# [R] Type III ANOVA of package car depends on factor level order

Xiaoxu LI lixiaoxu at gmail.com
Tue Nov 18 00:19:17 CET 2008

```## I got it. IV(s) of interaction should be orthogonal to main effect IV(s).
## Type III ANOVA / Interaction alone

x_interaction<-cbind(
(drug==2)&(disease==2)
,(drug==3)&(disease==2)
,(drug==4)&(disease==2)
,(drug==2)&(disease==3)
,(drug==3)&(disease==3)
,(drug==4)&(disease==3));
x_interaction <- residuals(lm(x_interaction~disease+drug));

## replace level order
x_interaction_2<-cbind(
(drug==2)&(disease==2)
,(drug==3)&(disease==2)
,(drug==1)&(disease==2)
,(drug==2)&(disease==1)
,(drug==3)&(disease==1)
,(drug==1)&(disease==1));
x_interaction_2 <- residuals(lm(x_interaction~disease+drug));

## Compare to http://www.otago.ac.nz/sas/stat/chap30/sect52.htm
## Type III SS: 2997.471860, 415.873046, 707.266259
rbind(
c(	anova(lm(y~x_interaction+disease),lm(y~disease*drug))\$'Sum of Sq'[2]
,	anova(lm(y~x_interaction+drug),lm(y~disease*drug))\$'Sum of Sq'[2]
,	anova(lm(y~disease+drug),lm(y~disease*drug))\$'Sum of Sq'[2])
,	c(	anova(lm(y~x_interaction_2+disease),lm(y~disease*drug))\$'Sum of Sq'[2]
,	anova(lm(y~x_interaction_2+drug),lm(y~disease*drug))\$'Sum of Sq'[2]
,	anova(lm(y~disease+drug),lm(y~disease*drug))\$'Sum of Sq'[2])
)

On Tue, Nov 18, 2008 at 3:27 AM, Xiaoxu LI <lixiaoxu at gmail.com> wrote:
> ## Question1: How to define IV with interaction alone, without main effects?
> ## Question2: Should Type III ANOVA in package car be independent of
> the factor level order?
>
> ## data from http://www.otago.ac.nz/sas/stat/chap30/sect52.htm
> drug <-  c(t(t(rep(1,3)))%*%t(1:4));
> disease <- c(t(t(1:3)) %*% t(rep(1,4)));
> y <- t(matrix(c(
>         42     ,44     ,36     ,13     ,19     ,22
>        ,33     ,NA     ,26     ,NA     ,33     ,21
>        ,31     ,-3     ,NA     ,25     ,25     ,24
>        ,28     ,NA     ,23     ,34     ,42     ,13
>        ,NA     ,34     ,33     ,31     ,NA     ,36
>        ,3      ,26     ,28     ,32     ,4      ,16
>        ,NA     ,NA     ,1      ,29     ,NA     ,19
>        ,NA     ,11     ,9      ,7      ,1      ,-6
>        ,21     ,1      ,NA     ,9      ,3      ,NA
>        ,24     ,NA     ,9      ,22     ,-2     ,15
>        ,27     ,12     ,12     ,-5     ,16     ,15
>        ,22     ,7      ,25     ,5      ,12     ,NA
>        ),nrow=6));
> ## verify data with http://www.otago.ac.nz/sas/stat/chap30/sect52.htm
> (cbind(drug,disease,y));
> ## make a big table
> drug <- as.factor(rep(drug,6));
> disease <- as.factor(rep(disease,6));
> y <- c(y);
> ## verify data through type I ANOVA to
> http://www.otago.ac.nz/sas/stat/chap30/sect52.htm
> anova(lm(y~drug*disease));
>
> require(car);
> ## Type III ANOVA in package car is not Type III ANOVA in SAS
> Anova(lm(y~drug*disease),type='III');
> ## Type III ANOVA in package car equates to wishful
> ## "anova(lm(y~INTERACTION +disease),lm(y~drug*disease))"
> ## However in R, df of lm(y~drug:disease) is automatically df of
> lm(y~drug*disease)
> ## How to define IV with interaction alone, without main effects?
>
> ## Verify type III of package car to be (INTERACTION + disease) vs.
> (disease*drug)
> ## However at the 3rd row, replace the maximun levels(drug==4,
> disease=3) with the minimum levels (==1).
