Abstract
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 T 2-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 T 2-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.
Highlights
In many studies an objective is to assess whether an experimental therapy (E) versus control (C) has beneficial effects on multiple component outcomes
The primary objective is to evaluate the durability of glucose control over 3–6 years of treatment, the primary outcome being the time to a confirmed rise of HbA1c $7% using a logrank test
In cases where there is a mixture of quantitative variables with different dispersions or units, such as LDL measured in mg/dl and systolic blood pressure measured in mm Hg, it is more meaningful to compute a scale-invariant test using the average of the corresponding standardized differences
Summary
In many studies an objective is to assess whether an experimental therapy (E) versus control (C) has beneficial effects on multiple component outcomes. For a given S^, the vector L is obtained as L~B’S^ where B is the quadratic program solution to miny1⁄2y’S^{1y under the constraints that yi§0 Vi and y’J~1 This test will principally be required in cases where the null hypothesis applies, or the treatment is inferior for some of the component outcome measures. To illustrate the construction of the Wei-Lachin test, consider a large sample test for a difference between groups in the means of two outcomes where it is assumed that Xij*f (mij,y2ij) with some distribution f where y2ij~V (Xij) is the variance of the observations for the jth outcome in the ith group, or the residual variance after adjusting for other covariates, and yiab~Cov(Xia,Xib), i = E, C; j = a, b.
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