escalc(measure, ai, bi, ci, di, n1i, n2i, m1i, m2i, sd1i, sd2i,
xi, mi, ri, ni, data=NULL, add=1/2, to="only0", vtype="LS")m1i and m2i denote the means of the two groups, sd1i and sd2i the standard deviations of the scores in the two groups, and n1i and n2i the sample sizes of the two groups.
"MD": Theraw mean differenceis equal tom1i-m2i."SMD": Thestandardized mean differenceis equal to(m1i-m2i)/spi, wherespiis the pooled standard deviation of the two groups (which is calculated inside of the function). The standardized mean difference is automatically corrected for its slight positive bias within the function (see Hedges & Olkin, 1985). Whenvtype="LS", the sampling variances are calculated based on the large sample approximation. Alternatively, the unbiased estimates of the sampling variances can be obtained withvtype="UB".ai bi n1i
group 2 ci di n2i
}
where ai, bi, ci, and di denote the cell frequencies and n1i and n2i the row totals. For example, in a set of RCTs, group 1 and group 2 may refer to the treatment and placebo group, with outcome 1 denoting some event of interest and outcome 2 its complement. In a set of case-control studies, group 1 and group 2 may refer to the group of cases and the group of controls, with outcome 1 denoting, for example, exposure to some risk factor and outcome 2 non-exposure. The 2x2 table may also be the result of cross-sectional (i.e., multinomial) sampling, so that none of the table margins (except the total sample size) are fixed through the study design.
Depending on the type of design (sampling method), a meta-analysis of 2x2 table data can be based on one of several different outcome measures, including the odds ratio, the risk ratio (also called relative risk), the risk difference, and the arc-sine transformed risk difference. The phi coefficient, Yule's Q, and Yule's Y are additional measures of association for 2x2 table data (but they may not be the most ideal choices for meta-analyses of such data). For these measures, one needs to supply either ai, bi, ci, and di or alternatively ai, ci, n1i, and n2i. Note that the log is taken of the risk and the odds ratio, which makes these outcome measures symmetric about 0 and helps to make the distribution of these outcome measure closer to normal.
"RR": Thelog relative riskis equal to the log of(ai/n1i)/(ci/n2i)."OR": Thelog odds ratiois equal to the log of(ai*di)/(bi*di)."RD": Therisk differenceis equal to(ai/n1i)-(ci/n2i)."AS": Thearc-sine transformed risk differenceis equal toasin(sqrt(ai/n1i)) - asin(sqrt(ci/n2i)). See Ruecker et al. (2009) for a discussion of this and other outcome measures for 2x2 table data."PETO": Thelog odds ratio estimated with Peto's method(see Yusuf et al., 1985) is equal to(ai-si*n1i/ni)/((si*ti*n1i*n2i)/(ni^2*(ni-1))), wheresi=ai+ci,ti=bi+di, andni=n1i+n2i. Note that this measure technically assumes that the true odds ratio is equal to 1 in all tables."PHI": Thephi coefficientis equal to(ai*di-bi*ci)/sqrt(n1i*n2i*si*ti), wheresi=ai+ciandti=bi+di."YUQ":Yule's Qis equal to(oi-1)/(oi+1), whereoiis the odds ratio."YUY":Yule's Yis equal to(sqrt(oi)-1)/(sqrt(oi)+1), whereoiis the odds ratio.to="all", the value of add is added to each cell of the 2x2 tables in all $k$ tables. When to="only0", the value of add is added to each cell of the 2x2 tables only in those tables with at least one cell equal to 0. When to="if0all", the value of add is added to each cell of the 2x2 tables in all $k$ tables, but only when there is at least one 2x2 table with a zero entry. Setting to="none" or add=0 has the same effect: No adjustment to the observed table frequencies is made. Depending on the outcome measure and the data, this may lead to division by zero inside of the function (when this occurs, the resulting Inf value is recoded to NA).
Proportions and Transformations Thereof
When the studies provide data for a single group with respect to a dichotomous dependent variable, then the raw proportion, log transformed proportion, logit transformed proportion (i.e., log odds), the arc-sine transformed proportion, and the Freeman-Tukey double arc-sine transformed proportion are useful outcome measures. Here, one needs to specify xi and ni, denoting the number of individuals experiencing the event of interest and the total number of individuals, respectively. Instead of specifying ni, one can use mi to specify the number of individuals that do not experience the event of interest.
"PR": Theraw proportionis equal toxi/ni."PLN": Thelog transformed proportionis equal to the log ofxi/ni."PLO": Thelogit transformed proportionis equal to the log ofxi/(ni-xi)."PAS": The arc-sine transformation is a variance stabilizing transformation for proportions. Thearc-sine transformed proportionis equal toasin(sqrt(xi/ni))."PFT": Yet another variance stabilizing transformation for proportions was suggested by Freeman & Tukey (1950). TheFreeman-Tukey double arc-sine transformed proportionis equal to1/2*(asin(sqrt(xi/(ni+1))) + asin(sqrt((xi+1)/(ni+1)))).to="all", the value of add is added to xi and mi in all $k$ studies. When to="only0", the value of add is added only for studies where the xi or mi is equal to 0. When to="if0all", the value of add is added in all $k$ studies, but only when there is at least one study with a zero value for xi or mi. Setting to="none" or add=0 again means that no adjustment to the observed values is made.
Raw and Transformed Correlation Coefficient
Another frequently used outcome measure in meta-analyses is the correlation coefficient. Here, one needs to specify ri, the vector with the raw correlation coefficients, and ni, the corresponding sample sizes.
"COR": Theraw correlation coefficientis simply equal torias supplied to the function. Whenvtype="LS", the sampling variances are calculated based on the large sample approximation. Alternatively, an approximation to the unbiased estimates of the sampling variances can be obtained withvtype="UB"(see Hedges, 1989)."UCOR": Theunbiased estimate of the correlation coefficientis obtained by correcting the raw correlation coefficient for its slight negative bias (based on equation 2.7 in Olkin & Pratt, 1958). Again,vtype="LS"andvtype="UB"can be used to choose between the large sample approximations or the approximately unbiased estimates of the sampling variances."ZCOR": Fisher's r-to-z transformation is a variance stabilizing transformation for correlation coefficients with the added benefit of also being a rather effective normalizing transformation (Fisher, 1921). TheFisher's r-to-z transformed correlation coefficientis equal to1/2*log((1+ri)/(1-ri)).rma.uni, rma.mh, rma.peto### load BCG vaccine data
data(dat.bcg)
### calculate log risk ratios and corresponding sampling variances
dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)
### add log risk ratios and sampling variances to the data frame
dat <- cbind(dat.bcg, dat)
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