Cumulation of Variance Ratios

Abstract

In my article (Feingold, 1992a), I examined sex differences in variability of performance on cognitive tests by calculating the ratios of males' variances to females' variances. Thus, variance ratios (VRs) of 1.00 indicated homogeneity of variance, VRs greater than 1.00 indicated greater male variability, and VRs less than 1.00 indicated greater female variability. However, because each test was normed separately by grade and year, multiple male-female comparisons were conducted, yielding a set of VRs for each test. Thus, the VRs needed to be summarized to determine whether there was an overall sex difference in variability on each test. The median of a sample of VRs must be an unbiased estimate of the population VR when the null hypothesis is true (i.e., when there is homogeneity of variance in the population and when all variation among VRs in a sample drawn from that population is attributable to sampling error), because one group will-by chance-be more variable half of the time and the second group will be more variable the other half of the time. Thus, the expected value of a median VR, as of a single VR, is 1.00. However, VRs follow the Fdistribution (indeed, Fratios are variance ratios), which is positively skewed (Snedecor & Cochran, 1967). Therefore, when the null hypothesis is true, the mean (i.e., the arithmetic mean) VR is greater than the unbiased median VR, making the mean an inappropriate measure of central tendency for VRs. (The direction of the bias favors the group-males in my work-that is arbitrarily selected as the numerator of the VR.) A positively skewed distribution of scores (e.g., VRs) can be normalized via an appropriate transformation, such as a log transformation, that spreads out differences among values at the left tail of the distribution (Tukey, 1977). Thus, it is not wrong to calculate the mean of log-transformed VRs. However, because the mean and the median of a normal distribution of scores are the same, the mean logtransformed VR and the median log-transformed VR are comparable. Thus, both measures of central tendency yield essentially the same unbiased average in a sample of log-transformed VRs. Most important, a log transformation is a monotonic transformation-that is, a transformation that does not change the rank order of scores. Therefore, the median (which is a function of rank order) of a sample of logtransformed VRs equals the log-transformed median VR of the same sample. Thus, it is pointless to use a log transformation when VRs are summarized by medians because the outcome is identical whether or not the transformation is used (when the medians are expressed in a common metric). Moreover, because the mean and median are the same when the distribution is normal (and about the same when the distribution is nearly normal), the median log-transformed VR, the mean logtransformed VR, and the median VR are all about the same (when expressed in the same metric). Thus, there are no advantages in using the mean log-transformed VR instead of the median VR when the sole objective is to summarize VRs. Yet, there is one major disadvantage in using the mean log-transformed VR. The magnitude of