# Measurements necessary for repeated measures model

If I have an repeated measures model for a continuous parameter measured at n visits for each of the p participants in a study how big must p then be to perform the fit?

The repeated measures model will use the baseline value as a covariate, the visit as a factor, have an interaction between the baseline value and the visit and an unstructured (residual) covariance matrix.

I am particularly uncertain about the number of measurements required for the covariance matrix. I have been informed that $$p > n$$ is required for the covariance structure but, as it has $$N = n(n+1)/2$$ variance parameters, I would assume that $$p\cdot n \ge N$$ would be necessary (+ measurements to determine factor and covariant coefficients). Moreover I do not understand why $$p>n$$ would be required.

Any help with an explanation of the correct number would be appreciated.

If you additionally can offer an insight into the impact of missing data points in this context that would be even better.

Thank you.

• Welcome to Cross Validated! Although it might seem to make sense at first, it's not correct to evaluate change scores as an outcome while including the original value as a predictor in your model. See this page and this page, among others. Also, if you have no random effects then this isn't really a MMRM (mixed model repeated measures) but just a standard repeated measures model. Please edit the question to clarify; comments are easy to overlook and can be deleted.
– EdM
Apr 4 at 14:47
• @EdM: Thank you. I have adapted based on your suggestions. I agree that repeated measures model is more specific here and appreciate the insights on the change from baseline as an outcome. Apr 4 at 19:28

Chapter 7 of Frank Harrell's Regression Modeling Strategies covers this situation in much more detail than is possible in an answer here. A few suggestions follow, mostly drawn from that.

First, unless the number of visits n is very small, you will probably lose power by treating it as a fixed factor. That requires estimating n coefficients including the intercept (plus any interactions with other predictors). Modeling time continuously and flexibly, for example with a regression spline as described in Chapter 2 of the above reference, could require only estimating on the order of 3 coefficients (plus any interactions). That has the further advantage of allowing for different time intervals between visit numbers among individuals, very likely in practice.

Second, I'm not sure that an unstructured covariance matrix is a good or useful choice. The main interest in the covariance matrix is to correct adequately for within-subject correlations. Details of the matrix are typically of only secondary interest.

For data like this examined with generalized least squares and time modeled flexibly, one typically uses an "isotropic" correlation structure that depends only on the absolute difference in time between two observations. There are several possibilities, outlined in Section 7.5 of the above reference, which you can visualize with variograms (Section 7.6) and choose among as illustrated in the case study of Section 7.8.

Third, missing data should not be a big problem, at least if they are at some post-baseline time points for an individual. Generalized least squares (unlike classic repeated-measures ANOVA) can readily handle such missing data, if time is modeled continuously. An individual with missing time points will just provide less information than will others. If baseline values are missing, however, you should look into multiple imputation.

Fourth, the number of participants is best evaluated by a formal power analysis based on the nature of the outcome data (including its variability) and the magnitude of changes that you want to be able to detect. At a minimum to avoid overfitting with a continuous outcome, you need to have on the order of 15 observations per coefficient that you are estimating, but you might need a lot more to have a result that is useful in practice.

Finally, if there's a treatment involved then you certainly want an interaction of visit/time with that. It's not clear, however, what you are trying to accomplish with the interaction between the baseline value and visit/time.