ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 2 Issue 7 - 2015
Simultaneous Estimation of Adjusted Rate of Two Factors Using Method of Direct Standardization
Oyeka ICA1* and Okeh UM2
1Department of Statistics, Nnamdi Azikiwe University, Nigeria
2Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Nigeria
Received: August 24, 2015 | Published: October 27, 2015
*Corresponding author: Oyeka ICA, Department of Statistics, Nnamdi Azikiwe University, Awka, Nigeria, Email:
Citation: Oyeka ICA, Okeh UM (2015) Simultaneous Estimation of Adjusted Rate of Two Factors Using Method of Direct Standardization. Biom Biostat Int J 2(7): 00051. DOI: 10.15406/bbij.2015.02.00051

Abstract

This paper presents the use of standardization or adjustment of rates and ratios in comparing two populations using single indices rather than a series of specific rates or ratios. Here the overall adjusted crude rate or the unadjusted crude rate for two populations will have same estimate irrespective of the nature of the standard population distribution. These results are obtained in all cases whenever the two standard distributions are of the total sample. In these cases the overall adjusted crude rates based on the two sets of directly adjusted rates would be equal to each other, although not necessarily always equal to the overall unadjusted crude rate as is found to be the case here. However, if the standard population distribution chosen for a given population is different from that chosen for another, then the two resulting estimated adjusted or standardized crude rates would most likely not be equal to each other.

Keywords: Standardization; Adjusted specific; Unadjusted crude rate; Adjusted crude rate; Ratios

Introduction

Standardization or adjustment of rates and ratios is often necessary because it is usually easier in comparing two populations, say, to make the comparison using single summary indices rather than a series of specific rates or ratios. This approach also helps avoid the problem of small imprecise and sometime non-existence of specific rates and ratios [1-3].

Standardization of rates and ratios may be done for only one factor or several factors of classification of a criterion variable of interest. In particular if a criterion variable or condition is associated with each of two factors of classification which may by themselves also be associated with each other, then standardization of rates or ratios may sometimes be necessary for a clearer analysis and inter-presentation of results to simultaneously standardize or adjust the rates for the two factors of classification, first specific to the levels or categories of one of the factors across the levels of the other factor, and then also specific to the levels of the second factor say holding constant the levels, that is for all levels or categories of the first factor [4,5]. Research interest in this case would be to identify and measure the separate effects of the two factors of classification on the criterion variable or condition. This paper proposes, develops and presents a formatted systematic statistical method for this purpose.

The proposed method

Research interest here is using the direct method of standardization of rates to measure or estimate the separate effects of two factors of classification which may be associated on the variable being studied and to obtain sample estimate of unadjusted and adjusted crude rates specific to the levels of each of the factors holding the levels of the other factor of classification constant.

Now to develop the method of estimation of direct standardized or adjusted rate, suppose A and B are two variables of classification with ‘a’ and ‘b’ groups or levels respectively. Factors A and B may be associated or related. Research interest is to estimate the rates of occurrence of a criterion variable or condition specific to each of the levels of factor A across, that is for all levels of factor B and also the rates of occurrence of the specified condition specific to each of the levels of factor B for all levels of factor A as well as the corresponding marginal rates and overall rate. Suppose a total random sample of size N=N.. of subject are randomly drawn from an antecedent or predisposing population C for all levels of factors A and B of which Nij is the number of subjects at the ith level of factor A and jth level of factor B, for i=1,2,…aj and j=1,2,…..,b. Also suppose there are a total of n=n.. outcomes or cases in condition or set D of cases for all levels of factors A and B of which nij cases are at the ith level of factor A and jth level of factor B, for i=1,2,…aj and j=1,2,…,b where population D is possibly a subset of population C.

