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Typically, when one encounters continuous but skewed outcome measures in a longitudinal design (say, with one between-subjects effect) the common approach is to transform the outcome to normality. If the situation is extreme, such as with truncated observations, one might get fancy and use a Tobit growth curve model, or some such.

But I am at a loss when I see outcomes that are normally distributed at certain time points and then heavily skewed at others; transformation may plug one leak but spring another. What might you suggest in such a case? Are there "non-parametric" versions of mixed effects models that I am not aware of?

Note: an applied example would be knowledge test scores pre/post a series of educational interventions. Scores begin normal but then cluster at the high end of the scale later on.

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The example is interesting because it occurs all the time. There are well-known transformations to deal with it, such as Tukey's "folded" power transformations. These make little change in the middle of the scale yet cure skewness at both ends. I have found that folded roots and logs work very well for standardized pre/post test comparisons. – whuber Nov 30 '11 at 23:16
Thank you, Whuber. I will look into the folded transformation approach. – Brenden Dufault Dec 1 '11 at 16:01
For a definition and examples, Brenden, see For instruction on their use, see the last few chapters in Tukey's book EDA. – whuber Dec 1 '11 at 16:06
An added note - remember that assumptions are made about the residuals of the model, not the actual variables involved. – Peter Flom Feb 19 '12 at 22:44

Assuming that the problem occurs in your residuals (as the distribution of the outcome variable itself is usually not a problem), I would be looking to investigate the cause of the problem rather than trying to "fix" it via a transformation or application of a nonparametric model.

If it is the case that there seems to be a trend (e.g., progressively getting more or less normal), or, a clear break between when it goes from normal to not normal, then it suggests a "regime change" of some kind in your data (i.e., the data generating mechanism is changing over time) or some type of missing variable problem.

If it is the case that there is no obvious pattern (e.g., time periods 1 and 3 look normal and time periods 2 and 4 do not) I'd be looking very carefully for a data integrity problem.

A simple way of checking to see if you do have a regime change is to estimate model using only the "normal" time periods and then re-estimate using the other time periods and see what difference occurs. A more complicated approach is to use a latent class model, perhaps with time as a concomitant variable.

As regards your question about nonparametric mixed effects models it kind of depends upon what you mean by nonparametric. If you mean models that do not assume a numeric dependent variable then there are lots of such models (e.g., LIMDEP has quite a few). Also, keep in mind that the violation of the normality assumption will probably only be problematic from an inference perspective if your sample size is small. One way of investigating this would be to try the various transformations discussed in other comments and answers and see if it makes much of an impact on your conclusions.

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+1 Thank you, Tim. I appreciate your suggestions regarding latent class models and LIMDEP. These approaches are growing in appeal to me as I start to learn more about them. – Brenden Dufault Feb 28 '13 at 21:44

There are the Box-Cox transformations which raise the variable to a power lambda where lambda is included in the model parameter estimation. I am not familiar with Tukey's folded power transformation, so i don't know if we are talking about the same thing. In orde to estimate lambda you need multiple points in the fit. Do you want to fit a different distribution at each time point where the distribution is defined on a set of subjects taking the test at each time point? Even if that is the case if you know that some time points should have the same distribution you might want to combine them in a single fit.

Another approach which is nonparametric and does not involve transformations to normality would be to apply the bootstrap at each time point or at each combined set of time points.

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