ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 5 - 2016
Dummy Variable Multiple Regression Analysis of Matched Samples
Okeh UM1*, Oyeka ICA2
1Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Nigeria
2Department of Statistics, Nnamdi Azikiwe University, Nigeriash
Received:April 03, 2016 | Published: May 23, 2016
*Corresponding author: Okeh UM, Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki Nigeria, Email:
Citation: Okeh UM, Oyeka ICA (2016) Dummy Variable Multiple Regression Analysis of Matched Samples. Biom Biostat Int J 3(5): 00077. DOI:10.15406/bbij.2016.03.00077

Abstract

This paper presents and discuses the use of dummy variable multiple regression techniques in the analysis of samples drawn from several related or dependent populations ordinarily appropriate for random effects and mixed effects derived from the two factor analysis of variances model with one observation per-cell or treatment combinations. Using the extra sum of squares principle the method develops necessary sums of squares, degrees of freedom and the F-ratios required to test the significance of factor level effects thereby helping to resolve the problem of one observation per treatment combination, encountered in the usual two factor analysis of variance models with one observation per cell. The method provides estimates of the overall and factor mean effects comparable to those obtained with the two factor analysis of variance methods. In addition, the method also provides estimates of the total or absolute effects as well as the direct and indirect effects of the independent variables or factors on the dependent or criterion variable which are not ordinarily obtainable with the usual analysis of variance techniques. The proposed method compares favorably with the usual Friedman’s two-way analysis of variance test by ranks using some sample data.

Keywords: Friedman’s two-way ANOVA; Mixed–effects ANOVA; Dummy variable; Regression; Extra sum of square; Treatment

Introduction

Dummy variable analysis of variance technique is an alternative approach to the non-parametric Friedman’s two-way analysis of variance test by ranks used to analyze sample data appropriate for use in parametric statistics for two factor random and mixed effects or analysis of variance models with one replication or observation per treatment combinations [1,2].

To develop a non-parametric alternative method for the analysis of matched samples that are appropriate for use with two factor random and mixed-effects analysis of variance models with only one observation per cell or treatment combination, we may suppose that a researcher has collected a random sample of size ’a’ observations randomly drawn from a population ‘A’ of subjects or blocks of subjects exposed to or observed at some ‘c’ time periods, points in space, experimental conditions, tests, or treatments that are either fixed or randomly drawn from population B experimental conditions, points in time, tests or experiments comprising numerical measurements.

The proposed method

Let y ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaaa@3A3C@  be the i th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAam aaCaaabeqcfasaaiaadshacaWGObaaaaaa@39A8@  observation drawn from population A, that is the observation on the i th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAam aaCaaabeqcfasaaiaadshacaWGObaaaaaa@39A8@  subject or block of subjects exposed to or observed at the j th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOAam aaCaaabeqcfasaaiaadshacaWGObaaaaaa@39A9@  level of factor B that is j th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOAam aaCaaabeqcfasaaiaadshacaWGObaaaaaa@39A9@  treatment or time period for i=1,2,…,a; j=1,2,…,c.

Now to set up a dummy variable multiple regression model for use with a two factor analysis of variance problem, we as usual present each factor or the so called parent independent variable with one dummy variable of 1s and 0s less than the number of its categories or levels [2]. Thus factor A, namely subject or block of subjects with ‘a’ levels is represented with a-1 dummy variables of 1s and 0s, while factor B with c levels is represented by c-1 dummy variables of 1s and 0s.

Hence we may let

x i;A ={ 1,if y ij isanobservationontheithsubjectorblockof subjectsandjthleveloffactorB(treatment) 0,otherwise fori=1,2,...,a1;andallj=1,2,...,c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WG4bWaaSbaaKqbGeaacaWGPbGaai4oaiaadgeaaKqbagqaaiabg2da 9maaceaabaqbaeqabiqaaaabaeqabaGaaGymaiaacYcacaWGPbGaam OzaiaaysW7caWG5bWaaSbaaeaacaWGPbGaamOAaaqabaGaaGjbVlaa dMgacaWGZbGaaGjbVlaadggacaWGUbGaaGjbVlaad+gacaWGIbGaam 4CaiaadwgacaWGYbGaamODaiaadggacaWG0bGaamyAaiaad+gacaWG UbGaaGjbVlaad+gacaWGUbGaaGjbVlaadshacaWGObGaamyzaiaays W7caWGPbGaamiDaiaadIgacaaMe8Uaam4CaiaadwhacaWGIbGaamOA aiaadwgacaWGJbGaamiDaiaaysW7caWGVbGaamOCaiaaysW7caWGIb GaamiBaiaad+gacaWGJbGaam4AaiaaysW7caWGVbGaamOzaiaaysW7 aeaacaWGZbGaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0bGaam 4CaiaaysW7caWGHbGaamOBaiaadsgacaaMe8UaamOAaiaadshacaWG ObGaaGjbVlaadYgacaWGLbGaamODaiaadwgacaWGSbGaaGPaVlaad+ gacaWGMbGaaiOzaiaacggacaGGJbGaaiiDaiaac+gacaGGYbGaaGjb VlaadkeacaaMe8UaaiikaiaadshacaWGYbGaamyzaiaadggacaWG0b GaamyBaiaadwgacaWGUbGaamiDaiaacMcaaaqaaiaaicdacaGGSaGa am4BaiaadshacaWGObGaamyzaiaadkhacaWG3bGaamyAaiaadohaca WGLbaaaaGaay5EaaaakeaajuaGcaWGMbGaam4BaiaadkhacaaMe8Ua amyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlai aac6cacaGGSaGaamyyaiabgkHiTiaaigdacaGG7aGaamyyaiaad6ga caWGKbGaaGjbVlaadggacaWGSbGaamiBaiaaysW7caWGQbGaeyypa0 JaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaacYca caWGJbaaaaa@D2B4@ (1)

Alsolet x j;B ={ 1,if y ij isanobservationorresponseatthejthlevel offactorB(treatment)andithleveloffactor A(subject,orblockofsubjects) 0,otherwise forj=1,2,...c1;andalli=1,2,...,a. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGbbGaamiBaiaadohacaWGVbGaaGjbVlaadYgacaWGLbGaamiDaaqa aiaadIhadaWgaaqcfasaaiaadQgacaGG7aGaamOqaaqcfayabaGaey ypa0ZaaiqaaeaafaqabeGabaaaeaqabeaacaaIXaGaaiilaiaadMga caWGMbGaaGjbVlaadMhadaWgaaqaaiaadMgacaWGQbaabeaacaaMe8 UaamyAaiaadohacaaMe8Uaamyyaiaad6gacaaMe8Uaam4Baiaadkga caWGZbGaamyzaiaadkhacaWG2bGaamyyaiaadshacaWGPbGaam4Bai aad6gacaaMe8Uaam4BaiaadkhacaaMe8UaamOCaiaadwgacaWGZbGa amiCaiaad+gacaWGUbGaam4CaiaadwgacaaMe8Uaamyyaiaadshaca aMe8UaamiDaiaadIgacaWGLbGaaGjbVlaadQgacaWG0bGaamiAaiaa ysW7caWGSbGaamyzaiaadAhacaWGLbGaamiBaiaaykW7aeaacaWGVb GaamOzaiaacAgacaGGHbGaai4yaiaacshacaGGVbGaaiOCaiaaysW7 caWGcbGaaGjbVlaacIcacaWG0bGaamOCaiaadwgacaWGHbGaamiDai aad2gacaWGLbGaamOBaiaadshacaGGPaGaaGjbVlaadggacaWGUbGa amizaiaaysW7caWGPbGaamiDaiaadIgacaaMe8UaamiBaiaadwgaca WG2bGaamyzaiaadYgacaaMe8Uaam4BaiaadAgacaaMe8UaamOzaiaa dggacaWGJbGaamiDaiaad+gacaWGYbGaaGjbVdqaaiaadgeacaGGOa Gaam4CaiaadwhacaWGIbGaamOAaiaadwgacaWGJbGaamiDaiaacYca caWGVbGaamOCaiaaysW7caWGIbGaamiBaiaad+gacaWGJbGaam4Aai aaysW7caWGVbGaamOzaiaaysW7caWGZbGaamyDaiaadkgacaWGQbGa amyzaiaadogacaWG0bGaam4CaiaacMcaaaqaaiaaicdacaGGSaGaam 4BaiaadshacaWGObGaamyzaiaadkhacaWG3bGaamyAaiaadohacaWG LbaaaaGaay5EaaaakeaajuaGcaWGMbGaam4BaiaadkhacaaMe8Uaam OAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaa c6cacaWGJbGaeyOeI0IaaGymaiaacUdacaWGHbGaamOBaiaadsgaca aMe8UaamyyaiaadYgacaWGSbGaaGjbVlaadMgacqGH9aqpcaaIXaGa aiilaiaaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaadggaca GGUaaaaaa@F968@ (2)

