaajuaibaGaaGymaiaaik daaKqbagqaaaaacqGHsisldaWcaaqaaiaadAgadaqhaaqcfasaaiaa ikdacaaIYaaabaGaey4kaScaaaqcfayaaiaad6gadaWgaaqcfasaai aaikdacaaIYaaajuaGbeaaaaGaeyypa0ZaaSaaaeaacaWGxbWaaSba aKqbGeaacaaIXaGaaGOmaaqcfayabaaabaGaamOBamaaBaaajuaiba GaaGymaiaaikdaaKqbagqaaaaacqGHsisldaWcaaqaaiaadEfadaWg aaqcfasaaiaaikdacaaIYaaajuaGbeaaaeaacaWGUbWaaSbaaKqbGe aacaaIYaGaaGOmaaqcfayabaaaaaaa@69AA@    (21)

The corresponding sample estimate of the variance of π ^ 12 + π ^ 22 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaaigdacaaIYaaabaGaey4kaScaaKqbakab gkHiTiqbec8aWzaajaWaa0baaKqbGeaacaaIYaGaaGOmaaqaaiabgU caRaaaaaa@40EC@  is

Var( π ^ 12 + π ^ 22 + )=Var( P 12 P 22 )=Var( W 12 n 12 W 22 n 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aadggacaWGYbWaaeWaaeaacuaHapaCgaqcamaaDaaajuaibaGaaGym aiaaikdaaeaacqGHRaWkaaqcfaOaeyOeI0IafqiWdaNbaKaadaqhaa qcfasaaiaaikdacaaIYaaabaGaey4kaScaaaqcfaOaayjkaiaawMca aiabg2da9iaadAfacaWGHbGaamOCamaabmaabaGaamiuamaaBaaaju aibaGaaGymaiaaikdaaKqbagqaaiabgkHiTiaadcfadaWgaaqcfasa aiaaikdacaaIYaaajuaGbeaaaiaawIcacaGLPaaacqGH9aqpcaWGwb GaamyyaiaadkhadaqadaqaamaalaaabaGaam4vamaaBaaajuaibaGa aGymaiaaikdaaKqbagqaaaqaaiaad6gadaWgaaqcfasaaiaaigdaca aIYaaajuaGbeaaaaGaeyOeI0YaaSaaaeaacaWGxbWaaSbaaKqbGeaa caaIYaGaaGOmaaqcfayabaaabaGaamOBamaaBaaajuaibaGaaGOmai aaikdaaKqbagqaaaaacaaMe8oacaGLOaGaayzkaaaaaa@6713@     (22)

It is easily shown using the specification of equations 13-15 that Cov( W 12 , W 22 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aad+gacaWG2bWaaeWaaeaacaWGxbWaaSbaaKqbGeaacaaIXaGaaGOm aaqcfayabaGaaiilaiaadEfadaWgaaqcfasaaiaaikdacaaIYaaaju aGbeaaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@4395@ .

