Shrinkage Estimation Methods for Subgroup Analyses

Abstract

Subgroup analyses increasingly gain importance for pharmaceutical investigations. Conventional approaches for treatment effect estimation are controversial because of multiplicity and small sample sizes within the subsets. Hence, we consider shrinkage estimators, which combine the overall effect estimate with the estimate within a given subgroup by using a Bayesian framework. This article contains a short introduction to two shrinkage estimation approaches proposed by Dixon and Simon and by Simon. Our key contribution is to present methodical extensions to enlarge the applicabilities and provide solutions for some computational issues. Besides an application to a real dataset, we perform an extensive simulation study, in which the conventional and the shrinkage approaches are compared under different models and scenarios of a typical clinical phase III design. The simulation results clearly show that the shrinkage approaches provide much better estimates than the conventional approaches according to the mean square error and the interval range under nearly all considered investigation cases. Exceeding advantages can be observed in the case of small sample sizes and low interaction effects. Some issues occur to the width and coverage probability of the credibility intervals concerning particular variants of the shrinkage estimators.

Keywords:

• Bayesian analysis
• Clinical trials
• Interactions
• Reparameterization

Supplementary Materials

The supplementary materials contain at the beginning a short derivation of the correction term c used in Section 2.5. In addition, a model with a trichotomous covariate is used to show essential steps to prove, following Section 2.6, that the results of the Simon approach depend on the choice of the reference level. This is followed by the derivation of the two systems of linear equations used in Section 2.6. Furthermore, it is shown how the condition mentioned in Section 3.1 concerning the determination of prevalences and correlations is obtained. Finally, two tables on the quality of the interval estimators resulting from the simulation study are presented (with reference to Section 3.3).

Acknowledgments

We would like to thank Bayer AG and in particular the Biostatistics Innovation Center for the intensive support of our research.

Disclosure Statement

Arno Fritsch is fulltime employee of Bayer AG Pharmaceuticals. The authors report there are no further competing interests to declare.

Funding

The author(s) reported there is no funding associated with the work featured in this article.