Description Options Details Note Author(s) References See Also Examples
Alternative options for the (co)variance structure of the random effects random effects in metaanalytical models, usually defined through the argument bscov
of the function mixmeta
.
Assuming a metaanalysis or metaregression based on k outcomes, for each grouping level with q randomeffects predictors the matrix can be specified in various forms listed below. For multivariate models with multiple predictors, the order implies a sequence of q parameters for each k outcomes. These are the options:
unstr
: an unstructured form for a general positivedefinite matrix. The matrix is represented by kq(kq+1)/2 unrestricted parameters defined as the upper triangular entries of its Cholesky decomposition.
diag
: a diagonal positivedefinite matrix. The matrix is represented by kq unrestricted parameters defined as the logarithm of the diagonal values.
id
: a multiple of the identity positivedefinite matrix. The matrix is represented by a single unrestricted parameter defined as the logarithm of the diagonal value.
cs
: a positivedefinite matrix with compound symmetry structure. The matrix is represented by 2 unrestricted parameters defined as the logarithm of the identical diagonal value and the transformed correlation. The latter is parameterized so to obtain a correlation value between 1/(kq1) and 1, in order to ensure positivedefiniteness.
hcs
: a positivedefinite matrix with heterogeneous compound symmetry structure. The matrix is represented by kq+1 unrestricted parameters defined as the logarithm of the diagonal values and the transformed correlation. The latter is parameterized so to obtain a correlation value between 1/(kq1) and 1, in order to ensure positivedefiniteness.
ar1
: a positivedefinite matrix with autoregressive structure of first order. The matrix is represented by 2 unrestricted parameters defined as the logarithm of the identical diagonal value and the logistic transformed correlation. The latter is parameterized so to obtain a correlation value between 1 and 1.
har1
: a positivedefinite matrix with heterogeneous autoregressive structure of first order. The matrix is represented by kq+1 unrestricted parameters defined as the logarithm of the diagonal value and the logistic transformed correlation. The latter is parameterized so to obtain a correlation value between 1 and 1.
prop
: a positivedefinite matrix proportional to that provided by the user through the argument Psifix
in the control list (see mixmeta.control
). The matrix is represented by 1 unrestricted parameter defined as the logarithm of the multiplier.
cor
: a positivedefinite matrix with correlation structure provided by the user through the argument Psifix
(with cov2cor
) in the control list (see mixmeta.control
). The matrix is represented by k unrestricted parameters defined as the logarithm of the diagonal values.
fixed
: a known matrix provided by the user through the argument Psifix
in the control list (see mixmeta.control
). The matrix is known and no parameters are needed to represent it.
Structures other than unstr
are only available for models estimated through (restricted) maximum likelihood.
The unrestricted parameters defining the randomeffects (co)variance matrix (or matrices for multilvel models) are estimated in the iterative optimization algorithm (see mixmeta.ml
). Although rarely needed and not recommeded, the user can provided a starting value of the (co)variance matrix, from which the parameters are derived (see mixmeta.control
).
The choice of structures can affect the performance of the optimization procedure, determining forms of likelihood surfaces which induce convergence to local maxima. In particular, structures such as multiple of identity or proportional to a fixed matrix are based on strong assumptions and should be used with caution.
Antonio Gasparrini <antonio.gasparrini@lshtm.ac.uk> and Francesco Sera <francesco.sera@lshtm.ac.uk>
Sera F, Armstrong B, Blangiardo M, Gasparrini A (2019). An extended mixedeffects framework for metaanalysis.Statistics in Medicine. 2019;38(29):54295444. [Freely available here].
Pinheiro JC and Bates DM (2000). MixedEffects Models in S and SPLUS. New York, Springer Verlag.
See mixmeta
. See lm
or glm
for standard regression functions. See mixmetapackage
for an overview of this modelling framework.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  # UNSTRUCTURED AND STRUCTURED BETWEENSTUDY (CO)VARIANCE
y < as.matrix(smoking[11:13])
S < as.matrix(smoking[14:19])
mod1 < mixmeta(y, S)
summary(mod1)
mod1$Psi
# DIAGONAL
mod2 < mixmeta(y, S, bscov="diag")
summary(mod2)
mod2$Psi
# HETEROGENEOUS COMPOUND SYMMETRY
mod3 < mixmeta(y, S, bscov="hcs")
summary(mod3)
mod3$Psi
# PROPORTIONAL
mod4 < mixmeta(y, S, bscov="prop", control=list(Psifix=diag(3)+1))
summary(mod4)
mod4$Psi
# CORRELATION
Psicor < matrix(0.2, 3, 3) ; diag(Psicor) < 1
mod5 < mixmeta(y, S, bscov="cor", control=list(Psifix=Psicor))
summary(mod5)
mod5$Psi

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