Hence

Var( π ^ 12 + π ^ 22 + )=Var ( W 12 ) n 12 2 +Var ( W 22 ) n 22 2 = π ^ 12 + ( 1 π ^ 12 + ) n 12 + π ^ 22 + ( 1 π ^ 22 + ) n 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aadggacaWGYbWaaeWaaeaacuaHapaCgaqcamaaDaaajuaibaGaaGym aiaaikdaaeaacqGHRaWkaaqcfaOaeyOeI0IafqiWdaNbaKaadaqhaa qcfasaaiaaikdacaaIYaaabaGaey4kaScaaaqcfaOaayjkaiaawMca aiabg2da9iaadAfacaWGHbGaamOCamaalaaabaWaaeWaaeaacaWGxb WaaSbaaKqbGeaacaaIXaGaaGOmaaqcfayabaaacaGLOaGaayzkaaaa baGaamOBamaaDaaajuaibaGaaGymaiaaikdaaeaacaaIYaaaaaaaju aGcqGHRaWkcaWGwbGaamyyaiaadkhadaWcaaqaamaabmaabaGaam4v amaaBaaajuaibaGaaGOmaiaaikdaaKqbagqaaaGaayjkaiaawMcaaa qaaiaad6gadaqhaaqcfasaaiaaikdacaaIYaaabaGaaGOmaaaaaaqc faOaeyypa0ZaaSaaaeaacuaHapaCgaqcamaaDaaajuaibaGaaGymai aaikdaaeaacqGHRaWkaaqcfa4aaeWaaeaacaaIXaGaeyOeI0IafqiW daNbaKaadaqhaaqcfasaaiaaigdacaaIYaaabaGaey4kaScaaaqcfa OaayjkaiaawMcaaaqaaiaad6gadaWgaaqcfasaaiaaigdacaaIYaaa juaGbeaaaaGaey4kaSYaaSaaaeaacuaHapaCgaqcamaaDaaajuaiba GaaGOmaiaaikdaaeaacqGHRaWkaaqcfa4aaeWaaeaacaaIXaGaeyOe I0IafqiWdaNbaKaadaqhaaqcfasaaiaaikdacaaIYaaabaGaey4kaS caaaqcfaOaayjkaiaawMcaaaqaaiaad6gadaWgaaqcfasaaiaaikda caaIYaaajuaGbeaaaaaaaa@81BA@     (23)

The null hypothesis H0 in equation 20 may now be treated using the chi-square test statistic

χ 2 = ( π ^ 12 + π ^ 22 + ) 2 Var( π ^ 12 + π ^ 22 + ) = ( W 12 n 12 W 22 n 22 ) 2 Var( π ^ 12 + π ^ 22 + ) = n 12 . n 22 ( π ^ 12 + π ^ 22 + ) 2 n 22 . π ^ 12 + ( 1 π ^ 12 + )+ n 12 . π ^ 22 + ( 1 π ^ 22 + ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaamaabmaa baGafqiWdaNbaKaadaqhaaqcfasaaiaaigdacaaIYaaabaGaey4kaS caaKqbakabgkHiTiqbec8aWzaajaWaa0baaKqbGeaacaaIYaGaaGOm aaqaaiabgUcaRaaaaKqbakaawIcacaGLPaaadaahaaqabKqbGeaaca aIYaaaaaqcfayaaiaadAfacaWGHbGaamOCamaabmaabaGafqiWdaNb aKaadaqhaaqcfasaaiaaigdacaaIYaaabaGaey4kaScaaKqbakabgk HiTiqbec8aWzaajaWaa0baaKqbGeaacaaIYaGaaGOmaaqaaiabgUca RaaaaKqbakaawIcacaGLPaaaaaGaeyypa0ZaaSaaaeaadaqadaqaam aalaaabaGaam4vamaaBaaabaGaaGymaiaaikdaaeqaaaqaaiaad6ga daWgaaqaaiaaigdacaaIYaaabeaaaaGaeyOeI0YaaSaaaeaacaWGxb WaaSbaaeaacaaIYaGaaGOmaaqabaaabaGaamOBamaaBaaabaGaaGOm aiaaikdaaeqaaaaaaiaawIcacaGLPaaadaahaaqabeaacaaIYaaaaa qaaiaadAfacaWGHbGaamOCamaabmaabaGafqiWdaNbaKaadaqhaaqc fasaaiaaigdacaaIYaaabaGaey4kaScaaKqbakabgkHiTiqbec8aWz aajaWaa0baaKqbGeaacaaIYaGaaGOmaaqaaiabgUcaRaaaaKqbakaa wIcacaGLPaaaaaGaeyypa0ZaaSaaaeaacaWGUbWaaSbaaKazfa4=ba GaaGymaiaaikdaaKqbagqaaiaac6cacaWGUbWaaSbaaKqbGeaacaaI YaGaaGOmaaqcfayabaWaaeWaaeaacuaHapaCgaqcamaaDaaajuaiba GaaGymaiaaikdaaeaacqGHRaWkaaqcfaOaeyOeI0IafqiWdaNbaKaa daqhaaqcfasaaiaaikdacaaIYaaabaGaey4kaScaaaqcfaOaayjkai aawMcaamaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaamOBamaaBaaa juaibaGaaGOmaiaaikdaaKqbagqaaiaac6cacuaHapaCgaqcamaaDa aajuaibaGaaGymaiaaikdaaeaacqGHRaWkaaqcfa4aaeWaaeaacaaI XaGaeyOeI0IafqiWdaNbaKaadaqhaaqcfasaaiaaigdacaaIYaaaba Gaey4kaScaaaqcfaOaayjkaiaawMcaaiabgUcaRiaad6gadaWgaaqc fasaaiaaigdacaaIYaaajuaGbeaacaGGUaGafqiWdaNbaKaadaqhaa qcfasaaiaaikdacaaIYaaabaGaey4kaScaaKqbaoaabmaabaGaaGym aiabgkHiTiqbec8aWzaajaWaa0baaKazfa4=baGaaGOmaiaaikdaaK qbGeaacqGHRaWkaaaajuaGcaGLOaGaayzkaaaaaaaa@B369@    (24)

