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 <j...@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.dalga...@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.dalga...@biostat.ku.dk) FAX: (+45) 35327907
>
> ______________________________________________
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--
----------------------------------------------
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Tal Galili
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