1. Fill in the missing values in the following ANOVA table for a completely rand
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1. Fill in the missing values in the following ANOVA table for a completely randomized design with two factors crossed factors (say A and B). A has 2 levels and B has 3 levels The number of experimental units per treatment n - 3 dl valeur p source Facteur A Facteur B nteractionS error total ms 5.0139 1.0906 0.05 7.9294 (a) Test for the significance of the interaction effects (b) Compute the eta square for each of the main effects and for the interaction effects. (c) Based on the results from parts (a) and (b), is it reasonable to ignore the inter- actions in the description of the main effects. (d) Pool the ss for the interactions and the error, to obtain a table that corresponds to an additive model. With this modified table, test for the significance of the A main effects and test for the significance of the B main effects.Explanation / Answer
Source dl ss ms F valeur p
Facteur A 2-1=1 5.0139 5.0139/1=5.0139 5.0139/0.05 P(F>100.278|F~F1,12)
=100.278 =3.5x10-7
Facteur B 3-1=2 1.0906x2=2.1812 1.0906 1.0906/0.05 P(F>21.812|F~F2,12)
=21.812 =0.0001
Interaction (2-1)(3-1)=2 7.9294-5.0139- 0.1343/2=0.06715 0.06715/0.05 P(F>1.343|F~F2,12)
2.1812 -0.6=0.1343 =1.343 =0.2976>0.05
Error 2x3x(3-1)=12 0.05x12=0.6 0.05
Total 2x3x3-1=17 7.9294
(a) Since p-value corresponding Interaction effects=0.2976 (>0.05) , interaction effect is insignificant.
(b) eta square= SS effect/SSTotal.
Hence eta square for factor A=5.0139/7.9294= 0.6323
eta square for factor B=2.1812/7.9294=0.2751
eta square for Interaction effect=0.1343/7.9294=0.0169
(c) Yes, it is reasonable to ignore the interaction effect.
(d)
ANOVA TABLE where pool the ss for the interactions and the error:
Source dl ss ms F valeur p
Facteur A 1 5.0139 5.0139 5.0139/0.05245=95.5939 P(F>95.5939|F~F1,14)=1.23x10-7
Facteur B 2 2.1812 1.0906 1.0906/0.05245=20.7931 P(F>20.7931|F~F2,14)=6.43x10-5
(Interaction 2+ 12=14 0.1343+ 0.7343/14
+Error) =Error 0.6=0.7343 =0.05245
Total 17 7.9294
Since p-value for main effect A=1.23x10-7<0.05, hence main effect of A is significant i.e. 2 levels of A are not equal.
Since p-value for main effect B=6.43x10-5<0.05, hence main effect of B is significant i.e. 3 levels of B are not all equal.
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