Which under the null hypothesis H0 of equation 20 has approximately the chi-square distribution with 1 degree of freedom for sufficiently large values of n 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaaIXaGaaGOmaaqabaaaaa@388C@ . π ^ l2 + = p 12 ,forl=1,2and n 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaadYgacaaIYaaabaGaey4kaScaaKqbakab g2da9iaadchadaWgaaqcfasaaiaaigdacaaIYaaajuaGbeaacaGGSa GaamOzaiaad+gacaWGYbGaamiBaiabg2da9iaaigdacaGGSaGaaGOm aiaaygW7caaMe8Uaamyyaiaad6gacaWGKbGaaGjbVlaad6gadaWgaa qcfasaaiaaikdacaaIYaaajuaGbeaaaaa@525F@ . The null hypothesis H0 of equation 20 is rejected at the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ level of significance if equation 12 is satisfied; otherwise H0 is accepted.

Illustrative Example

A researcher clinician is interested in comparing the effectiveness of two malaria drugs, D1 and D2 in the treatment of malaria using two period crossover designs in a controlled clinical trial. She collected 40 random samples of matched pairs of malaria patients, matched by age, sex and body weight. She administered treatment D1 first to a randomly selected patient in each pair of patients and also administered the remaining drug D2 first to the other patient in the pair. After the dry out period she repeated a drug administration in the reverse order. But this time she administered drug D1 to only those patients who fail to improve, that is who fail to respond positive when administered drug D2 first, and also administered drug D2 now to only those patients who fail to recover when administered drug D1 first. The results are presented in Table

Now from Table 1 we have that f 11 + =20; f 11 =20; f 21 + =15and f 21 =25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaaGymaiaaigdaaeaacqGHRaWkaaqcfaOaeyypa0Ja aGOmaiaaicdacaGG7aGaamOzamaaDaaajuaibaGaaGymaiaaigdaae aacqGHsislaaqcfaOaeyypa0JaaGOmaiaaicdacaGG7aGaamOzamaa DaaajuaibaGaaGOmaiaaigdaaeaacqGHRaWkaaqcfaOaeyypa0JaaG ymaiaaiwdacaaMe8UaaGjbVlaadggacaWGUbGaamizaiaaysW7caWG MbWaa0baaKqbGeaacaaIYaGaaGymaaqaaiabgkHiTaaajuaGcqGH9a qpcaaIYaGaaGynaaaa@5A00@ .