Now the rate of occurrence of cases in population D as a function of cases in population C specific to the ith level of factor A and jth level factor B is
r ij = n ij N ij ........1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaiaadQgaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaWGUbqcfa4aaSbaaSqaaKqzadGaamyAai aadQgaaSqabaaakeaajugibiaad6eajuaGdaWgaaWcbaqcLbmacaWG PbGaamOAaaWcbeaaaaqcLbsacaGGUaGaaiOlaiaac6cacaGGUaGaai Olaiaac6cacaGGUaGaaiOlaiaaigdaaaa@4EF1@
For i=1,2,…aj; j=1,2,…b.
Let
N i. = j=1 b N ij ; N .j = i=1 a N ij .........2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob qcfa4aaSbaaSqaaKqzadGaamyAaiaac6caaSqabaqcLbsacqGH9aqp juaGdaaeWbGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaamyAai aadQgaaSqabaqcLbsacaGG7aGaamOtaKqbaoaaBaaaleaajugWaiaa c6cacaWGQbaaleqaaKqzGeGaeyypa0tcfa4aaabCaOqaaKqzGeGaam OtaKqbaoaaBaaaleaajugWaiaadMgacaWGQbaaleqaaKqzGeGaaiOl aiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUa GaaGOmaaWcbaqcLbmacaWGPbGaeyypa0JaaGymaaWcbaqcLbmacaWG HbaajugibiabggHiLdaaleaajugWaiaadQgacqGH9aqpcaaIXaaale aajugWaiaadkgaaKqzGeGaeyyeIuoaaaa@686D@
be respectively the total or marginal number of subjects or observations in population C at the  ith level  of factor A and jth level of factor B.
Similarly let
r i.;unadj = n i. N i. ; r .j;unadj = n .j N .j ........4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaiaac6cacaGG7aGaamyDaiaad6ga caWGHbGaamizaiaadQgaaSqabaqcLbsacqGH9aqpjuaGdaWcaaGcba qcLbsacaWGUbqcfa4aaSbaaSqaaKqzadGaamyAaiaac6caaSqabaaa keaajugibiaad6eajuaGdaWgaaWcbaqcLbmacaWGPbGaaiOlaaWcbe aaaaqcLbsacaGG7aGaamOCaKqbaoaaBaaaleaajugWaiaac6cacaWG QbGaai4oaiaadwhacaWGUbGaamyyaiaadsgacaWGQbaaleqaaKqzGe Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaamOBaKqbaoaaBaaaleaajugW aiaac6cacaWGQbaaleqaaaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaK qzadGaaiOlaiaadQgaaSqabaaaaKqzGeGaaiOlaiaac6cacaGGUaGa aiOlaiaac6cacaGGUaGaaiOlaiaac6cacaaI0aaaaa@6B3D@
be respectively the total or marginal number of  cases or outcomes in population D at the ith level of factor A and jth level of factor B. Then the estimated unadjusted crude rates of occurrence of cases or outcomes in population D as a function of outcomes in population C specific to the ith level of factor A for all levels of factor B for all levels of factor A are respectively the ratios
N= N .. = i=1 a n i. = j=1 b N .j = j=1 b i=1 a N ij ...........(5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtai abg2da9iaad6eadaWgaaqcfasaaiaac6cacaGGUaaajuaGbeaacqGH 9aqpdaaeWbqaaiaad6gadaWgaaqcfasaaiaadMgacaGGUaaajuaGbe aacqGH9aqpdaaeWbqaaiaad6eadaWgaaqcfasaaiaac6cacaWGQbaa juaGbeaacqGH9aqpdaaeWbqaamaaqahabaGaamOtamaaBaaajuaiba GaamyAaiaadQgaaKqbagqaaaqcfasaaiaadMgacqGH9aqpcaaIXaaa baGaamyyaaqcfaOaeyyeIuoaaKqbGeaacaWGQbGaeyypa0JaaGymaa qaaiaadkgaaKqbakabggHiLdaajuaibaGaamOAaiabg2da9iaaigda aeaacaWGIbaajuaGcqGHris5aaqcfasaaiaadMgacqGH9aqpcaaIXa aabaGaamyyaaqcfaOaeyyeIuoacaGGUaGaaiOlaiaac6cacaGGUaGa aiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaacIcaca aI1aGaaiykaaaa@6C7F@
