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Cooper, H. and Hedges, L. V. (Eds.) 1994. The Handbook ofResearch Synthesis. New York: Russell Sage Foundation 16 PARAMETRIC MEASURES OF EFFECT SIZE ROBERT ROSENTHAL Harvard University CONTENTS 1. Defining Research Results 1.1 Effect Size and Tests of Significance 1.2 Two Families of Effect Sizes 1.2.1 The r family 1.2.2 The d family 1.3 Effect Sizes for the One-Sample Case 1.4 Effect Sizes for Comparing Effect Sizes 1.4.1 The two-sample case 1.4.2 The one-sample case 1.5 Comparing the r and d Families 1.5.1 Generality of interpretation 1.5.2 Consistency of meaning in the one-sample case 1.5.3 Simplicity of interpretation 2. The Computation of Effect Sizes 2.1 Direct Computation 2.2 From Significance Tests 2.2.1 For r 2.2.2 For Cohen's d 2.2.3 For Hedges's g 2.2.4 When X2 df> 1 or F df> 1 in the numerator 2.3 From Significance Levels 2.3.1 r from t 2.3.2 Cohen's d from t 2.3.3 Hedges's g from t 2.4 From Other Effect Size Indices 2.4.1 r from d and g 2.4.2 Cohen's d from r and g 232 232 234 234 234 235 235 235 235 236 236 236 236 236 236 236 237 238 238 238 239 239 239 239 239 239 239 231 232 STATISTICALLY DESCRIBING AND COMBINING STUDIES 2.4.3 Hedges's g from rand d 239 2.5 Choosing an Approach 240 3. Adjusting Effect Size Estimates 240 3.1 The Fisher and Hedges Adjustments 240 3.1.1 Adjusting Zr and r 240 3.1.2 Adjusting g and .:l 240 3.2 The Hunter and Schmidt Adjustments 240 3.3 The Glass, McGaw, and Smith Adjustments 241 4. Effect Sizes for Correlated Dependent Variables 241 5. The Interpretation of Effect Sizes 5.1 The Physicians' Aspirin Study 5.2 The Binomial Effect Size Display 5.3 Additional Results 6. References 1. DEFINING RESEARCH RESULTS 1.1 Effect Size and Tests of Significance The heart of the enterprise of synthesizing research consists of comparing and combining the results of individual studies of a particular, focused research question . In Chapter 15, the emphasis was on one type of result of an individual study-the test of significance. In the present chapter, the emphasis will be on a different type of result of an individual study-the size of the effect of an independent variable on a dependent variable , or, more generally, the size of the relationship between any two variables (Rosenthal 1991). These two types of results, the test of significance (test statistic) and the size of the effect (effect size estimate), are related to each other in a simple, direct way: Test of significance = size of effect x size of study. Table 16.1 gives useful specific examples of this general equation. Equation (16-1) shows that X2 on df= 1 is the product of the size of the effect expressed by 2 x N 16-2 Z x yIN r 16-3 Vl-r2 x V{lf (M1~M2r 16-4 x e -+n1 n2 16-5 (M1~M2r x 16-6 (M1:M2r x [~ vlif] --x df (n1 +n2) 16-7 d x V{lf 2 16-8 Fe r2 l-r2 x dferror Fd eta2 dferror 16-9 1- eta2 X dfmeans 16-10 Fd S2 means S2 X n r 16-11 te Vl-r2 x V{lf 16-12 te D x Vn SD 16-13 te d x V{lf NOTE: N = total study sample size, n = sample size for each condition, and df=N - 2 for most applications. aAlso called g (Hedges 1981,1982). b Also called d (Cohen 1969, 1977, 1988). eNumerator df= 1. dNumerator dfmay take on any value. eCorrelated observations. Another alternative when the S's of the two groups differ greatly is to transform the data to make the S's more similar. Such transformations (e.g., logs, square roots) of course require our having access to the original data, but that is also often required to compute S separately for the control group. When only a mean square error from an analysis of variance is available, we must be content to use its square root (S) as our standardizing denominator in any case. Or if only the results...

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