Hence

π ^ 11 + = P 11 = 20 40 =0.50; π ^ 11 =1 P 11 =1 20 40 =10.50=0.50; π ^ 21 + = P 21 = 15 40 =0.375;and π ^ 21 =1 P 21 = 25 40 =0.625 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGcu aHapaCgaqcamaaDaaajuaibaGaaGymaiaaigdaaeaacqGHRaWkaaqc faOaeyypa0JaamiuamaaBaaajuaibaGaaGymaiaaigdaaKqbagqaai abg2da9maalaaabaGaaGOmaiaaicdaaeaacaaI0aGaaGimaaaacqGH 9aqpcaaIWaGaaiOlaiaaiwdacaaIWaGaai4oaiqbec8aWzaajaWaa0 baaKqbGeaacaaIXaGaaGymaaqaaiabgkHiTaaajuaGcqGH9aqpcaaI XaGaeyOeI0IaamiuamaaBaaajuaibaGaaGymaiaaigdaaKqbagqaai abg2da9iaaigdacqGHsisldaWcaaqaaiaaikdacaaIWaaabaGaaGin aiaaicdaaaGaeyypa0JaaGymaiabgkHiTiaaicdacaGGUaGaaGynai aaicdacqGH9aqpcaaIWaGaaiOlaiaaiwdacaaIWaGaai4oaaGcbaqc faOafqiWdaNbaKaadaqhaaqcfasaaiaaikdacaaIXaaabaGaey4kaS caaKqbakabg2da9iaadcfadaWgaaqcfasaaiaaikdacaaIXaaajuaG beaacqGH9aqpdaWcaaqaaiaaigdacaaI1aaabaGaaGinaiaaicdaaa Gaeyypa0JaaGimaiaac6cacaaIZaGaaG4naiaaiwdacaGG7aGaamyy aiaad6gacaWGKbGaaGjbVlqbec8aWzaajaWaa0baaKqbGeaacaaIYa GaaGymaaqaaiabgkHiTaaajuaGcqGH9aqpcaaIXaGaeyOeI0Iaamiu amaaBaaajuaibaGaaGOmaiaaigdaaKqbagqaaiabg2da9maalaaaba GaaGOmaiaaiwdaaeaacaaI0aGaaGimaaaacqGH9aqpcaaIWaGaaiOl aiaaiAdacaaIYaGaaGynaaaaaa@8E63@

To test the null hypothesis H0 of equation 8 we have from equation 11 that

χ 2 = 40 ( 0.500.375 ) 2 ( 0.50 )( 0.50 )+( 0.375 )( 0.625 ) = 0.625 0.250+0.234 =1.291( Pvalue=0.1208 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaaiaaisda caaIWaWaaeWaaeaacaaIWaGaaiOlaiaaiwdacaaIWaGaeyOeI0IaaG imaiaac6cacaaIZaGaaG4naiaaiwdaaiaawIcacaGLPaaadaahaaqa bKqbGeaacaaIYaaaaaqcfayaamaabmaabaGaaGimaiaac6cacaaI1a GaaGimaaGaayjkaiaawMcaamaabmaabaGaaGimaiaac6cacaaI1aGa aGimaaGaayjkaiaawMcaaiabgUcaRmaabmaabaGaaGimaiaac6caca aIZaGaaG4naiaaiwdaaiaawIcacaGLPaaadaqadaqaaiaaicdacaGG UaGaaGOnaiaaikdacaaI1aaacaGLOaGaayzkaaaaaiabg2da9maala aabaGaaGimaiaac6cacaaI2aGaaGOmaiaaiwdaaeaacaaIWaGaaiOl aiaaikdacaaI1aGaaGimaiabgUcaRiaaicdacaGGUaGaaGOmaiaaio dacaaI0aaaaiabg2da9iaaigdacaGGUaGaaGOmaiaaiMdacaaIXaWa aeWaaeaacaWGqbGaeyOeI0IaamODaiaadggacaWGSbGaamyDaiaadw gacqGH9aqpcaaIWaGaaiOlaiaaigdacaaIYaGaaGimaiaaiIdaaiaa wIcacaGLPaaaaaa@7A17@

