Dear Henrik,
On Mon, 23 Jul 2012 00:56:16 +0200
Henrik Singmann <henrik.singm...@psychologie.uni-freiburg.de> wrote:
> Dear John,
>
> indeed, you are very right. Including the covariate as is, doesn't make any
> sense. The only correct way would be to center it on the mean beforehands. So
> actually the examples in my first and second mail are bogus (I add a
> corrected example at the end) and the reported test do not make much sense.
>
> Let me try to explain why I want to discard the interactions of the covariate
> with the within-factors. The reason I want to exclude them is that I want to
> stay within the ANCOVA framework. I looked at the three books on experimental
> design I have on my desk (Winer, 1971; Kirk, 1982; Maxwell & Delaney, 2003)
> and they unanimously define the ANCOVA as the ANOVA on the responses
> controlled for the covariate only (i.e., not controlled for the covariate and
> the interactions with the other effects).
> However, as you say, adding or removing an interaction with the orthogonal
> within-subject factors does indeed not alter the results (example at the
> end), so one could just use the output and discard the unwanted effects,
> although admittedly this seems sketchy given significant effects.
>
> Unfortunately, my involvement with this issue has led me to another question.
> Winer and Kirk both discuss a split-plot ANCOVA in which one has measured a
> covariate for each observation. That is a second matrix alike the original
> data matrix, e.g. the body temperature of each person at each measurement for
> the OBrienKaiser dataset:
>
> OBK.cov <- OBrienKaiser
> OBK.cov[,-(1:2)] <- runif(16*15, 36, 41)
>
> Would it be possible to fit the data using this temperature matrix as a
> covariate using car::Anova (I thought about this but couldn't find any idea
> of how to specify the imatrix)?
I'm afraid that Anova() won't handle repeated measures on covariates. I agree
that it would be desirable to do so, and this capability is on my list of
features to add to Anova(), but I can't promise when, or if, I'll get to it.
Sorry,
John
>
> Thanks a lot for the helpful responses,
> Henrik
>
>
> PS: Better examples:
> # compare the treatment and the phase effect across models.
> require(car)
> set.seed(1)
>
> # using scale for the covariate:
> n.OBrienKaiser <- within(OBrienKaiser, age <- scale(sample(18:35, size = 16,
> replace = TRUE), scale = FALSE))
>
> 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)
>
> # Full ANCOVA model:
> mod.1 <- 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 + age,
> data=n.OBrienKaiser)
> (av.1 <- Anova(mod.1, idata=idata, idesign=~phase*hour, type = 3))
>
> # Df test stat approx F num Df den Df Pr(>F)
> # (Intercept) 1 0.968 269.4 1 9 0.000000052
> ***
> # treatment 2 0.443 3.6 2 9 0.0719
> .
> # gender 1 0.305 3.9 1 9 0.0782
> .
> # age 1 0.054 0.5 1 9 0.4902
> # treatment:gender 2 0.222 1.3 2 9 0.3232
> # phase 1 0.811 17.2 2 8 0.0013
> **
> # ...
>
> # removing the between-subject interaction does alter the lower order effects:
> mod.2 <- 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 + age,
> data=n.OBrienKaiser)
> (av.2 <- Anova(mod.2, idata=idata, idesign=~phase*hour, type = 3))
>
> # Type III Repeated Measures MANOVA Tests: Pillai test statistic
> # Df test stat approx F num Df den Df Pr(>F)
> # (Intercept) 1 0.959 254.5 1 11 0.0000000059 ***
> # treatment 2 0.428 4.1 2 11 0.04644 *
> # gender 1 0.271 4.1 1 11 0.06832 .
> # age 1 0.226 3.2 1 11 0.10030
> # phase 1 0.792 19.0 2 10 0.00039 ***
> # ...
