Experimental control over the shape of the response distribution provides opportunities to increase power. Some learning tasks, for example, may be designed to measure either errors in a fixed number of trials or errors to some specified criterion. The latter measure, being unbounded, might be questionable with young children or patients because a few might persevere with some inappropriate strategy, yielding extremely high error scores. In the cited example of easy multiple choice tests, skewness could be reduced by making the test of intermediate difficulty. Among the many vital functions of pilot work, the importance of distribution shape should not be overlooked.
Even after the data are in, their shape can still be changed by applying a statistical transformation (Section 12.4). A long-tailed distribution of response times, for example, becomes more normal by taking reciprocals, that is, by transforming time to speed. Even more useful may be the trimmed Anova of Section 12.1.
The Experimental Pyramid implies that the shape of the data depends primarily on empirical determinants. Task, procedure, and measurement constitute an empirical transformation that determines the shape of the data. Statistical theory, in contrast, concentrates on statistical transformations after the data have been obtained. Although statistical transformations can be helpful, they involve only the tip of the Pyramid. For empirical investigators, the main concern over the shape of the data should instead be focused at the lower levels of the Pyramid in planning the investigation.
In this equation, v1 and v2 are the df for numerator and denominator, respectively. The constant c is a complicated function of v1 and v2 that makes the area under this F distribution equal to 1.
The curve in Figure 3.1 labeled “H0 true” was obtained from this formula: Prob(F) is the vertical elevation of the curve, as a function of F on the horizontal axis. Different df give different curves of different shape.