Function to fit meta-analytic fixed- and random/mixed-effects models with or without moderators via generalized linear (mixed-effects) models. See below and the documentation of the metafor-package for more details on these models.
rma.glmm(ai, bi, ci, di, n1i, n2i, x1i, x2i, t1i, t2i, xi, mi, ti, ni,
mods, measure, intercept=TRUE, data, slab, subset,
add=1/2, to="only0", drop00=TRUE, vtype="LS",
model="UM.FS", method="ML", test="z",
level=95, digits, btt, nAGQ=7, verbose=FALSE, control, …)see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
see below and the documentation of the escalc function for more details.
optional argument to include one or more moderators in the model. A single moderator can be given as a vector of length k specifying the values of the moderator. Multiple moderators are specified by giving a matrix with k rows and as many columns as there are moderator variables. Alternatively, a model formula can be used to specify the model. See ‘Details’.
character string to specify the outcome measure to use for the meta-analysis. Possible options are the odds ratio ("OR"), the incidence rate ratio ("IRR"), the logit transformed proportion ("PLO"), or the log transformed incidence rate ("IRLN").
logical to specify whether an intercept should be added to the model (the default is TRUE).
optional data frame containing the data supplied to the function.
optional vector with labels for the k studies.
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, counts, or frequencies when calculating the observed effect sizes or outcomes of the individual studies. See below and the documentation of the escalc function for more details.
character string to specify when the values under add should be added (either "only0", "all", "if0all", or "none"). See below and the documentation of the escalc function for more details.
logical to specify whether studies with no cases/events (or only cases) in both groups should be dropped. See the documentation of the escalc function for more details.
character string to specify the type of sampling variances to calculate when calculating the observed effect sizes or outcomes. See the documentation of the escalc function for more details.
character string to specify the general model type to use for the analysis (either "UM.FS" (the default), "UM.RS", "CM.EL", or "CM.AL"). See ‘Details’.
character string to specify whether a fixed- or a random/mixed-effects model should be fitted. A fixed-effects model (with or without moderators) is fitted when using method="FE". Random/mixed-effects models are fitted by setting method="ML" (the default). See ‘Details’.
character string to specify how test statistics and confidence intervals for the fixed effects should be computed. By default (test="z"), Wald-type tests and CIs are obtained, which are based on a standard normal distribution. When test="t", a t-distribution is used instead. See ‘Details’.
numeric value between 0 and 100 to specify the confidence interval level (the default is 95).
integer to specify the number of decimal places to which the printed results should be rounded. If unspecified, the default is 4.
optional vector of indices to specify which coefficients to include in the omnibus test of moderators. Can also be a string to grep for. See ‘Details’.
positive integer to specify the number of points per axis for evaluating the adaptive Gauss-Hermite approximation to the log-likelihood. The default is 7. Setting this to 1 corresponds to the Laplacian approximation. See ‘Note’.
logical to specify whether output should be generated on the progress of the model fitting (the default is FALSE). Can also be an integer. Values > 1 generate more verbose output. See ‘Note’.
optional list of control values for the estimation algorithms. If unspecified, default values are defined inside the function. See ‘Note’.
additional arguments.
An object of class c("rma.glmm","rma"). The object is a list containing the following components:
estimated coefficients of the model.
standard errors of the coefficients.
test statistics of the coefficients.
corresponding p-values.
lower bound of the confidence intervals for the coefficients.
upper bound of the confidence intervals for the coefficients.
variance-covariance matrix of the estimated coefficients.
estimated amount of (residual) heterogeneity. Always 0 when method="FE".
estimated amount of study level variability (only for model="UM.RS").
number of studies included in the analysis.
number of coefficients in the model (including the intercept).
number of coefficients included in the omnibus test of moderators.
Wald-type test statistic of the test for (residual) heterogeneity.
corresponding p-value.
likelihood ratio test statistic of the test for (residual) heterogeneity.
corresponding p-value.
test statistic of the omnibus test of moderators.
corresponding p-value.
value of I^2.
value of H^2.
logical that indicates whether the model is an intercept-only model.
the vector of outcomes, the corresponding sampling variances, and the model matrix.
a list with the log-likelihood, deviance, AIC, BIC, and AICc values.
some additional elements/values.
The results of the fitted model are formatted and printed with the print.rma.glmm function. If fit statistics should also be given, use summary.rma (or use the fitstats.rma function to extract them).