For i=1,2,….a; j=1,2,…b.
Note that
n= n .. = i=1 a n i. = j=1 b n .j = j=1 b i=1 a n ij ..............(6) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb Gaeyypa0JaamOBaKqbaoaaBaaaleaajugWaiaac6cacaGGUaaaleqa aKqzGeGaeyypa0tcfa4aaabCaOqaaKqzGeGaamOBaKqbaoaaBaaale aajugWaiaadMgacaGGUaaaleqaaaqaaKqzadGaamyAaiabg2da9iaa igdaaSqaaKqzadGaamyyaaqcLbsacqGHris5aiabg2da9Kqbaoaaqa hakeaajugibiaad6gajuaGdaWgaaWcbaqcLbmacaGGUaGaamOAaaWc beaajugibiabg2da9KqbaoaaqahakeaajuaGdaaeWbGcbaqcLbsaca WGUbqcfa4aaSbaaSqaaKqzadGaamyAaiaadQgaaSqabaaabaqcLbma caWGPbGaeyypa0JaaGymaaWcbaqcLbmacaWGHbaajugibiabggHiLd aaleaajugWaiaadQgacqGH9aqpcaaIXaaaleaajugWaiaadkgaaKqz GeGaeyyeIuoacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caca GGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaa cIcacaaI2aGaaiykaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcba qcLbmacaWGIbaajugibiabggHiLdaaaa@8198@
And
n= n .. = i=1 a n i. = j=1 b n .j = j=1 b i=1 a n ij ..............(6) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb Gaeyypa0JaamOBaKqbaoaaBaaaleaajugWaiaac6cacaGGUaaaleqa aKqzGeGaeyypa0tcfa4aaabCaOqaaKqzGeGaamOBaKqbaoaaBaaale aajugWaiaadMgacaGGUaaaleqaaaqaaKqzadGaamyAaiabg2da9iaa igdaaSqaaKqzadGaamyyaaqcLbsacqGHris5aiabg2da9Kqbaoaaqa hakeaajugibiaad6gajuaGdaWgaaWcbaqcLbmacaGGUaGaamOAaaWc beaajugibiabg2da9KqbaoaaqahakeaajuaGdaaeWbGcbaqcLbsaca WGUbqcfa4aaSbaaSqaaKqzadGaamyAaiaadQgaaSqabaaabaqcLbma caWGPbGaeyypa0JaaGymaaWcbaqcLbmacaWGHbaajugibiabggHiLd aaleaajugWaiaadQgacqGH9aqpcaaIXaaaleaajugWaiaadkgaaKqz GeGaeyyeIuoacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caca GGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaa cIcacaaI2aGaaiykaaWcbaqcLbmacaWGQbGaeyypa0JaaGymaaWcba qcLbmacaWGIbaajugibiabggHiLdaaaa@8198@    
Therefore the overall unadjusted crude rate of occurrence of event D as a function of event C for all levels of factors A and B is
r unadj =r= n .. N .. ...........(7) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyDaiaad6gacaWGHbGaamizaiaadQga aSqabaqcLbsacqGH9aqpcaWGYbGaeyypa0tcfa4aaSaaaOqaaKqzGe GaamOBaKqbaoaaBaaaleaajugWaiaac6cacaGGUaaaleqaaaGcbaqc LbsacaWGobqcfa4aaSbaaSqaaKqzadGaaiOlaiaac6caaSqabaaaaK qzGeGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaa c6cacaGGUaGaaiOlaiaac6cacaGGOaGaaG4naiaacMcaaaa@563F@
As noted above research interest is to obtain standardized or adjusted crude rate specific to each level of factor A for all levels of factor B and also specific to each level of factor B for all levels of factor A as well as the overall adjusted or standardized crude rate. To obtain estimates of adjusted or standardized crude rates specific to each level of factor B for all levels of factor A we use the proportionate distribution of total number of observations N .. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaGGUaGaaiOlaaqabaaaaa@3859@ across the ‘a’ levels or groups of factor A, namely Pis the waiting factor, for i=1,2,..,a.
Thus
