A New View of Statistics

© 1997 Will G Hopkins

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Generalizing to a Population:

 Other Repeated-Measures Models
I deal with several here: an extra between-subject effect (example: male and female subjects), two or more within-subject factors (example: the same subjects get several treatments at several time points), a general type of within-subject model (you fit data to each subject separately, then combine the fits), inclusion of covariates in the model to analyze for individual responses, trends in repeated sets of trials, and mechanisms, and finally troublesome variables that might require transformation or more advanced approaches.

An Extra Between-Subject Effect
model: numeric <= (subject)  trial  group  trial*group
                                               sex  sex*trial  sex*group  sex*trial*group
or simply: numeric <= (subject)  trial | group | sex
example: jumphgt <= (athlete)  time | group | sex
I've used the first example from the previous page. The only difference is that we now know our subjects are a mixture of males and females. The analysis gets complicated, because sex could affect everything--that's why there are so many interaction terms in the model. But we're usually interested only in the extent to which the sexes differ for the effect of the treatment, so that means comparing the appropriate levels of sex*trial*group. In short, find the levels for trial*group that tell you what you want to know, then find the difference between those for the females and those for the males in the sex*trial*group term.

A word of warning! The term trial*group gives you the overall effect of the treatment between the experimental and control groups, but with sex in the model it's the average of the effect on the females and males. In other words, it's the expected effect of the treatment with equal numbers of males and females, even though your sample may have had unequal numbers.

The simple notation of a vertical bar ( | ) between effects is what the Statistical Analysis System uses to indicate that you want to include all possible main effects and interactions in the analysis. I usually leave them all in, because I subscribe to the idea that all independent variables have some effect on the outcome variable, however small. The only harm that can come from including all the interaction terms is loss of degrees of freedom, and therefore widening of the confidence intervals. On the other hand, if the effects of sex on the treatment are substantial, then inclusion of sex*trial*group will actually make the confidence intervals smaller, because it will eliminate the variability in the effect of the treatment that was due to sex.

In any case, if the effect of sex is substantial, you will want to know about it! For that reason, if the outcome of your study is as important for males as for females, you should try to have equal numbers of male and female subjects. Your confidence interval for the overall effect on females and males combined will be only a bit wider than if you had subjects of one sex, but of course the effect will be the average of the effect on (equal numbers of) females and males. The sex*trial*group interaction will tell you how different the males are from the females, although the confidence interval for the comparison of the effect on females and males will be about twice as wide as that for the overall effect. So, to properly delimit the difference between females and males, you will need four times as many subjects as for a single-sex study. That's the bad news. The good news is that you will end up with a wonderfully narrow confidence interval for the overall effect.

Two or More Within-Subject Effects
model: numeric <= (subject)  trial  condition  trial*condition
example: jumphgt <= (athlete)  time  test  time*test
Imagine that the intervention is a program aimed at training athletes to use visualization before a jump (and thereby jump higher). At weekly intervals the athletes perform a jump with and without visualization. The figure shows a possible outcome, in which visualization starts to work after a couple of weeks of training:

The model has the same form as in the previous example, but I've replaced group (representing separate measurements on different subjects) with condition (representing separate measurements on the same subjects). The interpretation of trial and condition in the model is the same as that for trial and group. Coaxing your stats program to deal with two or more within-subject effects will be a challenge!

In the example, I've renamed trial and condition to time and test to make things a bit clearer. The interpretation of time is obvious. Test represents the test of jump height, and it has two levels: visualization and no visualization. The interaction effect time*test tells us about the difference between the two time courses, and contrasts between the different levels of time and test for this effect tell us when the effect of visualization differs from no visualization. A polynomial contrast would show a substantial difference in linear and quadratic components, indicating that jumping with visualization shows a more rapid improvement in jump height initially (the linear effect), but that the gap is closing by Week 4 (the quadratic effect). You'll have to think really hard about this one.

An unsophisticated approach to these data would be to perform a series of paired t tests for each time point. For example, you might find that the difference between visualization and no visualization is significant at Weeks 2 and 3, but not at the other times. This approach does not take into account any difference between the tests at Week 0, so it's not valid. An acceptable fix is to subtract the jump height at Week 0 from that at each of the other weeks (for the two tests separately), then do a paired t test to compare visualization with no visualization at Weeks 1 through 4.

Within-Subject Modeling
This is a name I've devised for an approach that reduces or avoids the complexity of repeated-measures analyses. Basically, you derive a single measurement from the repeated measurements on each subject, then apply an appropriate simple analysis to the single measurement. The post-pre change score is the simplest example of a single measurement, and you would analyze the change scores with the unequal-variances version of the unpaired t statistic.

