George V, Johnson W D, Shahane A, Nick T G
Division of Biostatistics, Medical College of Wisconsin, Milwaukee 53226, USA.
Biometrics. 1997 Mar;53(1):49-59.
We are often faced with the statistical problem of evaluating the effect of a treatment in the extreme of a population. This requires taking measurements on truncated random variables and, hence, it becomes necessary to take proper account of the effect of regression toward the mean. The usual statistical procedures are inappropriate for testing treatment effect in the presence of regression toward the mean. Likelihood ratio and score tests based on truncated distributions should provide valid statistical inferences in these situations. We conducted simulation studies to investigate the properties of these methods and found that the likelihood ratio test performs well even when the sample size is moderate, whereas the score test does not seem to control the nominal significance level. We compared the likelihood ratio test to a regression-based t-test, assuming the mean of the baseline distribution to be known, and found the likelihood ratio test more powerful. In the case where the baseline mean is unknown, we also investigated Wald's test and compared it with the likelihood ratio test and score test with respect to validity and power using simulation. Wald's test and the score test do not control the nominal significance level unless the sample size is extremely large. Overall, the likelihood ratio test has the best performance among all the methods studied. The proposed likelihood ratio test is illustrated using an example of a cholesterol study.
我们常常面临评估总体极端情况下治疗效果的统计问题。这需要对截断随机变量进行测量，因此，有必要适当考虑均值回归的影响。在存在均值回归的情况下，通常的统计程序不适用于检验治疗效果。基于截断分布的似然比检验和得分检验应能在这些情况下提供有效的统计推断。我们进行了模拟研究以考察这些方法的性质，发现即使样本量适中，似然比检验的表现也很好，而得分检验似乎无法控制名义显著性水平。我们将似然比检验与基于回归的t检验进行比较（假设基线分布的均值已知），发现似然比检验更具功效。在基线均值未知的情况下，我们还研究了 Wald 检验，并通过模拟将其与似然比检验和得分检验在有效性和功效方面进行比较。除非样本量极大，否则 Wald 检验和得分检验无法控制名义显著性水平。总体而言，在所有研究的方法中，似然比检验表现最佳。通过一项胆固醇研究的例子来说明所提出的似然比检验。
