Maximin 39

Many studies aim to assess whether a therapy has a beneficial effect on multiple outcomes simultaneously relative to a control. Often the joint null hypothesis of no difference for the set of outcomes is tested using separate tests with a correction for multiple tests, or using a multivariate T2-like MANOVA or global test. However, a more powerful test in this case is a multivariate one-sided or one-directional test directed at detecting a simultaneous beneficial treatment effect on each outcome, though not necessarily of the same magnitude. The Wei-Lachin test is a simple 1 df test obtained from a simple sum of the component statistics that was originally described in the context of a multivariate rank analysis. Under mild conditions this test provides a maximin efficient test of the null hypothesis of no difference between treatment groups for all outcomes versus the alternative hypothesis that the experimental treatment is better than control for some or all of the component outcomes, and not worse for any. Herein applications are described to a simultaneous test for multiple differences in means, proportions or life-times, and combinations thereof, all on potentially different scales. The evaluation of sample size and power for such analyses is also described. For a test of means of two outcomes with a common unit variance and correlation 0.5, the sample size needed to provide 90% power for two separate one-sided tests at the 0.025 level is 64% greater than that needed for the single Wei-Lachin multivariate one-directional test at the 0.05 level. Thus, a Wei-Lachin test with these operating characteristics is 39% more efficient than two separate tests. Likewise, compared to a T2-like omnibus test on 2 df, the Wei-Lachin test is 32% more efficient. An example is provided in which the Wei-Lachin test of multiple components has superior power to a test of a composite outcome.

We consider two frequently used PK/PD models and provide closed form descriptions of locally optimal designs for estimating individual parameters. In a novel way, we use these optimal designs and construct locally standardized maximin optimal designs for estimating any subset of the model parameters of interest. We do this by maximizing the minimal efficiency of the estimates across all relevant parameters so that these optimal designs are less dependent on the individual parameter or parameters of interest. Additionally, robust designs are proposed to further reduce the dependence on the nominal values of the parameters. We compare efficiencies of our proposed optimal designs with locally optimal designs and designs used in four real studies from the literature and show that our proposed designs provide advantages over those used in practice.

Abortion critics have argued that one should err on the side of life and prohibit abortion since the status of the fetus is uncertain. David Boonin has criticized this precautionary argument, but his criticism has been ignored. The aim is to elaborate on the precautionary argument by responding to Boonin's criticism. Boonin considers three versions of the precautionary argument-the disaster avoidance argument, the maximin argument, and the expected utility argument; yet all three are judged unsuccessful for the same reasons: they lead to unacceptable implications, they lead to conclusions that are too weak, and they undermine the integrity of moral reasoning. I respond to this criticism by arguing that one can avoid unacceptable implications by considering a criterion of realism, that the weaker conclusions are rather an advantage, and that the application of the precautionary principle makes room for considerations which maintain the integrity of moral reasoning. I also consider some criticism beyond Boonin's objections.