Differences

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--- analyses:morris2008 [2021/01/16 09:55] – Wolfgang Viechtbauer
+++ analyses:morris2008 [2022/08/03 17:05] (current) – Wolfgang Viechtbauer
@@ 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.
-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:
@@ Line 99: / Line 99: @@
 ==== 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>
-rma(yi, vi, data=dat, method="FE", digits=2)
+rma(yi, vi, data=dat, method="EE", digits=2)
 </code>
 <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
 Model Results:
-estimate       se     zval     pval    ci.lb    ci.ub
+estimate    se  zval  pval  ci.lb  ci.ub
-.95     0.14     6.62     <.01     0.67     1.23      ***
+.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
 </code>
 Note that these results are slightly different than the ones in Table 5 due to the different ways of estimating the sampling variances.
@@ Line 127: / Line 130: @@
 <code rsplus>
 sd_pool <- sqrt((with(datT, (ni-1)*sd_pre^2) + with(datC, (ni-1)*sd_pre^2)) / (datT$ni + datC$ni - 2))
-dat <- data.frame(yi = metafor:::.cmicalc(datT$ni + datC$ni - 2) * (with(datT, m_post - m_pre) - with(datC, m_post - m_pre)) / sd_pool)
+dat <- data.frame(yi = metafor:::.cmicalc(datT$ni + datC$ni - 2) *
+                       (with(datT, m_post - m_pre) - with(datC, m_post - m_pre)) / sd_pool)
 dat$vi <- 2*(1-datT$ri) * (1/datT$ni + 1/datC$ni) + dat$yi^2 / (2*(datT$ni + datC$ni))
 round(dat, 2)