Testing for order-restricted hypotheses in longitudinal data

In many biomedical studies, we are interested in comparing treatment effects with an inherent ordering. We propose a quadratic score test (QST) based on a quadratic inference function for detecting an order in treatment effects for correlated data. The quadratic inference function is similar to the negative of a log-likelihood, and it provides test statistics in the spirit of a$\chi^{2}-test$for testing nested hypotheses as well as for assessing the goodness of fit of model assumptions. Under the null hypothesis of no order restriction, it is shown that the QST statistic has a Wald-type asymptotic representation and that the asymptotic distribution of the QST statistic is a weighted$\chi^{2}-distribution$. Furthermore, an asymptotic distribution of the QST statistic under an arbitrary convex cone alternative is provided. The performance of the QST is investigated through Monte Carlo simulation experiments. Analysis of the polyposis data demonstrates that the QST outperforms the Wald test when data are highly correlated with a small sample size and there is a significant amount of missing data with a small number of clusters. The proposed test statistic accommodates both time-dependent and time-independent covariates in a model.