Then the resulting dummy variable multiple regression model fitting or regressing the dependent or criterion variable y ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaaa@3A3C@ on the dummy variables representing factors A (subject or block of subjects) and B (treatment) is

y l = β 0 + β 1;A x l1;A + β 2;A x l2;A +......+ β α1;A x lα1;A + β 1;B x l1;B + β 2;B x l2;B +....+ β c1;B x lc1;B + e i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamiBaaqcfayabaGaeyypa0JaeqOSdi2aaSbaaKqb GeaacaaIWaaajuaGbeaacqGHRaWkcqaHYoGydaWgaaqcfasaaiaaig dacaGG7aGaamyqaaqcfayabaGaamiEamaaBaaajuaibaGaamiBaiaa igdacaGG7aGaamyqaaqcfayabaGaey4kaSIaeqOSdi2aaSbaaKqbGe aacaaIYaGaai4oaiaadgeaaKqbagqaaiaadIhadaWgaaqcfasaaiaa dYgacaaIYaGaai4oaiaadgeaaKqbagqaaiabgUcaRiaac6cacaGGUa GaaiOlaiaac6cacaGGUaGaaiOlaiabgUcaRiabek7aInaaBaaajuai baGaeqySdeMaeyOeI0IaaGymaiaacUdacaWGbbaajuaGbeaacaWG4b WaaSbaaKqbGeaacaWGSbGaeqySdeMaeyOeI0IaaGymaiaacUdacaWG bbaajuaGbeaacqGHRaWkcqaHYoGydaWgaaqcfasaaiaaigdacaGG7a GaamOqaaqcfayabaGaamiEamaaBaaajuaibaGaamiBaiaaigdacaGG 7aGaamOqaaqcfayabaGaey4kaSIaeqOSdi2aaSbaaKqbGeaacaaIYa Gaai4oaiaadkeaaKqbagqaaiaadIhadaWgaaqcfasaaiaadYgacaaI YaGaai4oaiaadkeaaKqbagqaaiabgUcaRiaac6cacaGGUaGaaiOlai aac6cacqGHRaWkcqaHYoGydaWgaaqcfasaaiaadogacqGHsislcaaI XaGaai4oaiaadkeaaKqbagqaaiaadIhadaWgaaqcfasaaiaadYgaca WGJbGaeyOeI0IaaGymaiaacUdacaWGcbaajuaGbeaacqGHRaWkcaWG LbWaaSbaaKqbGeaacaWGPbaajuaGbeaaaaa@92AB@ (3)

For l=1,2,,n=a. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBai abg2da9iaabgdacaGGSaGaaeOmaiaacYcacqGHMacVcaGGSaGaaeOB aiabg2da9iaabggacaGGUaGaae4yaiaabccaaaa@4298@  sample observations where y l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamiBaaqcfayabaaaaa@3950@  is the l th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBam aaCaaabeqcfasaaiaadshacaWGObaaaaaa@39AB@  response or observation on the criterion or dependent variable; x ls MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaiaadohaaKqbagqaaaaa@3A47@ are dummy variables of 1s and 0s representing levels of factors A and B; β ls MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaWGSbGaam4Caaqcfayabaaaaa@3AEB@  are partial regression coefficients and e ls MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyzam aaBaaajuaibaGaamiBaiaadohaaKqbagqaaaaa@3A34@  are error terms, with E( e i )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaamyzamaaBaaajuaibaGaamyAaaqcfayabaaacaGLOaGa ayzkaaGaeyypa0JaaGimaaaa@3D4C@ ,for l=1,2,,n=a. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBai abg2da9iaabgdacaGGSaGaaeOmaiaacYcacqGHMacVcaGGSaGaaeOB aiabg2da9iaabggacaGGUaGaae4yaiaabccaaaa@4298@ . Note that since there are only one observation per row by column, that is factor A (subject or block of subjects) by factor B (treatment) combination; for one to be able to have an estimate for the error sum of squares for the regression model, and hence be able to test desired hypotheses, it is necessary to assume that there are no factors A by B interactions or that such interactions have been removed by an appropriate data transformation. Also note that an advantage of the present method over the extended median test for dependent or matched samples and also over the Friedmans two –way analysis of variance test by ranks is that the problem of tied observations within subjects or blocks of subjects does not arise, and hence unlike in the other two non-parametric methods under reference there is no need to find ways to adjust for or break ties between scores within blocks of subjects [3]. The expected or mean value of the criterion variable is from equation 3.

E( y l )= β 0 + β 1;A x l1;A + β 2;A x l2;A +....+ β α1;A x lα1;A + β 1;B x l1;B + β 2;B x l2;B +....+ β c1;B x lc1;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaamyEamaaBaaajuaibaGaamiBaaqcfayabaaacaGLOaGa ayzkaaGaeyypa0JaeqOSdi2aaSbaaKqbGeaacaaIWaaajuaGbeaacq GHRaWkcqaHYoGydaWgaaqcfasaaiaaigdacaGG7aGaamyqaaqcfaya baGaamiEamaaBaaajuaibaGaamiBaiaaigdacaGG7aGaamyqaaqcfa yabaGaey4kaSIaeqOSdi2aaSbaaKqbGeaacaaIYaGaai4oaiaadgea aKqbagqaaiaadIhadaWgaaqcfasaaiaadYgacaaIYaGaai4oaiaadg eaaKqbagqaaiabgUcaRiaac6cacaGGUaGaaiOlaiaac6cacqGHRaWk cqaHYoGydaWgaaqcfasaaiabeg7aHjabgkHiTiaaigdacaGG7aGaam yqaaqcfayabaGaamiEamaaBaaajuaibaGaamiBaiabeg7aHjabgkHi TiaaigdacaGG7aGaamyqaaqcfayabaGaey4kaSIaeqOSdi2aaSbaaK qbGeaacaaIXaGaai4oaiaadkeaaKqbagqaaiaadIhadaWgaaqcfasa aiaadYgacaaIXaGaai4oaiaadkeaaKqbagqaaiabgUcaRiabek7aIn aaBaaajuaibaGaaGOmaiaacUdacaWGcbaajuaGbeaacaWG4bWaaSba aKqbGeaacaWGSbGaaGOmaiaacUdacaWGcbaajuaGbeaacqGHRaWkca GGUaGaaiOlaiaac6cacaGGUaGaey4kaSIaeqOSdi2aaSbaaKqbGeaa caWGJbGaeyOeI0IaaGymaiaacUdacaGGcbaajuaGbeaacaWG4bWaaS baaKqbGeaacaWGSbGaam4yaiabgkHiTiaaigdacaGG7aGaamOqaaqc fayabaaaaa@9002@  (4)

To find the expected or mean effect of any of the factors or parent independent variables, we set all the dummy variables representing that factor equal to 1 and all the other dummy variables found in equation 4 equal to 0.Thus for example the expected or mean effect or value of factor A (subject or block of subjects) on the dependent variable is obtained by setting x l;A =1and x j;B =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaiaacUdacaWGbbaajuaGbeaacqGH9aqpcaaI XaGaaGjbVlaadggacaWGUbGaamizaiaaysW7caWG4bWaaSbaaKqbGe aacaWGQbGaai4oaiaadkeaaKqbagqaaiabg2da9iaaicdaaaa@4880@  in equation 4 for l=1,2,,a1;j=1,2,,c1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBai abg2da9iaabgdacaGGSaGaaeOmaiaacYcacqGHMacVcaGGSaGaamyy aiabgkHiTiaabgdacaGG7aGaaeOAaiabg2da9iaabgdacaGGSaGaae OmaiaacYcacqGHMacVcaGGSaGaam4yaiabgkHiTiaabgdaaaa@4A4B@ .

Similarly the expected or mean value of factor B (treatment) is obtained by setting x l;B =1and x j;A =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaiaacUdacaWGcbaajuaGbeaacqGH9aqpcaaI XaGaaGjbVlaadggacaWGUbGaamizaiaaysW7caWG4bWaaSbaaKqbGe aacaWGQbGaai4oaiaadgeaaKqbagqaaiabg2da9iaaicdaaaa@4880@  in equation 4 for l=1,2,,c1;j=1,2,,a1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBai abg2da9iaabgdacaGGSaGaaeOmaiaacYcacqGHMacVcaGGSaGaam4y aiabgkHiTiaabgdacaGG7aGaaeOAaiabg2da9iaabgdacaGGSaGaae OmaiaacYcacqGHMacVcaGGSaGaamyyaiabgkHiTiaabgdaaaa@4A4B@  thereby obtaining

E( y l;A )= β 0 + l=1 a1 β l;A andE( y l;B )= β 0 + l=1 c1 β l;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaamyEamaaBaaajuaibaGaamiBaiaacUdacaWGbbaajuaG beaaaiaawIcacaGLPaaacqGH9aqpcqaHYoGydaWgaaqcfasaaiaaic daaKqbagqaaiabgUcaRmaaqahabaGaeqOSdi2aaSbaaKqbGeaacaWG SbGaai4oaiaadgeaaKqbagqaaiaaysW7caWGHbGaamOBaiaadsgaca aMe8UaamyraiaacIcacaWG5bWaaSbaaKqbGeaacaWGSbGaai4oaiaa dkeaaKqbagqaaiaacMcacqGH9aqpcqaHYoGydaWgaaqcfasaaiaaic daaKqbagqaaiabgUcaRmaaqahabaGaeqOSdi2aaSbaaKqbGeaacaWG SbGaai4oaiaadkeaaKqbagqaaaqcfasaaiaadYgacqGH9aqpcaaIXa aabaGaam4yaiabgkHiTiaaigdaaKqbakabggHiLdaajuaibaGaamiB aiabg2da9iaaigdaaeaacaWGHbGaeyOeI0IaaGymaaqcfaOaeyyeIu oaaaa@6DF0@  (5)

Now the dummy variable multiple regression model of equation 3 can equivalently be expressed in matrix form as

y _ =X β _ + e _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWG5baaaiabg2da9iaadIfadaadaaqaaiabek7aIbaacqGHRaWk daadaaqaaiaadwgaaaaaaa@3D02@  (6)

Where y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWG5baaaaaa@3792@  is an nx1 column vector of observations or scores on the dependent or criterion variables; X is an nxr design matrix of ‘r’ dummy variables of 1s and 0s; β _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacqaHYoGyaaaaaa@3835@  is an rx1 column vector of partial regression coefficients; and e _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWGLbaaaaaa@377E@  is on nx1 column vector of error terms, with E( e _ )= 0 _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aacIcadaadaaqaaiaadwgaaaGaaiykaiabg2da9maamaaabaGaaGim aaaaaaa@3B71@  where ‘n’=a.c observations and ‘n’=(a-1)+(c-1)=a+c-2 dummy variables of 1s and 0s included in the regression model.

Similarly the expected value of y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWG5baaaaaa@3792@  is from equation 4.