Specifying the Data
The function can be used in conjunction with the following effect size or outcome measures:
measure="OR" for odds ratios (analyzed in log units)
measure="IRR" for incidence rate ratios (analyzed in log units)
measure="PLO" for logit transformed proportions (i.e., log odds)
measure="IRLN" for log transformed incidence rates.
The escalc function describes the data/arguments that should be specified/used for these measures.
Specifying the Model
A variety of model types are available when analyzing 2 22x2 table data (i.e., when measure="OR") or two-group event count data (i.e., when measure="IRR"):
model="UM.FS" for an unconditional generalized linear mixed-effects model with fixed study effects
model="UM.RS" for an unconditional generalized linear mixed-effects model with random study effects
model="CM.AL" for a conditional generalized linear mixed-effects model (approximate likelihood)
model="CM.EL" for a conditional generalized linear mixed-effects model (exact likelihood).
For measure="OR", models "UM.FS" and "UM.RS" are essentially (mixed-effects) logistic regression models, while for measure="IRR", these models are (mixed-effects) Poisson regression models. A choice must be made on how to model study level variability (i.e., differences in outcomes across studies irrespective of group membership). One can choose between using fixed study effects (which means that k dummy variables are added to the model) or random study effects (which means that random effects corresponding to the levels of the study factor are added to the model).
The conditional model (model="CM.EL") avoids having to model study level variability by conditioning on the total numbers of cases/events in each study. For measure="OR", this leads to a non-central hypergeometric distribution for the data within each study and the corresponding model is then a (mixed-effects) conditional logistic model. Fitting this model can be difficult and computationally expensive. When the number of cases in each study is small relative to the group sizes, one can approximate the exact likelihood by a binomial distribution, which leads to a regular (mixed-effects) logistic regression model (model="CM.AL"). For measure="IRR", the conditional model leads directly to a binomial distribution for the data within each study and the resulting model is again a (mixed-effects) logistic regression model (no approximate likelihood model is needed here).
When analyzing proportions (i.e., measure="PLO") or incidence rates (i.e., measure="IRLN") of individual groups, the model type is always a (mixed-effects) logistic or Poisson regression model, respectively (i.e., the model argument is not relevant here).
Aside from choosing the general model type, one has to decide whether to fit a fixed- or random-effects model to the data. A fixed-effects model is fitted by setting method="FE". A random-effects model is fitted by setting method="ML" (the default). Note that random-effects models with dichotomous data are often referred to as ‘binomial-normal’ models in the meta-analytic literature. Analogously, for event count data, such models could be referred to as ‘Poisson-normal’ models.
One or more moderators can be included in all of these models via the mods argument. A single moderator can be given as a (row or column) vector of length k specifying the values of the moderator. Multiple moderators are specified by giving an appropriate model matrix (i.e., X) with k rows and as many columns as there are moderator variables (e.g., mods = cbind(mod1, mod2, mod3), where mod1, mod2, and mod3 correspond to the names of the variables for three moderator variables). The intercept is added to the model matrix by default unless intercept=FALSE.
Alternatively, one can use standard formula syntax to specify the model. In this case, the mods argument should be set equal to a one-sided formula of the form mods = ~ model (e.g., mods = ~ mod1 + mod2 + mod3). Interactions, polynomial terms, and factors can be easily added to the model in this manner. When specifying a model formula via the mods argument, the intercept argument is ignored. Instead, the inclusion/exclusion of the intercept is controlled by the specified formula (e.g., mods = ~ mod1 + mod2 + mod3 - 1 would lead to the removal of the intercept).
Fixed-, Saturated-, and Random/Mixed-Effects Models
When fitting a particular model, actually up to three different models are fitted within the function:
the fixed-effects model (i.e., where ^2 is set to 0),
the saturated model (i.e., the model with a deviance of 0), and
the random/mixed-effects model (i.e., where ^2 is estimated) (only if method="ML").
The saturated model is obtained by adding as many dummy variables to the model as needed so that the model deviance is equal to zero. Even when method="ML", the fixed-effects and saturated models are fitted, as they are used to compute the test statistics for the Wald-type and likelihood ratio tests for (residual) heterogeneity (see below).
Omnibus Test of Moderators
For models including moderators, an omnibus test of all model coefficients is conducted that excludes the intercept (the first coefficient) if it is included in the model. If no intercept is included in the model, then the omnibus test includes all of the coefficients in the model including the first. Alternatively, one can manually specify the indices of the coefficients to test via the btt argument. For example, with btt=c(3,4), only the third and fourth coefficient from the model would be included in the test (if an intercept is included in the model, then it corresponds to the first coefficient in the model). Instead of specifying the coefficient numbers, one can specify a string for btt. In that case, grep will be used to search for all coefficient names that match the string. The omnibus test is called the Q_M-test and follows, under the assumptions of the model, a chi-square distribution with m degrees of freedom (with m denoting the number of coefficients tested) under the null hypothesis (that the true value of all coefficients tested is equal to 0).