P is = N i. N .. .........(8) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzadGaamyAaiaadohaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaamyAai aac6caaSqabaaakeaajugibiaad6eajuaGdaWgaaWcbaqcLbmacaGG UaGaaiOlaaWcbeaaaaqcLbsacaGGUaGaaiOlaiaac6cacaGGUaGaai Olaiaac6cacaGGUaGaaiOlaiaac6cacaGGOaGaaGioaiaacMcaaaa@5014@
Similarly to obtain estimates of adjusted or standardized crude rate specific to each level of factor A for all levels of factor B we use the proportionate distributions N.. across the ‘b’ levels or groups of factor B, namely Psj the waiting factor, for j=1,2,…,’b.’ Thus
P sj = N .j N .. ..........(9) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzadGaam4CaiaadQgaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaaiOlai aadQgaaSqabaaakeaajugibiaad6eajuaGdaWgaaWcbaqcLbmacaGG UaGaaiOlaaWcbeaaaaqcLbsacaGGUaGaaiOlaiaac6cacaGGUaGaai Olaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiikaiaaiMdacaGG Paaaaa@50C9@
Hence the adjusted or standardized crude rate of condition D as a function of population C specific to the jth level of factor B for all levels of factor A is
r .j;adj = i=1 a p is r ij ...........(10) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaadQgacaGG7aGaamyyaiaadsga caWGQbaaleqaaKqzGeGaeyypa0tcfa4aaabCaOqaaKqzGeGaamiCaK qbaoaaBaaaleaajugWaiaadMgacaWGZbaaleqaaKqzGeGaamOCaKqb aoaaBaaaleaajugWaiaadMgacaWGQbaaleqaaaqaaKqzadGaamyAai abg2da9iaaigdaaSqaaKqzadGaamyyaaqcLbsacqGHris5aiaac6ca caGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlai aac6cacaGGUaGaaiikaiaaigdacaaIWaGaaiykaaaa@5E96@
Similarly the adjusted or standardized crude rate of condition D as a function of population C specific to the ith level of factor A for all levels of factor B is
r i.;adj = j=1 b p sj r ij ..........(11) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaiaac6cacaGG7aGaamyyaiaadsga caWGQbaaleqaaKqzGeGaeyypa0tcfa4aaabCaOqaaKqzGeGaamiCaK qbaoaaBaaaleaajugWaiaadohacaWGQbaaleqaaKqzGeGaamOCaKqb aoaaBaaaleaajugWaiaadMgacaWGQbaaleqaaaqaaKqzadGaamOAai abg2da9iaaigdaaSqaaKqzadGaamOyaaqcLbsacqGHris5aiaac6ca caGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlai aac6cacaGGOaGaaGymaiaaigdacaGGPaaaaa@5DE7@
We then obtain the sample estimate of the overall adjusted crude rate of condition D as a function of population C for all levels of factors A and B as
r s;adj = r ..adj = i=1 a p is. r i. = j=1 b p sj . r .j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaam4CaiaacUdacaWGHbGaamizaiaadQga aSqabaqcLbsacqGH9aqpcaWGYbqcfa4aaSbaaSqaaKqzadGaaiOlai aac6cacaWGHbGaamizaiaadQgaaSqabaqcLbsacqGH9aqpjuaGdaae WbGcbaqcLbsacaWGWbqcfa4aaSbaaSqaaKqzadGaamyAaiaadohaca GGUaaaleqaaKqzGeGaamOCaKqbaoaaBaaaleaajugWaiaadMgacaGG UaaaleqaaaqaaKqzadGaamyAaiabg2da9iaaigdaaSqaaKqzadGaam yyaaqcLbsacqGHris5aiabg2da9Kqbaoaaqahakeaajugibiaadcha juaGdaWgaaWcbaqcLbmacaWGZbGaamOAaaWcbeaajugibiaac6caca WGYbqcfa4aaSbaaSqaaKqzadGaaiOlaiaadQgaaSqabaaabaqcLbma caWGQbGaeyypa0JaaGymaaWcbaqcLbmacaWGIbaajugibiabggHiLd aaaa@720E@ …………(12)
These results are summarized in Table 1.