E( y _ )=X. β _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aacIcadaadaaqaaiaadMhaaaGaaiykaiabg2da9iaadIfacaGGUaWa aWaaaeaacqaHYoGyaaaaaa@3DFB@  (7)

Application of the usual methods of least squares to either equation 3 or 6 yields an unbiased estimate of the regression parameter β _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacqaHYoGyaaaaaa@3835@  as

β ^ _ = b _ = ( X X ) 1 X y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacuaHYoGygaqcaaaacqGH9aqpdaadaaqaaiaadkgaaaGaeyypa0Za aeWaaeaaceWGybGbauaacaWGybaacaGLOaGaayzkaaWaaWbaaeqaju aibaGaeyOeI0IaaGymaaaajuaGceWGybGbauaadaadaaqaaiaadMha aaaaaa@4314@ (8)

Where ( X X ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aaceWGybGbauaacaWGybaacaGLOaGaayzkaaWaaWbaaeqajuaibaGa eyOeI0IaaGymaaaaaaa@3BCB@  is the inverse matrix of the non-singular variance-covariance matrix X X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiway aafaGaamiwaaaa@384A@ . A hypothesis that is usually of research interest is that the regression model of either equation 3 or 6 fits, or equivalently that the independent variables or factors have no effects on the dependent or criterion variable, meaning that the partial regression coefficient is equal to zero stated symbolically that we have the null hypothesis.

H 0 : β _ = 0 _ versus H 1 : β _ 0 _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaGaaiOoamaamaaabaGaeqOSdiga aiabg2da9maamaaabaGaaGimaaaacaaMe8UaamODaiaadwgacaWGYb Gaam4CaiaadwhacaWGZbGaaGjbVlaadIeadaWgaaqcfasaaiaaigda aKqbagqaaiaacQdadaadaaqaaiabek7aIbaacqGHGjsUdaadaaqaai aaicdaaaaaaa@4D6C@ (9)

As in equation 3 this null hypothesis is tested using the usual F-test presented in an analysis of variance Table where the total sum of squares is calculated in the usual way as

SSTotal= y _ y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGubGaam4BaiaadshacaWGHbGaamiBaiabg2da9maamaaa baGabmyEayaafaaaamaamaaabaGaamyEaaaacqGHsislcaWGUbGaai OlaiqadMhagaqeamaaCaaabeqcfasaaiaaikdaaaqcfaOaaGjbVdaa @46CE@ (10)

With n-1=a.c-1 degrees of freedom where y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraaaaa@379A@  is the mean value of the dependent variables.

Similarly the treatment sum of squares in analysis of variance parlance which is the same as the regression sum of squares in regression models is calculated as

SSTreatment=SSR= b _ . X . y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGubGaamOCaiaadwgacaWGHbGaamiDaiaad2gacaWGLbGa amOBaiaadshacqGH9aqpcaWGtbGaam4uaiaadkfacqGH9aqpdaadaa qaaiqadkgagaqbaaaacaGGUaGabmiwayaafaGaaiOlamaamaaabaGa amyEaaaacqGHsislcaWGUbGaaiOlaiqadMhagaqeamaaCaaabeqcfa saaiaaikdaaaaaaa@4E3A@ (11)

With (a-1)+(c-1) =a+c-2 degrees of freedom. The error sum of squares SSE indicates the difference between the total sum of squares, SST and the sum of squares regression SSR; thus,

SSE=SSTSSR= y _ y _ b _ X . y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGfbGaeyypa0Jaam4uaiaadofacaWGubGaeyOeI0Iaam4u aiaadofacaWGsbGaeyypa0ZaaWaaaeaaceWG5bGbauaaaaWaaWaaae aacaWG5baaaiabgkHiTmaamaaabaGabmOyayaafaaaaiqadIfagaqb aiaac6cadaadaaqaaiaadMhaaaaaaa@47C8@ (12)

With (a.c1)( (a1)+(c1) )=(a1)(c1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aadggacaGGUaGaam4yaiabgkHiTiaaigdacaGGPaGaeyOeI0YaaeWa aeaacaGGOaGaamyyaiabgkHiTiaaigdacaGGPaGaey4kaSIaaiikai aadogacqGHsislcaaIXaGaaiykaaGaayjkaiaawMcaaiabg2da9iaa cIcacaWGHbGaeyOeI0IaaGymaiaacMcacaGGOaGaam4yaiabgkHiTi aaigdacaGGPaaaaa@5003@  degrees of freedom.

These results are summarized in an analysis of variance Table (Table 1)

The null hypotheses H0 of Equation 13 is tested using the F-ratio of Table 1. The null hypothesis is rejected at the if the calculated F-ratio is greater than the tabulated or critical F-ratio at a specified α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ -level of significance, otherwise the null hypothesis H0 is accepted.

If the model fits, that if not all the elements of β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  are equal to zero, that is if the null hypothesis H0 of equation 9 is rejected, then one may proceed to test further hypothesis concerning factor level effects, that is one may proceed to test the null hypothesis that factors A (subject or block of subjects) and B (treatment) separately have no effects on the dependent or criterion variable. In other words, the null hypotheses

H 0 : β _ A = 0 _ versus H 1 : β _ A 0 _ and H 0 : β _ B = 0 _ versus H 1 : β _ B 0 _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGibWaaSbaaKqbGeaacaaIWaaajuaGbeaacaGG6aWaaWaaaeaacqaH YoGyaaWaaSbaaKqbGeaacaWGbbaajuaGbeaacqGH9aqpdaadaaqaai aaicdaaaGaaGjbVlaadAhacaWGLbGaamOCaiaadohacaWG1bGaam4C aiaaysW7caWGibWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGG6aWaaW aaaeaacqaHYoGyaaWaaSbaaKqbGeaacaWGbbaajuaGbeaacqGHGjsU daadaaqaaiaaicdaaaaabaGaamyyaiaad6gacaWGKbaakeaajuaGca WGibWaaSbaaKqbGeaacaaIWaaajuaGbeaacaGG6aWaaWaaaeaacqaH YoGyaaWaaSbaaKqbGeaacaWGcbaajuaGbeaacqGH9aqpdaadaaqaai aaicdaaaGaaGjbVlaadAhacaWGLbGaamOCaiaadohacaWG1bGaam4C aiaaysW7caWGibWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGG6aWaaW aaaeaacqaHYoGyaaWaaSbaaKqbGeaacaWGcbaajuaGbeaacqGHGjsU daadaaqaaiaaicdaaaaaaaa@6E44@ (13, 14)

Where β _ A and β _ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacqaHYoGyaaWaaSbaaKqbGeaacaWGbbaajuaGbeaacaaMe8Uaamyy aiaad6gacaWGKbGaaGjbVpaamaaabaGaeqOSdigaamaaBaaajuaiba GaamOqaaqcfayabaaaaa@4309@  are respectively the (a-1) and (c-1) vectors of partial regression coefficients or effects of factor A (subject or block of subjects) and B (treatment) on the criterion or dependent variable. However a null hypothesis that is usually of greater interest here is that of equation 14, that is that treatments, points in time or space of tests or experiments do not have differential effects on subjects.

Source of Variation

Sum of Squares

Degrees of Freedom

Mean sum of Squares

F-ratio

Regression(treatment)

SSR= b _ . X . y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGsbGaeyypa0ZaaWaaaeaaceWGIbGbauaaaaGaaiOlaiqa dIfagaqbaiaac6cadaadaaqaaiaadMhaaaGaeyOeI0IaamOBaiaac6 caceWG5bGbaebadaahaaqabKqbGeaacaaIYaaaaaaa@4323@

a+c-2

MSR= SSR a+c2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGsbGaeyypa0ZaaSaaaeaacaWGtbGaam4uaiaadkfaaeaa caWGHbGaey4kaSIaam4yaiabgkHiTiaaikdaaaaaaa@40FB@

MSR MSE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGnbGaam4uaiaadkfaaeaacaWGnbGaam4uaiaadweaaaaaaa@3B89@

Error

SSE= y _ y _ b _ X . y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGfbGaeyypa0ZaaWaaaeaaceWG5bGbauaaaaWaaWaaaeaa caWG5baaaiabgkHiTmaamaaabaGabmOyayaafaaaaiqadIfagaqbai aac6cadaadaaqaaiaadMhaaaaaaa@40C5@

(a-1)(c-1)

MSE= SSE (a1)(c1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGfbGaeyypa0ZaaSaaaeaacaWGtbGaam4uaiaadweaaeaa caGGOaGaamyyaiabgkHiTiaaigdacaGGPaGaaiikaiaadogacqGHsi slcaaIXaGaaiykaaaaaaa@4458@

Total

SST= y _ y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGubGaeyypa0ZaaWaaaeaaceWG5bGbauaaaaWaaWaaaeaa caWG5baaaiabgkHiTiaad6gacaGGUaGabmyEayaaraWaaWbaaeqaju aibaGaaGOmaaaaaaa@40EF@

(a.c)-1

Table 1: Two factor analysis of variance Table for the full model of Equation 6.

Now to obtain appropriate test statistics for use in testing these null hypothesis we apply the extra sum of squares principle to partition the treatment or regression sum of squares SSR into its two component parts namely, the sum of squares due to factor A (subject or block of subjects), SSA and the sum of squares due to factor B (treatment), SSB, to enable the calculation of the appropriate F-ratios.

