Type II and III sum of squares (R and SPSS)

To whom it may concern

I made some analysis with R using the command Anova. However, I found 
some problmes with the output obtained by selecting type II o type III 
sum of squares.

M3l <- reshape(M3, direction="long", varying=c("b1","b2","b3"),sep="")
summary(aov(b~fattA*factor(time)+ Error(factor(id)), M3l))

Briefly, I have to do a 2x3 mixed model anova, wherein the first factor 
is a between factor and the second factor is a within factor. I use the 
command Anova in the list below, because I want to obtain also the sum 
of squares of the linear and quadratic contrast between the levels of 
the within factor.




Below I report the list of commands used in R ("fattA" is the beteween 
factor and "fB" is the within factor):

a1b1<-c(10,9,8,7)
a1b2<-c(7,6,4,5)
a1b3<-c(3,2,3,4)
a2b1<-c(9,9,8,7)
a2b2<-c(8,7,9,7)
a2b3<-c(7,8,8,6)

M3<-matrix(0,8,4)
M3[,1]<-cbind(a1b1,a2b1)
M3[,2]<-cbind(a1b2,a2b2)
M3[,3]<-cbind(a1b3,a2b3)
M3[,4]<-rep(c(1,2),each=4)

colnames(M3)<-c("b1","b2","b3","fattA")

M3<-as.data.frame(M3)

require(car)

Loading required package: car
Loading required package: MASS
Loading required package: nnet

f5<-lm(cbind(b1,b2,b3)~fattA,data=M3)
a5<-Anova(f5)

f6<-lm(c(b1,b2,b3)~rep(fattA,3),data=M3)


fB<-factor(c(1:3))
d2<-data.frame(fB)
a6<-Anova(f5,idata=d2,idesign=~fB,type=2)

summary(a6,multivariate=F)

Univariate Type II Repeated-Measures ANOVA Assuming Sphericity

                 SS num Df Error SS den Df       F    Pr(>F)
(Intercept) 1080.04      1   9.5833      6 676.200 2.133e-07 ***
fattA         26.04      1   9.5833      6  16.304  0.006819 **
fB            42.33      2  10.6667     12  23.812 6.645e-05 ***
fattA:fB      20.33      2  10.6667     12  11.438  0.001660 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Mauchly Tests for Sphericity

         Test statistic p-value
fB              0.87891  0.7242
fattA:fB        0.87891  0.7242


Greenhouse-Geisser and Huynh-Feldt Corrections
 for Departure from Sphericity

          GG eps Pr(>F[GG])
fB       0.89199  0.0001474 ***
fattA:fB 0.89199  0.0026452 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

         HF eps Pr(>F[HF])
fB       1.2438  6.645e-05 ***
fattA:fB 1.2438    0.00166 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Warning message:
In summary.Anova.mlm(a6, multivariate = F) : HF eps > 1 treated as 1


I repated the anlysis by setting type III sum of squares and I obtained:

a6<-Anova(f5,idata=d2,idesign=~fB,type=3)
summary(a6,multivariate=F)

Univariate Type III Repeated-Measures ANOVA Assuming Sphericity

                SS num Df Error SS den Df      F   Pr(>F)
(Intercept) 30.817      1   9.5833      6 19.294 0.004606 **
fattA       26.042      1   9.5833      6 16.304 0.006819 **
fB          40.133      2  10.6667     12 22.575 8.57e-05 ***
fattA:fB    20.333      2  10.6667     12 11.438 0.001660 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Mauchly Tests for Sphericity

         Test statistic p-value
fB              0.87891  0.7242
fattA:fB        0.87891  0.7242


Greenhouse-Geisser and Huynh-Feldt Corrections
 for Departure from Sphericity

          GG eps Pr(>F[GG])
fB       0.89199  0.0001851 ***
fattA:fB 0.89199  0.0026452 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

         HF eps Pr(>F[HF])
fB       1.2438   8.57e-05 ***
fattA:fB 1.2438    0.00166 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Warning message:
In summary.Anova.mlm(a6, multivariate = F) : HF eps > 1 treated as 1


As you can see, the sum of squares of the within factor "fB" and of the 
intercept obtained by setting type II sum of squares are dofferent form 
those obtained by setting type III sum of squares. I repeated the 
analysis by using SPPS (type II and III) and i obtained the same sum of 
squares for both types., which I report below:

within factor and interaction
source             Sum of squares (type II and III)
fB                        42.33333333
fB * fattA             20.33333333
Error(fattB)       10.66666667

between factor
Source            Sum of squares (type II and III)
intercept        1080.041667
fattA                    26.04166667
Error                      9.583333333

The most strange thing, for me, is not only that R gives different 
outputs both for type II and III sum of squares, but that the output 
obtained with type II sum of squares in R coincides with the output 
obtained with type III  of SPSS.

As I remember, with balanced design, type II and III sum of squares 
should give the same output.


Is there anybody that can help me about this point?


thank you for your kind attention.

     Marco Tommasi, Ph/D.


     Department of Neuroscience and Imaging
     "G. D'Annunzio" University of Chieti-Pescara
     Via dei Vestini 31
     66100 Chieti
     ITALY

e-mail: marco.tommasi at unich.it </mc/compose?to=marco.tommasi at unich.it>
tel.: +39 0871 3555890 / fax: + 39 0871 3554163
Web site: www.dni.unich.it



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