Single Factor Studies KNNL Chapter 16 Single Factor Models Independent Variable can be qualitative or quantitative If Quantitative we typically assume a linear polynomial or no structural relation If Qualitative we typically have no structural relation Balanced designs have equal numbers of replicates at each level of the independent variable When no structure is assumed we refer to models as Analysis of Variance models and use indicator variables for treatments in regression model Single Factor ANOVA Model Model Assumptions for Model Testing All probability distributions are normal All probability distributions have equal variance Responses are random samples from their probability distributions and are independent Analysis Procedure Test for differences among factor level means Follow up post hoc comparisons among pairs or groups of factor level means Cell Means Model r of levels of the study factor ni of replicates cases units for the i th level of the study factor r n1 nr ni nT overall sample size number of cases i 1 Yij mi eij i 1 r j 1 ni Yij Response for j th case within the i th level of the study factor mi Population mean for the i th level of the study factor E Y m eij NID 0 s 2 where NID Normally and Independently Distributed s 2 Yij s 2 ij i Yij are independent N mi s 2 Cell Means Model Regression Form Suppose r 3 and n1 n2 n3 2 Y11 Y 12 Y Y 21 Y22 Y31 Y32 1 1 0 X 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 E Y11 1 E Y12 1 E Y 0 E Y 21 X E Y22 0 E Y31 0 E Y32 0 2 0 0 X X 0 2 0 0 0 2 m1 m 2 m3 0 0 1 1 0 0 e11 e 12 e 21 e22 e31 e32 s 2 s2 0 0 0 0 0 2 0 s 0 0 0 0 2 0 0 s 0 0 0 2 s I 2 0 0 0 s 0 0 2 0 0 0 0 s 0 2 0 0 0 0 0 s m1 0 m 0 1 m1 m2 0 m 2 m2 0 m 3 m3 1 m3 1 Y11 Y12 X Y Y Y 21 22 Y Y 31 32 m1 Y 1 0 5 0 0 Y11 Y12 1 X X X Y 0 0 5 0 Y21 Y22 Y 2 m2 0 0 5 Y31 Y32 Y 3 0 m3 Model Interpretations Factor Level Means Observational Studies The i represent the population means among units from the populations of factor levels Experimental Studies The i represent the means of the various factor levels had they been assigned to a population of experimental units Fixed and Random Factors Fixed Factors All levels of interest are observed in study Random Factors Factor levels included in study represent a sample from a population of factor levels Fitting ANOVA Models ni Y ij ni Notation Yi Yij Y i j 1 j 1 ni ni r r Y i ni ni Y Yij Y ij Y i 1 j 1 i 1 j 1 nT Y r ni Y i nT i 1 nT Least Squares and Maximum Likelihood Estimation r ni ni r Error Sum of Squares Q e Yij mi 2 ij i 1 j 1 2 i 1 j 1 nk Q 2 Ykj mk mk j 1 nk Q Setting 0 mk nk Y kj Y kj nk mk mk j 1 nk j 1 Likelihood L m1 mr s 2 Y11 Yrnr 1 2ps 2 Y k k 1 r 1 r ni 2 exp Yij mi n 2 2 s i 1 j 1 r ni maximizing Likelihood wrt m1 mr minimizing Yij mi i 1 j 1 Fitted values Y ij Y i Residuals eij Yij Y ij Yij Y i 2 k 1 r Analysis of Variance Yij Y Yij Y i Y i Y 14 2 43 14 2 43 1 4 2 43 Total Deviation r ni Y Deviation of trt mean from overall mean r i 1 j 1 ni r ni Y i Y i Y Y i Y Yij Y i 0 ij Deviation from trt mean residual 2 i 1 ni r j 1 ni r 2 Yij Y Yij Y i Y i Y i 1 j 1 i 1 j 1 i 1 j 1 r ni 2 Total Corrected Sum of Squares SSTO Yij Y i 1 j 1 ni r 2 2 r dfTO nT 1 Treatment Sum of Squares SSTR Y i Y ni Y i Y i 1 j 1 r ni Error Sum of Squares SSE Yij Y i i 1 j 1 Note SSTO SSTR SSE ni 2 i s Yij Y i j 1 ni 1 2 i 1 2 dfTR r 1 df E nT r dfTO dfTR df E Useful result 2 Mean Squares MSTR ni ni 1 s Yij Y i 2 i j 1 SSTR r 1 MSE SSE nT r 2 r SSE ni 1 si2 i 1 r df E nT r ni 1 i 1 ANOVA Table Source df SS MS E MS r Treatments r 1 r i 1 nT r Error ni r nT 1 r ni 2 MSE SSTO Yij Y i 1 j 1 r SSE Yij Y i i 1 j 1 Total SSTR ni Y i Y 2 i Note SSTR ni Y nT Y r 2 SSE nT r s2 2 ni SSE Y i 1 ni mi m SSTR 2 MSTR s i 1 r 1 r 1 2 2 2 ij i 1 j 1 r 2 ni Y i i 1 r ni 2 r s Yij s E Y m s E Yij ni mi2 nT s 2 i 1 i 1 j 1 r 2 2 s2 s2 r 2 2 s Y i E Y i mi E ni Y i ni mi2 rs 2 ni ni i 1 i 1 E Yij mi 2 2 2 ij 2 E Y i mi 2 i r E Y n m i i 1 nT i m s 2 Y s2 nT E Y 2 s2 m nT 2 2 2 E nT Y nT m s 2 F Test for H0 1 r H 0 m1 mr H A Not all mi are equal MSTR MSE Under null hypothesis and independence and normality of errors SSTR SSE 2 2 c c r 1 nT r and are independent independent even if H 0 false 2 2 s s SSTR r 1 MSTR 2 s F r 1 nT r SSE MSE n r T 2 s MSTR Decision Rule Reject H 0 if F F 1 a r 1 nT r MSE Test Statistic F General Linear Test of Equal Means H 0 m1 mr mc mc Common Mean Reduced Model H A Not all mi are equal Complete Model Reduced Model mc Y Y ij r ni 2 n n 2 r i …