Impute a patterned block-diagonal covariance matrix — pattern_covariance

pattern_covariance_matrix calculates a block-diagonal covariance matrix, given the marginal variances, the block structure, and an assumed correlation structure defined by a patterned correlation matrix.

Usage

pattern_covariance_matrix(
  vi,
  cluster,
  pattern_level,
  r_pattern,
  r,
  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.
pattern_level: Vector of categories for each effect size, used to determine which entry of the pattern matrix will be used to impute a correlation.
r_pattern: Patterned correlation matrix with row and column names corresponding to the levels of pattern.
r: Vector or numeric value of assumed constant correlation(s) between effect size estimates from each study.
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 correlation structure, defined by a patterned correlation matrix. 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 $j$. Let $p_{ij}$ be the level of the pattern variable for effect $i$ in cluster $j$, taking a value in $1,...,C$. A patterned correlation matrix is defined as a set of correlations between pairs of effects taking each possible combination of patterns. Formally, let $r_{cd}$ be the correlation between effects in categories $c$ and $d$, respectively, where $r_{cd} = r_{dc}$. Then the covariance between effects $h$ and $i$ in cluster $j$ is taken to be $$C_{hij} = \sqrt{v_{hj} v_{ij}} \times r_{p_{hj} p_{ij}}.$$

Correlations between effect sizes within the same category are defined by the diagonal values of the pattern matrix, which may take values less than one.

Combinations of pattern levels that do not occur in the patterned correlation matrix will be set equal to r.

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


pkgs_available <- 
  requireNamespace("metafor", quietly = TRUE) & 
  requireNamespace("robumeta", quietly = TRUE)
  
if (pkgs_available) {
library(metafor)

data(oswald2013, package = "robumeta")
dat <- escalc(data = oswald2013, measure = "ZCOR", ri = R, ni = N)
subset_ids <- unique(dat$Study)[1:20]
dat <- subset(dat, Study %in% subset_ids)

# make a patterned correlation matrix 

p_levels <- levels(dat$Crit.Cat)
r_pattern <- 0.7^as.matrix(dist(1:length(p_levels)))
diag(r_pattern) <- seq(0.75, 0.95, length.out = 6)
rownames(r_pattern) <- colnames(r_pattern) <- p_levels

# impute the covariance matrix using patterned correlations
V_list <- pattern_covariance_matrix(vi = dat$vi, 
                                    cluster = dat$Study, 
                                    pattern_level = dat$Crit.Cat,
                                    r_pattern = r_pattern,
                                    smooth_vi = TRUE)
                                    
# fit a model using imputed covariance matrix

MVFE <- rma.mv(yi ~ 0 + Crit.Cat, V = V_list, 
               random = ~ Crit.Cat | Study,
               data = dat)
               
conf_int(MVFE, vcov = "CR2")

}
#>                           Coef. Estimate       SE d.f. Lower 95% CI
#>          Crit.CatBrain Activity   0.1425 4.49e-01 2.84     -1.33431
#>  Crit.CatInterpersonal Behavior   0.0167 3.65e-03 1.63     -0.00307
#>           Crit.CatMicrobehavior   0.1308 3.43e-02 3.64      0.03178
#>       Crit.CatPerson Perception   0.1544 1.94e-02 8.08      0.10963
#>       Crit.CatPolicy Preference   0.0864 5.10e-02 1.98     -0.13561
#>           Crit.CatResponse Time   0.2988 8.48e-14 1.00      0.29882
#>  Upper 95% CI
#>        1.6193
#>        0.0364
#>        0.2298
#>        0.1991
#>        0.3084
#>        0.2988