  In many psychometric applications, the relationship between the mean of an
outcome and a quantitative covariate is too complex to be described by simple
parametric functions; instead, flexible nonlinear relationships can be
incorporated using penalized splines. Penalized splines can be conveniently
represented as a linear mixed effects model (LMM), where the coefficients of
the spline basis functions are random effects. The LMM representation of
penalized splines makes the extension to multivariate outcomes relatively
straightforward. In the LMM, no effect of the quantitative covariate on the
outcome corresponds to the null hypothesis that a fixed effect and a variance
component are both zero. Under the null, the usual asymptotic chi-square
distribution of the likelihood ratio test for the variance component does not
hold. Therefore, we propose three permutation tests for the likelihood ratio
test statistic: one based on permuting the quantitative covariate, the other
two based on permuting residuals. We compare via simulation the Type I error
rate and power of the three permutation tests obtained from joint models for
multiple outcomes, as well as a commonly used parametric test. The tests are
illustrated using data from a stimulant use disorder psychosocial clinical
trial.

Methodology;Applications