ABSTRACT
It is well known that, when a null hypothesis has been rejected, it simply means that the assertion made in the null hypothesis (H 0) is unlikely to be true in the population under study. When that assertion is that the independent variables has no effect on the dependent variable, then the rejection of Ho tells us only that the effect is not likely to be zero in the relevant population. It does not suggest that the effect is large or even nontrivial but simply that it is nonzero. Hence, the rejection of Ho "cries out" for estimating the magnitude of the effect in question. This, combined with another well-known fact that the larger our sample size is, the easier it is (other things being equal) to reject a null hypothesis, makes it reasonable to give a general, conceptual definition of effect size (as Rosenthal does in chap. 20) in the following way:
Effect size = [significance-test statistic]/[sample size] (I) It can be said we are "cutting the test statistic down to size," if the reader will pardon the pun. (Here "sample size" is to be understood in a more general way than "number of cases observed": It may be some function of the sizes of the two samples involved. This is probably why Rosenthal referred to "size of study" rather than "sample size." Examples of functions of the two sample sizes may be seen in his Equations 3 and 4.) The quantity stemming from Equation I may be called either a measure or an estimate of effect size, depending on whether it refers to the particular sample at hand or to the population under study. Example. As a simple example of the previous definition of effect size, consider the following. We are interested in the extent of relationship, if any, between educational background (Graduated from College vs. High School Only) and job satisfaction. Suppose that we chose 180 individuals at random and asked each of
= 7.60
which, as a x2-variate with 1 df, is significant beyond the I % level (p = .006). So it is very unlikely that educational background is unrelated to (i.e., has no effect on) job satisfaction. But just how large is the size of this effect? From Equation I we get
The reader may recognize that X2/N gives us the squared phi coefficient, <1>2, Thus, our effect-size measure here is
<1>2 = .0422,
or, if <I> itself is preferred,
According to Cohen (1988, pp. 477-478), this effect size would be considered at best only "medium," or even rather small. We have, in shifting our focus from the significance test to effect size, cut down our "exalted" chi-square value, which was significant beyond the I % level, to an effect-size of <1>2 = .0422 (or <I> = .2055).
