rma.mh: Meta-Analysis via the Mantel-Haenszel Method

Description

Function to fit fixed-effects models to $2 \times 2$ table and person-time data via the Mantel-Haenszel method. See below and the documentation of the metafor-package for more details on these models.

Usage

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)

Arguments

vector to specify the $2 \times 2$ table frequencies (upper left cell). See below and the documentation of the escalc function for more details.

vector to specify the $2 \times 2$ table frequencies (upper right cell). See below and the documentation of the escalc function for more details.

vector to specify the $2 \times 2$ table frequencies (lower left cell). See below and the documentation of the escalc function for more details.

vector to specify the $2 \times 2$ table frequencies (lower right cell). See below and the documentation of the escalc function for more details.

n1i

vector to specify the group sizes or row totals (first group). See below and the documentation of the escalc function for more details.

n2i

vector to specify the group sizes or row totals (second group). See below and the documentation of the escalc function for more details.

x1i

vector to specify the number of events (first group). See below and the documentation of the escalc function for more details.

x2i

vector to specify the number of events (second group). See below and the documentation of the escalc function for more details.

t1i

vector to specify the total person-times (first group). See below and the documentation of the escalc function for more details.

t2i

vector to specify the total person-times (second group). See below and the documentation of the escalc function for more details.

measure

character string indicating the outcome measure to use for the meta-analysis. Possible options are the odds ratio ("OR"), the relative risk ("RR"), the risk difference ("RD"), the incidence rate ratio ("IRR"

data

optional data frame containing the data supplied to the function.

slab

optional vector with unique labels for the $k$ studies.

subset

optional vector indicating the subset of tables that should be used for the analysis. This can be a logical vector of length $k$ or a numeric vector indicating the indices of the tables to include.

add

non-negative number indicating the amount to add to zero cells, counts, or frequencies when calculating the individual outcomes. Can also be a vector of two numbers, where the first number is used in the calculation of the individual outcomes and the seco

character string indicating when the values under add should be added (either "only0", "all", "if0all", or "none"). Can also be a character vector, where the first string again applies when

drop00

logical indicating whether studies with no cases/events (or only cases) in both groups should be dropped when calculating the observed outcomes of the individual studies (the outcomes for such studies are set to NA). See below and the documen

correct

logical indicating whether to apply a continuity correction when computing the Cochran-Mantel-Haenszel test statistic.

level

numerical value between 0 and 100 specifying the confidence interval level (default is 95).

digits

integer specifying the number of decimal places to which the printed results should be rounded (default is 4).

verbose

logical indicating whether output should be generated on the progress of the model fitting (default is FALSE).

Value

An object of class c("rma.mh","rma"). The object is a list containing the following components:
baggregated log odds ratio, log relative risk, risk difference, log rate ratio, or rate difference.
sestandard error of the aggregated value.
zvaltest statistics of the aggregated value.
pvalp-value for the test statistic.
ci.lblower bound of the confidence interval.
ci.ubupper bound of the confidence interval.
QEtest statistic for the test of heterogeneity.
QEpp-value for the test of heterogeneity.
MHCochran-Mantel-Haenszel test statistic (measure="OR") or Mantel-Haenszel test statistic (measure="IRR").
MHpcorresponding p-value.
TATarone's heterogeneity test statistic (only when measure="OR").
TApcorresponding p-value (only when measure="OR").
knumber of tables included in the analysis.
yi, vithe vector of individual outcomes and corresponding sampling variances.
fit.statsa list with the log-likelihood, deviance, AIC, BIC, and AICc values under the unrestricted and restricted likelihood.
...some additional elements/values.
The results of the fitted model are neatly 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'Abbe, and Baujat plots can be obtained with forest.rma, funnel.rma, radial.rma, labbe.rma, and baujat.rma.mh. 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 assessor functions include coef.rma, vcov.rma, logLik.rma, deviance.rma, AIC.rma, and BIC.rma.

Details

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 $2 \times 2$ tables of the form: lccc{ outcome 1 outcome 2 total group 1 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 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: lcc{ events person-time group 1 x1i t1i group 2 x2i t2i } where 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. 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). The Mantel-Haenszel method itself does not require the calculation of the individual outcome values 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 $2 \times 2$ 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 counts (or the number of events) when there are zero cells/events. However, for plotting and various other functions, it is necessary to calculate the individual outcome values for the $k$ studies. Here, zero cells/events can be problematic, so adding a constant value to the cell counts (or the number of events) 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 frequencies (or the number of events) and under what circumstances when calculating the individual outcome values and when applying the Mantel-Haenszel method. The documentation of the escalc function explains how the add and to arguments work. If only one value for these arguments is specified, then these values are used when calculating the individual outcomes and no adjustment to the cell counts (or the number of events) is made when applying the Mantel-Haenszel method. Alternatively, when specifying two values for these arguments, the first value applies when calculating the individual outcomes and the second value when applying the Mantel-Haenszel method. Note that drop00 is set to TRUE by default, since studies where ai=ci=0 or bi=di=0 or studies where x1i=x2i=0 are also automatically dropped when applying the Mantel-Haenszel method.

References

Cochran, W. G. (1954). Some methods for strengthening the common $\chi^2$ tests. Biometrics, 10, 417--451. 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/.

Examples

Run this code

### load BCG vaccine data
data(dat.bcg)

### meta-analysis of the (log) odds ratios using the Mantel-Haenszel method
rma.mh(ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg, measure="OR")

### meta-analysis of the (log) relative risks using the Mantel-Haenszel method
rma.mh(ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg, measure="RR")

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