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)?
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
______________________________________________
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.
