# [R] Type I SS and Type III SS problem

Peter Dalgaard P.Dalgaard at biostat.ku.dk
Fri Sep 19 10:27:02 CEST 2008

```leeznar wrote:
> Dear all:
> I m a newer on R.  I have some problem when I use anova function.  I use anova function to get Type I SS results, but I also need to get Type III SS results.  However, in my code, there is some different between the result of Type I SS and Type III SS.  I don’t know why the “seqe” factor disappeared in the result of Type III SS.  How can I do?
>
>
Well, first make sure that you copy/paste correctly. Your drop1()
results appear not to come from the model M!!

> drop1(M, test="F")
Single term deletions

Model:
Cmax ~ seqe + subj:seqe + per + drug
Df Sum of Sq     RSS     AIC F value  Pr(F)
<none>                  438395     302
per        1     63175  501570     304  1.7293 0.2131
drug       1     58149  496544     304  1.5917 0.2311
seqe:subj 12    634325 1072719     303  1.4469 0.2660

So things are not massively different, which is because this is a nicely
balanced cross-over trial. In an unbalanced trial, the type I anova
would be order-dependent.

Now, seqe is nested in subj:seqe, in fact it is even nested in subj
(because each subject gets the two treatments in only one order,
right?), so taking seqe out of the model doesn't change the fit, as long
as, and this is the important bit, subj:seqe. The drop1() function knows
this and doesn't do the test. In contrast, anova looks at the effect
_before_ subj:seqe was in the model.

Since subj:seqe is equivalent to subj, it can be illuminating to write
the model in this (equivalent) way and notic the differences:

> M<- lm(Cmax ~ subj + seqe + per + drug , data=KK)
> anova(M)
Analysis of Variance Table

Response: Cmax
Df Sum Sq Mean Sq F value Pr(>F)
subj      13 634910   48839  1.3369 0.3109
per        1  63175   63175  1.7293 0.2131
drug       1  58149   58149  1.5917 0.2311
Residuals 12 438395   36533
> drop1(M)
Single term deletions

Model:
Cmax ~ subj + seqe + per + drug
Df  Sum of Sq     RSS     AIC
<none>                438395     302
subj   12     634325 1072719     303
seqe    0 -1.630e-09  438395     302
per     1      63175  501570     304
drug    1      58149  496544     304

And (I think I've said this before on the list): When things are nicely
balanced, aov() can be preferable:

> summary(aov(Cmax ~ per*drug + Error(subj) , data=KK))

Error: subj
Df Sum Sq Mean Sq F value Pr(>F)
per:drug   1    585     585  0.0111  0.918
Residuals 12 634325   52860

Error: Within
Df Sum Sq Mean Sq F value Pr(>F)
per        1  63175   63175  1.7293 0.2131
drug       1  58149   58149  1.5917 0.2311
Residuals 12 438395   36533

> Here is my example and result.
> a<-c(1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14)
> b<-c(1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2)
> c<-c(2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2)
> d<-c(2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1)
> e<-c(1739,1633,1481,1837,1780,2073,1374,1629,1555,1385,1756,1522,1566,1643,
>      1939,1615,1475,1759,1388,1483,1127,1682,1542,1247,1235,1605,1598,1718 )
> KK<-data.frame(subj=as.factor(a), drug=as.factor(b), per=as.factor(c),  seqe=as.factor(d), Cmax=e)
> M<- lm(Cmax ~ seqe+ subj:seqe + per + drug , data=KK)
> anova(M)
> drop1(M, test="F")
>
> The result of Type I SS:
> Analysis of Variance Table
> Response: Cmax
>           Df Sum Sq Mean Sq F value Pr(>F)
> seqe       1    585     585  0.0160 0.9014
> per        1  63175   63175  1.7293 0.2131
> drug       1  58149   58149  1.5917 0.2311
> seqe:subj 12 634325   52860  1.4469 0.2660
> Residuals 12 438395   36533
>
> The result of Type III SS:
> Single term deletions
> Model:
> AUCt ~ seqe + subj:seqe + per + drug
>           Df Sum of Sq      RSS      AIC F value  Pr(F)
> <none>                 63208187      442
> per        1   2100484 65308672      441  0.3988 0.5396
> drug       1   4714183 67922370      442  0.8950 0.3628
> seqe:subj 12  35813062 99021249      430  0.5666 0.8308
>
> Best regards,
> HY Lee
>
>
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--
O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
(*) \(*) -- University of Copenhagen   Denmark      Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)              FAX: (+45) 35327907

```

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