Which with 1 degree of freedom is not statistical significant ( Pvalue=0.1208 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGqbGaeyOeI0IaamODaiaadggacaWGSbGaamyDaiaadwgacqGH 9aqpcaaIWaGaaiOlaiaaigdacaaIYaGaaGimaiaaiIdaaiaawIcaca GLPaaaaaa@43EA@ . Further research interest would now be to administer treatment T1(drug D2) to subject who fail to respond positive when administered treatment T2(drug D2) first, and also to administer treatment T2(drug D2) to subjects who fail to respond positive when administered treatment T1(drug D1) first and compare the positive responds rates for the two groups of subjects. The results are shown in Table 2.

Pair(i)

u i1l | u i2l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaamyAaiaaigdacaWGSbaajuaGbeaadaabbaqaaiaa dwhadaWgaaqcfasaaiaadMgacaaIYaGaamiBaaqcfayabaaacaGLhW oaaaa@40FB@

Pair(i)

u i1l | u i2l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaamyAaiaaigdacaWGSbaajuaGbeaadaabbaqaaiaa dwhadaWgaaqcfasaaiaadMgacaaIYaGaamiBaaqcfayabaaacaGLhW oaaaa@40FB@

Pair(i)

u i1l | u i2l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaamyAaiaaigdacaWGSbaajuaGbeaadaabbaqaaiaa dwhadaWgaaqcfasaaiaadMgacaaIYaGaamiBaaqcfayabaaacaGLhW oaaaa@40FB@

1

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@     T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

15

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@     T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

29

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@    T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

2

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@     T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

16

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@     T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

30

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@    T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

3

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@     T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

17

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@     T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

31

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@   T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

4

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@   T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

18

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@    T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

32

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@     T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

5

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@    T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

19

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@    T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

33

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@     T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

6

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@     T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

20

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@    T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

34

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@    T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

7

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@    T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

21

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@    T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

35

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@    T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

8

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@     T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

22

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@      T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

36

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@   T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

9

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@      T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

23

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@     T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa aaleaacaaIYaaabaGaey4kaScaaaaa@389A@

37

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@     T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

10

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@       T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

24

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@    T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

38

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@     T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

11

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@      T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

25

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@     T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

39

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@    T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

12

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@      T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

26

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@   T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

40

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@    T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

13

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@     T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

27

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@    T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

f l1 + = W l1 π ^ l1 + = P l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGMbWaa0baaKqbGeaacaWGSbGaaGymaaqaaiabgUcaRaaajuaGcqGH 9aqpcaWGxbWaaSbaaKqbGeaacaWGSbGaaGymaaqcfayabaaakeaaju aGcuaHapaCgaqcamaaDaaajuaibaGaamiBaiaaigdaaeaacqGHRaWk aaqcfaOaeyypa0JaamiuamaaBaaajuaibaGaamiBaiaaigdaaKqbag qaaaaaaa@4982@

14

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@       T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

28

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@   T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

Table 1: Responses (+,-) by subjects in Randomly Selected Matched pairs Administered Treatment T l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamiBaaqcfayabaaaaa@392B@ first ( u il1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG1bWaaSbaaKqbGeaacaWGPbGaamiBaiaaigdaaKqbagqaaaGa ayjkaiaawMcaaaaa@3C7E@ .