Categorical Moderators
Categorical moderator variables can be included in the model via the mods argument in the same way that appropriately (dummy) coded categorical independent variables can be included in linear models. One can either do the dummy coding manually or use a model formula together with the factor function to let R handle the coding automatically (note that string/character variables in a model formula are automatically converted to factors).
Tests and Confidence Intervals
By default, tests of individual coefficients in the model (and the corresponding confidence intervals) are based on a standard normal distribution, while the omnibus test is based on a chi-square distribution (see above). As an alternative, one can set test="t", in which case tests of individual coefficients and confidence intervals are based on a t-distribution with k-p degrees of freedom, while the omnibus test statistic then uses an F-distribution with m and k-p degrees of freedom (with k denoting the total number of estimates included in the analysis and p the total number of model coefficients including the intercept if it is present). Note that test="t" is not the same as test="knha" in rma.uni, as no adjustment to the standard errors of the estimated coefficients is made.
Tests for (Residual) Heterogeneity
Two different tests for (residual) heterogeneity are automatically carried out by the function. The first is a Wald-type test, which tests the coefficients corresponding to the dummy variables added in the saturated model for significance. The second is a likelihood ratio test, which tests the same set of coefficients, but does so by computing -2 times the difference in the log-likelihood of the fixed-effects and the saturated model. These two tests are not identical for the types of models fitted by the rma.glmm function and may even lead to conflicting conclusions.
Observed Effect Sizes or Outcomes of the Individual Studies
The various models do not require the calculation of the observed effect sizes or outcomes of the individual studies (e.g., the observed odds ratios of the k studies) and directly make 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 or when there are no events in any of the studies). 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 or 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 effect sizes or outcomes. The documentation of the escalc function explains how the add and to arguments work. 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 uninformative about the size of the effect.
Agresti, A. (2002). Categorical data analysis (2nd. ed). Hoboken, NJ: Wiley.
Bagos, P. G., & Nikolopoulos, G. K. (2009). Mixed-effects Poisson regression models for meta-analysis of follow-up studies with constant or varying durations. The International Journal of Biostatistics, 5(1). https://doi.org/10.2202/1557-4679.1168
van Houwelingen, H. C., Zwinderman, K. H., & Stijnen, T. (1993). A bivariate approach to meta-analysis. Statistics in Medicine, 12(24), 2273--2284. https://doi.org/10.1002/sim.4780122405
Liao, J. G., & Rosen, O. (2001). Fast and stable algorithms for computing and sampling from the noncentral hypergeometric distribution. American Statistician, 55(4), 366--369. https://doi.org/10.1198/000313001753272547
Stijnen, T., Hamza, T. H., & Ozdemir, P. (2010). Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29(29), 3046--3067. https://doi.org/10.1002/sim.4040
Turner, R. M., Omar, R. Z., Yang, M., Goldstein, H., & Thompson, S. G. (2000). A multilevel model framework for meta-analysis of clinical trials with binary outcomes. Statistics in Medicine, 19(24), 3417--3432. https://doi.org/10.1002/1097-0258(20001230)19:24<3417::aid-sim614>3.0.co;2-l
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.mh, rma.peto, and rma.mv for other model fitting functions.
dat.nielweise2007, dat.nielweise2008, dat.collins1985a, and dat.pritz1997 for further examples of the use of the rma.glmm function.
# NOT RUN {
### random-effects model using rma.uni() (standard RE model analysis)
rma(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg, method="ML")
### random-effects models using rma.glmm() (require 'lme4' package)
### unconditional model with fixed study effects
# }
# NOT RUN {
rma.glmm(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg, model="UM.FS")
# }
# NOT RUN {
### unconditional model with random study effects
# }
# NOT RUN {
rma.glmm(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg, model="UM.RS")
# }
# NOT RUN {
### conditional model with approximate likelihood
# }
# NOT RUN {
rma.glmm(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg, model="CM.AL")
# }
# NOT RUN {
### conditional model with exact likelihood
### note: fitting this model may take a bit of time, so be patient
# }
# NOT RUN {
rma.glmm(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg, model="CM.EL")
# }
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