Bootstrap hypothesis testing in generalized additive models for comparing curves of treatments in longitudinal studies

Abstract

The study of the effect of a treatment may involve the evaluation of a variable at a number of moments. When assuming a smooth curve for the mean response along time, estimation can be afforded by spline regression, in the context of generalized additive models. The novelty of our work lies in the construction of hypothesis tests to compare two curves of treatments in any interval of time for several types of response variables. The within-subject correlation is not modeled but is considered to obtain valid inferences by the use of bootstrap. We propose both semiparametric and nonparametric bootstrap approaches, based on resampling vectors of residuals or responses, respectively. Simulation studies revealed a good performance of the tests, considering, for the outcome, different distribution functions in the exponential family and varying the correlation between observations along time. We show that the sizes of bootstrap tests are close to the nominal value, with tests based on a standardized statistic having slightly better size properties. The power increases as the distance between curves increases and decreases when correlation gets higher. The usefulness of these statistical tools was confirmed using real data, thus allowing to detect changes in fish behavior when exposed to the toxin microcystin-RR.

Keywords:

• correlated data
• spline
• semiparametric
• nonparametric
• size
• power

Supplementary material

The file supplementarydata.pdf contains the proof of Lemma 3.1 (Appendix 1) and the derivation of the equivalence between nonparametric and semi-parametric bootstrap approaches (Appendix 2). It also includes additional simulations conducted to analyze the properties of the tests (Appendix 3). The file supplementarydata.docx contains the R function created to perform the bootstrap tests, along with a description of its usage, and the fish behavior data. Supplemental data for this article can be accessed at 10.1080/02664763.2015.1078301 description of location.

Acknowledgments

The authors thank D. A. Wunderlin, M. A. Bistoni and J. Cazenave for providing the fish behavior data and for valuable discussions. The authors are grateful to Dr. Geert Molenberghs for his revision of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This study was partially supported by Agencia Nacional de Promoción Científica y Tecnológica, under FONCyT Grants PICT/99 13-7143 and PICTR/03 13-327. Nores' research was supported by a fellowship provided by CONICET. The findings and conclusions in this report are those of the authors and do not necessarily represent the views of Universidad Nacional de Córdoba.