Now the nxr matrix X for the full model of equation 6 can be partitioned into its two component sub-matrices namely X A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaajuaibaGaamyqaaqcfayabaaaaa@3904@ , an nx(a-1) design matrix of a-1 dummy variables of 1s and 0s representing the included a-1 levels of factor A (subject or block of subjects) and X B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaajuaibaGaamOqaaqcfayabaaaaa@3905@ , an nx(c-1) matrix of the c-1 dummy variables of 1s and 0s representing the included c-1 levels of factor B (treatment). The partial regression coefficient b _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWGIbaaaaaa@377B@ , estimated being an rx1 column vector of regression effects of equation 8 can also be partitioned into the corresponding partial regression coefficients estimated such as, b _ A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWGIbaaamaaBaaajuaibaGaamyqaaqcfayabaaaaa@391E@ ,which is an (a-1)x1 column vector of partial regression coefficients or effects of factor A and b _ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWGIbaaamaaBaaajuaibaGaamOqaaqcfayabaaaaa@391F@ which is a (c-1)x1 column vector of the effects of factor B on the dependent variable. Hence the treatment sum of squares SST, that is the sum of squares regression SSR of equation 11 can be equivalently expressed as

SSTreatment=SSR= b _ X y _ n. y ¯ 2 =(X b _ ) . y _ n. y ¯ 2 ; equivalentlyas SSR= ( ( X A X B )| b _ A b _ B | ) y _ n. y ¯ 2 =( b _ A . X A . y _ + b _ B . X B . y _ )n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGtbGaam4uaiaadsfacaWGYbGaamyzaiaadggacaWG0bGaamyBaiaa dwgacaWGUbGaamiDaiabg2da9iaadofacaWGtbGaamOuaiabg2da9m aamaaabaGabmOyayaafaaaaiqadIfagaqbamaamaaabaGaamyEaaaa cqGHsislcaWGUbGaaiOlaiqadMhagaqeamaaCaaabeqcfasaaiaaik daaaqcfaOaeyypa0JaaiikaiaadIfadaadaaqaaiaadkgaaaGabiyk ayaafaGaaiOlamaamaaabaGaamyEaaaacqGHsislcaWGUbGaaiOlai qadMhagaqeamaaCaaabeqcfasaaiaaikdaaaqcfaOaai4oaaqaaiaa dwgacaWGXbGaamyDaiaadMgacaWG2bGaamyyaiaadYgacaWGLbGaam OBaiaadshacaWGSbGaamyEaiaaysW7caWGHbGaam4CaaGcbaqcfaOa am4uaiaadofacaWGsbGaeyypa0ZaaeWaaeaacaGGOaGaamiwamaaBa aajuaibaGaamyqaaqcfayabaGaaGjbVlaaysW7caaMe8UaaGjbVlaa dIfadaWgaaqcfasaaiaadkeaaKqbagqaaiaacMcadaabdaqaauaabe qaceaaaeaadaadaaqaaiaadkgaaaWaaSbaaKqbGeaacaWGbbaajuaG beaaaeaadaadaaqaaiaadkgaaaWaaSbaaKqbGeaacaWGcbaajuaGbe aaaaaacaGLhWUaayjcSdaacaGLOaGaayzkaaWaaWbaaeqabaGamai4 gkdiIcaadaadaaqaaiaadMhaaaGaeyOeI0IaamOBaiaac6caceWG5b GbaebadaahaaqabKqbGeaacaaIYaaaaKqbakabg2da9maabmaabaWa aWaaaeaaceWGIbGbauaaaaWaaSbaaKqbGeaacaWGbbaajuaGbeaaca GGUaGabmiwayaafaWaaSbaaKqbGeaacaWGbbaajuaGbeaacaGGUaWa aWaaaeaacaWG5baaaiabgUcaRmaamaaabaGabmOyayaafaaaamaaBa aajuaibaGaamOqaaqcfayabaGaaiOlaiqadIfagaqbamaaBaaajuai baGaamOqaaqcfayabaGaaiOlamaamaaabaGaamyEaaaaaiaawIcaca GLPaaacqGHsislcaWGUbGaaiOlaiqadMhagaqeamaaCaaabeqcfasa aiaaikdaaaqcfaOaaGjbVdaaaa@A5BA@ (15)

or equivalently

SSR= b _ X y _ n. y ¯ 2 =( b _ A . X A . y _ n. y ¯ 2 )+( b _ B . X B . y _ n. y ¯ 2 )+n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGsbGaeyypa0ZaaWaaaeaaceWGIbGbauaaaaGabmiwayaa faWaaWaaaeaacaWG5baaaiabgkHiTiaad6gacaGGUaGabmyEayaara WaaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaqadaqaamaamaaa baGabmOyayaafaaaamaaBaaajuaibaGaamyqaaqcfayabaGaaiOlai qadIfagaqbamaaBaaajuaibaGaamyqaaqcfayabaGaaiOlamaamaaa baGaamyEaaaacqGHsislcaWGUbGaaiOlaiqadMhagaqeamaaCaaabe qcfasaaiaaikdaaaaajuaGcaGLOaGaayzkaaGaey4kaSYaaeWaaeaa daadaaqaaiqadkgagaqbaaaadaWgaaqcfasaaiaadkeaaKqbagqaai aac6caceWGybGbauaadaWgaaqcfasaaiaadkeaaKqbagqaaiaac6ca daadaaqaaiaadMhaaaGaeyOeI0IaamOBaiaac6caceWG5bGbaebada ahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaiabgUcaRiaa d6gacaGGUaGabmyEayaaraWaaWbaaeqajuaibaGaaGOmaaaaaaa@65BE@ (16)

Which when interpreted is the same as the statement
SSTreatment=SSR=SSA+SSB+SS( y ¯ = μ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGubGaamOCaiaadwgacaWGHbGaamiDaiaad2gacaWGLbGa amOBaiaadshacqGH9aqpcaWGtbGaam4uaiaadkfacqGH9aqpcaWGtb Gaam4uaiaadgeacqGHRaWkcaWGtbGaam4uaiaadkeacqGHRaWkcaWG tbGaam4uaiaacIcaceWG5bGbaebacqGH9aqpcuaH8oqBgaqcaiaacM caaaa@52C4@ (17)

Where SSR is the sum of squares of regression for the full model with r=a+c-2 degrees of freedom; SSA is the sum of squares due to factor A (subject or block of subject); with a-1 degrees of freedom; SSB is the sum of squares due to factor B (treatment) with c-1 degrees of freedom; and SS( y ¯ = μ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaGGOaGabmyEayaaraGaeyypa0JafqiVd0MbaKaacaGGPaaa aa@3D6F@  is an additive correction factor due to mean effect. These sums of squares namely SSR, SSA and SSB are obtained by separately fitting the full model of equations 6 with X, and the reduced regression models of X A and X B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaajuaibaGaamyqaaqcfayabaGaaGjbVlaadggacaWGUbGaamiz aiaaysW7caWGybWaaSbaaKqbGeaacaWGcbaajuaGbeaaaaa@4161@ again separately on the criterion or dependent variable y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWG5baaaaaa@3792@ .

Now if the full model of equation 6 fits, that is if the null hypothesis of equation 9 is rejected, then the additional null hypotheses of equations 13 and 14 may be tested using the extra sum of squares principle [4,5]. If we denote the sums of squares due to the full model of equation 6 and the reduced models due to the fitting of the criterion variables y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca WG5baaaaaa@3704@  to any of the reduced design matrices X A and X B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaajuaibaGaamyqaaqcfayabaGaaGjbVlaadggacaWGUbGaamiz aiaaysW7caWGybWaaSbaaKqbGeaacaWGcbaajuaGbeaaaaa@4161@  by SS(F) and SS(R) respectively then following the extra sum of squares principle [4,5] the extra sum of squares due to a given factor is calculated as

ESS=SS( F )SS( R )  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbGaam4uaiaadofacqGH9aqpcaWGtbGaam4ua8aadaqa daqaa8qacaWGgbaapaGaayjkaiaawMcaa8qacqGHsislcaWGtbGaam 4ua8aadaqadaqaa8qacaWGsbaapaGaayjkaiaawMcaa8qacaGGGcaa aa@44C5@ (18)

With degrees of freedom obtained as the difference between the degrees of freedom of SS(F) and SS(R); that is as Edf=df(F)-df(R). Thus the extra sums of squares for factors A (subject or block of subjects) and B (treatment) are obtained as follows respectively

ESSA=SSRSSA;ESSB=SSRSSB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aadofacaWGtbGaamyqaiabg2da9iaadofacaWGtbGaamOuaiabgkHi TiaadofacaWGtbGaamyqaiaacUdacaWGfbGaam4uaiaadofacaWGcb Gaeyypa0Jaam4uaiaadofacaWGsbGaeyOeI0Iaam4uaiaadofacaWG cbaaaa@4BA5@ (19)

With (a1)+(c1)(a1)=b1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aadggacqGHsislcaaIXaGaaiykaiabgUcaRiaacIcacaWGJbGaeyOe I0IaaGymaiaacMcacqGHsislcaGGOaGaamyyaiabgkHiTiaaigdaca GGPaGaeyypa0JaamOyaiabgkHiTiaaigdaaaa@479F@  degrees of freedom and (a1)+(b1)(b1)=a1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aadggacqGHsislcaaIXaGaaiykaiabgUcaRiaacIcacaWGIbGaeyOe I0IaaGymaiaacMcacqGHsislcaGGOaGaamOyaiabgkHiTiaaigdaca GGPaGaeyypa0JaamyyaiabgkHiTiaaigdaaaa@479E@  degrees of freedom.

