  If an effect measure is more homogeneous than others, then its value is more
likely to be stable across different subgroups or subpopulations. Therefore, it
is of great importance to find a more homogeneous effect measure that allows
for transportability of research results. For a binary outcome, applied
researchers often claim that the risk difference is more heterogeneous than the
risk ratio or odds ratio, because they find, based on evidence from surveys of
meta-analyses, that the null hypotheses of homogeneity are rejected more often
for the risk difference than for the risk ratio and odds ratio. However, the
evidence for these claims are far from satisfactory, because of different
statistical powers of the homogeneity tests under different effect scales. For
binary treatment, covariate and outcome, we theoretically quantify the
homogeneity of different effect scales. Because when homogeneity holds the four
outcome probabilities lie in a three dimensional sub-space of the four
dimensional space, we can use results from differential geometry to compute the
volumes of these three dimensional spaces to compare the relative homogeneity
of the risk difference, risk ratio, and odds ratio. We demonstrate that the
homogeneity space for the risk difference has the smallest volume, and the
homogeneity space for the odds ratio has the largest volume, providing some
further evidence for the previous claim that the risk difference is more
heterogeneous than the risk ratio and odds ratio.
