Testing statistical hypotheses about rat liver foci

Tests of statistical hypotheses concerning treatment effect on the development of hepatocellular foci can be carried out directly on two-dimensional observations made on histologic sections or on estimates of the density and volume of foci in three dimensions. Inferences about differences in the density or size of foci from tests based on two-dimensional observations, however, can be misleading. This is because both the number of focus cross-sections observed in a tissue section and the percent area occupied by foci can be expressed in terms of the number of foci per unit volume of liver tissue and the mean focus size. As a consequence, a treatment difference may be caused by a difference in the density of foci, their average size, or both. Of more serious concern is the possibility that failure to detect a treatment effect may occur not only when there is no treatment effect but also when the density and size of foci differ between treatments in such a way that their product is unchanged. This can happen if the effect of treatment is to increase the number of foci and decrease their average size, or vice versa. A similar difficulty of interpretation is associated with hypothesis tests based on average focus cross-section area. Tests based on estimates of the number of foci per unit volume and mean focus volume allow direct inference about the quantities of interest, but these estimates are unstable because they have large variances. Empirical estimates of statistical power for the Wilcoxon rank sum test and the t-test from data on control rats suggest power may be limited in experiments with group sizes of ten and low observed numbers of focus cross-sections. If hypothesis tests based on estimates of the density and size of foci are to form the basis for a bioassay, then the power of statistical tests used to identify treatment effects should be investigated.