S/N of Subjects Responding Negative When Given Treatment T2 First

Subject Response to Treatment T1 When Given Later

u i12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaamyAaiaaigdacaaIYaaajuaGbeaaaaa@3AC0@

S/N of Subjects Responding Negative When Given Treatment T1 First

Subject Response to Treatment T2 When Given Later

u i22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaamyAaiaaikdacaaIYaaajuaGbeaaaaa@3AC1@

1

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

1

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

2

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

3

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

4

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

4

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

5

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

5

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

6

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

11

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

8

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

13

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

9

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

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T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

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T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

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18

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T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

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T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

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24

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

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19

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

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25

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

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20

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

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30

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

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T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

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31

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22

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33

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

25

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

34

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

26

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

37

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

27

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

38

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

28

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

39

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

29

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

40

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

T 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgUcaRaaaaaa@394B@

31

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgkHiTaaaaaa@3955@

 

 

 

32

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

 

 

 

37

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

 

 

 

38

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

 

 

 

40

T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGOmaaqaaiabgkHiTaaaaaa@3956@

T 1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaDaaajuaibaGaaGymaaqaaiabgUcaRaaaaaa@394A@

 

 

 

Table 2: Responses (+,-) to treatment T l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamiBaaqcfayabaaaaa@392B@ by Randomly Selected subjects from Matched Pairs of Subjects who fail to Respond positive when Treated with Treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamOAaaqcfayabaaaaa@3929@ first ( u il2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG1bWaaSbaaKqbGeaacaWGPbGaamiBaiaaikdaaKqbagqaaaGa ayjkaiaawMcaaaaa@3C7F@ .

Now from Table 2 we have that f 12 + =13, f 12 =12, f 22 + =12and f 22 =8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaaGymaiaaikdaaeaacqGHRaWkaaqcfaOaeyypa0Ja aGymaiaaiodacaGGSaGaamOzamaaDaaajuaibaGaaGymaiaaikdaae aacqGHsislaaqcfaOaeyypa0JaaGymaiaaikdacaGGSaGaamOzamaa DaaajuaibaGaaGOmaiaaikdaaeaacqGHRaWkaaqcfaOaeyypa0JaaG ymaiaaikdacaaMe8Uaamyyaiaad6gacaWGKbGaaGjbVlaadAgadaqh aaqcfasaaiaaikdacaaIYaaabaGaeyOeI0caaKqbakabg2da9iaaiI daaaa@57A0@ .

Hence

π ^ 12 + = P 12 = 13 25 =0.52; π ^ 12 =1 P 12 = 12 25 =0.48; π ^ 22 + = P 22 = 12 20 =0.60and π ^ 22 + =1 P 22 = 8 20 =0.40 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaaigdacaaIYaaabaGaey4kaScaaKqbakab g2da9iaadcfadaWgaaqcfasaaiaaigdacaaIYaaajuaGbeaacqGH9a qpdaWcaaqaaiaaigdacaaIZaaabaGaaGOmaiaaiwdaaaGaeyypa0Ja aGimaiaac6cacaaI1aGaaGOmaiaacUdacuaHapaCgaqcamaaDaaaju aibaGaaGymaiaaikdaaeaacqGHsislaaqcfaOaeyypa0JaaGymaiab gkHiTiaadcfadaWgaaqcfasaaiaaigdacaaIYaaajuaGbeaacqGH9a qpdaWcaaqaaiaaigdacaaIYaaabaGaaGOmaiaaiwdaaaGaeyypa0Ja aGimaiaac6cacaaI0aGaaGioaiaacUdacuaHapaCgaqcamaaDaaaju aibaGaaGOmaiaaikdaaeaacqGHRaWkaaqcfaOaeyypa0Jaamiuamaa BaaajuaibaGaaGOmaiaaikdaaKqbagqaaiabg2da9maalaaabaGaaG ymaiaaikdaaeaacaaIYaGaaGimaaaacqGH9aqpcaaIWaGaaiOlaiaa iAdacaaIWaGaaGjbVlaadggacaWGUbGaamizaiaaysW7cuaHapaCga qcamaaDaaajuaibaGaaGOmaiaaikdaaeaacqGHRaWkaaqcfaOaeyyp a0JaaGymaiabgkHiTiaadcfadaWgaaqcfasaaiaaikdacaaIYaaaju aGbeaacqGH9aqpdaWcaaqaaiaaiIdaaeaacaaIYaGaaGimaaaacqGH 9aqpcaaIWaGaaiOlaiaaisdacaaIWaaaaa@8520@ .

