Calculating EffectSizes


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1 Calculating EffectSizes David B. Wilson, PhD George Mason University August 2011 The Heart and Soul of Metaanalysis: The Effect Size Metaanalysis shifts focus from statistical significance to the direction and magnitude of effect Key to this is the effect size It is the dependent variable of metaanalysis Encodes research findings on a numerical scale Different types of effect sizes for different research situations Each type may have multiple methods of computation 1
2 Overview Main types of Effect Sizes Logic of the Standardized Mean Difference Methods of Computing the Standardized Mean Difference Logic of the Oddsratio and Riskratio Methods of Computing the Oddsratio Adjustments, such as for baseline differences Issues related to the variance estimate Most Common Effect Size Indexes Standardized mean difference (d or g) Group contrast (e.g., treatment versus control) Inherently continuous outcome construct Oddsratio and Riskratio (OR and RR) Group contrast (e.g., treatment versus control) Inherently dichotomous (binary) outcome construct Correlation coefficient (r ) Two inherently continuous constructs 2
3 Necessary Characteristics of Effect Sizes Numerical values produced must be comparable across studies Must be able to compute its standard error Must not be a direct function of sample size The Standardized Mean Difference Compares the means of two groups Standardized this difference 3
4 Visual Example of dtype Effect Size Small Sample Size Bias Adjustment d is slightly upwardly bias when based on small samples. This can be fixed with the adjustment below: It is common practice to apply this adjustment whenever using d. 4
5 Standard Error and Variance of d The standard error (and variance) of d is mostly a function of sample size: Methods of Computing d Lots of methods of computing d Goal is to reproduce what you would get with means, standard deviations, and sample sizes Some methods are straightforward 5
6 Computing d from a ttest Formula for d: Formula for t: Therefore: Computing d from a pvalue from a ttest An exact pvalue from a ttest can be convert into a tvalue, assuming you have the degreesoffreedom Then proceed as above 6
7 Computing d from dichotomous outcomes Some studies may report outcome dichotomously d can be estimated from these data Several methods Logit Cox Logit Probit Arcsine Logit method of computing d Logistic distribution similar to the normal distribution Logged oddsratios conform to the logistic distribution Standard deviation of the logistic distribution is Thus, we can rescale a logged OR into d Monte Carlo simulations suggest that this performs fairly well; some advantages to the Cox logit method (divide by 1.65) 7
8 More complicated estimates of d Can think of d as having two parts Numerator: mean difference Denominator: pooled standard deviation Trick is to get reasonable estimates of each Often have one but not the other Estimates of numerator of d Covariate adjusted means Difference in gain score means Unstandardized regression coefficient for treatment dummy 8
9 Estimates of denominator of d Gain score standard deviation Overall standard deviation for the outcome Standard error Oneway ANOVA Be care: don't just use any standard deviation laying around See Online Effect Size Calculator On CEBCP Website: On Campbell Website: 9
10 Logic of the OddsRatio and RiskRatio Oddsratio is the ratio of odds An odds is the probability of failure over the probability of success Riskratio is the ratio of risks A risk is simply the probability of failure An oddsratio or riskratio of 1 indicates no difference between the groups Oddsratio more difficult to interpret than riskratios Risk ratios are sensitive to base event rate; oddsratios are not Logic of the OddsRatio and RiskRatio Oddsratios and riskratios contrast two groups on dichotomous outcome I.e., a 2 by 2 frequency table where a, b, c, and d, are cell frequencies 10
11 Computing the OddsRatio and RiskRatio Oddsratio is computed as: Riskratio is computed as Computing the OddsRatio and RiskRatio The oddsratio can also be computed from the proportion successful in each group (p). Oddsratio is computed as: 11
12 Computing the Standard Error and Variance of the OddsRatio and Risk Ratio Standard error of the logged Oddsratio: Standard error of the logged Riskratio: Variance: Computing the OddsRatio from Other Data Only so many was you can report binary data Makes computing oddsratios and risk ratios relatively easy Generally have Full twobytwo table Percent of events (successes/failures) in each group Proportion of events (successes/failures) in each group Actual oddsratio or riskratio Convert dtype effect size Chisquare 12
13 Computing the OddsRatio from a χ 2 Oddsratio can be computed from a χ 2 if you have the overall event rate for the sample (total number of successes or failures) and the sample size in each group Computations reproduce the twobytwo table See online calculator or Lipsey/Wilson Practical Metaanalysis Appendix B Adjustments to Effect Size Hunter and Schmidt  Psychometric metaanalysis Measurement unreliability, invalidity Range restriction Artificial dichotomization Hunter and Schmidt adjustments generally not done in a Campbell or Cochrane review Pretest of outcome For quasiexperimental designs, may remove some bias Must still use posttest standard deviation in denominator Adds additional error (variance estimate should be bigger) Sensitivity analysis recommended 13
14 Issues Related to Variance Estimate Method of computing effect size may affect variance estimate For example d computed from dichotomous data numerator of d is adjusted for baseline or other covariates Study data may involve clusters Possible solution: rescale standard error from correct model provided in study (if available) where z is either a z or t test associate with the treatment effect from a complex statistical model. Questions? 14
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