# [R] two way ANOVA with unequal sample sizes

Ben Bolker ben at zoo.ufl.edu
Tue Oct 16 21:59:12 CEST 2001

```  Is this a problem (perhaps) with the dreaded "SAS type III sums of
squares"?  I don't know the reference, but the authors may be estimating
the effect of dropping the main effects while the interaction terms are
still included in the model (which is at the very least controversial, and
probably wrong).
The sums of squares estimated for unbalanced linear models necessarily
vary according to the order in which the factors are added or dropped.
What do the authors say about the order they're using?

See

http://finzi.psych.upenn.edu/R/Rhelp/archive/0913.html
http://finzi.psych.upenn.edu/R/Rhelp/archive/3612.html

and responses to them ...

Ben Bolker

On Tue, 16 Oct 2001, julien claude wrote:

> Hi,
>
> I am trying a two way anova with unequal sample sizes but results are not
> as expected:
>
> I take the example from Applied Linear Statistical Models (Neter et al.
> pp889-897, 1996)
>
> growth rate		gender	bone development
> 1.4			1		1
> 2.4			1		1
> 2.2			1		1
> 2.4			1		2
> 2.1			2		1
> 1.7			2		1
> 2.5			2		2
> 1.8			2		2
> 2			2		2
> 0.7			3		1
> 1.1			3		1
> 0.5			3		2
> 0.9			3		2
> 1.3			3		2
>
> expected results are
>
> source of variation	SS	df	MS	F
> gender			0.12	1	0.12	0.74
> bone development	4.1897	2	2.0949	12.89**
> interaction		0.0754	2	0.377	0.23
> Error			1.3	8	0.1625
>
> # I use
> aov (growrate ~ gender * bonedevelopment)->m
> summary(m)
>
>    						  Df      Sum Sq    Mean Sq 	F value   Pr(>F)
> as.factor(gender)                     			2 	4.3063  	2.1531 		13.2501
> 0.002891 **
> as.factor(bonedevlopment)            		1 	0.0926  	0.0926  		0.5697
> 0.472022
> as.factor(gender:bonedevlopment)  		 2 	0.0754 	 0.0377 	 	0.2321	 0.798034
> Residuals                            			8 	1.3000 	 0.1625
>
> #if I change the order of factors, results are different
> aov (growrate ~ bonedevelopment * gender)->m
> summary(m)
>
>                                       		Df 	Sum Sq Mean Sq 	F value
> Pr(>F)
> as.factor(bonedevlopment)             	1 	0.0029  0.0029 		 0.0176
> 0.897785
> as.factor(gender)                    		2 	4.3960  2.1980 		13.5262 0.002713 **
> as.factor(gender:bonedevlopment)  	2 	0.0754  0.0377  		0.2321   0.798034
> Residuals                           		8 	1.3000  0.1625
>
> #In the both cases, results for main effects differ from those expected in
> Neter et al.
> However interaction and residuals are well estimated.
> Can anyone help, either I am wrong in the formula, or either is there an
> other problem? Is there a mean to conduct easily the  test as in it is in
> Neter et al. ?
> The same problems occurs with anova(lm(....))?
>
> thank you very much
>
> julien CLAUDE
> -------------------------------
>
> CLAUDE julien
> Université Montpellier II
> Institut des Sciences de l'Evolution de Montpellier
> Laboratoire de Paléontologie (Morphométrie), Cc64
> 2, Place Eugène Bataillon.
> 34095, Montpellier, cedex 5
> FRANCE
>
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318 Carr Hall                                bolker at zoo.ufl.edu
Zoology Department, University of Florida    http://www.zoo.ufl.edu/bolker
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