Restricted likelihood ratio tests in nonparametric longitudinal models

We assume that repeated measurements are taken on each of several subjects that are randomly sampled from some population. The observations on a particular subject are expressed as the sum of an average curve for the population and a deviation of the subject's curve from the average plus independent errors. Both curves are modeled nonparametrically as splines. We use roughness penalties on the splines, which is equivalent to assuming a linear mixed model. Within this linear mixed model, we consider likelihood ratio tests of several scientifically relevant hypotheses about the two curves, for example, that the subject deviations are all zero or that they are each constant. The large-sample null distributions of the test statistics are shown to be non-standard, but we develop bootstrap techniques that can compute the exact null distributions much more rapidly than a direct application of the bootstrap.