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Testing for order-restricted hypotheses in longitudinal data. (English) Zbl 1105.62044

Summary: 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 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 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.

MSC:

62G10 Nonparametric hypothesis testing
62F30 Parametric inference under constraints
62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
65C05 Monte Carlo methods
62P10 Applications of statistics to biology and medical sciences; meta analysis

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gee; R
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