We construct multiple comparisons procedures in k exponential populations. Exact theory and asymptotic theory of simultaneous confidence intervals and multiple comparisons tests are discussed. First, we consider multiple comparisons for the differences among parameters. We can give the Tukey-Kramer type multiple test procedure based on estimators of k means. However, the degree of conservativeness for the multiple tests depends on unknown mean parameters. Therefore, multiple tests based on the logarithm transformation of estimators are proposed. It is found that the degree of conservativeness for the proposed tests is controlled by the sample sizes. Furthermore, the closed testing procedure, more powerful than the REGW (Ryan/Einot-Gabriel/Welsch) tests, is proposed. Simultaneous confidence intervals for the differences among the logarithms of parameters are discussed. Next, for the multiple comparisons with a control, we propose the multiple test procedures. It is shown that the proposed multiple test is superior to the tests based on the Bonferroni inequality asymptotically. A sequentially rejective procedure is derived under unequal sample sizes. Last, we consider multiple comparisons for all parameters. The exact single-step multiple comparison procedures based on the upper 100α% points the χ2.distribution are proposed. The asymptotic theory for the multiple comparisons is discussed. Especially sequentially rejective procedures can be constructed in the asymptotic theory.
A cure rate model is a survival model incorporating the cure rate on the assumption that a population contains both uncured and cured individuals. It is a powerful statistical tool for cancer prognostic studies. In order to accurately predict long-term outcome the proportional hazards (PH) cure model requires variable selection methods. However, no specific variable selection method for the PH cure model has been established in practice. In this study, we present a stepwise variable selection method for the PH cure model with a logistic regression for the cure rate and a Cox regression for the hazard for uncured patients. We conducted simulation studies to evaluate the operating characteristics of the stepwise method in comparison to those of the best subset selection method based on Akaike information criterion and of the convenience variable selection method that puts all variables in the PH cure model and selects the significant ones. The results demonstrated that in many cases the stepwise method outperformed other methods with respect to false positive determinations and estimation bias for the survival curve. In addition, we demonstrated the usefulness of the stepwise method for the PH cure model by applying it to analyze clinical data on breast cancer patients.
Clinical trials often employ two or more primary endpoints because a single endpoint may not provide a comprehensive picture of the intervention’s effects. In such clinical trials, a decision is generally made as to whether it is desirable to evaluate the joint effects on all endpoints (i.e., co.primary endpoints) or at least one of the endpoints. This decision defines the alternative hypothesis to be tested and provides a framework for approaching trial design. In this article, we discuss recent statistical issues in clinical trials with multiple primary endpoints. Especially, we introduce the methods for power and sample size determinations in clinical trials with co-primary endpoints, considering the correlations among the endpoints into the calculations. We also discuss the methods to alleviate conservativeness of testing co-primary endpoints.