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221 12 EFFECT SIZES FOR CONTINUOUS DATA MICHAEL BORENSTEIN Biostat C O N T E N T S 12.1 Introduction 222 12.1.1 Effect Sizes and Treatment Effects 222 12.1.2 Effect Sizes Rather than p-Values 223 12.1.3 Effect-Size Parameters and Sample Estimates of Effect Sizes 223 12.1.4 The Variance of the Effect Size 223 12.1.5 Calculating Effect-Size Estimates from Reported Information 224 12.2 The Raw (Unstandardized) Mean Difference D 224 12.2.1 Computing D, Independent Groups 224 12.2.2 Computing D, Pre-Post Scores or Matched Groups 225 12.2.3 Including Different Study Designs in the Same Analysis 225 12.3 The Standardized Mean Difference d and g 225 12.3.1 Computing d and g, Independent Groups 226 12.3.2 Computing d and g, Pre-Post Scores or Matched Groups 227 12.3.3 Computing d and g, Analysis of Covariance 228 12.3.4 Including Different Study Designs in the Same Analysis 230 12.4 Correlation as an Effect Size 231 12.4.1 Computing r 231 12.5 Converting Effect Sizes from One Metric to Another 231 12.5.1 Converting from the Log Odds Ratio to d 232 12.5.2 Converting from d to the Log Odds Ratio 233 12.5.3 Converting from r to d 233 12.5.4 Converting from d to r 234 12.6 Interpreting Effect Sizes 234 12.7 Resources 234 12.8 Acknowledgments 234 12.9 References 235 222 STATISTICALLY DESCRIBING STUDY OUTCOMES 12.1 INTRODUCTION In any meta-analysis, we start with summary data from each study and use it to compute an effect size for the study. An effect size is a number that reflects the magnitude of the relationship between two variables. For example , if a study reports the mean and standard deviation for the treated and control groups, we might compute the standardized mean difference between groups. Or, if a study reports events and nonevents in two groups we might compute an odds ratio. It is these effect sizes that are then compared and combined in the meta-analysis. Consider figure 12.1, the forest plot of a fictional metaanalysis to assess the impact of an intervention. In this plot, each study is represented by a square, bounded on either side by a confidence interval. The location of each square on the horizontal axis represents the effect size for that study. The confidence interval represents the precision with which the effect size has been estimated, and the size of each square is proportional to the weight that will be assigned to the study when computing the combined effect. This figure also serves as the outline for this chapter, in which I discuss what these items mean and how they are computed. This chapter addresses effect sizes for continuous outcomes such as means and correlations (for effect sizes for binary outcomes, see chapter 13, this volume). 12.1.1 Effect Sizes and Treatment Effects Meta-analyses in medicine often refer to the effect size as a treatment effect, and this term is sometimes assumed to refer to odds ratios, risk ratios, or risk differences, which are common in meta-analyses that deal with medical interventions. Similarly, meta-analyses in the social sciences often refer to the effect size simply as an effect size, and this term is sometimes assumed to refer to standardized mean differences or to correlations, which are common in social science meta-analyses. In fact, though, both the terms effect size and treatment effect can refer to any of these indices, and the distinction between the terms lies not in the index but rather in the nature of the study. The term effect size is appropriate when the index is used to quantify the relationship between any two variables or a difference between any two groups. By contrast, the term treatment effect is appropriate only for an index used to quantify the impact of a deliberate intervention. Thus, the difference between males and females could be called an effect size only, while the difference between treated and control groups could be Figure 12.1 Fictional Meta-Analysis Showing Impact of an Intervention source: Author’s compilation. Impact of Intervention Study A 0.400 60 0.067 0.258 0.121 B 0.200 600 0.007 0.082 0.014 C 0.300 100 0.040 0.200 0.134 D 0.400 200...

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