>
> # removing the within-subject interaction does NOT alter the lower order
> effects:
> mod.3 <- 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 + age,
> data=n.OBrienKaiser)
> (av.3 <- Anova(mod.3, idata=idata, idesign=~phase+hour, type = 3))
> # Type III Repeated Measures MANOVA Tests: Pillai test statistic
> # Df test stat approx F num Df den Df Pr(>F)
> # (Intercept) 1 0.968 269.4 1 9 0.000000052 ***
> # treatment 2 0.443 3.6 2 9 0.0719 .
> # gender 1 0.305 3.9 1 9 0.0782 .
> # age 1 0.054 0.5 1 9 0.4902
> # treatment:gender 2 0.222 1.3 2 9 0.3232
> # phase 1 0.811 17.2 2 8 0.0013 **
> # ...
>
>
>
> Am 22.07.2012 23:25, schrieb John Fox:
> > Dear Henrik,
> >
> > The within-subjects contrasts are constructed by Anova() to be orthogonal
> > in the row-basis of the design, so you should be able to safely ignore the
> > effects in which (for some reason that escapes me) you are uninterested.
> > This would also be true (except for the estimated error) for the
> > between-subjects design if you used "type-II" tests. It's true that the
> > "type-III" between-subjects tests will be affected by the presence of an
> > interaction, but for these tests to make sense at all, you have to
> > formulate the model very carefully. For example, your type-III test for the
> > "main effect" of treatment with the interaction in the model is for the
> > treatment effect at age 0. Does that really make sense to you? Indeed, the
> > type-III tests for the ANOVA (not ANCOVA) model only make sense because I
> > was careful to use contrasts for the between-subjects factors that are
> > orthogonal in the basis of the design:
> >
> > > contrasts(OBrienKaiser$treatment)
> > [,1] [,2]
> > control -2 0
> > A 1 -1
> > B 1 1
> >
> >> contrasts(OBrienKaiser$gender)
> > [,1]
> > F 1
> > M -1
> >
> > Best,
> > John
> >
> > On Sun, 22 Jul 2012 22:06:58 +0200
> > Henrik Singmann <henrik.singm...@psychologie.uni-freiburg.de> wrote:
> >> Dear John,
> >>
> >> thanks for your response. But if I simply ignore the unwanted effects, the
> >> estimates of the main effects for the within-subjects factors are
> >> distroted (rationale see below). Or doesn't this hold for between-within
> >> interactions?
> >>
> >> Or put another way: Do you think this approach is the correct way of
> >> running an ANCOVA involving within-subject factors?
> >>
> >> As far as I understand ANCOVA, the covariate(s) should only be additive
> >> factors and do not interact with the factors of interest:
> >> "Suppose that differences in [the mean of the covariate] are due to
> >> sources of variation related to [the mean of the dependent variable], but
> >> not directly related to the treatment effects." (Winer, 1972, p. 753, the
> >> parts in squared bracktes exchange the mathematical symbols with the
> >> definition).
> >>
> >> Best,
> >> Henrik
> >>
> >> PS: Showing that adding the interaction term massively changes the main
> >> effect for a between-factor:
> >>
> >>> # The ANCOVA:
> >>> Anova(lm(pre.1 ~ treatment + age, data = n.OBrienKaiser), type = 3)
> >> Anova Table (Type III tests)
> >>
> >> Response: pre.1
> >> Sum Sq Df F value Pr(>F)
> >> (Intercept) 0.0 1 0.01 0.90
> >> treatment 0.3 2 0.06 0.94
> >> age 4.5 1 1.54 0.24
> >> Residuals 34.9 12
> >>>
> >>> # The ANOVA:
> >>> Anova(lm(pre.1 ~ treatment, data = n.OBrienKaiser), type = 3)
> >> Anova Table (Type III tests)
> >>
> >> Response: pre.1
> >> Sum Sq Df F value Pr(>F)
> >> (Intercept) 225.6 1 74.47 0.00000097 ***
> >> treatment 1.1 2 0.17 0.84
> >> Residuals 39.4 13
> >> ---
> >> Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
> >>>
> >>> # The model with interaction
> >>> Anova(lm(pre.1 ~ treatment * age, data = n.OBrienKaiser), type = 3)
> >> Anova Table (Type III tests)
> >>
> >> Response: pre.1
> >> Sum Sq Df F value Pr(>F)
> >> (Intercept) 3.01 1 1.40 0.264
> >> treatment 13.71 2 3.18 0.085 .
