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

peter dalgaard pdalgd at gmail.com
Wed Mar 21 18:27:17 CET 2012


On Mar 21, 2012, at 11:27 , Marco Tommasi wrote:

> 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.

Well, it would primarily concern the maintainer of the "car" package, which is the one containing the (capital-A) Anova() function. The type III SS don't look right to me either. With aov() we get

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

Error: factor(id)
          Df Sum Sq Mean Sq F value  Pr(>F)   
fattA      1 26.042  26.042    16.3 0.00682 **
Residuals  6  9.583   1.597                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Error: Within
                   Df Sum Sq Mean Sq F value   Pr(>F)    
factor(time)        2  42.33  21.167   23.81 6.65e-05 ***
fattA:factor(time)  2  20.33  10.167   11.44  0.00166 ** 
Residuals          12  10.67   0.889                     
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 



> 
> 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
> 
> 
> 
> 	[[alternative HTML version deleted]]
> 
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-- 
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com



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