                                                               FACTOR B

FACTOR A

1

2

………..

b

Total

Proportion

Un adjust

adjust

 

n 11 ( N 11 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaGymaiaaigdaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaaigdacaaIXaaaleqaaKqzGeGaai ykaaaa@4194@

n 12 ( N 12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaGymaiaaikdaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaaigdacaaIYaaaleqaaKqzGeGaai ykaaaa@4196@

……….

n 1b ( N 1b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaGymaiaadkgaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaaigdacaWGIbaaleqaaKqzGeGaai ykaaaa@41EC@

( n 1. ( N 1. ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaaigdacaGGUaaaleqa aKqzGeGaaiikaiaad6eajuaGdaWgaaWcbaqcLbmacaaIXaGaaiOlaa WcbeaajugibiaacMcaaOGaayjkaiaawMcaaaaa@43AD@

( p sj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiCaKqbaoaaBaaaleaajugWaiaadohacaWGQbaaleqa aaGccaGLOaGaayzkaaaaaa@3D7F@

( r .j;adj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOCaKqbaoaaBaaaleaajugWaiaac6cacaWGQbGaai4o aiaadggacaWGKbGaamOAaaWcbeaaaOGaayjkaiaawMcaaaaa@40B8@

( r .j;unadj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOCaKqbaoaaBaaaleaajugWaiaac6cacaWGQbGaai4o aiaadwhacaWGUbGaamyyaiaadsgacaWGQbaaleqaaaGccaGLOaGaay zkaaaaaa@42A5@

1

 

 

 

 

 

 

 

 

r .1;adj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaGGUaGaaGymaiaacUdacaWGHbGaamizaiaadQgaaeqaaaaa @3C03@

 

 

 

 

 

 

 

 

n .1 N .1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaac6cacaaIXaaaleqa aaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaaiOlaiaaigdaaS qabaaaaaaa@404C@

 

 

 

 

 

 

 

 

2

n 21 ( N 21 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaGOmaiaaigdaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaaikdacaaIXaaaleqaaKqzGeGaai ykaaaa@4196@

n 22 ( N 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaGOmaiaaikdaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaaikdacaaIYaaaleqaaKqzGeGaai ykaaaa@4198@

……..

n 2b ( N 2b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaGOmaiaadkgaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaaikdacaWGIbaaleqaaKqzGeGaai ykaaaa@41EE@

n 2. ( N 2. ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaGOmaiaac6caaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaaikdacaGGUaaaleqaaKqzGeGaai ykaaaa@4184@

p 2s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb qcfa4aaSbaaSqaaKqzadGaaGOmaiaadohaaSqabaaaaa@3B21@

n 2. N 2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaaikdacaGGUaaaleqa aaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaaGOmaiaac6caaS qabaaaaaaa@404E@

r 2.;adj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaGOmaiaac6cacaGG7aGaamyyaiaadsga caWGQbaaleqaaaaa@3E5A@

r .2;adj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaaikdacaGG7aGaamyyaiaadsga caWGQbaaleqaaaaa@3E5A@

r 21 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaGOmaiaaigdaaSqabaaaaa@3AE6@

r 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaGOmaiaaikdaaSqabaaaaa@3AE7@

………

r 2b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaGOmaiaadkgaaSqabaaaaa@3B12@

r 2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaGOmaiaac6caaSqabaaaaa@3ADD@

….

….

…..

n .2 N .2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaac6cacaaIYaaaleqa aaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaaiOlaiaaikdaaS qabaaaaaaa@404E@

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

n a1 ( N a1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaamyyaiaaigdaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaadggacaaIXaaaleqaaKqzGeGaai ykaaaa@41EA@

n a2 ( N a2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaamyyaiaaikdaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaadggacaaIYaaaleqaaKqzGeGaai ykaaaa@41EC@

……

n ab ( N ab ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaamyyaiaadkgaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaadggacaWGIbaaleqaaKqzGeGaai ykaaaa@4242@

p sa. n a. ( N a. ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamiCaKqbaoaaBaaaleaajugWaiaadohacaWGHbGaaiOl aaWcbeaaaOqaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaadggaca GGUaaaleqaaKqzGeGaaiikaiaad6eajuaGdaWgaaWcbaqcLbmacaWG HbGaaiOlaaWcbeaajugibiaacMcaaaaaaa@4891@

p as MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb qcfa4aaSbaaSqaaKqzadGaamyyaiaadohaaSqabaaaaa@3B4B@

n a. N a. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaadggacaGGUaaaleqa aaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaamyyaiaac6caaS qabaaaaaaa@40A2@

r a.;adj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyyaiaac6cacaGG7aGaamyyaiaadsga caWGQbaaleqaaaaa@3E84@

r .j;adj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaadQgacaGG7aGaamyyaiaadsga caWGQbaaleqaaaaa@3E8D@

r a1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqbaoaaBaaameaajugWaiaadggacaaIXaaameqa aaWcbeaaaaa@3BD7@

r a2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqbaoaaBaaameaajugWaiaadggacaaIYaaameqa aaWcbeaaaaa@3BD8@

