Two Factor Studies with Equal Replication KNNL Chapter 19 Two Factor Studies Factor A a levels Factor B b levels ab treatments with n replicates per treatment Controlled Experiments CRD Randomize abn experimental units to the ab treatments n units per trt Observational Studies Take random samples of n units from each population sub population One Factor at a Time Method Choose 1 level of one factor say A and compare levels of other factor B Choose best level factor B levels hold that constant and compare levels of factor A Not effective Poor randomization logistics no interaction tests Better Method Observe all combinations of factor levels ANOVA Model Notation Additive Model Halo Effect Study Factor A Essay Quality Good Poor Factor B Photo Attract Unatt None A EQ B Pic j 1 Attract j 2 Unatt j 3 None Row Average i 1 Good 11 25 12 18 13 20 1 21 i 2 Poor 21 17 22 10 23 12 2 13 Column Average 1 21 2 14 3 16 17 Additive Effects Model mij m a i b j a b 1 a b s t a i b j 0 m mij ab i 1 j 1 i 1 j 1 b 1 b 1 b 1 mi mij m a i b j bm ba i b j m a i m j m b j b j 1 b j 1 b j 1 1 a b 1 a 1 b a i mi m b j m j m m mij mi m j ab i 1 j 1 a i 1 b j 1 Halo Effect Example a1 m1 m 21 17 4 a 2 m2 m 13 17 4 a 1 a 2 0 b1 m 1 m 21 17 4 b2 m 2 m 14 17 3 b3 m 3 m 16 17 1 ANOVA Model Notation Interaction Model Halo Effect Study Factor A Essay Quality Good Poor Factor B Photo Attract Unatt None A EQ B Pic j 1 Attract j 2 Unatt j 3 None Row Average i 1 Good 11 23 12 20 13 20 1 21 i 2 Poor 21 19 22 8 23 12 2 13 2 14 3 16 17 Column Average 1 21 Interaction Model mij m a i b j ab ij a s t b a a b ab i i 1 j j 1 i 1 b ij ab ij 0 j 1 ab ij mij m a i b j mij m mi m m j m mij mi m j m Halo Effect Example ab 11 23 21 21 17 2 ab 21 19 23 21 17 2 ab 12 20 21 14 17 2 ab 22 8 13 14 17 2 ab 13 20 21 16 17 0 ab 23 12 13 16 17 0 Comments on Interactions Some interactions while present can be ignored and analysis of main effects can be conducted Plots with almost parallel means will be present In some cases a transformation can be made to remove an interaction Typically logarithmic square root square or reciprocal transformations may work In many settings particular interactions may be hypothesized or observed interactions can have interesting theoretical interpretations When factors have ordinal factor levels we may observe antagonistic or synergistic interactions Two Factor ANOVA Fixed Effects Cell Means Fixed Effects All factor levels of interest are used in the experiment Cell Means Model Yijk mij eijk i 1 a j 1 b k 1 n nT abn mij mean when Factor A at level i B at j eijk NID 0 s 2 Matrix Form a 2 b 2 n 2 Y111 e111 m11 e111 1 0 0 0 Y e m e 1 0 0 0 112 112 11 112 Y121 m11 e121 m12 e121 0 1 0 0 Y m e m e122 0 1 0 0 122 12 122 12 Y X Y211 e211 m21 e211 0 0 1 0 m21 Y212 m22 e212 m21 e212 0 0 1 0 Y221 e221 m22 e221 0 0 0 1 Y222 e222 m22 e222 0 0 0 1 2 Y 2 2 I nT Two Factor ANOVA Fixed Effects Factor Effects Fixed Effects All factor levels of interest are used in the experiment Factor Effects Model Yijk m a i b j ab ij eijk 1 a b m mij ab i 1 j 1 i 1 a a i mi m j 1 b k 1 n nT abn b j m j m ab ij mij mi m j m m overall mean a i main effect of i th level of A b j main effect of j th level of B ab ij interaction of effect at i th level of A and j th level of B a b a a b ab i i 1 j j 1 i 1 eijk NID 0 s 2 b ij ab ij 0 j 1 Yijk N m a i b j ab ij s 2 independent with mij m a i b j ab ij Analysis of Variance Least Squares ML Estimators Notation Observation when A i B j k th replicate Yijk Yij 1 n Sample mean when A i B j Y ij Yijk n k 1 n Y 1 b n Sample mean when A i Y i Yijk i bn j 1 k 1 bn Y 1 a n j Y Y j ijk an i 1 k 1 bn Sample mean when B j Y 1 a b n Overall Mean Y Yijk abn i 1 j 1 k 1 abn a b n a b n Error Sum of Squares Q eijk Yijk mij 2 i 1 j 1 k 1 Q 0 mij 2 i 1 j 1 k 1 Least squares and maximum likelihood estimators mij Y ij Fitted values Y ijk Y ij Residuals eijk Yijk Y ijk Yijk Y ij Factor Effects Model Estimators m Y a i Y i Y b j Y j Y ab ij Y ij Y i Y j Y Y ijk Y Y i Y Y j Y Y ij Y i Y j Y Y ij Analysis of Variance Sums of Squares Y Y Cell Means Model Yijk Y Yijk Y ij Y ij Y a b n Y ijk Y i 1 j 1 k 1 a 2 b n ijk Y ij i 1 j 1 k 1 2 a b n Y ij i 1 j 1 k 1 2 SSTO SSE SSTR dfTO abn 1 nT 1 df E ab n 1 nT ab dfTR ab 1 Factor Effects Model Yijk Y Yijk Y ij Y i Y Y j Y Y ij Y i Y j Y 2 2 2 Yijk Y Yijk Y ij Y i Y i j k Y i j k i j k 2 i k Y Y ij Y i Y j Y j i j k SSTO SSE SSA SSB SSAB dfTO abn 1 nT 1 df A a 1 j df B b 1 df E ab n 1 nT ab …