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analyses:morris2008 [2021/01/16 09:55]
Wolfgang Viechtbauer
analyses:morris2008 [2021/11/08 13:17]
Wolfgang Viechtbauer
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 Morris (2008) discusses various ways for computing a (standardized) effect size measure for pretest posttest control group designs, where the characteristic, response, or dependent variable assessed in the individual studies is a quantitative variable. Morris (2008) discusses various ways for computing a (standardized) effect size measure for pretest posttest control group designs, where the characteristic, response, or dependent variable assessed in the individual studies is a quantitative variable.
  
-As described by Becker (1988), we can compute the standardized mean change (with raw score standardization) for a treatment and control group with $$g_T = c(n_T-1) \frac{\bar{x}_{post,T} - \bar{x}_{pre,T}}{SD_{pre,T}}$$ and $$g_C = c(n_C-1) \frac{\bar{x}_{post,C} - \bar{x}_{pre,C}}{SD_{pre,C}},$$ where $\bar{x}_{pre,T}$ and $\bar{x}_{post,T}$ are the treatment group pretest and posttest means, $SD_{pre,T}$ is the standard deviation of the pretest scores, $c(m) = \sqrt{2/m} \Gamma[m/2] / \Gamma[(m-1)/2]$ is a bias-correction factor((The bias correction factor given on page 261 by Becker (1988) includes a slight error. See Morris (2000).)), $n_T$ is the size of the treatment group, and $\bar{x}_{pre,C}$, $\bar{x}_{post,C}$, $SD_{pre,C}$, and $n_C$ are the analogous values for the control group. Then the difference in the two standardized mean change values, namely $$g = g_T - g_C$$ indicates how much larger (or smaller) the change in the treatment group was (in standard deviation units) when compared to the change in the control group. Values of $g$ computed for a number of studies could then be meta-analyzed with standard methods.((Note that $g$ is used here to denote the bias-corrected value (as opposed to Becker (1988who uses $d$ to denote this).))+As described by Becker (1988), we can compute the standardized mean change (with raw score standardization) for a treatment and control group with $$g_T = c(n_T-1) \frac{\bar{x}_{post,T} - \bar{x}_{pre,T}}{SD_{pre,T}}$$ and $$g_C = c(n_C-1) \frac{\bar{x}_{post,C} - \bar{x}_{pre,C}}{SD_{pre,C}},$$ where $\bar{x}_{pre,T}$ and $\bar{x}_{post,T}$ are the treatment group pretest and posttest means, $SD_{pre,T}$ is the standard deviation of the pretest scores, $c(m) = \sqrt{2/m} \Gamma[m/2] / \Gamma[(m-1)/2]$ is a bias-correction factor((The bias correction factor given on page 261 by Becker (1988) includes a slight error. See Morris (2000).)), $n_T$ is the size of the treatment group, and $\bar{x}_{pre,C}$, $\bar{x}_{post,C}$, $SD_{pre,C}$, and $n_C$ are the analogous values for the control group. Then the difference in the two standardized mean change values, namely $$g = g_T - g_C$$ indicates how much larger (or smaller) the change in the treatment group was (in standard deviation units) when compared to the change in the control group. Values of $g$ computed for a number of studies could then be meta-analyzed with standard methods.((Note that $g$ is used here to denote the bias-corrected value (as opposed to Becker1988who uses $d$ to denote this). There was (and sometimes still is) some inconsistency in notation when referring to the biased and the bias-corrected version of standardized mean difference / change measures, but I would say the general trend has been to use $d$ for the biased version and $g$ for the bias-corrected version and this is the notation I am also using here.))
  
 Morris (2008) uses five studies from a meta-analysis on training effectiveness by Carlson and Schmidt (1999) to illustrate these computations. We can create the same dataset with: Morris (2008) uses five studies from a meta-analysis on training effectiveness by Carlson and Schmidt (1999) to illustrate these computations. We can create the same dataset with:
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 ==== The Actual Meta-Analysis ==== ==== The Actual Meta-Analysis ====
  
-For the actual meta-analysis part, we simply pass the ''yi'' and ''vi'' values to the ''rma()'' function. For example, a fixed-effects model can be fitted with:+For the actual meta-analysis part, we simply pass the ''yi'' and ''vi'' values to the ''rma()'' function. For example, an equal-effects model can be fitted with:
 <code rsplus> <code rsplus>
-rma(yi, vi, data=dat, method="FE", digits=2)+rma(yi, vi, data=dat, method="EE", digits=2)
 </code> </code>
 <code output> <code output>
-Fixed-Effects Model (k = 5)+Equal-Effects Model (k = 5)
  
-Test for Heterogeneity: +I^2 (total heterogeneity / total variability):   9.69% 
 +H^2 (total variability / sampling variability):  1.11 
 + 
 +Test for Heterogeneity:
 Q(df = 4) = 4.43, p-val = 0.35 Q(df = 4) = 4.43, p-val = 0.35
  
 Model Results: Model Results:
  
-estimate       se     zval     pval    ci.lb    ci.ub           +estimate    se  zval  pval  ci.lb  ci.ub  
-    0.95     0.14     6.62     <.01     0.67     1.23      *** +    0.95  0.14  6.62  <.01   0.67   1.23  *** 
  
 --- ---
-Signif. codes: ***’ 0.001 **’ 0.01 *’ 0.05 .’ 0.1 ‘ ’ 1+Signif. codes: '***0.001 '**0.01 '*0.05 '.0.1 ' ' 1
 </code> </code>
 Note that these results are slightly different than the ones in Table 5 due to the different ways of estimating the sampling variances. Note that these results are slightly different than the ones in Table 5 due to the different ways of estimating the sampling variances.
analyses/morris2008.txt · Last modified: 2021/11/08 13:17 by Wolfgang Viechtbauer