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

rma.mh(ai, bi, ci, di, n1i, n2i, x1i, x2i, t1i, t2i,
measure="OR", data, slab, subset,
correct=TRUE, level=95, verbose=FALSE, digits, ...)

## Arguments

ai

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

bi

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

ci

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

di

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

n1i

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

n2i

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

x1i

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

x2i

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

t1i

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

t2i

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

measure

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

data

optional data frame containing the data supplied to the function.

slab

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

subset

optional (logical or numeric) vector to specify the subset of studies that should be used for the analysis.

non-negative number to specify the amount to add to zero cells or even counts when calculating the observed effect sizes of the individual studies. Can also be a vector of two numbers, where the first number is used in the calculation of the observed effect sizes and the second number is used when applying the Mantel-Haenszel method. See below and the documentation of the escalc function for more details.

to

character string to specify 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 calculating the observed effect sizes or outcomes and the second string when applying the Mantel-Haenszel method. See below and the documentation of the escalc function for more details.

drop00

logical to specify whether studies with no cases/events (or only cases) in both groups should be dropped when calculating the observed effect sizes or outcomes (the outcomes for such studies are set to NA). Can also be a vector of two logicals, where the first applies to the calculation of the observed effect sizes or outcomes and the second when applying the Mantel-Haenszel method. See below and the documentation of the escalc function for more details.

correct

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

level

numeric value between 0 and 100 to specify the confidence interval level (the default is 95; see here for details).

verbose

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

digits

optional integer to specify the number of decimal places to which the printed results should be rounded. If unspecified, the default is 4. See also here for further details on how to control the number of digits in the output.

...

## Details

### Specifying the Data

When the outcome measure is either the risk ratio (measure="RR"), odds ratio (measure="OR"), or risk difference (measure="RD"), the studies are assumed to provide data in terms of $$2 \times 2$$ tables of the form:

 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/non-exposed group, respectively, 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 the cell frequencies via the ai, bi, ci, and di arguments (or alternatively, one can use the ai, ci, n1i, and n2i arguments).

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 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.

### Mantel-Haenszel Method

An approach for aggregating 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 an equal-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 risk ratios, odds ratios, or incidence rate ratios, the printed results are given both in terms of the log and the raw units (for easier interpretation).

### Observed Effect Sizes or Outcomes of the Individual Studies

The Mantel-Haenszel method itself does not require the calculation of the observed effect sizes of the individual studies (e.g., the observed log odds ratios of the $$k$$ studies) and directly makes use of the cell/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/event counts when there are zero cells/events.

However, for plotting and various other functions, it is necessary to calculate the observed effect sizes 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 effect sizes 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 effect sizes 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 effect sizes and the second value when applying the Mantel-Haenszel method.

Note that drop00 is set to TRUE by default. Therefore, the observed effect sizes 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).

## Value

An object of class c("rma.mh","rma"). The object is a list containing the following components:

beta

aggregated log risk ratio, log odds ratio, risk difference, log rate ratio, or rate difference.

se

standard error of the aggregated value.

zval

test statistics of the aggregated value.

pval

corresponding p-value.

ci.lb

lower bound of the confidence interval.

ci.ub

upper bound of the confidence interval.

QE

test statistic of the test for heterogeneity.

QEp

correspinding p-value.

MH

Cochran-Mantel-Haenszel test statistic (measure="OR") or Mantel-Haenszel test statistic (measure="IRR").

MHp

corresponding p-value.

TA

test statistic of Tarone's test for heterogeneity (only when measure="OR").

TAp

corresponding p-value (only when measure="OR").

k

number of studies included in the analysis.

yi, vi

the vector of outcomes and corresponding sampling variances.

fit.stats

a list with the log-likelihood, deviance, AIC, BIC, and AICc values under the unrestricted and restricted likelihood.

...

## Methods

The results of the fitted model are formatted and printed with the print function. If fit statistics should also be given, use summary (or use the fitstats function to extract them).

The residuals, rstandard, and rstudent functions extract raw and standardized residuals. Leave-one-out diagnostics can be obtained with leave1out.

Forest, funnel, radial, L'Abbé, and Baujat plots can be obtained with forest, funnel, radial, labbe, and baujat. The qqnorm function provides normal QQ plots of the standardized residuals. One can also just call plot on the fitted model object to obtain various plots at once.

A cumulative meta-analysis (i.e., adding one observation at a time) can be obtained with cumul.

Other extractor functions include coef, vcov, logLik, deviance, AIC, and BIC.

## Author

Wolfgang Viechtbauer (wvb@metafor-project.org, https://www.metafor-project.org).

## References

Cochran, W. G. (1954). Some methods for strengthening the common $$\chi^2$$ tests. Biometrics, 10(4), 417–451. https://doi.org/10.2307/3001616

Greenland, S., & Robins, J. M. (1985). Estimation of a common effect parameter from sparse follow-up data. Biometrics, 41(1), 55–68. https://doi.org/10.2307/2530643

Mantel, N., & Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22(4), 719–748. https://doi.org/10.1093/jnci/22.4.719

Nurminen, M. (1981). Asymptotic efficiency of general noniterative estimators of common relative risk. Biometrika, 68(2), 525–530. https://doi.org/10.1093/biomet/68.2.525

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(2), 311–323. https://doi.org/10.2307/2531052

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(4), 1323–1324. https://www.jstor.org/stable/2531784

Tarone, R. E. (1981). On summary estimators of relative risk. Journal of Chronic Diseases, 34(9-10), 463–468. https://doi.org/10.1016/0021-9681(81)90006-0

Tarone, R. E. (1985). On heterogeneity tests based on efficient scores. Biometrika, 72(1), 91–95. https://doi.org/10.1093/biomet/72.1.91

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1–48. https://doi.org/10.18637/jss.v036.i03

rma.uni, rma.glmm, rma.peto, and rma.mv for other model fitting functions.

## Examples

### 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)
#>
#> Equal-Effects Model (k = 13)
#>
#> I^2 (total heterogeneity / total variability):  92.68%
#> H^2 (total variability / sampling variability): 13.66
#>
#> Test for Heterogeneity:
#> Q(df = 12) = 163.9426, p-val < .0001
#>
#> Model Results (log scale):
#>
#> estimate      se      zval    pval    ci.lb    ci.ub
#>  -0.4734  0.0410  -11.5444  <.0001  -0.5538  -0.3930
#>
#> Model Results (OR scale):
#>
#> estimate   ci.lb   ci.ub
#>   0.6229  0.5748  0.6750
#>
#> Cochran-Mantel-Haenszel Test:    CMH = 135.6889, df = 1,  p-val < 0.0001
#> Tarone's Test for Heterogeneity: X^2 = 171.7567, df = 12, p-val < 0.0001
#>

### meta-analysis of the (log) risk ratios using the Mantel-Haenszel method
rma.mh(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)
#>
#> Equal-Effects Model (k = 13)
#>
#> I^2 (total heterogeneity / total variability):  92.13%
#> H^2 (total variability / sampling variability): 12.71
#>
#> Test for Heterogeneity:
#> Q(df = 12) = 152.5676, p-val < .0001
#>
#> Model Results (log scale):
#>
#> estimate      se      zval    pval    ci.lb    ci.ub
#>  -0.4537  0.0393  -11.5338  <.0001  -0.5308  -0.3766
#>
#> Model Results (RR scale):
#>
#> estimate   ci.lb   ci.ub
#>   0.6353  0.5881  0.6862
#>