Note that since each of the reduced models and the full model have the same total sum of squares SST, the extra sum of squares may alternatively be obtained as the difference between the error sum of squares of each reduced model and the error sum of squares of the full model. In other words, the extra sum of squares is equivalently calculated as ESS=SS(F)SS(R)=SSTSS(F)SSTSS(R)=SSE(R)SSE(F) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aadofacaWGtbGaeyypa0Jaam4uaiaadofacaGGOaGaamOraiaacMca cqGHsislcaWGtbGaam4uaiaacIcacaWGsbGaaiykaiabg2da9iaado facaWGtbGaamivaiabgkHiTiaadofacaWGtbGaaiikaiaadAeacaGG PaGaeyOeI0Iaam4uaiaadofacaWGubGaeyOeI0Iaam4uaiaadofaca GGOaGaamOuaiaacMcacqGH9aqpcaWGtbGaam4uaiaadweacaGGOaGa amOuaiaacMcacqGHsislcaWGtbGaam4uaiaadweacaGGOaGaamOrai aacMcacaaMe8oaaa@6000@ (20)

With degrees of freedom similarly obtained. Thus the extra sum of squares due to factors A (subject or block of subjects) and B (treatment) are alternatively obtained as follows respectively. ESSA=SSEASSE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aadofacaWGtbGaamyqaiabg2da9iaadofacaWGtbGaamyraiaadgea cqGHsislcaWGtbGaam4uaiaadweaaaa@4171@ (21)

With c-1 and a-1 degrees of freedom. Where SSR and SSE are respectively the regression sum of squares and the error sum of squares for the full model and SSEA and SSEB are respectively the error sums of squares for the reduced models for factors A and B. The null hypotheses of equations 13 and 14 are tested using the F-ratios

F A = MESA MSE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamyqaaqcfayabaGaeyypa0ZaaSaaaeaacaWGnbGa amyraiaadofacaWGbbaabaGaamytaiaadofacaWGfbaaaaaa@3FB6@ (22)

With a-1 and (a-1)(c-1) degrees of freedom where

MESA= ESSA c1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadweacaWGtbGaamyqaiabg2da9maalaaabaGaamyraiaadofacaWG tbGaamyqaaqaaiaadogacqGHsislcaaIXaaaaaaa@40A4@  (23)

Is the mean extra sum of squares due to factor A (subject or block of subjects) and

F B = MESB MSE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamOqaaqcfayabaGaeyypa0ZaaSaaaeaacaWGnbGa amyraiaadofacaWGcbaabaGaamytaiaadofacaWGfbaaaaaa@3FB8@ (24)

With a-1 and (a-1)(c-1) degrees of freedom where

MESB= ESSB a1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadweacaWGtbGaamOqaiabg2da9maalaaabaGaamyraiaadofacaWG tbGaamOqaaqaaiaadggacqGHsislcaaIXaaaaaaa@40A4@  (25)

Is the mean extra sum of squares due to factor B (treatment).These results are summarized in Table 2a which for ease of presentation also includes the sum of squares and other values of Table 1 for the full models.

If the various F–ratios and in particular the F-ratios based on the extra sums of squares of Table 2b indicate that the independent variables or factor levels have differential effects on the response, dependent, or criterion variable, that is if the null hypotheses of either equation 13 or 14 or both are rejected, then one may proceed further to estimate desired factor level effects and test hypotheses concerning them.

Source of Variation

Sum of Squares (SS)

Degrees of Freedom(DF)

Mean sum of Squares(MS)

F-ratio

Full model

Regression

SSR= b _ X y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGsbGaeyypa0ZaaWaaaeaacaWGIbaaamaaCaaabeqaaiad acUHYaIOaaGabmiwayaafaWaaWaaaeaacaWG5baaaiabgkHiTiaad6 gacaGGUaGabmyEayaaraWaaWbaaeqajuaibaGaaGOmaaaaaaa@44BD@

a+c-2

MSR= SSR a+c2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGsbGaeyypa0ZaaSaaaeaacaWGtbGaam4uaiaadkfaaeaa caWGHbGaey4kaSIaam4yaiabgkHiTiaaikdaaaaaaa@40FB@

F= MSR SSR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOrai abg2da9maalaaabaGaamytaiaadofacaWGsbaabaGaam4uaiaadofa caWGsbaaaaaa@3D6D@

Error

MCEP0028

(a-1)(c-1)

MSE= SSE (a1)(c1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGfbGaeyypa0ZaaSaaaeaacaWGtbGaam4uaiaadweaaeaa caGGOaGaamyyaiabgkHiTiaaigdacaGGPaGaaiikaiaadogacqGHsi slcaaIXaGaaiykaaaaaaa@4458@

Factor A (Subjects on block of subjects)

Regression

SSA= b _ A X A y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGbbGaeyypa0ZaaWaaaeaacaWGIbaaamaaCaaabeqaaiad acUHYaIOaaWaaSbaaeaacaWGbbaabeaaceWGybGbauaadaWgaaqaai aadgeaaeqaamaamaaabaGaamyEaaaacqGHsislcaWGUbGaaiOlaiqa dMhagaqeamaaCaaabeqaaiaaikdaaaaaaa@464C@

a-1

MSA= SSA a1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGbbGaeyypa0ZaaSaaaeaacaWGtbGaam4uaiaadgeaaeaa caWGHbGaeyOeI0IaaGymaaaaaaa@3F0E@

Error

SSEA= y _ y _ b _ A X A y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGfbGaamyqaiabg2da9maamaaabaGaamyEaaaadaahaaqa beaacWaGGBOmGikaamaamaaabaGaamyEaaaacqGHsisldaadaaqaai aadkgaaaWaaWbaaeqabaGamai4gkdiIcaadaWgaaqaaiaadgeaaeqa aiqadIfagaqbamaaBaaabaGaamyqaaqabaWaaWaaaeaacaWG5baaaa aa@48A3@

a(c-1)

MSEA= MSA a(c1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGfbGaamyqaiabg2da9maalaaabaGaamytaiaadofacaWG bbaabaGaamyyaiaacIcacaWGJbGaeyOeI0IaaGymaiaacMcaaaaaaa@4213@

Factor B(Treatment)

Regression

SSB= b _ B X B y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGcbGaeyypa0ZaaWaaaeaacaWGIbaaamaaCaaabeqaaiad acUHYaIOaaWaaSbaaeaacaWGcbaabeaaceWGybGbauaadaWgaaqaai aadkeaaeqaamaamaaabaGaamyEaaaacqGHsislcaWGUbGaaiOlaiqa dMhagaqeamaaCaaabeqaaiaaikdaaaaaaa@464F@

c-1

MSB= SSB c1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGcbGaeyypa0ZaaSaaaeaacaWGtbGaam4uaiaadkeaaeaa caWGJbGaeyOeI0IaaGymaaaaaaa@3F12@

F= MSB MSEB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOrai abg2da9maalaaabaGaamytaiaadofacaWGcbaabaGaamytaiaadofa caWGfbGaamOqaaaaaaa@3E11@

Error

SSEB= y _ y _ b _ B X B y _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadofacaWGfbGaamOqaiabg2da9maamaaabaGaamyEaaaadaahaaqa beaacWaGGBOmGikaamaamaaabaGaamyEaaaacqGHsisldaadaaqaai aadkgaaaWaaWbaaeqabaGamai4gkdiIcaadaWgaaqaaiaadkeaaeqa aiqadIfagaqbamaaBaaabaGaamOqaaqabaWaaWaaaeaacaWG5baaaa aa@48A6@

c(a-1)

MSEB= MSEB c(a1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGfbGaamOqaiabg2da9maalaaabaGaamytaiaadofacaWG fbGaamOqaaqaaiaadogacaGGOaGaamyyaiabgkHiTiaaigdacaGGPa aaaaaa@42DF@

Total

y _ y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWG5baaamaaCaaabeqaaiadacUHYaIOaaWaaWaaaeaacaWG5baa aiabgkHiTiaad6gacaGGUaGabmyEayaaraWaaWbaaeqajuaibaGaaG Omaaaaaaa@405E@

a.c-1

Table 2a: Table showing two factor Analysis of Variance for Sums of Squares for the full model and due to reduced models and other statistics.

Extra sum of Squares (ESS=SS(F)-SS(R)

Degrees of Freedom(DF)

Extra Mean sum of Squares (EMSA)

F-ratio

ESR=SSR

a+c2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyai abgUcaRiaadogacqGHsislcaaIYaaaaa@3ADD@

EMSR= SSR a+c2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aad2eacaWGtbGaamOuaiabg2da9maalaaabaGaam4uaiaadofacaWG sbaabaGaamyyaiabgUcaRiaadogacqGHsislcaaIYaaaaaaa@41C5@

F= MSR MSE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOrai abg2da9maalaaabaGaamytaiaadofacaWGsbaabaGaamytaiaadofa caWGfbaaaaaa@3D5A@

ESER=SSE

(a-1)(c-1)

EMSE= SSE (a1)(c1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aad2eacaWGtbGaamyraiabg2da9maalaaabaGaam4uaiaadofacaWG fbaabaGaaiikaiaadggacqGHsislcaaIXaGaaiykaiaacIcacaWGJb GaeyOeI0IaaGymaiaacMcaaaaaaa@4522@

Factor A

ESSA=SSR-SSA

c-1

EMSA= ESSA c1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbGaamytaiaadofacaWGbbGaeyypa0ZaaSaaaeaacaWG fbGaam4uaiaadofacaWGbbaabaGaam4yaiabgkHiTiaaigdaaaaaaa@40C4@

F A = EMSA MSE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamyqaaqcfayabaGaeyypa0ZaaSaaaeaacaWGfbGa amytaiaadofacaWGbbaabaGaamytaiaadofacaWGfbaaaaaa@3FB6@

ESSEA=SSEA-SSE=ESSA

c-1

EMSEA= ESSEA c1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbGaamytaiaadofacaWGfbGaamyqaiabg2da9maalaaa baGaamyraiaadofacaWGtbGaamyraiaadgeaaeaacaWGJbGaeyOeI0 IaaGymaaaaaaa@4258@

Factor B

ESSB=SSR-SSB

a-1

EMSB= ESSB a1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbGaamytaiaadofacaWGcbGaeyypa0ZaaSaaaeaacaWG fbGaam4uaiaadofacaWGcbaabaGaamyyaiabgkHiTiaaigdaaaaaaa@40C4@

F B = EMSB MSE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamOqaaqcfayabaGaeyypa0ZaaSaaaeaacaWGfbGa amytaiaadofacaWGcbaabaGaamytaiaadofacaWGfbaaaaaa@3FB8@

ESSEB=SSEB-SSE=ESSB

a-1

EMSEB= ESSEB a1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbGaamytaiaadofacaWGfbGaamOqaiabg2da9maalaaa baGaamyraiaadofacaWGtbGaamyraiaadkeaaeaacaWGHbGaeyOeI0 IaaGymaaaaaaa@4258@

y _ y _ n. y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaWaaae aacaWG5baaamaaCaaabeqaaiadacUHYaIOaaWaaWaaaeaacaWG5baa aiabgkHiTiaad6gacaGGUaGabmyEayaaraWaaWbaaeqajuaibaGaaG Omaaaaaaa@405E@

a.c-1

Table 2b: Two-factor Analysis of Variance Table for the Extra sums of Squares due to reduced models and other statistics (Continuation).

