Hello John and other R mailing list members.

I've been following your discussions regarding the Anova command for the SS
type 2/3 repeated measures Anova, and I have a question:

I found that when I go from using type II to using type III, the summary
model is suddenly added with an "intercept" term (example in the end of the
e-mail). So my question is
1) why is this "intercept" term added (in SS type "III" vs the type "II")?
2) Can/should this "intercept" term be removed ? (or how should it be
interpreted ?)

My purpose is to be able to use the Anova for analyzing an experiment with a
2 between and 3 within factors, where the between factors are not balanced,
and the within factors are (that is why I can't use the aov command).


#---code start
#---code start
#---code start
# (taken from the ?Anova help file)
phase <- factor(rep(c("pretest", "posttest", "followup"), c(5, 5, 5)),
    levels=c("pretest", "posttest", "followup"))
hour <- ordered(rep(1:5, 3))
idata <- data.frame(phase, hour)
idata
mod.ok <- lm(cbind(pre.1, pre.2, pre.3, pre.4, pre.5,
                     post.1, post.2, post.3, post.4, post.5,
                     fup.1, fup.2, fup.3, fup.4, fup.5) ~  treatment*gender,
                data=OBrienKaiser)

# now we have two options
# option one is to use type II:

(av.ok <- Anova(mod.ok, idata=idata, idesign=~phase*hour, type = "II"))

#output:
Type II Repeated Measures MANOVA Tests: Pillai test statistic
                            Df test stat approx F num Df den Df    Pr(>F)

treatment                    2    0.4809   4.6323      2     10 0.0376868 *

gender                       1    0.2036   2.5558      1     10 0.1409735

treatment:gender             2    0.3635   2.8555      2     10 0.1044692

phase                        1    0.8505  25.6053      2      9 0.0001930
***
treatment:phase              2    0.6852   2.6056      4     20 0.0667354 .

gender:phase                 1    0.0431   0.2029      2      9 0.8199968

treatment:gender:phase       2    0.3106   0.9193      4     20 0.4721498

hour                         1    0.9347  25.0401      4      7 0.0003043
***
treatment:hour               2    0.3014   0.3549      8     16 0.9295212

gender:hour                  1    0.2927   0.7243      4      7 0.6023742

treatment:gender:hour        2    0.5702   0.7976      8     16 0.6131884

phase:hour                   1    0.5496   0.4576      8      3 0.8324517

treatment:phase:hour         2    0.6637   0.2483     16      8 0.9914415

gender:phase:hour            1    0.6950   0.8547      8      3 0.6202076

treatment:gender:phase:hour  2    0.7928   0.3283     16      8 0.9723693

---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


# option two is to use type III, and then get an added intercept term:
 (av.ok <- Anova(mod.ok, idata=idata, idesign=~phase*hour, type = "III"))

# here is the output:
Type III Repeated Measures MANOVA Tests: Pillai test statistic
                            Df test stat approx F num Df den Df    Pr(>F)

(Intercept)                  1     0.967  296.389      1     10 9.241e-09
***
treatment                    2     0.441    3.940      2     10 0.0547069 .

gender                       1     0.268    3.659      1     10 0.0848003 .

treatment:gender             2     0.364    2.855      2     10 0.1044692

phase                        1     0.814   19.645      2      9 0.0005208
***
treatment:phase              2     0.696    2.670      4     20 0.0621085 .

gender:phase                 1     0.066    0.319      2      9 0.7349696

treatment:gender:phase       2     0.311    0.919      4     20 0.4721498

hour                         1     0.933   24.315      4      7 0.0003345
***
treatment:hour               2     0.316    0.376      8     16 0.9183275

gender:hour                  1     0.339    0.898      4      7 0.5129764

treatment:gender:hour        2     0.570    0.798      8     16 0.6131884

phase:hour                   1     0.560    0.478      8      3 0.8202673

treatment:phase:hour         2     0.662    0.248     16      8 0.9915531

gender:phase:hour            1     0.712    0.925      8      3 0.5894907

treatment:gender:phase:hour  2     0.793    0.328     16      8 0.9723693

---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1



#---code end
#---code end
#---code end


Thanks in advance for your help!
Tal Galili
















On Sun, Jan 25, 2009 at 3:08 AM, John Fox <jfox@mcmaster.ca> wrote:

