# The metafor Package

A Meta-Analysis Package for R

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analyses:morris2008

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 analyses:morris2008 [2021/01/16 09:55]Wolfgang Viechtbauer analyses:morris2008 [2021/11/08 13:17]Wolfgang Viechtbauer Both sides previous revision Previous revision 2021/11/08 13:17 Wolfgang Viechtbauer 2021/08/27 07:27 Wolfgang Viechtbauer 2021/01/16 09:55 Wolfgang Viechtbauer 2017/03/19 13:09 external edit Next revision Previous revision 2021/11/08 13:17 Wolfgang Viechtbauer 2021/08/27 07:27 Wolfgang Viechtbauer 2021/01/16 09:55 Wolfgang Viechtbauer 2017/03/19 13:09 external edit Line 5: Line 5: 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 (1988) who 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 Becker, 1988, who 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: Line 99: Line 99: ==== 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: - rma(yi, vi, data=dat, method="FE", digits=2) + rma(yi, vi, data=dat, method="EE", digits=2) - 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 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 + Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 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. 