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Abstract:
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Critical values and p-values of hypothesis tests are often derived using asymptotic approximations of sampling distributions. However, this can result in tests that are conservative (i.e., understate the frequency of an incorrectly rejected null hypothesis). Although computationally rigorous options (e.g., the bootstrap) are available for such situations, we illustrate that simple transformations can be used to improve both the size and power of such tests. Using a logarithmic transformation, we show that the transformed statistic is asymptotically equivalent to its untransformed analogue under the null hypothesis and is divergent from the untransformed version under the alternative (yielding a potentially substantial increase in power). The transformation is applied to several easily-accessible statistical hypothesis tests, a few of which are taught in introductory statistics courses. With theoretical arguments and simulations, we illustrate that the log transformation is preferable to other forms of correction (such as statistics that use a multiplier). Finally, we illustrate application of the method to a a well-known dataset.
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