I wonder if someone could explain the behavior of the anova() and lm()
functions in the following situation:
I have a standard 3x2 factorial design, factorA has 3 levels, factorB has 2
levels, they are fully crossed. I have a dependent variable DV.
Of course I can do the following to get the usual anova table:
> anova(lm(DV~factorA+factorB+factorA:factorB))
Analysis of Variance Table
Response: DV
Df Sum Sq Mean Sq F value Pr(>F)
factorA 2 7.4667 3.7333 4.9778 0.015546 *
factorB 1 2.1333 2.1333 2.8444 0.104648
factorA:factorB 2 9.8667 4.9333 6.5778 0.005275 **
Residuals 24 18.0000 0.7500
This is perfectly satisfactory for my situation, but as a pedagogical
exercise, I wanted to demonstrate the model comparison approach to analysis
of variance by using anova() to compare a full model that contains all
effects, to restricted models that contain all effects save for the effect
of interest.
The test of the interaction effect seems to be as I expected:
> fullmodel<-lm(DV~factorA+factorB+factorA:factorB)
> restmodel<-lm(DV~factorA+factorB)
> anova(fullmodel,restmodel)
Analysis of Variance Table
Model 1: DV ~ factorA + factorB + factorA:factorB
Model 2: DV ~ factorA + factorB
Res.Df RSS Df Sum of Sq F Pr(>F)
1 24 18.0000
2 26 27.8667 -2 -9.8667 6.5778 0.005275 **
As you can see the value of F (6.5778) is the same as in the anova table
above. All is well.
However, if I try to test a main effect, e.g. factorA, by testing the full
model against a restricted model that doesn't contain the main effect
factorA, I get something strange:
> restmodel<-lm(DV~factorB+factorA:factorB)
> anova(fullmodel,restmodel)
Analysis of Variance Table
Model 1: DV ~ factorA + factorB + factorA:factorB
Model 2: DV ~ factorB + factorA:factorB
Res.Df RSS Df Sum of Sq F Pr(>F)
1 24 18
2 24 18 0 0
upon inspection of each model I see that the Residuals are identical, which
is not what I was expecting:
> anova(fullmodel)
Analysis of Variance Table
Response: DV
Df Sum Sq Mean Sq F value Pr(>F)
factorA 2 7.4667 3.7333 4.9778 0.015546 *
factorB 1 2.1333 2.1333 2.8444 0.104648
factorA:factorB 2 9.8667 4.9333 6.5778 0.005275 **
Residuals 24 18.0000 0.7500
This looks fine, but then the restricted model is where things are not as I
expected:
> anova(restmodel)
Analysis of Variance Table
Response: DV
Df Sum Sq Mean Sq F value Pr(>F)
factorB 1 2.1333 2.1333 2.8444 0.104648
factorB:factorA 4 17.3333 4.3333 5.7778 0.002104 **
Residuals 24 18.0000 0.7500
I was expecting the Residuals in the restricted model (the one not
containing main effect of factorA) to be larger than in the full model
containing all three effects. In other words, the variance accounted for by
the main effect factorA should be added to the Residuals. Instead, it looks
like the variance accounted for by the main effect of factorA is being
soaked up by the factorA:factorB interaction term. Strangely, the degrees of
freedom are also affected.
I must be misunderstanding something here. Can someone point out what is
happening?
Thanks,
-Paul
--
Paul L. Gribble, Ph.D.
Associate Professor
Dept. Psychology
The University of Western Ontario
London, Ontario
Canada N6A 5C2
Tel. +1 519 661 2111 x82237
Fax. +1 519 661 3961
pgrib...@uwo.ca
http://gribblelab.org
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