Abstract

We improve students’ understanding of the F-test for linear hypotheses in a linear model by explaining elements that affect the power of the test. Including true restrictions in a joint null hypothesis affects test power in a way that is not generally known. Asking a student whether including the true restrictions in the null hypothesis will increase or decrease power, the student is likely to say: “I don’t know.” The student’s answer is not bad because the power depends on the noncentrality parameter and the degrees of freedom. We show that adding true restrictions to a linear hypothesis cannot decrease the noncentrality parameter of the F-statistic, a result many will find counterintuitive. Adding true restrictions can increase or decrease F-test power depending on the offsetting negative effect of reducing the numerator degrees of freedom. We provide illustrative examples of these results and prove them for the general case.

摘要

我们通过解释影响检验功效的元素来提高学生对线性模型中线性假设的F检验的理解。在联合零假设中包含真正的限制会以一种不为人知的方式影响检验功效。当问学生是否在零假设中包含真正的限制会增加或减少功效时，学生可能会说：“我不知道。” 学生的答案还不错，因为功效取决于非中心性参数和自由度。我们表明，向线性假设添加真正的限制不能降低F统计量的非中心性参数，许多人会发现这一结果违反直觉。添加真正的限制可以增加或减少F-测试功效取决于减少分子自由度的抵消负面影响。我们提供了这些结果的说明性示例，并在一般情况下证明了它们。