> >> age 11.56 1 5.37 0.043 *
> >> treatment:age 13.37 2 3.11 0.089 .
> >> Residuals 21.53 10
> >> ---
> >> Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
> >>>
> >>
> >>
> >> Am 22.07.2012 16:59, schrieb John Fox:
> >>> Dear Henrik,
> >>>
> >>> As you discovered, entering the covariate age additively into the
> >>> between-subject model doesn't prevent Anova() from reporting tests for
> >>> the interactions between age and the within-subjects factors. I'm not
> >>> sure why you would want to do so, but you could simply ignore these tests.
> >>>
> >>> I hope this helps,
> >>> John
> >>>
> >>> --------------------------------
> >>> John Fox
> >>> Senator William McMaster
> >>> Professor of Social Statistics
> >>> Department of Sociology
> >>> McMaster University
> >>> Hamilton, Ontario, Canada
> >>> http://socserv.mcmaster.ca/jfox
> >>>
> >>>
> >>>
> >>>> -----Original Message-----
> >>>> From: r-help-boun...@r-project.org [mailto:r-help-bounces@r-
> >>>> project.org] On Behalf Of Henrik Singmann
> >>>> Sent: July-21-12 1:29 PM
> >>>> To: r-h...@stat.math.ethz.ch
> >>>> Subject: [R] car::Anova - Can it be used for ANCOVA with repeated-
> >>>> measures factors.
> >>>>
> >>>> Dear list,
> >>>>
> >>>> I would like to run an ANCOVA using car::Anova with repeated measures
> >>>> factors, but I can't figure out how to do it. My (between-subjects)
> >>>> covariate always interacts with my within-subject factors.
> >>>> As far as I understand ANCOVA, covariates usually do not interact with
> >>>> the effects of interest but are simply additive (or am I wrong here?).
> >>>>
> >>>> More specifically, I can add a covariate as a factor to the between-
> >>>> subjects part when fitting the MLM that behaves like expected (i.e.,
> >>>> does not interact with the other factors), but when calling Anova on
> >>>> the model, I don't know how I can specify the between-within design
> >>>> (i.e., which parts of the model should interact with the repeated
> >>>> measures factors).
> >>>>
> >>>> As far as I understand it, neither the idesign, icontrasts or imatrix
> >>>> arguments, nor the linearHypothesis function can specify the within-
> >>>> between design (as far as I get it they all specify the within or
> >>>> intra-subject design, see John Fox's slides from User 2011:
> >>>> http://web.warwick.ac.uk/statsdept/useR-
> >>>> 2011/TalkSlides/Contributed/17Aug_1705_FocusV_4-Multivariate_1-
> >>>> Fox.pdf).
> >>>>
> >>>> If this it is not possible using car::Anova, is there another way to
> >>>> achiebve what I want or is it plainly wrong?
> >>>> I have the feeling that using R's "New Functions for Multivariate
> >>>> Analysis" (Dalgaard, 2007, R News) this could be possible, but some
> >>>> advice on how, would be greatly appreciated, as this does not seem to
> >>>> be the most straight forward way.
> >>>>
> >>>> Below is an example using the car::OBrienKaiser dataset adding an age
> >>>> covariate. The example is merely an adoption from ?Anova with miniml
> >>>> changes and includes e.g. age:phase:hour which I don't want to have.
> >>>>
> >>>> Note that I posted this question to stackoverflow two days ago
> >>>> (http://stackoverflow.com/q/11567446/289572) and did not receive any
> >>>> responses. Please excuse my "crossposting", but I think R-help may be
> >>>> the better place.
