The package mixmeta consists of a collection of functions to perform various meta-analytical models in R through a unified mixed-effects framework, including standard univariate fixed and random-effects meta-analysis and meta-regression, and non-standard extensions such as multivariate, multilevel, longitudinal, and dose-response models.
Standard applications of meta-analysis amount to the pooling of estimates of a single effect size, here defined generally as outcome, collected as unique observations in a set of independent studies, together with a measure of uncertainty (usually standard errors). Fixed-effects models do not assume heterogeneity across studies, and the estimates are conditional on the set of studies collected in the meta-analysis. Random-effects meta-analysis, instead, allows a degree of heterogeneity among studies, assuming the (true but unobserved) study-specific outcomes as randomly sampled from a (usually hypothetical) population of studies. Meta-regression extends both fixed and random-effects methods by allowing the pooled outcome to depend on study-level meta-predictors.
However, this traditional setting can be limited for many modern applications of meta-analysis. For instance, studies can provide estimates of different outcomes. Alternatively, studies can collect multiple estimates of the same outcome, either longitudinally or referring to different groups or levels of a continuous variable. Similarly, studies can be clustered, or being characterized by a hierarchical structure (i.e., by country). In all these instances, the key assumption of independence across estimates is not met, and basic models must be extended to consider potentially complex correlation structures within and between studies. This leads to extension to multivariate, multilevel, longitudinal, or dose-response models for meta-analysis, among others.
A unified modelling framework can be defined by casting the meta-analytical problem as a linear mixed model. In general terms, we assume that there is a set of \(n\) observations of \(k\) different outcomes, representing units of analysis aggregated in \(i = 1,\dots,m\) groups that are considered independent. An extended random-effects meta-regression model for the \(\boldsymbol{y}_i\) outcomes in group \(i\) can be generally written as:
$$\boldsymbol{y}_i = \boldsymbol{X}_i\boldsymbol{\beta} + \boldsymbol{Z}_i\boldsymbol{b} + \boldsymbol{\epsilon}_i$$ $$\boldsymbol{b} \sim \mbox{N}(\boldsymbol{0},\boldsymbol{\Psi}) , \boldsymbol{\epsilon}_i \sim \mbox{N}(\boldsymbol{0},\boldsymbol{S}_i)$$
Here, \(\boldsymbol{X}_i\boldsymbol{\beta}\) defines the fixed effects that represent the population-averaged outcomes in terms of \(p\) unit-level predictors in the design matrix \(\boldsymbol{X}_i\) with fixed-effects coefficients \(\boldsymbol{\beta}\). The random part of the model \(\boldsymbol{Z}_i\boldsymbol{b}\) describes the deviation from the population averages in terms of \(q\) unit-level predictors in the design matrix \(\boldsymbol{Z}_i\) and random-effects coefficients \(\boldsymbol{b}\). The marginal (co)variance matrix \(\boldsymbol{\Sigma}_i = \boldsymbol{Z}\boldsymbol{\Psi}\boldsymbol{Z}^t + \boldsymbol{S}_i\) is given by the sum of within (assumed known) and between-group contributions, defined by (co)variance matrices \(\boldsymbol{S}_i\) and \(\boldsymbol{\Psi}\), respectively.
All the models mentioned above, and other extensions, can be described as special cases of this unified framework. Specifically, in the standard random-effects univariate meta-analysis or meta-regression, each group represents a study with a single observation (\(n=m\)), where \(\boldsymbol{Z_i=1}\) (\(q=1\)), and \(\boldsymbol{y}_i\), \(\boldsymbol{S}_i\) and \(\boldsymbol{\Psi}\) are scalars. In fixed-effects models, \(\boldsymbol{\Psi}\) and \(\boldsymbol{Z_i}\) do not exist. In multivariate models, the \(k\)-dimensional \(\boldsymbol{y}_i\) represents the different outcomes from study \(i\), \(\boldsymbol{X}_i\) is Kronecker-expanded to \(k \times kp\), and \(\boldsymbol{S}_i\) and \(\boldsymbol{\Psi}\) are \(k \times k\) matrices representing within and between-study correlations among outcomes, respectively. In multilevel models, where additional inner levels of grouping exist within each of the \(m\) outer-level groups, \(q\) is the sum of level-specific meta-predictors, while \(\boldsymbol{\Psi}\) and \(\boldsymbol{Z}_i\) have a block-diagonal and column-binded (and expanded) forms, respectively, with each part referring to a different level. In longitudinal and dose-response models, repeated measures are accomodated in a similar way through random-effects grouping.
The aim is to estimate the \(kp\) coefficients \(\boldsymbol{\beta}\) and, for random-effects models, the components of the between-group (co)variance matrix \(\boldsymbol{\Psi}\). The parameters for the random part depend on the number of random-effects levels, and for each of them, on the number of random-effects meta-predictors and the structure of the related part of the (co)variance matrix, with a maximum of \(kq(kq+1)/2\) for single-level unstructured \(\boldsymbol{\Psi}\).