> Dear Peter and Nils,
>
> In my initial message, I stated misleadingly that the contrast coding
> didn't
> matter for the "type-III" tests here since there is just one
> between-subjects factor, but that's not right: The between type-III SS is
> correct using contr.treatment(), but the within SS is not. As is generally
> the case, to get reasonable type-III tests (i.e., tests of reasonable
> hypotheses), it's necessary to have contrasts that are orthogonal in the
> row-basis of the design, such as contr.sum(),  contr.helmert(), or
> contr.poly(). The "type-II" tests, however, are insensitive to the contrast
> parametrization. Anova() always uses an orthogonal parametrization for the
> within-subjects design.
>
> The general advice in ?Anova is, "Be very careful in formulating the model
> for type-III tests, or the hypotheses tested will not make sense."
>
> Thanks, Peter, for pointing this out.
>
> John
>
> ------------------------------
> John Fox, Professor
> Department of Sociology
> McMaster University
> Hamilton, Ontario, Canada
> web: socserv.mcmaster.ca/jfox
>
>
> > -----Original Message-----
> > From: Peter Dalgaard [mailto:p.dalgaard@biostat.ku.dk]
> > Sent: January-24-09 6:31 PM
> > To: Nils Skotara
> > Cc: John Fox; r-help@r-project.org; 'Michael Friendly'
> > Subject: Re: [R] Anova and unbalanced designs
> >
> > Nils Skotara wrote:
> > > Dear John,
> > >
> > > thank you again! You replicated the type III result I got in SPSS! When
> I
> > > calculate Anova() type II:
> > >
> > > Univariate Type II Repeated-Measures ANOVA Assuming Sphericity
> > >
> > >                     SS num Df Error SS den Df      F  Pr(>F)
> > > between         4.8000      1   9.0000      8 4.2667 0.07273 .
> > > within          0.2000      1  10.6667      8 0.1500 0.70864
> > > between:within  2.1333      1  10.6667      8 1.6000 0.24150
> > > ---
> > > Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> > >
> > > I see the exact same values as you had written.
> > > However, and now I am really lost, type III (I did not change anything
> > else)
> > > leads to the following:
> > >
> > > Univariate Type III Repeated-Measures ANOVA Assuming Sphericity
> > >
> > >                               SS num Df Error SS den Df       F
> Pr(>F)
> > > (Intercept)               72.000      1    9.000      8 64.0000
> 4.367e-05
> > ***
> > > between                    4.800      1    9.000      8  4.2667
> 0.07273 .
> > > as.factor(within)          2.000      1   10.667      8  1.5000
> 0.25551
> > > between:as.factor(within)  2.133      1   10.667      8  1.6000
> 0.24150
> > > ---
> > > Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> > >
> > > How is this possible?
> >
> > This looks like a contrast parametrization issue: If we look at the
> > per-group mean within-differences and their SE, we get
> >
> >  > summary(lm(within1-within2~between - 1))
> > ..
> > Coefficients:
> >           Estimate Std. Error t value Pr(>|t|)
> > between1  -1.0000     0.8165  -1.225    0.256
> > between2   0.3333     0.6667   0.500    0.631
> > ..
> >  > table(between)
> > between
> > 1 2
> > 4 6
> >
> > Now, the type II F test is based on weighting the two means as you would
> > after testing for no interaction
> >
> >  > (4*-1+6*.3333)^2/(4^2*0.8165^2+6^2*0.6667^2)
> > [1] 0.1500205
> >
> > and type III is to weight them as if there had been equal counts
> >
> >  > (5*-1+5*.3333)^2/(5^2*0.8165^2+5^2*0.6667^2)
> > [1] 0.400022
> >
> > However, the result above corresponds to looking at group1 only
> >
> >  > (-1)^2/(0.8165^2)
> > [1] 1.499987
> >
> > It helps if you choose orhtogonal contrast parametrizations:
> >
> >  > options(contrasts=c("contr.sum","contr.helmert"))
> >  > betweenanova <- lm(values ~ between)> Anova(betweenanova, idata=with,
> > idesign= ~as.factor(within), type = "III" )
> >
> > Type III Repeated Measures MANOVA Tests: Pillai test statistic
> >                            Df test stat approx F num Df den Df    Pr(>F)
> > (Intercept)                1     0.963  209.067      1      8 5.121e-07
> ***
> > between                    1     0.348    4.267      1      8   0.07273 .
> > as.factor(within)          1     0.048    0.400      1      8   0.54474
> > between:as.factor(within)  1     0.167    1.600      1      8   0.24150
> > ---
> > Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> >
> >
> >
> >
> > --
> >     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@biostat.ku.dk)              FAX: (+45) 35327907
>
> ______________________________________________
> R-help@r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>



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