In fact an additional advantage of using dummy variable regression models in two factor or multiple factor analysis of variance type problems is that the method also more easily enables the estimation of factor level effects separately of several factors on a specified dependent or criterion variable. For example it enables the estimation of the total or absolute effect, the partial regression coefficient or the so called direct effect of a given independent variable here referred to as the parent independent variable on the dependent variable through the effect of its representative dummy variables as well as the indirect effect of that parent independent variable through the mediation of other independent variables in the model [6]. The total or absolute effect of a parent independent variable on a dependent variable is estimated as the simple regression coefficient of that independent variable represented by codes assigned to its various categories when regressed on the dependent variable. The direct effect of a parent independent variable on a dependent variable is the weighted sum of the partial regression coefficients or effects of the dummy variables representing that parent independent variable on the dependent variable where the weights are the simple regression coefficients of each representative dummy variable regressing on the specified parent independent variable represented by codes. The indirect effect of a given parent independent variable on a dependent variable is then simply the difference between its total and direct effects [6].

Now the direct effect or partial regression coefficient of a given parent independent variable on a dependent variable is obtained by taking the partial derivative of the expected value of the corresponding regression model with respect to that parent independent variable. For example the direct effect of the parent independent variable ‘A’ say on the dependent variable Y is obtained from equation 5 as

β A dir= dE( y i ) d A = l=1 a1 β l;A . dE( x l;A ) d A + l β l;Z . dE( x l;Z ) d A or β A dir= l=1 a1 β l;A . dE( x l;A ) d A since l β l;Z . dE( x l;Z ) d A =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGcq aHYoGydaWgaaqcfasaaiaadgeaaKqbagqaaiaadsgacaWGPbGaamOC aiabg2da9maalaaabaGaamizaiaadweacaGGOaGaamyEamaaBaaaju aibaGaamyAaaqcfayabaGaaiykaaqaaiaadsgadaWgaaqcfasaaiaa dgeaaKqbagqaaaaacqGH9aqpdaaeWbqaaiabek7aInaaBaaajuaiba GaamiBaiaacUdacaWGbbaajuaGbeaacaGGUaWaaSaaaeaacaWGKbGa amyraiaacIcacaWG4bWaaSbaaKqbGeaacaWGSbGaai4oaiaadgeaaK qbagqaaiaacMcaaeaacaWGKbWaaSbaaKqbGeaacaWGbbaajuaGbeaa aaGaey4kaSYaaabuaeaacqaHYoGydaWgaaqcfasaaiaadYgacaGG7a GaamOwaaqcfayabaGaaiOlamaalaaabaGaamizaiaadweacaGGOaGa amiEamaaBaaajuaibaGaamiBaiaacUdacaWGAbaajuaGbeaacaGGPa aabaGaamizamaaBaaajuaibaGaamyqaaqcfayabaaaaaqcfasaaiaa dYgaaKqbagqacqGHris5aaqcfasaaiaadYgacqGH9aqpcaaIXaaaba GaamyyaiabgkHiTiaaigdaaKqbakabggHiLdaabaGaam4Baiaadkha aeaacqaHYoGydaWgaaqcfasaaiaadgeaaKqbagqaaiaadsgacaWGPb GaamOCaiabg2da9maaqahabaGaeqOSdi2aaSbaaKqbGeaacaWGSbGa ai4oaiaadgeaaKqbagqaaiaac6cadaWcaaqaaiaadsgacaWGfbGaai ikaiaadIhadaWgaaqcfasaaiaadYgacaGG7aGaamyqaaqcfayabaGa aiykaaqaaiaadsgadaWgaaqcfasaaiaadgeaaKqbagqaaaaaaKqbGe aacaWGSbGaeyypa0JaaGymaaqaaiaadggacqGHsislcaaIXaaajuaG cqGHris5aaGcbaqcfaOaci4CaiaacMgacaGGUbGaam4yaiaadwgada aeqbqaaiabek7aInaaBaaajuaibaGaamiBaiaacUdacaWGAbaajuaG beaaaKqbGeaacaWGSbaajuaGbeGaeyyeIuoacaGGUaWaaSaaaeaaca WGKbGaamyraiaacIcacaWG4bWaaSbaaKqbGeaacaWGSbGaai4oaiaa dQfaaKqbagqaaiaacMcaaeaacaWGKbWaaSbaaKqbGeaacaWGbbaaju aGbeaaaaGaeyypa0JaaGimaaaaaa@AE9B@ (26)

For all other independent variable ‘z’ in the model different from ‘A’.

The weight α l;A = dE( x l;A ) d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaWGSbGaai4oaiaadgeaaKqbagqaaiabg2da9maa laaabaGaamizaiaadweacaGGOaGaamiEamaaBaaajuaibaGaamiBai aacUdacaWGbbaajuaGbeaacaGGPaaabaGaamizamaaBaaajuaibaGa amyqaaqcfayabaaaaaaa@4674@  is estimated by fitting a simple regression line of dummy variable. x l;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaiaacUdacaWGbbaajuaGbeaaaaa@3AD4@ regressing on its parent independent variable, A represented by codes and taking the derivative of its expected value with respect to ‘A’. Thus, if the expected value of the dummy variable x l;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaiaacUdacaWGbbaajuaGbeaaaaa@3AD4@ regressing on its parent independent variable ‘A’ is expressed as E( x l;A )= α 0 + α l;A .A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaamiEamaaBaaajuaibaGaamiBaiaacUdacaWGbbaajuaG beaaaiaawIcacaGLPaaacqGH9aqpcqaHXoqydaWgaaqcfasaaiaaic daaKqbagqaaiabgUcaRiabeg7aHnaaBaaajuaibaGaamiBaiaacUda caWGbbaajuaGbeaacaGGUaGaamyqaaaa@48AF@

Then the derivative of this expected value with respect to A is

dE( x l;A ) d A = α l;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaamyraiaacIcacaWG4bWaaSbaaKqbGeaacaWGSbGaai4o aiaadgeaaKqbagqaaiaacMcaaeaacaWGKbWaaSbaaKqbGeaacaWGbb aajuaGbeaaaaGaeyypa0JaeqySde2aaSbaaKqbGeaacaWGSbGaai4o aiaadgeaaKqbagqaaiaaykW7aaa@47FF@ (27)

Hence using Equation 27 in Equation 26 gives the direct effect of the parent independent variable A on the dependent variable Y as

β A dir= l=1 a1 α l;A . β l;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaWGbbaajuaGbeaacaWGKbGaamyAaiaadkhacqGH 9aqpdaaeWbqaaiabeg7aHnaaBaaajuaibaGaamiBaiaacUdacaWGbb aajuaGbeaacaGGUaGaeqOSdi2aaSbaaKqbGeaacaWGSbGaai4oaiaa dgeaaKqbagqaaaqcfasaaiaadYgacqGH9aqpcaaIXaaabaGaamyyai abgkHiTiaaigdaaKqbakabggHiLdaaaa@505A@ (28)

Whose sample estimate is from Equation 8

β ^ A dir= b A dir= l=1 a1 α l;A . b l;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaiaadgeaaKqbagqaaiaadsgacaWGPbGaamOC aiabg2da9iaadkgadaWgaaqcfasaaiaadgeaaKqbagqaaiaadsgaca WGPbGaamOCaiabg2da9maaqahabaGaeqySde2aaSbaaKqbGeaacaWG SbGaai4oaiaadgeaaKqbagqaaiaac6cacaWGIbWaaSbaaKqbGeaaca WGSbGaai4oaiaadgeaaKqbagqaaaqcfasaaiaadYgacqGH9aqpcaaI XaaabaGaamyyaiabgkHiTiaaigdaaKqbakabggHiLdaaaa@560E@ (29)

The total or absolute effect of ‘A’ on ‘Y’ is estimated as the simple regression coefficient or effect of the parent independent variable ‘A’ represented by codes on the dependent variable ‘Y’ as

β ^ A = b A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaiaadgeaaKqbagqaaiabg2da9iaadkgadaWg aaqcfasaaiaadgeaaKqbagqaaaaa@3D68@ (30)

Where b A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOyam aaBaaajuaibaGaamyqaaqcfayabaaaaa@390E@  is the estimated simple regression coefficient or effect of ‘A’ on ‘Y’. The indirect effect of ‘A’ on ‘Y’ is then estimated as the difference between b A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOyam aaBaaajuaibaGaamyqaaqcfayabaaaaa@390E@ and b A dir MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOyam aaBaaajuaibaGaamyqaaqcfayabaGaamizaiaadMgacaWGYbaaaa@3BDC@ , that is as

β ^ A indir= b A indir= b A b A dir MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaiaadgeaaKqbagqaaiaadMgacaWGUbGaamiz aiaadMgacaWGYbGaeyypa0JaamOyamaaBaaajuaibaGaamyqaaqcfa yabaGaamyAaiaad6gacaWGKbGaamyAaiaadkhacqGH9aqpcaaMe8Ua amOyamaaBaaajuaibaGaamyqaaqcfayabaGaeyOeI0IaamOyamaaBa aajuaibaGaamyqaaqcfayabaGaamizaiaadMgacaWGYbaaaa@5228@ (31)

The total, direct and indirect effects of factor B are similarly estimated.

Illustrative Example 1

The body weights of a random sample of 10 Broilers here termed “ subject or block of subjects” regarded as factor ‘A’ with ten levels and types of weighing machine here termed “treatment” regarded as factor ‘B’ with five levels are shown below.
To set up a dummy variable regression model of body weight (y) regressing on “subject or block of subjects” here termed factor ‘A’ with ten levels and types of weighing machine, here termed “treatments” treated as factor ‘B’ with five levels, we as usual represent factor ‘A’ with nine dummy variables of 1s and 0s and factor ‘B’ with four dummy variables of 1s and 0s, using Equation 1.