Different estimators are implemented in the package mixmeta. The options available in the current version are:
Fixed-effects
Maximum likelihood (ML)
Restricted maximum likelihood (REML)
Method of moments
Variance components
The fixed-effects model is fitted through generalized least squares (GLS), assuming the (co)variance structure, composed by the within-study errors only, as completely known. Likelihood-based random-effects estimators, ML and REML, represent the most comprehensive implementation of the modelling framework, and allow the specification all the various models described in the previous section through a flexible definition of the random-effects structure. They rely on two alternative iterative optimization procedures, based on Newton-type and (restricted) iterative generalized least squares (IGLS and RIGLS) algorithms, respectively. Estimators based on semiparametric alternatives such as the non-iterative method of moments or the iterative variance components are also included, although they are only available for models with a basic random-effects structure. Further details on estimation methods are given in the related help pages.
The main function in the package is mixmeta, which performs the various models illustrated above. This function resembles standard regression functions in R, and specifies the model through regression formulae for fixed and random-effects (see mixmetaFormula). The function returns a list object of class "mixmeta" (see mixmetaObject).
The estimation is carried out internally through mixmeta.fit, a wrapper which accepts data in a specific format, then prepares the various data components and calls ad hoc estimation functions for fitting the models. Specifically, mixmeta.fixed is applied for fixed-effects models, while estimators for random-effects models are implemented in the functions mixmeta.ml and mixmeta.reml for (restricted) maximum likelihood, mixmeta.mm for the method of moments, and mixmeta.vc for variance components. For likelihood-based methods, alternative iterative optimizations methods are provided in two sets of functions implementing Newton-type and (R)IGLS algorithms used for maximizing the (restricted) likelihood. The former method applies specific likelihood functions. Various types of likelihood-based models are defined by separate regression formulae for fixed and random-effects (see mixmetaFormula). Specific (co)variance structures for the between-group random effects at single or multiple levels are available. Fitting parameter options are set by mixmeta.control.
Method functions are available for objects of class "mixmeta" (see mixmetaObject for a complete list). The method summary produces a list of class "summary.mixmeta" for summarizing the fit of the model and providing additional results. The method function predict computes predicted values, optionally for a set of new values of the predictors. blup gives the (empirical) best linear unbiased predictions for the set of studies used for estimation. Other default or specific method functions for regression can be used on objects of class "mixmeta", such as fitted and residuals, logLik, AIC and BIC, or drop1 and add1, among others.
Methods for model.frame, model.matrix, and terms are used to extract or construct the model frame, the design matrix, or the terms of the regression meta-analytical model, respectively. These specific methods for objects of class "mixmeta" are needed to appropriately deal with missing values and to account for model frames that include terms for both the fixed and random parts. In particular, methods for na.omit and na.exclude are used to handle correctly missing values.
Simulations can be produced using the function mixmetaSim and the method function simulate, which return one or multiple sets of simulated outcomes for a group of studies. The function inputna and inputcov are used internally to augment the missing data values and to input missing correlations, respectively.
The method function qtest.mixmeta (producing an object with class of the same name) performs the (multivariate) Cochran Q test for (residual) heterogeneity. For multivariate models, the function returns both an overall estimate and those for each single outcome. The generic method function is qtest.
Printing functions for the objects of classes defined above are also provided. Other functions are used internally in the source code, and not exported in the namespace. For users interested in getting into details of the package structure, these functions can be displayed using the triple colon (':::') operator. For instance, mixmeta:::glsfit displays the code of the function glsfit. Also, some comments are added in the original source code.
The package includes several datasets used for applications of the extended meta-analytical framework. The related help pages provide examples of specific models, and fully demonstrate the flexibility of the extended meta-analytical framework. In particular:
Standard meta-analysis is illustrated using the dataset bcg, including examples of meta-regression.
Multivariate meta-analysis is performed using various datasets, including bivariate models (berkey98, hyp, p53) and multivariate models with three or more outcomes (fibrinogen and hsls). The examples describe also how to deal with missing data or missing within-group correlations, and multivariate meta-regression.
Network meta-analysis is shown in the dataset smoking. The examples illustrate an indirect mixed-treatment comparison including consistency and inconsistency models.
Multilevel meta-analysis is displayed in the examples of the datasets school and thrombolytic, and include data with multiple nested levels of grouping and/or repeated measures within each group.
Dose-response meta-analysis is illustrated in the dataset alcohol, using the recently proposed one-stage approach.
Longitudinal meta-analysis is performed using the datasets dbs and gliomas. The two sets of examples present different cases using data in wide and long format, respectively.
The mixmeta package is available on the Comprehensive R Archive Network (CRAN), with info at the related web page (CRAN.R-project.org/package=mixmeta). A development website is available on GitHub (github.com/gasparrini/mixmeta). General information on the development and applications of this extended meta-analytical modelling framework, together with an updated version of the R scripts for running the examples in published papers, can be found in GitHub (github.com/gasparrini) or at the personal web page of the package maintainer (www.ag-myresearch.com).
The package mixmeta is an extension of the package mvmeta, previously developed to perform multivariate meta-analytical models. The latter now depends on the former, and while both are still maintained, users are encouraged to switch to mixmeta as it represents a more general and updated option. A list of changes included in the current and previous versions of mixmeta can be found by typing:
news(package="mixmeta")
Use citation("mixmeta") to cite this package.
This release of the package mixmeta has been tested with different simulated and real datasets. The functions generally perform well under several scenarios, and comparisons with alternative software implementations show good agreement. However, bugs and bad performance under un-tested conditions may not be excluded. Please report any error or unexpected behaviour to the e-mail address below.
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