…….

r ab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqbaoaaBaaameaajugWaiaadggacaWGIbaameqa aaWcbeaaaaa@3C03@

r a. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaamaaBaaameaajugWaiaadggacaGGUaaameqaaaWc beaaaaa@3B40@

 

 

 

Total

n .1 ( N .1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaiOlaiaaigdaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaac6cacaaIXaaaleqaaKqzGeGaai ykaaaa@4182@

n .2 ( N .2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaiOlaiaaikdaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaac6cacaaIYaaaleqaaKqzGeGaai ykaaaa@4184@

 

n .b ( N .b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzadGaaiOlaiaadkgaaSqabaqcLbsacaGGOaGa amOtaKqbaoaaBaaaleaajugWaiaac6cacaWGIbaaleqaaKqzGeGaai ykaaaa@41DA@

 

 

 

 

( n .j ( N .j ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaac6cacaWGQbaaleqa aKqzGeGaaiikaiaad6eajuaGdaWgaaWcbaqcLbmacaGGUaGaamOAaa WcbeaajugibiaacMcaaOGaayjkaiaawMcaaaaa@4415@

r .1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaaigdaaSqabaaaaa@3ADC@

r .2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaaikdaaSqabaaaaa@3ADD@

 

r .b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaadkgaaSqabaaaaa@3B08@

 

 

 

 

proportion

 

 

 

 

 

 

 

 

( p sj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiCaKqbaoaaBaaaleaajugWaiaadohacaWGQbaaleqa aaGccaGLOaGaayzkaaaaaa@3D7F@

p s1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb qcfa4aaSbaaSqaaKqzadGaam4CaiaaigdaaSqabaaaaa@3B20@

p s2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb qcfa4aaSbaaSqaaKqzadGaam4CaiaaikdaaSqabaaaaa@3B21@

……..

p sb MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb qcfa4aaSbaaSqaaKqzadGaam4CaiaadkgaaSqabaaaaa@3B4C@

….

…..

….

 

Un adjust ( r .j;adj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOCaKqbaoaaBaaaleaajugWaiaac6cacaWGQbGaai4o aiaadggacaWGKbGaamOAaaWcbeaaaOGaayjkaiaawMcaaaaa@40B8@

n .1 N .1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaac6cacaaIXaaaleqa aaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaaiOlaiaaigdaaS qabaaaaaaa@404C@

n .1 N .1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaac6cacaaIXaaaleqa aaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaaiOlaiaaigdaaS qabaaaaaaa@404C@

…..

n .b N .b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaac6cacaWGIbaaleqa aaGcbaqcLbsacaWGobqcfa4aaSbaaSqaaKqzadGaaiOlaiaadkgaaS qabaaaaaaa@40A4@

……

r .unadj = n .. N .. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaadwhacaWGUbGaamyyaiaadsga caWGQbaaleqaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamOBaK qbaoaaBaaaleaajugWaiaac6cacaGGUaaaleqaaaGcbaqcLbsacaWG obqcfa4aaSbaaSqaaKqzadGaaiOlaiaac6caaSqabaaaaaaa@4AA5@

 

Adjust ( r .j;unadj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOCaKqbaoaaBaaaleaajugWaiaac6cacaWGQbGaai4o aiaadwhacaWGUbGaamyyaiaadsgacaWGQbaaleqaaaGccaGLOaGaay zkaaaaaa@42A5@

r .1;adj. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaaigdacaGG7aGaamyyaiaadsga caWGQbGaaiOlaaWcbeaaaaa@3F0B@

r .2;adj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaaikdacaGG7aGaamyyaiaadsga caWGQbaaleqaaaaa@3E5A@

……

r .b;adj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaadkgacaGG7aGaamyyaiaadsga caWGQbaaleqaaaaa@3E85@

 

 

 

r ..;adj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaiOlaiaac6cacaGG7aGaamyyaiaadsga caWGQbaaleqaaaaa@3E50@

Table 1: Data format for Estimation of Unadjusted and Adjusted Rates in two Factor Standardization by Direct method.

In table 1 the entries in each of the cells are the number of cases in condition D the number of observations in population D and the ratios of these numbers.