The resulting design matrix ‘X’ for the full model is presented in Table 3 where x 1;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaiaacUdacaWGbbaajuaGbeaaaaa@3A9E@  represents level 1 or broiler No.1; x 2;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGOmaiaacUdacaWGbbaajuaGbeaaaaa@3A9F@  represents levels 9 or broiler No.9 and so on. Similarly x 1;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaiaacUdacaWGcbaajuaGbeaaaaa@3A9F@  represents weighing machine No.1 or treatment 1, x 2;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGOmaiaacUdacaWGcbaajuaGbeaaaaa@3AA0@  represents weighing machine No.2 or treatment 2 and so on, until x 4;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGinaiaacUdacaWGcbaajuaGbeaaaaa@3AA2@ represents weighing machine No.4 or treatment 4.

Using the design matrix X of Table 3 for the full model of Equation 6 we obtain the fitted regression Equation expressing the dependent of broiler body weight on, that is as a function of broiler (subject) treated as factor A and type of weighing machine (treatment) treated as factor B, both represented by dummy variables of 1s and 0s, as

y ^ l =2.3020.593 x l 1 ;A +3.175 x l 2 ;A +0.212 x l 3 ;A 2.023 x l 4 ;A 1.491 x l 5 ;A +0.352 x l 6 ;A 1.219 x l 7 ;A +0.123 x l 8 ;A 2.185 x l 9 ;A 0.094 x l 1 ;B 0.235 x l 2 ;B +2.329 x l 3 ;B 0.029 x l 4 ;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGce WG5bGbaKaadaWgaaqcfasaaiaadYgaaKqbagqaaiabg2da9iaaikda caGGUaGaaG4maiaaicdacaaIYaGaeyOeI0IaaGimaiaac6cacaaI1a GaaGyoaiaaiodacaWG4bWaaSbaaKqbGeaacaWGSbqcfa4aaSbaaKqb GeaacaaIXaaabeaacaGG7aGaamyqaaqcfayabaGaey4kaSIaaG4mai aac6cacaaIXaGaaG4naiaaiwdacaWG4bWaaSbaaKqbGeaacaWGSbqc fa4aaSbaaKqbGeaacaaIYaaabeaacaGG7aGaamyqaaqcfayabaGaey 4kaSIaaGimaiaac6cacaaIYaGaaGymaiaaikdacaWG4bWaaSbaaKqb GeaacaWGSbqcfa4aaSbaaKqbGeaacaaIZaaabeaacaGG7aGaamyqaa qcfayabaGaeyOeI0IaaGOmaiaac6cacaaIWaGaaGOmaiaaiodacaWG 4bWaaSbaaKqbGeaacaWGSbqcfa4aaSbaaKqbGeaacaaI0aaabeaaca GG7aGaamyqaaqcfayabaGaeyOeI0IaaGymaiaac6cacaaI0aGaaGyo aiaaigdacaWG4bWaaSbaaKqbGeaacaWGSbqcfa4aaSbaaKqbGeaaca aI1aaabeaacaGG7aGaamyqaaqcfayabaGaey4kaSIaaGimaiaac6ca caaIZaGaaGynaiaaikdacaWG4bWaaSbaaKqbGeaacaWGSbqcfa4aaS baaKqbGeaacaaI2aaabeaacaGG7aGaamyqaaqcfayabaaakeaajuaG caaMe8UaaGjbVlaaysW7cqGHsislcaaIXaGaaiOlaiaaikdacaaIXa GaaGyoaiaadIhadaWgaaqcfasaaiaadYgajuaGdaWgaaqcfasaaiaa iEdaaeqaaiaacUdacaWGbbaajuaGbeaacqGHRaWkcaaIWaGaaiOlai aaigdacaaIYaGaaG4maiaadIhadaWgaaqcfasaaiaadYgajuaGdaWg aaqcfasaaiaaiIdaaeqaaiaacUdacaWGbbaajuaGbeaacqGHsislca aIYaGaaiOlaiaaigdacaaI4aGaaGynaiaadIhadaWgaaqcfasaaiaa dYgajuaGdaWgaaqcfasaaiaaiMdaaeqaaiaacUdacaWGbbaajuaGbe aacqGHsislcaaIWaGaaiOlaiaaicdacaaI5aGaaGinaiaadIhadaWg aaqcfasaaiaadYgajuaGdaWgaaqcfasaaiaaigdaaeqaaiaacUdaca WGcbaajuaGbeaacqGHsislcaaIWaGaaiOlaiaaikdacaaIZaGaaGyn aiaadIhadaWgaaqcfasaaiaadYgajuaGdaWgaaqcfasaaiaaikdaae qaaiaacUdacaWGcbaajuaGbeaacqGHRaWkcaaIYaGaaiOlaiaaioda caaIYaGaaGyoaiaadIhadaWgaaqcfasaaiaadYgajuaGdaWgaaqcfa saaiaaiodaaeqaaiaacUdacaWGcbaajuaGbeaacqGHsislcaaIWaGa aiOlaiaaicdacaaIYaGaaGyoaiaadIhadaWgaaqcfasaaiaadYgaju aGdaWgaaqcfasaaiaaisdaaeqaaiaacUdacaWGcbaajuaGbeaaaaaa @CB82@

Now to estimate the total or absolute effect of type of weighing machine (treatment), ‘B; or body weight y of broilers, we regress y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamyAaaqcfayabaaaaa@394D@  on ‘B’ represented by codes to obtain β ^ B = b B =0.054 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaiaadkeaaKqbagqaaiabg2da9iaadkgadaWg aaqcfasaaiaadkeaaKqbagqaaiabg2da9iaaicdacaGGUaGaaGimai aaiwdacaaI0aaaaa@4213@ . The weights α j;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaWGQbGaai4oaiaadkeaaKqbagqaaaaa@3B75@  to be applied to Equation 6 to determine the direct effect are obtained as explained above by taking the derivative with respect to ‘B’ of the expected value of the simple regression equation expressing the dependence of the dummy variable x ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaaa@3A3B@  of 1s and 0s on its parent variable ‘B’ represented by codes yielding

α 1;B =0.20; α 2;B =0.10; α 3;B =0.00and α 4;B =0.10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaaIXaGaai4oaiaadkeaaKqbagqaaiabg2da9iab gkHiTiaaicdacaGGUaGaaGOmaiaaicdacaGG7aGaeqySde2aaSbaaK qbGeaacaaIYaGaai4oaiaadkeaaKqbagqaaiabg2da9iabgkHiTiaa icdacaGGUaGaaGymaiaaicdacaGG7aGaeqySde2aaSbaaKqbGeaaca aIZaGaai4oaiaadkeaaKqbagqaaiabg2da9iaaicdacaGGUaGaaGim aiaaicdacaaMe8Uaamyyaiaad6gacaWGKbGaaGjbVlabeg7aHnaaBa aajuaibaGaaGinaiaacUdacaWGcbaajuaGbeaacqGH9aqpcaaIWaGa aiOlaiaaigdacaaIWaaaaa@624E@  .

Using these values in Equation 6, we obtain with Equation 6 the partial or the so called direct effect of type of weighing machine (treatment) ‘B’ on body weight ‘y’ of broilers as

β ^ B dir= b B dir=( 0.094×0.2 )+( 0.235×0.10 )+( 0.00×2.329 )+( 0.029×0.10 ) β ^ B dir= b B dir=0.0394 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaiaadkeaaKqbagqaaiaadsgacaWGPbGaamOC aiabg2da9iaadkgadaWgaaqcfasaaiaadkeaaKqbagqaaiaadsgaca WGPbGaamOCaiabg2da9maabmaabaGaeyOeI0IaaGimaiaac6cacaaI WaGaaGyoaiaaisdacqGHxdaTcqGHsislcaaIWaGaaiOlaiaaikdaai aawIcacaGLPaaacqGHRaWkdaqadaqaaiabgkHiTiaaicdacaGGUaGa aGOmaiaaiodacaaI1aGaey41aqRaeyOeI0IaaGimaiaac6cacaaIXa GaaGimaaGaayjkaiaawMcaaiabgUcaRmaabmaabaGaaGimaiaac6ca caaIWaGaaGimaiabgEna0kaaikdacaGGUaGaaG4maiaaikdacaaI5a aacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacqGHsislcaaIWaGaaiOl aiaaicdacaaIYaGaaGyoaiabgEna0kaaicdacaGGUaGaaGymaiaaic daaiaawIcacaGLPaaacuaHYoGygaqcamaaBaaajuaibaGaamOqaaqc fayabaGaamizaiaadMgacaWGYbGaeyypa0JaamOyamaaBaaajuaiba GaamOqaaqcfayabaGaamizaiaadMgacaWGYbGaaGjbVlabg2da9iaa icdacaGGUaGaaGimaiaaiodacaaI5aGaaGinaaaa@86B0@

Hence the corresponding indirect effect is estimated using Equation 6 as

β ^ B indir= b B indir=0.0146 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaadaWgaaqcfasaaiaadkeaaKqbagqaaiaadMgacaWGUbGaamiz aiaadMgacaWGYbGaeyypa0JaamOyamaaBaaajuaibaGaamOqaaqcfa yabaGaamyAaiaad6gacaWGKbGaamyAaiaadkhacqGH9aqpcaaIWaGa aiOlaiaaicdacaaIXaGaaGinaiaaiAdaaaa@4C2D@ .

The total or absolute, direct and indirect effects of the subjects or block of subjects called factor A are similarly calculated.

It would for comparative purpose be instructive to also analyze the data of example 1 using Friedman two-way analysis of variance test by ranks.

To do this we first rank for each broiler (subject) the body weight as obtained using the five weighing machines (treatment) from the smallest ranked ‘1’ to the largest ranked ‘5’. All tied body weights for each broiler are as usual assigned their mean ranks. The results are presented in Table 4.