The approach works well for more complex derived measurements, too. For example, imagine you're looking at the effect of overtraining on recovery of heart rate following a standard bout of exercise. Let's say you record heart rate at half-minute intervals for three minutes. OK, that makes seven repeated measurements. Do you use repeated-measures ANOVA on these heart rates? Well, you could, I suppose. But what about if you want to fit an exponential decay curve on the heart rates, and extract the time constant. I defy anyone to deal with that within a repeated-measures model. It's only possible if you fit an exponential curve to each subject's data separately, extract a time constant for each subject, then use the time constant in your subsequent analyses. Hence the name within-subject modeling: you fit the same model to each subject and extract one or more parameters, which you then use for further analysis. r.

In fitting the model to each subject, you don't have to worry about distributions of residuals. Subsequent modeling with the parameters does need to be done properly, though. That modeling could be cross-sectional or longitudinal (repeated measures). For example, if the seven measurements of heart rate are taken on only one occasion for each subject, subsequent analysis of the parameter(s) describing the change in heart rate will be cross-sectional. But if the seven measurements are taken on several occasions, each subject provides several estimates of the parameter(s), so repeated-measures modeling will be necessary.

A real advantage of within-subject modeling is that most research students can do it without complex statistical analyses. Another advantage is that the analyses require fewer assumptions about the repeated-measures structure of the data, so the p values and confidence limits are more trustworthy. But mixed modeling, properly applied, is more powerful, especially when you want to include covariates in the analysis. See the slideshow on repeated measures for more examples and explanations of within-subject modeling.

Covariates in Repeated-Measures Analyses
By adding terms called covariates to the usual (fixed-effects) model, you can analyze for the following: the extent to which a subject characteristic accounts for individual responses to a treatment, the effect of the treatment on trends in repeated sets of trials, and the extent to which the effect of the treatment was due to changes in a putative mechanism variable. All is explained in the slideshow on repeated measures that I referred to on the first page on repeated measures. Click here to download the slideshow, which is an updated and extended version of an earlier slideshow on covariates in repeated measures.

Repeated-Measures Analysis of Troublesome Variables
In earlier pages on non-repeated-measures models, I showed how to deal with dependent variables that don't produce uniform normally distributed residuals. The same approaches apply to repeated-measures models. Thus, you will often need to log-transform or rank-transform a variable before analyzing it. When you rank-transform, make sure you do it to all the observations in one shot, not to each repeated measurement separately. The data will need to be in the form of one row per trial (as for mixed modeling), not one row per subject (as for ANOVA), for you to do the rank transformation correctly within an Excel spreadsheet.

An exact analysis of ordinal variables, such as those derived from Likert scales, requires repeated-measures logistic regression, but the analyses are difficult for newbies and the outcome statistics (odds ratios) are hard to interpret. For most variables, including even those with only two levels (yes or no, injured or not...), you can code each level of the variable as consecutive integers (0 and 1; 1, 2, 3, 4, and 5; and so on) and analyze it as if it was a well-behaved continuous normally distributed variable. Sure, the residuals are anything but normal, but as before, you can count on the central limit theorem to make the sampling distribution of the effect statistic normal, so the confidence limits or p values will be trustworthy. If responses for one or more groups are severely stacked up at one end of the scale, you will need a large sample size (possibly >20) for the central limit theorem to do its thing. I can't say exactly how many, but I hope to do some simulations to get an idea. The unequal-variances t test came through with flying colors for modest sample sizes (10-20) in my simulations with non-repeated-measures ordinal variables, and it will probably do equally well when applied to change scores derived from ordinal variables. We can't assume that mixed modeling will perform as well, because its method of estimation is different from that in the t test. Simulation will reveal all.

Nominal dependent variables can be analyzed by repeated-measures categorical modeling, if you want outcomes expressed as odds ratios. Otherwise treat each level of the nominal variable as a separate variable coded 0 or 1 (as I suggested under categorical modeling), then analyze each variable with conventional repeated-measures approaches. For example, you get schoolkids to tick one of four boxes representing the most important reason for playing sport. You collect the questionnaire, then show half of them a video aimed at convincing them that winning is (or isn't) everything. Finally you give them a fresh copy of the questionnaire to fill in. To what extent did the video change their attitude? You might even administer the questionnaire again a month later to see how the changes lasted. To do the analysis, treat each reason as a separate variable, code it 1 if the kid ticked it, or 0 if not, then use the unequal-variances t statistic to investigate differences in the changes between the group who saw the video and those who didn't. The magnitude of the outcome is the proportion of kids who changed their choice of the given reason.

Variables representing proportions or counts require root or arcsine-root transformation before you give them the usual repeated-measures analysis. The more exact approach is to use binomial or Poisson regression. Proc Genmod in SAS does it for repeated measures.

The next page deals specifically with the use of mixed modeling in the Statistical Analysis System.

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Last updated 8 June 03