Therefore the resulting difference in positive response rates by those two populations of subjects is estimated as π ^ 12 + π ^ 22 + = P 12 P 22 =0.520.60=0.08 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaaigdacaaIYaaabaGaey4kaScaaKqbakab gkHiTiqbec8aWzaajaWaa0baaKqbGeaacaaIYaGaaGOmaaqaaiabgU caRaaajuaGcqGH9aqpcaWGqbWaaSbaaKqbGeaacaaIXaGaaGOmaaqc fayabaGaeyOeI0IaamiuamaaBaaajuaibaGaaGOmaiaaikdaaKqbag qaaiabg2da9iaaicdacaGGUaGaaGynaiaaikdacqGHsislcaaIWaGa aiOlaiaaiAdacaaIWaGaeyypa0JaeyOeI0IaaGimaiaac6cacaaIWa GaaGioaaaa@565B@ .

To test the null hypothesis H0 of equation 20 that subjects who fail to respond positive when administered treatment T2(D2) first but respond positive when administered treatment T1(D1) first are equally likely to experience the same level of positive responds this time around as subject who fail to respond positive when administered treatment T1(D1) first but respond positive when administered treatment T2(D2) later, we obtain from equation 24 that the required chi-square test statistics as

χ 2 = ( 25 )( 20 ) ( 0.08 ) 2 20(0.52)(0.48)+25(0.60)(0.40) = 3.20 10.992 =0.291 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaamaabmaa baGaaGOmaiaaiwdaaiaawIcacaGLPaaadaqadaqaaiaaikdacaaIWa aacaGLOaGaayzkaaWaaeWaaeaacaaIWaGaaiOlaiaaicdacaaI4aaa caGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaaaKqbagaacaaIYa GaaGimaiaacIcacaaIWaGaaiOlaiaaiwdacaaIYaGaaiykaiaacIca caaIWaGaaiOlaiaaisdacaaI4aGaaiykaiabgUcaRiaaikdacaaI1a GaaiikaiaaicdacaGGUaGaaGOnaiaaicdacaGGPaGaaiikaiaaicda caGGUaGaaGinaiaaicdacaGGPaaaaiabg2da9maalaaabaGaaG4mai aac6cacaaIYaGaaGimaaqaaiaaigdacaaIWaGaaiOlaiaaiMdacaaI 5aGaaGOmaaaacqGH9aqpcaaIWaGaaiOlaiaaikdacaaI5aGaaGymaa aa@68DD@

Which with 1 degree of freedom is not statistically significant again leading to an acceptance of the null hypothesis of equal population proportions of positive responds by subjects or patients?

Conclusion

We have in this paper proposed and presented a chi-square statistical method for the analysis of response from one period cross over design for two sample data in which the sampled populations may be measurements that are numeric (assuming real values) and non-numeric assuming only values on the nominal scale. Test statistics developed were used in testing the null hypothesis that subjects who receive each of the treatments first do not differ in their response leading to the acceptance of the null hypothesis of no difference as well as the null hypothesis that subjects exposed to one of the treatment or experimental conditions first do not on the average differ in their responses with those exposed to the other treatment or experimental condition first also leading to the acceptance of the null hypothesis of no difference. Estimates of the proportions responding positive; experiencing no change in response or responding negative are provided for subjects exposed to each treatment first as well as for the two treatments together.

The proposed method was illustrated with some sample non-numeric data here and is shown to be at least as powerful as the traditional two sample small ’t’ test.

References

  1. Meiser P, Free SM, Jackson GL (1958) Re-moderation of methodology in studies of pains relief. Biometrics 14: 330-342.
  2.  Gart JJ (1969) An exact test for comparing matched proportions in crossover designs. Biometrika 56(1): 75-80.
  3. Grizzle JE (1965) The two-period change-over design and its use in clinical trials. Biometrics 21: 467-480.
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