Repeated Measurement Designs with Random Selections

Abstract

We consider repeated measurement designs with a single group as well as with multiple groups. Conventionally, the repeated measurements are made at fixed, often evenly spaced times. The first and second moment conditions needed for an exact F test are not satisfied in general, but with random permutation of the times, a probability measure results that does satisfy these moment conditions, unconditionally. As a consequence, unfortunately, we lose the asymptotic normality that is also essential to justify an F test. We introduce a class of designs and estimators for which both the moment conditions and asymptotic normality are satisfied. The times are the same for all subjects and they are chosen in such a way that they are mutually independent and identically distributed. It is important to understand that our conditions are satisfied only unconditionally--that is, if we do not condition by the randomly chosen times. There is a small probability that the randomly sampled times could be very unevenly allocated. Since we are not conditioning by the times, strictly speaking, this does not matter. It is possible to impose a mild condition of "spread," as we show, at small cost in terms of the P-value. Our method of design makes it possible to do regression analysis, interval estimation, and hypothesis testing on the mean response as a function of time. Because we have mutual independence and identical distributions of times we can form confidence bands.