# Meta-Analysis of Proportions Using Two-Step Methods Performs a meta-analysis of proportions using conventional two-step methods with various data transformations. ```r maprop.twostep(e, n, data, link = "logit", method = "ML", alpha = 0.05, pop.avg = TRUE, int.approx = 10000, b.iter = 1000, seed = 1234) ``` ## Arguments - `e`: a numeric vector specifying the event counts in the collected studies. - `n`: a numeric vector specifying the sample sizes in the collected studies. - `data`: an optional data frame containing the meta-analysis dataset. If `data` is specified, the previous arguments, `e` and `n`, should be specified as their corresponding column names in `data`. - `link`: a character string specifying the data transformation for each study's proportion used in the two-step method, which can be one of `"log"` (log transformation), `"logit"` (logit transformation, the default), `"arcsine"` (arcsine transformation), and `"double.arcsine"` (Freeman--Tukey double-arcsine transformation). - `method`: a character string specifying the method to perform the meta-analysis, which is passed to the argument `method` in the function `rma.uni` in the package `metafor`. It can be one of `"ML"` (maximum likelihood, the default), `"REML"` (restricted maximum likelihood), and many other options; see more details in the manual of `metafor`. The default is set to `"ML"` for consistency with the function `maprop.glmm`, where generalized linear mixed models are often estimated via the maximum likelihood approach. For the two-step method, users might also use `"REML"` because the restricted maximum likelihood estimation may have superior performance in many cases. - `alpha`: a numeric value specifying the statistical significance level. - `pop.avg`: a logical value indicating whether the population-averaged proportion and its confidence interval are to be produced. This quantity is the marginal mean of study-specific proportions, while the commonly-reported overall proportion usually represents the median (or interpreted as a conditional measure); see more details about this quantity in Section 13.2.3 in Agresti (2013), Chu et al.(2012), Lin and Chu (2020), and Zeger et al.(1988). If `pop.avg` = `TRUE` (the default), the bootstrap resampling is used to produce the confidence interval of the population-averaged proportion; the confidence interval of the commonly-reported median proportion will be also produced, in addition to its conventional confidence interval (by back-transforming the Wald-type confidence interval derived on the scale specified by `link`). - `int.approx`: an integer specifying the number of independent standard normal samples for numerically approximating the integration involved in the calculation of the population-averaged proportion; see details in Lin and Chu (2020). It is only used when `pop.avg` = `TRUE`. For the commonly-used data transformations available for `link`, the population-averaged proportion does not have a closed form. - `b.iter`: an integer specifying the number of bootstrap iterations; it is only used when `pop.avg` = `TRUE`. - `seed`: an integer for specifying the seed of the random number generation for reproducibility during the bootstrap resampling (and numerical approximation for the population-averaged proportion); it is only used when `pop.avg` = `TRUE`. ## Returns This function returns a list containing the point and interval estimates of the overall proportion. Specifically, `prop.c.est` is the commonly-reported median (or conditional) proportion, and `prop.c.ci` is its confidence interval. If `pop.avg` = `TRUE`, the following additional elements will be also in the produced list: `prop.c.ci.b` is the bootstrap confidence interval of the commonly-reported median (conditional) proportion, `prop.m.est` is the point estimate of the population-averaged (marginal) proportion, `prop.m.ci.b` is the bootstrap confidence interval of the population-averaged (marginal) proportion, and `b.w.e` is a vector of two numeric values, indicating the counts of warnings and errors occurred during the bootstrap iterations. Moreover, if the Freeman--Tukey double-arcsine transformation (`link` = `"double.arcsine"`) is used, the back-transformation will be implemented at four values as the overall sample size: the harmonic, geometric, and arithmetic means of the study-specific sample sizes, and the inverse of the synthesized result's variance. See details in Barendregt et al.