rma.mh(ai, bi, ci, di, n1i, n2i, x1i, x2i, t1i, t2i, measure="OR", data, slab, subset, add=1/2, to="only0", drop00=TRUE, correct=TRUE, level=95, digits=4, verbose=FALSE)escalc function for more details.escalc function for more details.escalc function for more details.escalc function for more details.escalc function for more details.escalc function for more details.escalc function for more details.escalc function for more details.escalc function for more details.escalc function for more details."OR"), the relative risk ("RR"), the risk difference ("RD"), the incidence rate ratio ("IRR"), or the incidence rate difference ("IRD").escalc function for more details.add should be added (either "only0", "all", "if0all", or "none"). Can also be a character vector, where the first string again applies when calculating the observed outcomes and the second string when applying the Mantel-Haenszel method. See below and the documentation of the escalc function for more details.NA). Can also be a vector of two logicals, where the first applies to the calculation of the observed outcomes and the second when applying the Mantel-Haenszel method. See below and the documentation of the escalc function for more details.FALSE).c("rma.mh","rma"). The object is a list containing the following components:The results of the fitted model are formated and printed with the print.rma.mh function. If fit statistics should also be given, use summary.rma (or use the fitstats.rma function to extract them).The residuals.rma, rstandard.rma.mh, and rstudent.rma.mh functions extract raw and standardized residuals. Leave-one-out diagnostics can be obtained with leave1out.rma.mh.Forest, funnel, radial, L'Abbé, and Baujat plots can be obtained with forest.rma, funnel.rma, radial.rma, labbe.rma, and baujat. The qqnorm.rma.mh function provides normal QQ plots of the standardized residuals. One can also just call plot.rma.mh on the fitted model object to obtain various plots at once.A cumulative meta-analysis (i.e., adding one obervation at a time) can be obtained with cumul.rma.mh.Other extractor functions include coef.rma, vcov.rma, logLik.rma, deviance.rma, AIC.rma, and BIC.rma.
Specifying the Data
When the outcome measure is either the odds ratio (measure="OR"), relative risk (measure="RR"), or risk difference (measure="RD"), the studies are assumed to provide data in terms of $2x2$ tables of the form:
| outcome 1 | outcome 2 | |
| total | group 1 | ai |
bi |
n1i |
ai, bi, ci, and di denote the cell frequencies and n1i and n2i the row totals. For example, in a set of randomized clinical trials (RCTs) or cohort studies, group 1 and group 2 may refer to the treatment (exposed) and placebo/control (not exposed) group, with outcome 1 denoting some event of interest (e.g., death) 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. For these outcome measures, one needs to specify either ai, bi, ci, and di or alternatively ai, ci, n1i, and n2i. Alternatively, when the outcome measure is the incidence rate ratio (measure="IRR") or the incidence rate difference (measure="IRD"), the studies are assumed to provide data in terms of tables of the form:
| events | |
| person-time | group 1 |
x1i |
t1i |
x1i and x2i denote the number of events in the first and the second group, respectively, and t1i and t2i the corresponding total person-times at risk.Mantel-Haenszel Method
An approach for aggregating table data of these types was suggested by Mantel and Haenszel (1959) and later extended by various authors (see references). The Mantel-Haenszel method provides a weighted estimate under a fixed-effects model. The method is particularly advantageous when aggregating a large number of studies with small sample sizes (the so-called sparse data or increasing strata case).
When analyzing odds ratios, the Cochran-Mantel-Haenszel (CMH) test (Cochran, 1954; Mantel & Haenszel, 1959) and Tarone's test for heterogeneity (Tarone, 1985) are also provided (by default, the CMH test statistic is computed with the continuity correction; this can be switched off with correct=FALSE). When analyzing incidence rate ratios, the Mantel-Haenszel (MH) test (Rothman et al., 2008) for person-time data is also provided (again, the correct argument controls whether the continuity correction is applied). When analyzing odds ratios, relative risks, or incidence rate ratios, the printed results are given both in terms of the log and the raw units (for easier interpretation).
Observed Outcomes of the Individual Studies
The Mantel-Haenszel method itself does not require the calculation of the observed outcomes of the individual studies (e.g., the observed odds or incidence rate ratios of the $k$ studies) and directly makes use of the table/event counts. Zero cells/events are not a problem (except in extreme cases, such as when one of the two outcomes never occurs in any of the $2x2$ tables or when there are no events for one of the two groups in any of the tables). Therefore, it is unnecessary to add some constant to the cell/event counts when there are zero cells/events.
However, for plotting and various other functions, it is necessary to calculate the observed outcomes for the $k$ studies. Here, zero cells/events can be problematic, so adding a constant value to the cell/event counts ensures that all $k$ values can be calculated. The add and to arguments are used to specify what value should be added to the cell/event counts and under what circumstances when calculating the observed outcomes and when applying the Mantel-Haenszel method. Similarly, the drop00 argument is used to specify how studies with no cases/events (or only cases) in both groups should be handled. The documentation of the escalc function explains how the add, to, and drop00 arguments work. If only a single value for these arguments is specified (as per default), then these values are used when calculating the observed outcomes and no adjustment to the cell/event counts is made when applying the Mantel-Haenszel method. Alternatively, when specifying two values for these arguments, the first value applies when calculating the observed outcomes and the second value when applying the Mantel-Haenszel method.
Note that drop00 is set to TRUE by default. Therefore, the observed outcomes for studies where ai=ci=0 or bi=di=0 or studies where x1i=x2i=0 are set to NA. When applying the Mantel-Haenszel method, such studies are not explicitly dropped (unless the second value of drop00 argument is also set to TRUE), but this is practically not necessary, as they do not actually influence the results (assuming no adjustment to the cell/event counts are made when applying the Mantel-Haenszel method).
Greenland, S., & Robins, J. M. (1985). Estimation of a common effect parameter from sparse follow-up data. Biometrics, 41, 55--68.
Mantel, N., & Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22, 719--748.
Nurminen, M. (1981). Asymptotic efficiency of general noniterative estimators of common relative risk. Biometrika, 68, 525--530.
Robins, J., Breslow, N., & Greenland, S. (1986). Estimators of the Mantel-Haenszel variance consistent in both sparse data and large-strata limiting models. Biometrics, 42, 311--323.
Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern epidemiology (3rd ed.). Philadelphia: Lippincott Williams & Wilkins.
Sato, T., Greenland, S., & Robins, J. M. (1989). On the variance estimator for the Mantel-Haenszel risk difference. Biometrics, 45, 1323--1324.
Tarone, R. E. (1981). On summary estimators of relative risk. Journal of Chronic Diseases, 34, 463--468.
Tarone, R. E. (1985). On heterogeneity tests based on efficient scores. Biometrika, 72, 91--95.
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1--48. http://www.jstatsoft.org/v36/i03/.
rma.uni, rma.glmm, rma.peto, and rma.mv for other model fitting functions.
### meta-analysis of the (log) odds ratios using the Mantel-Haenszel method
rma.mh(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)
### meta-analysis of the (log) relative risks using the Mantel-Haenszel method
rma.mh(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)
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