Impute a block-diagonal covariance matrix

impute_covariance_matrix calculates a block-diagonal covariance matrix, given the marginal variances, the block structure, and an assumed correlation structure. Can be used to create compound-symmetric structures, AR(1) auto-correlated structures, or combinations thereof.

Usage

impute_covariance_matrix(
  vi,
  cluster,
  r,
  ti,
  ar1,
  smooth_vi = FALSE,
  subgroup = NULL,
  return_list = identical(as.factor(cluster), sort(as.factor(cluster))),
  check_PD = TRUE
)

Arguments

vi

Vector of variances

cluster

Vector indicating which effects belong to the same cluster. Effects with the same value of `cluster` will be treated as correlated.

r

Vector or numeric value of assumed constant correlation(s) between effect size estimates from each study.

ti

Vector of time-points describing temporal spacing of effects, for use with auto-regressive correlation structures.

ar1

Vector or numeric value of assumed AR(1) auto-correlation(s) between effect size estimates from each study. If specified, then ti argument must be specified.

smooth_vi

Logical indicating whether to smooth the marginal variances by taking the average vi within each cluster. Defaults to FALSE.

subgroup

Vector of category labels describing sub-groups of effects. If non-null, effects that share the same category label and the same cluster will be treated as correlated, but effects with different category labels will be treated as uncorrelated, even if they come from the same cluster.

return_list

Optional logical indicating whether to return a list of matrices (with one entry per block) or the full variance-covariance matrix.

check_PD

Optional logical indicating whether to check whether each covariance matrix is positive definite. If TRUE (the default), the function will display a warning if any covariance matrix is not positive definite.

Value

If cluster is appropriately sorted, then a list of matrices, with one entry per cluster, will be returned by default. If cluster

is out of order, then the full variance-covariance matrix will be returned by default. The output structure can be controlled with the optional

return_list argument.

Details

A block-diagonal variance-covariance matrix (possibly represented as a list of matrices) with a specified structure. The structure depends on whether the r argument, ar1 argument, or both arguments are specified. Let \(v_{ij}\) denote the specified variance for effect \(i\) in cluster \(j\) and \(C_{hij}\) be the covariance between effects \(h\) and \(i\) in cluster \(j\).

• If only r is specified, each block of the variance-covariance matrix will have a constant (compound symmetric) correlation, so that $$C_{hij} = r_j \sqrt{v_{hj} v_{ij},}$$ where \(r_j\) is the specified correlation for cluster \(j\). If only a single value is given in r, then it will be used for every cluster.

• If only ar1 is specified, each block of the variance-covariance matrix will have an AR(1) auto-correlation structure, so that $$C_{hij} = \phi_j^{|t_{hj} - t_{ij}|} \sqrt{v_{hj} v_{ij},}$$ where \(\phi_j\) is the specified auto-correlation for cluster \(j\) and \(t_{hj}\) and \(t_{ij}\) are specified time-points corresponding to effects \(h\) and \(i\) in cluster \(j\). If only a single value is given in ar1, then it will be used for every cluster.

• If both r and ar1 are specified, each block of the variance-covariance matrix will have combination of compound symmetric and an AR(1) auto-correlation structures, so that $$C_{hij} = \left[r_j + (1 - r_j)\phi_j^{|t_{hj} - t_{ij}|}\right] \sqrt{v_{hj} v_{ij},}$$ where \(r_j\) is the specified constant correlation for cluster \(j\), \(\phi_j\) is the specified auto-correlation for cluster \(j\) and \(t_{hj}\) and \(t_{ij}\) are specified time-points corresponding to effects \(h\) and \(i\) in cluster \(j\). If only single values are given in r or ar1, they will be used for every cluster.

If smooth_vi = TRUE, then all of the variances within cluster \(j\) will be set equal to the average variance of cluster \(j\), i.e., $$v'_{ij} = \frac{1}{n_j} \sum_{i=1}^{n_j} v_{ij}$$ for \(i=1,...,n_j\) and \(j=1,...,k\).

Examples

if (requireNamespace("metafor", quietly = TRUE)) {

library(metafor)

# Constant correlation
data(SATcoaching)
V_list <- impute_covariance_matrix(vi = SATcoaching$V, cluster = SATcoaching$study, r = 0.66)
MVFE <- rma.mv(d ~ 0 + test, V = V_list, data = SATcoaching)
conf_int(MVFE, vcov = "CR2", cluster = SATcoaching$study)

}
#>       Coef. Estimate     SE d.f. Lower 95% CI Upper 95% CI
#>    testMath    0.132 0.0376 13.2       0.0506        0.213
#>  testVerbal    0.122 0.0250 16.9       0.0686        0.174