Illustrative Example
We now illustrate the proposed method with the sample data of Table 2 on premature and live births by birth order and age of mother in a certain population.

                                                                      Birth Order

Maternal Age

1

2

3

4

5+

Total
( n i. ( N 1. ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaadMgacaGGUaaaleqa aKqzGeGaaiikaiaad6eajuaGdaWgaaWcbaqcLbmacaaIXaGaaiOlaa WcbeaajugibiaacMcaaOGaayjkaiaawMcaaaaa@43E0@

Proportion of total births ( p is ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamiCaKqbaoaaBaaaleaajugWaiaadMgacaWGZbaaleqaaKqzGeGa aiykaaaa@3D3B@

Under 20

11(23)

3(72)

3(32)

1(43)

0(33)

18(203)

 

 

0.478

0.042

0.094

0.023

0.000

0.089

0.066

20-24

14(329)

15(327)

7(176)

3(69)

8(67)

47(968)

 

 

0.043

0.046

0.040

0.043

0.119

0.049

0.012

25-29

6(115)

11(209)

11(207)

6(132)

6(123)

40(786)

 

 

0.052

0.053

0.053

0.045

0.049

0.051

0.254

30-34

4(78)

8(83)

10(117)

9(98)

12(150)

43(526)

 

 

0.051

0.096

0.085

0.092

0.080

0.082

0.170

35-39

4(42)

8(56)

11(90)

14(56)

3(104)

40(348)

 

 

0.095

0.143

0.122

0.050

0.029

0.115

0.112

40 and above

3(34)

4(457)

8(72)

10(48)

4(68)

29(267)

 

 

0.088

0.089

0.111

0.208

0.059

0.109

0.086

Total ( n .j ( N .j ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOBaKqbaoaaBaaaleaajugWaiaac6cacaWGQbaaleqa aKqzGeGaaiikaiaad6eajuaGdaWgaaWcbaqcLbmacaGGUaGaamOAaa WcbeaajugibiaacMcaaOGaayjkaiaawMcaaaaa@4415@

42(621)

49(792)

47(694)

45(446)

33(545)

217(3098)

 

 

0.068

0.010

0.068

0.096

0.060

0.070

0.070

Proportion of total births
( p sj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamiCaKqbaoaaBaaaleaajugWaiaadohacaWGQbaaleqaaKqzGeGa aiykaaaa@3D3C@

0.200

0.256

0.256

0.224

0.144

0.176

 

Table 2: Sample Data on Premature and Live births by Birth order and Maternal age in a population.

The data of Table 2 is used to obtain estimates of the unadjusted and adjusted crude rate specific to each of the levels or groups of the two factors of classification.

Specifically to estimate adjusted or standardized crude rates specific to birth order, we apply the proportionate distribution of the total life births across maternal age as the standard population, namely p is MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb qcfa4aaSbaaSqaaKqzadGaamyAaiaadohaaSqabaaaaa@3B53@ in the last column of Table 2 to each of the columns of rates, r ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaiaadQgaaSqabaaaaa@3B4C@ of the Table, for j=1,2,3,4,5.Similarly to estimate adjusted or standardize crude rate specific to Maternal age we apply the proportionate distribution of total life births across birth order as the standard population, namely p sj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb qcfa4aaSbaaSqaaKqzadGaam4CaiaadQgaaSqabaaaaa@3B54@ in the last row of Table 2 to each of the rows of rates, r ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaiaadQgaaSqabaaaaa@3B4C@ of the Table, for i=1,2,3,4,5,6.The results are presented in Table 3.

                                    Birth Order

Maternal Age

Proportion of total birth ( p is ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadc hadaWgaaWcbaGaamyAaiaadohaaeqaaOGaaiykaaaa@3A60@

r i1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaiaaigdaaSqabaaaaa@3B18@

1

r i2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaqaaKqzadGaamyAaiaaikdaaSqabaaaaa@3A8B@

2

r i3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaiaaiodaaSqabaaaaa@3B1A@

3

r i4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbGaaGinaaqabaaaaa@38C5@

4

r i5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaiaaiwdaaSqabaaaaa@3B1C@

