These functions compute the value of the log-likelihood and the related vectors of first partial derivatives for random-effects multivariate and univariate meta-analysis and meta-regression, in terms of model parameters. They are meant to be used internally and not directly run by the users.
mlprof.fn(par, Xlist, ylist, Slist, nalist, k, m, p, nall, bscov, ctrl)
mlprof.gr(par, Xlist, ylist, Slist, nalist, k, m, p, nall, bscov, ctrl)remlprof.fn(par, Xlist, ylist, Slist, nalist, k, m, p, nall, bscov, ctrl)
remlprof.gr(par, Xlist, ylist, Slist, nalist, k, m, p, nall, bscov, ctrl)
iter.igls(Psi, Xlist, ylist, Slist, nalist, k, m)
a vector representing the random-effects parameters defining the between-study (co)variance matrix.
a \(k \times k\) matrix representing the current estimate of the between-study (co)variance matrix.
a \(m\)-dimensional list of study-specific design matrices for the fixed-effects part of the model. Rows corresponding to missing outcomes have been excluded.
a \(m\)-dimensional list of study-specific of vectors of estimated outcomes. Entries corresponding to missing outcomes have been excluded.
a \(m\)-dimensional list of within-study (co)variance matrices of estimated outcomes. Rows and columns corresponding to missing outcomes have been excluded.
a \(m\)-dimensional list of \(k\)-dimensional study-specific logical vectors, identifying missing outcomes.
numeric scalars: number of outcomes, number of studies included in estimation (equal to the length of lists above), number of predictors (including the intercept), number of observations (excluding missing).
a string defining the between-study (co)variance structure in likelihood based models. See Details.
list of parameters for controlling the fitting process, usually internally set to default values by mvmeta.control. The name is chosen to avoid conflicts with the argument control in optim.
mlprof.fn and remlprof.fn return the value of the (restricted) log-likelihood for a given set of parameters in par. mlprof.gr and remlprof.gr return instead the related vector of first partial derivatives. iter.igls returns an updated estimate of Psi given its initial value or the value at the previous iteration.
These functions are called internally by the fitting functions mvmeta.ml and mvmeta.reml to perform iterative optimization algorithms for estimating random effects meta-analytical models.
The maximization of the (restricted) likelihood starts with few runs of an iterative generalized least square algorithm implemented in iter.igls. This can be regarded as a fast and stable way to get starting values close to the maximum for the Quasi-Newton iterative algorithm, implemented in optim. Alternatively, starting values can be provided by the user in the control list (see mvmeta.control). The function optim requires the algorithms to compute the value of the (restricted) likelihood and (optionally) the vector of its first partial derivatives, provided by the related likelihood functions.
These functions actually specify the profiled version of the (restricted) likelihood, expressed only in terms of random-effects parameters, while the estimate of the fixed-effects coefficients is provided at each iteration by the internal function glsfit, based on the current value of the between-study (co)variance matrix. At convergence, the value of this profiled version is identical to the full (restricted) likelihood. This approach is computationally efficient, as it reduces the number of parameters in the optimization routine, especially for meta-regression models.
The random-effects parameters in par depends on the chosen structure for the between-study (co)variance matrix. The parameterization ensures the positive-definiteness of the estimated matrix. A Cholesky decomposition is then performed on the marginal (co)variance matrix in order to re-express the problem as standard least square equations, an approach which speeds up the computation of matrix inverses and determinants. These equations are finally solved through a QR decomposition, which guarantees stability. More details are provided in the references below.
Some parameters of the fitting procedures are determined through mvmeta.control. Specifically, the user can obtain the Hessian matrix of the estimated parameters (appropriately transformed, see mvmetaCovStruct) in the optimization function by setting hessian=TRUE, and specific control settings in the optimization process can be defined by the control list argument optim. These values are passed to the optimization function optim.
Sera F, Armstrong B, Blangiardo M, Gasparrini A (2019). An extended mixed-effects framework for meta-analysis.Statistics in Medicine. 2019;38(29):5429-5444. [Freely available here].
Gasparrini A, Armstrong B, Kenward MG (2012). Multivariate meta-analysis for non-linear and other multi-parameter associations. Statistics in Medicine. 31(29):3821--3839. [Freely available here].
Goldstein H (1986). Multilevel mixed linear model analysis using iterative generalized least squares. Biometrika. 73(1):43.
Lindstrom MJ and Bates DM (1988). Newton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data. Journal of the American Statistical Association. 83(404):1014--1022.
Pinheiro JC and Bates DM (2000). Mixed-Effects Models in S and S-PLUS. New York, Springer Verlag.
See mvmeta.fit and mvmeta.ml for additional info on the fitting procedures. See mvmeta.control to determine specific parameters of the fitting procedures. See mvmetaCovStruct for (co)variance structures. See chol and qr for info on the Cholesky and QR decomposition.
See mvmeta-package for an overview of the package and modelling framework.