> rbind(Anova(lm(y~drug*disease),type='III')\$'Sum Sq'[2:4]
> ,       c(
>                anova(
>                        lm(y~ 1
>                                +       (       I((drug==2)&(disease==2))
>                                        +       I((drug==3)&(disease==2))
>                                        +       I((drug==4)&(disease==2))
>                                        +       I((drug==2)&(disease==3))
>                                        +       I((drug==3)&(disease==3))
>                                        +       I((drug==4)&(disease==3))
>                                        )
>                                +       (       I(disease==2) + I(disease==3)))
>                        ,lm(y~drug*disease))\$'Sum of Sq'[2]
>        ,       anova(
>                        lm(y~ 1
>                                +       (       I((drug==2)&(disease==2))
>                                        +       I((drug==3)&(disease==2))
>                                        +       I((drug==4)&(disease==2))
>                                        +       I((drug==2)&(disease==3))
>                                        +       I((drug==3)&(disease==3))
>                                        +       I((drug==4)&(disease==3))
>                                        )
>                                +       (       I(drug==2) + I(drug==3) +  I(drug==4)))
>                        ,lm(y~drug*disease))\$'Sum of Sq'[2]
>        ,       anova(lm(y~drug+disease),lm(y~drug*disease))\$'Sum of Sq'[2]
>        )
> ,
>        c(
>                anova(
>                        lm(y~ 1
>                                +       (       I((drug==2)&(disease==2))
>                                        +       I((drug==3)&(disease==2))
>                                        +       I((drug==1)&(disease==2))
>                                        +       I((drug==2)&(disease==1))
>                                        +       I((drug==3)&(disease==1))
>                                        +       I((drug==1)&(disease==1))
>                                        )
>                                +       (       I(disease==2) + I(disease==1)))
>                        ,lm(y~drug*disease))\$'Sum of Sq'[2]
>        ,       anova(
>                        lm(y~ 1
>                                +       (       I((drug==2)&(disease==2))
>                                        +       I((drug==3)&(disease==2))
>                                        +       I((drug==1)&(disease==2))
>                                        +       I((drug==2)&(disease==1))
>                                        +       I((drug==3)&(disease==1))
>                                        +       I((drug==1)&(disease==1))
>                                        )
>                                +       (       I(drug==2) + I(drug==3) +  I(drug==1)))
>                        ,lm(y~drug*disease))\$'Sum of Sq'[2]
>        ,       anova(lm(y~drug+disease),lm(y~drug*disease))\$'Sum of Sq'[2]
>        )
> )
>
> ## I don't know whether the problem is in anova(lm1,lm2) or lm, or Anova
>
> ## while type II is independent of level order --
> rbind(Anova(lm(y~drug*disease),type='II')\$'Sum Sq'[1:3]
> ,       c(
>                anova(
>                        lm(y~ 1+disease)
>                        ,lm(y~ 1
>                                +       (       I(drug==2)
>                                        +       I(drug==3)
>                                        +       I(drug==4)
>                                        )
>                                +       (       I(disease==2) + I(disease==3)))
>                        ,lm(y~drug*disease))\$'Sum of Sq'[2]
>        ,       anova(
>                        lm(y~ 1+drug)
>                        ,lm(y~ 1
>                                +       (       I(drug==2)
>                                        +       I(drug==3)
>                                        +       I(drug==4)
>                                        )
>                                +       (       I(disease==2) + I(disease==3)))
>                        ,lm(y~drug*disease))\$'Sum of Sq'[2]
>
>        ,       anova(lm(y~drug+disease),lm(y~drug*disease))\$'Sum of Sq'[2]
>        )
> ,
>        c(
>                anova(
>                        lm(y~ 1+disease)
>                        ,lm(y~ 1
>                                +       (       I(drug==2)
>                                        +       I(drug==3)
>                                        +       I(drug==1)
>                                        )
>                                +       (       I(disease==2) + I(disease==1)))
>                        ,lm(y~drug*disease))\$'Sum of Sq'[2]
>        ,       anova(
>                        lm(y~ 1+drug)
>                        ,lm(y~ 1
>                                +       (       I(drug==2)
>                                        +       I(drug==3)
>                                        +       I(drug==1)
>                                        )
>                                +       (       I(disease==2) + I(disease==1)))
>                        ,lm(y~drug*disease))\$'Sum of Sq'[2]
>
>        ,       anova(lm(y~drug+disease),lm(y~drug*disease))\$'Sum of Sq'[2]
>        )
> )
>
> ## LI, Xiaoxu
> ## School of Humanities & Social Sciences, Shenzhen Graduate School of
> Peking Univ.
> ## http://lixiaoxu.lxxm.com/
>

```