> >>>>
> >>>> Best,
> >>>> Henrik
> >>>>
> >>>> PS: I know that the posting guide says "No questions about contributed
> >>>> packages" but there are some questions about car on R-help, so I
> >>>> thought this would be the correct place.
> >>>>
> >>>> ###### Example follows #####
> >>>>
> >>>> require(car)
> >>>> set.seed(1)
> >>>>
> >>>> n.OBrienKaiser <- within(OBrienKaiser, age <- sample(18:35, size = 16,
> >>>> replace = TRUE))
> >>>>
> >>>> 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)
> >>>>
> >>>> 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 +
> >>>> age, data=n.OBrienKaiser) (av.ok <- Anova(mod.ok, idata=idata,
> >>>> idesign=~phase*hour, type = 3))
> >>>>
> >>>> # Type II Repeated Measures MANOVA Tests: Pillai test statistic
> >>>> # Df test stat approx F num Df den Df
> >>>> Pr(>F)
> >>>> # (Intercept) 1 0.971 299.9 1 9
> >>>> 0.000000032 ***
> >>>> # treatment 2 0.492 4.4 2 9
> >>>> 0.04726 *
> >>>> # gender 1 0.193 2.1 1 9
> >>>> 0.17700
> >>>> # age 1 0.045 0.4 1 9
> >>>> 0.53351
> >>>> # treatment:gender 2 0.389 2.9 2 9
> >>>> 0.10867
> >>>> # phase 1 0.855 23.6 2 8
> >>>> 0.00044 ***
> >>>> # treatment:phase 2 0.696 2.4 4 18
> >>>> 0.08823 .
> >>>> # gender:phase 1 0.079 0.3 2 8
> >>>> 0.71944
> >>>> # age:phase 1 0.140 0.7 2 8
> >>>> 0.54603
> >>>> # treatment:gender:phase 2 0.305 0.8 4 18
> >>>> 0.53450
> >>>> # hour 1 0.939 23.3 4 6
> >>>> 0.00085 ***
> >>>> # treatment:hour 2 0.346 0.4 8 14
> >>>> 0.92192
> >>>> # gender:hour 1 0.286 0.6 4 6
> >>>> 0.67579
> >>>> # age:hour 1 0.262 0.5 4 6
> >>>> 0.71800
> >>>> # treatment:gender:hour 2 0.539 0.6 8 14
> >>>> 0.72919
> >>>> # phase:hour 1 0.663 0.5 8 2
> >>>> 0.80707
> >>>> # treatment:phase:hour 2 0.893 0.3 16 6
> >>>> 0.97400
> >>>> # gender:phase:hour 1 0.700 0.6 8 2
> >>>> 0.76021
> >>>> # age:phase:hour 1 0.813 1.1 8 2
> >>>> 0.56210
> >>>> # treatment:gender:phase:hour 2 1.003 0.4 16 6
> >>>> 0.94434
> >>>> # ---
> >>>> # Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
> >>>>
> >>>>
> >>>> --
> >>>> Dipl. Psych. Henrik Singmann
> >>>> PhD Student
> >>>> Albert-Ludwigs-Universität Freiburg
> >>>> http://www.psychologie.uni-freiburg.de/Members/singmann
> >>>>
> >>>> ______________________________________________
> >>>> 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.
> >>>
> >>> ______________________________________________
> >>> 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.
> >>>
> >>
> >> --
> >> Dipl. Psych. Henrik Singmann
> >> PhD Student
> >> Albert-Ludwigs-Universität Freiburg
> >> http://www.psychologie.uni-freiburg.de/Members/singmann
> >>
> >> ______________________________________________
> >> 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.
> >
>
> --
> Dipl. Psych. Henrik Singmann
> PhD Student
> Albert-Ludwigs-Universität Freiburg, Germany
> http://www.psychologie.uni-freiburg.de/Members/singmann
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