[R] repeated measures ANOVA

[John Maindonald]

There seem to several issues here:
1) In the analysis that has a (1|Subject) error term, there is a large
negative correlation between the parameter estimates for time and
time:group. Overall, the effect of time is significant, as can be seen
from

time.lme <- lme ( p.pa ~ time * group, random = ~ 1 | subject,  
method="ML")
 > notime.lme <- lme ( p.pa ~ group, random = ~ 1 | subject,  
method="ML")
 > anova(time.lme, notime.lme)
            Model df   AIC   BIC logLik   Test L.Ratio p-value
time.lme       1  6 245.0 253.4 -116.5
notime.lme     2  4 254.0 259.6 -123.0 1 vs 2   12.95  0.0015

What is uncertain is how this time effect should be divided up, between
a main effect of slope and the interaction.

2) What the interaction plot makes clear, and what the change in  
treatment
(for group 1 only?) for time point 3 should have suggested is that  
the above
analysis is not really appropriate.  There are two comparisons:
(i) at time points 1 and 2; and (ii) at time point 3.

(3) The above does not allow for a random group to group change in
slope, additional to the change that can be expected from random
variation about the line.  Models 3 and 4 in your account do this, and
allow also for a group:subject and group:time random effects that make
matters more complicated still.  The fitting of such a model has the
consequence that between group differences in slope are entirely
explained by this random effect.  Contrary to what the lmer() output
might suggest, no degrees of freedom are left with which to estimate
the time:group interaction.
(Or you can estimate the interaction, and no degrees of freedom are
left for either the time or time:group random effect).  All you can talk
about is the average and the difference of the time effects for these
two specific groups.

Thus, following on from (3), I do not understand how lmer() is able to
calculate a t-statistic. There seems to me to be double dipping.
Certainly, I noted a convergence problem.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Mathematical Sciences Institute, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.


On 28 Feb 2006, at 10:00 PM, Christian Gold wrote:

> From: Christian Gold <c.gold at magnet.at>
> Date: 28 February 2006 2:15:04 AM
> To: r-help at stat.math.ethz.ch
> Subject: [R] repeated measures ANOVA
>
>
> Dear list members:
>
> I have the following data:
> group <- rep(rep(1:2, c(5,5)), 3)
> time <- rep(1:3, rep(10,3))
> subject <- rep(1:10, 3)
> p.pa <- c(92, 44, 49, 52, 41, 34, 32, 65, 47, 58, 94, 82, 48, 60, 47,
> 46, 41, 73, 60, 69, 95, 53, 44, 66, 62, 46, 53, 73, 84, 79)
> P.PA <- data.frame(subject, group, time, p.pa)
>
> The ten subjects were randomly assigned to one of two groups and
> measured three times. (The treatment changes after the second time
> point.)
>
> Now I am trying to find out the most adequate way for an analysis of
> main effects and interaction. Most social scientists would call this
> analysis a repeated measures ANOVA, but I understand that mixed- 
> effects
> model is a more generic term for the same analysis. I did the analysis
> in four ways (one in SPSS, three in R):
>
> 1. In SPSS I used "general linear model, repeated measures",  
> defining a
> "within-subject factor" for the three different time points. (The data
> frame is structured differently in SPSS so that there is one line for
> each subject, and each time point is a separate variable.)
> Time was significant.
>
> 2. Analogous to what is recommended in the first chapter of Pinheiro &
> Bates' "Mixed-Effects Models" book, I used
> library(nlme)
> summary(lme ( p.pa ~ time * group, random = ~ 1 | subject))
> Here, time was NOT significant. This was surprising not only in
> comparison with the result in SPSS, but also when looking at the  
> graph:
> interaction.plot(time, group, p.pa)
>
> 3. I then tried a code for the lme4 package, as described by Douglas
> Bates in RNews 5(1), 2005 (p. 27-30). The result was the same as in 2.
> library(lme4)
> summary(lmer ( p.pa ~ time * group + (time*group | subject), P.PA ))
>
> 4. The I also tried what Jonathan Baron suggests in his "Notes on the
> use of R for psychology experiments and questionnaires" (on CRAN):
> summary( aov ( p.pa ~ time * group + Error(subject/(time * group)) ) )
> This gives me yet another result.
>
> So I am confused. Which one should I use?
>
> Thanks
>
> Christian