(2013) and Schwarzer et al.(2019). ## References Agresti A (2013). **Categorical Data Analysis**. Third edition. John Wiley & Sons, Hoboken, NJ. Barendregt JJ, Doi SA, Lee YY, Norman RE, Vos T (2013). "Meta-analysis of prevalence." **Journal of Epidemiology and Community Health**, 67 (11), 974--978. Freeman MF, Tukey JW (1950). "Transformations related to the angular and the square root." **The Annals of Mathematical Statistics**, 21 (4), 607--611. Lin L, Chu H (2020). "Meta-analysis of proportions using generalized linear mixed models." **Epidemiology**, 31 (5), 713--717. Miller JJ (1978). "The inverse of the Freeman--Tukey double arcsine transformation." **The American Statistician**, 32 (4), 138. Schwarzer G, Chemaitelly H, Abu-Raddad LJ, Rucker G (2019). "Seriously misleading results using inverse of Freeman-Tukey double arcsine transformation in meta-analysis of single proportions." **Research Synthesis Methods**, 10 (3), 476--483. Viechtbauer W (2010). "Conducting meta-analyses in R with the metafor package." **Journal of Statistical Software**, 36 , 3. Zeger SL, Liang K-Y, Albert PS (1988). "Models for longitudinal data: a generalized estimating equation approach." **Biometrics**, 44 (4), 1049--1060. ## Note This function implements the two-step method for the meta-analysis of proportions via the `rma.uni` function in the package `metafor`. It is possible that the algorithm of the maximum likelihood or restricted maximum likelihood estimation may not converge for some bootstrapped meta-analyses when `pop.avg` = `TRUE`, and the `rma.uni` function may report warnings or errors about the convergence issue. The bootstrap iterations are continued until `b.iter` replicates without any warnings or errors are obtained; those replicates with any warnings or errors are discarded. ## See Also `maprop.glmm` ## Examples ```r # chorioamnionitis data data("dat.chor") # two-step method with the logit transformation out.chor.twostep.logit <- maprop.twostep(e, n, data = dat.chor, link = "logit", b.iter = 10, seed = 1234) out.chor.twostep.logit # not calculating the population-averaged (marginal) proportion, # without bootstrap resampling out.chor.twostep.logit <- maprop.twostep(e, n, data = dat.chor, link = "logit", pop.avg = FALSE) out.chor.twostep.logit # increases the number of bootstrap iterations to 1000, # taking longer time out.chor.twostep.logit <- maprop.twostep(e, n, data = dat.chor, link = "logit", b.iter = 1000, seed = 1234) out.chor.twostep.logit # two-step method with the log transformation out.chor.twostep.log <- maprop.twostep(e, n, data = dat.chor, link = "log", b.iter = 10, seed = 1234) out.chor.twostep.log # two-step method with the arcsine transformation out.chor.twostep.arcsine <- maprop.twostep(e, n, data = dat.chor, link = "arcsine", b.iter = 10, seed = 1234) out.chor.twostep.arcsine # two-step method with the Freeman--Tukey double-arcsine transformation out.chor.twostep.double.arcsine <- maprop.twostep(e, n, data = dat.chor, link = "double.arcsine", b.iter = 10, seed = 1234) out.chor.twostep.double.arcsine # depression data data("dat.beck17") out.beck17.twostep.log <- maprop.twostep(e, n, data = dat.beck17, link = "log", b.iter = 10, seed = 1234) out.beck17.twostep.log out.beck17.twostep.logit <- maprop.twostep(e, n, data = dat.beck17, link = "logit", b.iter = 10, seed = 1234) out.beck17.twostep.logit out.beck17.twostep.arcsine <- maprop.twostep(e, n, data = dat.beck17, link = "arcsine", b.iter = 10, seed = 1234) out.beck17.twostep.arcsine out.beck17.twostep.double.arcsine <- maprop.twostep(e, n, data = dat.beck17, link = "double.arcsine", b.iter = 10, seed = 1234) out.beck17.twostep.double.arcsine ```

maprop.twostep function