Using the ranks shown in Table 4, we calculate the Friedmans test statistic as
χ 2 = 12 rc(c+1) j=1 c R .j 2 3r(c+1) = 12( 13 2 + 33 2 + 27 2 + 40.5 2 + 36.5 2 ) (10)(5)(5+1) 3(10)(5+1)=198.38180=17.38 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaaiaaigda caaIYaaabaGaamOCaiaadogacaGGOaGaam4yaiabgUcaRiaaigdaca GGPaaaamaaqahabaGaamOuamaaDaaajuaibaGaaiOlaiaadQgaaeaa caaIYaaaaKqbakabgkHiTiaaiodacaWGYbGaaiikaiaadogacqGHRa WkcaaIXaGaaiykaaqcfasaaiaadQgacqGH9aqpcaaIXaaabaGaam4y aaqcfaOaeyyeIuoacqGH9aqpdaWcaaqaaiaaigdacaaIYaWaaeWaae aacaaIXaGaaG4mamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIa aG4maiaaiodadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaik dacaaI3aWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI0aGa aGimaiaac6cacaaI1aWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRa WkcaaIZaGaaGOnaiaac6cacaaI1aWaaWbaaeqajuaibaGaaGOmaaaa aKqbakaawIcacaGLPaaaaeaacaGGOaGaaGymaiaaicdacaGGPaGaai ikaiaaiwdacaGGPaGaaiikaiaaiwdacqGHRaWkcaaIXaGaaiykaaaa cqGHsislcaaIZaGaaiikaiaaigdacaaIWaGaaiykaiaacIcacaaI1a Gaey4kaSIaaGymaiaacMcacqGH9aqpcaaIXaGaaGyoaiaaiIdacaGG UaGaaG4maiaaiIdacqGHsislcaaIXaGaaGioaiaaicdacqGH9aqpca aIXaGaaG4naiaac6cacaaIZaGaaGioaaaa@8B2A@

Which with c-1=5-1=4 degrees of freedom is statistically significant ( χ 0.99;4 2 =13.277 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHhpWydaqhaaqcfasaaiaaicdacaGGUaGaaGyoaiaaiMdacaGG 7aGaaGinaaqaaiaaikdaaaqcfaOaeyypa0JaaGymaiaaiodacaGGUa GaaGOmaiaaiEdacaaI3aaacaGLOaGaayzkaaaaaa@453B@ ,indicating that weighing machines probability differ in the values of body weights of broilers obtained using them. This is the same conclusion that is also reached using the present method.

S/no (l)
Body weight (yi)
x l o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaam4BaaqabaaajuaG beaaaaa@3B20@
x l 1 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGymaaqabaGaai4o aiaadgeaaKqbagqaaaaa@3C6C@
1
x l 2 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGOmaaqabaGaai4o aiaadgeaaKqbagqaaaaa@3C6D@
2
x l 3 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqRbjuaGca WG4bWaaSbaaKqbGeaacaWGSbqcfa4aaSbaaKqbGeaacaaIZaaabeaa caGG7aGaamyqaaqcfayabaaaaa@3D2B@
3
x l 4 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWDbjuaGca WG4bWaaSbaaKqbGeaacaWGSbqcfa4aaSbaaKqbGeaacaaI0aaabeaa caGG7aGaamyqaaqcfayabaaaaa@3D53@
4
x l 5 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGSbWaaSbaaWqaaiaaiwdaaeqaaSGaai4oaiaadgeaaeqa aaaa@3A8C@
5
x l 6 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGOnaaqabaGaai4o aiaadgeaaKqbagqaaaaa@3C71@
6
x l 7 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaG4naaqabaGaai4o aiaadgeaaKqbagqaaaaa@3C72@
7
x l 8 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGioaaqabaGaai4o aiaadgeaaKqbagqaaaaa@3C73@
8
x l 9 ;A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGyoaaqabaGaai4o aiaadgeaaKqbagqaaaaa@3C74@
9
x l 1 ;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGymaaqabaGaai4o aiaadkeaaKqbagqaaaaa@3C6D@
1
x l 2 ;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGOmaaqabaGaai4o aiaadkeaaKqbagqaaaaa@3C6E@
2
x l 3 ;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaG4maaqabaGaai4o aiaadkeaaKqbagqaaaaa@3C6F@
3
x l 4 ;B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamiBaKqbaoaaBaaajuaibaGaaGinaaqabaGaai4o aiaadkeaaKqbagqaaaaa@3C70@
4
1
1.9
1
1
0
0
0
0
0
0
0
0
1
0
0
0
2
2
1
1
0
0
0
0
0
0
0
0
0
1
0
0
3
2.1
1
1
0
0
0
0
0
0
0
0
0
0
1
0
4
2.1
1
1
0
0
0
0
0
0
0
0
0
0
0
1
5
1.9
1
1
0
0
0
0
0
0
0
0
0
0
0
0
6
1.7
1
0
1
0
0
0
0
0
0
0
1
0
0
0
7
2
1
0
1
0
0
0
0
0
0
0
0
1
0
0
8
1.8
1
0
1
0
0
0
0
0
0
0
0
0
1
0
9
2.1
1
0
1
0
0
0
0
0
0
0
0
0
0
1
10
2
1
0
1
0
0
0
0
0
0
0
0
0
0
0
11
1.9
1
0
0
1
0
0
0
0
0
0
1
0
0
0
12
2.2
1
0
0
1
0
0
0
0
0
0
0
1
0
0
13
1.9
1
0
0
1
0
0
0
0
0
0
0
0
1
0
14
2.2
1
0
0
1
0
0
0
0
0
0
0
0
0
1
15
2.2
1
0
0
1
0
0
0
0
0
0
0
0
0
0
16
1.8
1
0
0
0
1
0
0
0
0
0
1
0
0
0
17
2.2
1
0
0
0
1
0
0
0
0
0
0
1
0
0
18
2.1
1
0
0
0
1
0
0
0
0
0
0
0
1
0
19
2
1
0
0
0
1
0
0
0
0
0
0
0
0
1
20
2.1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
21
1.9
1
0
0
0
0
1
0
0
0
0
1
0
0
0
22
1.8
1
0
0
0
0
1
0
0
0
0
0
1
0
0
23
1.9
1
0
0
0
0
1
0
0
0
0
0
0
1
0
24
2.2
1
0
0
0
0
1
0
0
0
0
0
0
0
1
25
2.1
1
0
0
0
0
1
0
0
0
0
0
0
0
0
26
1.8
1
0
0
0
0
0
1
0
0
0
1
0
0
0
27
2
1
0
0
0
0
0
1
0
0
0
0
0
0
0
28
2.1
1
0
0
0
0
0
1
0
0
0
0
1
0
0
29
2.1
1
0
0
0
0
0
1
0
0
0
0
0
1
0
30
2.1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
31
1.8
1
0
0
0
0
0
0
1
0
0
1
0
0
0
32
2.1
1
0
0
0
0
0
0
1
0
0
1
1
0
0
33
1.9
1
0
0
0
0
0
0
1
0
0
0
0
1
0
34
2.2
1
0
0
0
0
0
0
1
0
0
0
0
0
1
35
2
1
0
0
0
0
0
0
1
0
0
0
0
0
0
36
1.7
1
0
0
0
0
0
0
0
1
0
1
0
0
0
37
2.1
1
0
0
0
0
0
0
0
1
0
0
1
0
0
38
1.9
1
0
0
0
0
0
0
0
1
0
0
0
1
0
39
1.9
1
0
0
0
0
0
0
0
1
0
0
0
0
1
40
2.1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
41
1.8
1
0
0
0
0
0
0
0
0
1
1
0
0
0
42
1.9
1
0
0
0
0
0
0
0
0
1
0
1
0
0
43
2
1
0
0
0
0
0
0
0
0
1
0
0
1
0
44
2.1
1
0
0
0
0
0
0
0
0
1
0
0
0
1
45
2.1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
46
2
1
0
0
0
0
0
0
0
0
0
1
0
0
0
47
2.1
1
0
0
0
0
0
0
0
0
0
0
1
0
0
48
2
1
0
0
0
0
0
0
0
0
0
0
0
1
0
49
2.1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
50
2.1
1
0
0
0
0
0
0
0
0
0
0
0
0
0

Table 3: Design matrix for the sample data of example 1.

Body Weight(Treatment)

Broiler(Subject)

1

2

3

4

5

1

1.5

3

4.5

4.5

1.5

2

1

3.5

2

5

3.5

3

1.5

4

1.5

4

4

4

1

5

3.5

2

3.5

5

2.5

1

2.5

5

4

6

1

2

4

4

4

7

1

4

2

5

3

8

1

4.5

2.5

2.5

4.5

9

1

2

3

4.5

4.5

10

1.5

4

1.5

4

4

Total

13

33

27

40.5

36.5

Table 4: Ranks of body weights of broilers in Table 1.

Summary and Conclusion

This paper has proposed the use of dummy variable multiple regression methods for the analysis of several related or dependent samples appropriate for random effects and mixed effects two factor analysis of variance with one observation per cell or treatment combination.

Using the extra sum of squares principle, the method developed necessary sums of squares, degrees of freedom and the F-ratios required in testing for the significance of factor level effects.

The method provided estimates of the overall and factor mean effects comparable to those obtained with the two factor analysis of variance method. In addition the method also provided estimates of the total or absolute effects as well as the direct and indirect effects of the independent variables or factors on the dependent or criterion variable which are not ordinarily obtainable with the usual analysis of variance techniques. The proposed method is illustrated with some sample data and shown to compare favorably with the usual Friedmans two-way analysis of variance test by ranks often used for the same purpose.

References

  1. Oyeka ICA (2013) Ties Adjusted Two way Analysis of Variance tests with unequal observations per cell. Science Journal of Mathematics & Statistics 2013(2013): 1-6.
  2. Boyle, Richard P (1970) Path Analysis and Ordinal Data. American Journal of Sociology 75(4): 461-480.
  3. Oyeka ICA, Afuecheta EO, Ebuh GU, Nnanatu CC, Obiora Ilouno HO (2012) Partitioning the total chi-square for matched Dichotomous Data. International Journal of Mathematics and Computations. 16(3): 41-50.
  4. Draper NR, Smith H (1966) Applied Regression Analysis. John Wiley & sons, New York, USA.
  5. Neter J, Wasserman W, Christopher N, Michael K (1974). Applied Linear Statistical Models. New York, USA.
  6. Wright, Sewall (1934) The Methods of Path Coefficients. The Annals of Mathematical Statistics 5(3): 161-215.
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