5+

Unadjusted crude rate
( r 1.;unadj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamOCaKqbaoaaBaaaleaajugWaiaaigdacaGGUaGaai4oaiaadwha caWGUbGaamyyaiaadsgacaWGQbaaleqaaKqzGeGaaiykaaaa@422E@

Adjusted crude rate ( r 1.;adj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamOCaKqbaoaaBaaaleaajugWaiaaigdacaGGUaGaai4oaiaadgga caWGKbGaamOAaaWcbeaajugibiaacMcaaaa@4041@

less than 20

0.066

0.478

 

0.042

 

0.094

 

0.023

 

0.000

 

0.089

0.131

20-24

0.312

0.042

 

0.046

 

0.040

 

0.043

 

0.119

 

0.049

0.057

25-29

0.254

0.052

 

0.052

 

0.053

 

0.045

 

0.049

 

0.050

0.051

30-34

0.170

0.051

 

0.096

 

0.085

 

0.092

 

0.080

 

0.082

0.081

35-39

0.112

0.095

 

0.143

 

0.122

 

0.250

 

0.029

 

0.115

0.157

40 and over

0.086

0.088

 

0.008

 

0.111

 

0.208

 

0.059

 

0.109

0.594

Proportion of total birth ( p .j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamiCaKqbaoaaBaaaleaajugWaiaac6cacaWGQbaaleqaaKqzGeGa aiykaaaa@3CF6@

 

 

 

 

 

 

 

 

 

 

 

 

 

Unadjusted cruderate ( r .j;unadj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamOCaKqbaoaaBaaaleaajugWaiaac6cacaWGQbGaai4oaiaadwha caWGUbGaamyyaiaadsgacaWGQbaaleqaaKqzGeGaaiykaaaa@4262@

 

0.068

 

0.062

 

0.068

 

0.096

 

0.061

 

0.070

 

Adjusted crude rate ( r .j;adj ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaamOCaKqbaoaaBaaaleaajugWaiaac6cacaWGQbGaai4oaiaadgga caWGKbGaamOAaaWcbeaajugibiaacMcaaaa@4075@

 

 

0.086

 

0.070

 

0.069

 

0.088

 

0.068

 

0.070

Table 3: Simultaneous Estimates of Unadjusted Adjusted Premature Birth rates by maternal age and Birth order: Direct Standardization

Summary and Conclusion

The adjusted crude rate of premature births specific to birth order for all age groups shown in the last row of Table 3 are estimated using equation 10,while the corresponding adjusted crude rate specific to maternal age for all birth orders shown in the last column of Table 3 are estimated using equation 11. Thus the last two rows of Table 3 show rates specific for birth order and directly adjusted for maternal age, with the standard maternal age distribution of births being that of the total sample of births. The last two columns of the Table show rates specific for maternal age and directly adjusted for birth order, with the standard birth order distribution of birth being that of total sample of births.

The estimated adjusted specific premature birth rate of Table 3 seem to indicate that incidence of premature births may not be strongly associated with birth order, but may probably be some how associated with increasing maternal age, especially from age 25 years. The overall adjusted crude premature birth rate is estimated to be severally 70 per 1000 live births whether the standard population distribution is either the proportionate distribution of total birth by birth order or by maternal age. The unadjusted crude rate is also here estimated to be 70 per 1000 live births. These results are usually the case whenever the two standard distributions are those of the total sample. In these cases the overall adjusted crude rates based on the two sets of directly adjusted rates would be equal to each other, although not necessarily always equal to the overall unadjusted crude rate as is found to be the case here. However, if the standard population distribution chosen for population A (here maternal age)is different from that chosen for factor B(here birth order),then the two resulting estimated adjusted or standardized crude rates would most lively not be equal to each other.

References

  1. Fleiss JL (1981) Statistical Method for Rates and proportions (2nd ed), New York:John Wiley & Sons.
  2. Pepe MS (2003) The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford statistical series 28, Oxford: University Press, U.K.;
  3. Greenberg RS, Daniels SR, Flanders WD, Eley JW and Boring JR (2001) Medical Epidemiology, London: Lange-McGraw-Hill.
  4. Cochran WG (1950) ”The comparison of percentages in matched samples”, Biometrika 37: 256-266.
  5. Gibbon JD (1971) Non Parametric Statistical Inference McGraw Hill Inc.
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