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", tdist=FALSE,
level=95, digits=4, btt, nAGQ=7, verbose=FALSE, control)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.escalc function for more details.escalc function for more details.escalc function for more details.escalc function for more details."OR"), the incidence rate ratio ("IRR"), the logit transformed proportion ("PLO"), or the log transfoTRUE).escalc function for more details.add should be added (either "only0", "all", "if0all", or "none"). See below and the documentation of the escalc function for more details.escalc function for more details."UM.FS" (the default), "UM.RS", "CM.EL", or "CM.AL"). See method="FE". Random/mixed-effects models are fitted by setting methodFALSE, the default) or the t-distribution (when TRUE). See FALSE). Can also be an integer. Values > 1 generate more verbose output. See c("rma.glmm","rma"). The object is a list containing the following components:0 when method="FE".model="UM.RS").print.rma.uni function. If fit statistics should also be given, use summary.rma (or use the fitstats.rma function to extract them).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.escalc function describes the data/arguments that should be specified/used for these measures.
Specifying the Model
A variety of general model types are available when analyzing $2 \times 2$ 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 effectsmodel="UM.RS"for an unconditional generalized linear mixed-effects model with random study effectsmodel="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).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 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 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., $\mathbf{X}$) with $k$ rows and $p'$ columns (e.g., using mods = cbind(mod1, mod2, mod3), where mod1, mod2, and mod3 correspond to the names of the variables for the three moderator variables). The intercept is added to the model matrix by default unless intercept=FALSE.
Alternatively, one can use the 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 term is controlled by the specified formula (e.g., mods = ~ mod1 + mod2 + mod3 - 1 would lead to the removal of the intercept term). With moderators, a fixed-effects with moderators model is then fitted by setting method="FE", while a mixed-effects model is fitted by setting method="ML".
Fixed-, Saturated-, and Random/Mixed-Effects Models
When fitting a particular model, actually up to three different models are fitted within the function:
method="ML").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 Parameters
For models including moderators, an omnibus test of all the 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, use btt=c(3,4) to only include the third and fourth coefficient from the model in the test (if an intercept is included in the model, then it corresponds to the first coefficient in the model).
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 Rhandle the coding automatically.
Tests and Confidence Intervals
By default, the test statistics of the individual coefficients in the model (and the corresponding confidence intervals) are based on the normal distribution, while the omnibus test is based on a chi-square distribution with $m$ degrees of freedom ($m$ being the number of coefficients tested). As an alternative, one can set tdist=TRUE, which slightly mimics the Knapp and Hartung (2003) method by using a t-distribution with $k-p$ degrees of freedom for tests of individual coefficients and confidence intervals and an F-distribution with $m$ and $k-p$ degrees of freedom ($p$ being the total number of model coefficients including the intercept if it is present) for the omnibus test statistic.
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.
Individual Outcomes
The various models do not require the calculation of the individual outcome values 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 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. 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 (the counts for such studies are set to NA).rma.uni, rma.mh, rma.peto, rma.mv### load BCG vaccine data
data(dat.bcg)
### 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="REML", verbose=TRUE)
### random-effects models using rma.glmm() (requires 'lme4' package)
### unconditional model with fixed study effects
rma.glmm(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg,
data=dat.bcg, model="UM.FS", verbose=TRUE)
### unconditional model with random study effects
rma.glmm(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg,
data=dat.bcg, model="UM.RS", verbose=TRUE)
### conditional model with approximate likelihood
rma.glmm(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg,
data=dat.bcg, model="CM.AL", verbose=TRUE)
### conditional model with exact likelihood
### note: fitting this model is very slow, so be patient
rma.glmm(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg,
data=dat.bcg, model="CM.EL", verbose=TRUE)Run the code above in your browser using DataLab