A New View of Statistics  
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REPEATED MEASURES MODELS
So far, all the models we have looked have been
for data from crosssectional or descriptive studies. These are
studies in which each person is observed only once, so for each variable you have
only one value per person. To put it another way, each row in the data set is
for a different subject.
Now, what about longitudinal studies, in which people are observed more than once? In particular, what about interventions or experiments, where you compare values of a dependent variable before and after you try something like a training program or a potentially active drug? You can analyze data from these studies with the procedures used for crosssectional data only if you can assume that the residuals are uniformhave the same standard deviationfor each of the repeated measurements. But in general, you can't assume such uniformity: subjects will show more variation on some repeated measurements than on others, usually because of differences between measurements in the effects of time or the treatment. So you have to use repeatedmeasures models.
We'll start on this page with the simple case of only two trials for only one group of subjects (no betweensubject effect). On the next page I'll extend it to several groups (a betweensubjects effect, e.g. an experimental and control group). Then I'll deal with more than two trials, first without a betweensubjects effect, then with a betweensubjects effect, before I deal with other repeatedmeasures models including the simple, robust approach of withinsubject modeling. Then there is a page on how to use the mixed procedure in the Statistical Analysis System, with links to . Finally, I devote a page to a problem that can arise in repeatedmeasures analyses, regression to the mean. But first, some other resources I have created since writing these pages: a slideshow, a standalone article, and some spreadsheets.
Slideshow on Repeated Measures
For a Powerpoint slideshow (340 kB) dealing
with most aspects of repeatedmeasures analyses, click
here. I presented this talk at the 2003 annual meeting of the American College
of Sports Medicine. The sections are Basics (analysis by ANOVA, withinsubject
modeling, and mixed modeling; fixed and random effects; individual responses
and asphericity), Accounting for Individual Responses, Analyzing for Patterns
of Responses, and Analyzing for Mechanisms. The information in the slide show
complements the information on these pages. Read both.
Articles and Spreadsheets for Straightforward
Repeated Measures
I have created spreadsheets for analysis
of repeatedmeasures data from controlled trials and crossovers. You add the
raw observations, the spreadsheet does the rest. I have also written articles
at the Sportscience site explaining important issues in such analyses, and how
the spreadsheets address them. (Links to the Sportscience articles will not
work if you are using a copy of these pages offline.)
Click to view the 2006 article, which explains the use of a covariate and has links to earlier articles. See also an article on the different kinds of controlled trials in the 2005 issue, which explains the names I have used below for the spreadsheets.
Click to download the spreadsheet
for prepost parallelgroups trials, the spreadsheet
for postonly crossovers (which also works for the paired ttest model on
this page), and the spreadsheet
for prepost crossovers. The following links will download earlier version
that do not include the covariate and other enhancements: spreadsheet
for controlled trials, spreadsheet
for crossovers, and fully controlled
crossovers.

Paired T Test or RepeatedMeasures ANOVA with two trials and no betweensubjects effect 
Don't try to understand the model yet. Just look at the example in the figure, which shows individual values on the left and means and standard deviations on the right. There is one measurement on each of eight athletes before (pre) and after (post) a training program aimed at increasing jump height, with no control group. This sort of design is sometimes described as one in which the subjects "act as their own controls", although this description fits any longitudinal study, whether or not there is a control group.
The results can be displayed as shown in the lefthand panel, with pre and post heights linked for each subject. The righthand panel shows the more usual way of connecting the means by a line. By the way, it's wrong to use a bar graph, because the pre and post data are from the same subjects.
It doesn't look anything like it, but this model is actually a twoway ANOVA. If I'd drawn bars instead of points for the pre and post heights, you might have seen that it is at least a oneway ANOVA, time being the nominal effect (with two levels, pre and post), and height the dependent numeric variable. So let's get started with jumphgt <= time.
The other effect is hidden in the righthand figure, but it's
clear in the lefthand side: the identity of the subjects. We
introduce this variable as a way to link each subject's measurement
of height at the pre and post times. Hence the full model:
jumphgt <= (athlete) time. In the general
model, one term in the ANOVA is the identity of the subjects, and the
other term is the identity of the time points or trials.
Hang on. Why (athlete) rather than athlete? Well, the variable representing
the identity of the subjects is a bit different from all the other variables
we've met so far. The subjects are usually a random sample of a population,
so this variable is known as a random effect. If we repeated the study,
we could have a different sample of subjects, each with different values drawn
randomly from the population. In contrast, the identity of the time points is
a fixed effect, because this variable would have the same values and
levels (pre and post) in any repeat of the study. Look back at the nominal variables
in the other models we've dealt with and you'll see that they are all fixed
effects. For example, sex always has values male or female in every sample,
and we assume the effect of maleness or femaleness is the same for every male
or female. For more information on fixed and random effects, see the slideshow
on repeated measures. If you want to work with mixed models, make sure you
get familiar with my "hats" metaphor for random effects, as explained
in the slideshow.
So, I've put parentheses around the subject term to indicate that it's a random effect, and to let you know that stats programs don't normally include the subject term in the model in the way that I have here. If I left the parentheses out, I would imply that the subject term is a fixed effect. It is possible to analyze your data as a straightforward nonrepeatedmeasures ANOVA with the subject term as a fixed effect, but the results you get are appropriate only for repeatedmeasures data that have uniformity of residuals. I deal with that later under the heading sphericity or covariance structure.
We don't have the interaction term athlete*time in the model, partly because athlete is a random effect, and partly because we would need multiple measurements for subjects at the pre and post time points for the interaction term to make any sense. Let's leave aside this complexity.
It all sounds awfully complicated, but in practice it's straightforward. You have two lots of measurements performed on the same subjects, and all you want to know is how the means have changed. Most stats programs can do that for you without you having to worry about models like the above. All you do is click up a paired t test, which produces a p value for the difference in the means, and hopefully a confidence interval. The paired t test has the same internal workings as the unpaired t test, which is why they share the same name.
On the next page we'll add a control group. After all, the athletes might jump higher in the post test simply because they have learned how to do the test, not because they responded to your training program. A group that does everything the same as the experimental group, other than the training program, "controls" for this and other problems. But the main reason I'm talking about a control group now is to explain the terminology in the heading for this page. Having a control group in a repeatedmeasures design is an example of a betweensubjects effect, because there are different subjects in the control and experimental groups. Hence no betweensubjects effect in the title of this section. Time or trial is a withinsubjects effect, because the same subjects experience the different levels of that effect.