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.

翻译：在许多心理测量学应用中，结局均值与定量协变量之间的关系过于复杂，难以用简单参数函数描述；此时，可使用惩罚样条纳入灵活的非线性关系。惩罚样条可方便地表示为线性混合效应模型（LMM），其中样条基函数的系数为随机效应。惩罚样条的LMM表示使得向多变量结局的扩展相对直接。在LMM中，定量协变量对结局无效应对应于固定效应和方差分量均为零的原假设。在原假设下，对似然比检验的方差分量进行通常的渐近卡方分布不再成立。因此，我们提出了三种针对似然比检验统计量的置换检验：一种基于置换定量协变量，另外两种基于置换残差。我们通过模拟比较了从多结局联合模型中获得的三种置换检验的I型错误率和功效，以及一种常用的参数检验。这些检验方法通过兴奋剂使用障碍心理社会临床试验的数据进行了示例说明。