Computing Variance-Covariance Matrices for Standardized Mean Differences

The function lgOR.vcov computes effect sizes and variance-covariance matrix for multivariate meta-analysis when the effect sizes of interest are all measured by standardized mean difference. See mix.vcov for effect sizes of the same or different types.

smd.vcov(nt, nc, d, r, n_rt = NA, n_rc = NA, name = NULL)

Arguments

nt	A \(N \times p\) matrix storing sample sizes in the treatment group reporting the \(p\) outcomes. nt[i,j] is the sample size from study \(i\) reporting outcome \(j\).
nc	A matrix defined in a similar way as nt for the control group.
d	A \(N \times p\) matrix or data frame with standard mean differences (SMD) from the \(N\) studies. d[i,j] is the value from study \(i\) for outcome \(j\).
r	A \(N\)-dimensional list of \(p \times p\) correlation matrices for the \(p\) outcomes from the \(N\) studies. r[[k]][i,j] is the correlation coefficient between outcome \(i\) and outcome \(j\) from study \(k\).
n_rt	A \(N\)-dimensional list of \(p \times p\) matrices storing sample sizes in the treatment group reporting pairwise outcomes in the off-diagonal elements. n_rt[[k]][i,j] is the sample size reporting both outcome \(i\) and outcome \(j\) from study \(k\). Diagonal elements of these matrices are discarded. The default value is NA, which means that the smaller sample size reporting the corresponding two outcomes is imputed: i.e. n_rt[[k]][i,j]=min(nt[k,i],nt[k,j]).
n_rc	A list defined in a similar way as n_rt for the control group.
name	Names for the outcomes.

Author

Min Lu

Value

ef	A \(N \times p\) data frame that transforms the input argument d into Hedges's g (Wei and Higgins, 2013).
list.vcov	A \(N\)-dimensional list of \(p(p+1)/2 \times p(p+1)/2\) variance-covariance matrices for Hedges's g (Wei and Higgins, 2013).
matrix.vcov	A \(N \times p(p+1)/2\) whose rows are computed variance-covariance vectors for Hedges's g (Wei and Higgins, 2013).
list.dvcov	A \(N\)-dimensional list of \(p(p+1)/2 \times p(p+1)/2\) variance-covariance matrices for SMD (Olkin and Gleser, 2009).
matrix.dvcov	A \(N \times p(p+1)/2\) matrix whose rows are computed variance-covariance vectors for SMD (Olkin and Gleser, 2009).

References

Ahn, S., Lu, M., Lefevor, G.T., Fedewa, A. & Celimli, S. (2016). Application of meta-analysis in sport and exercise science. In N. Ntoumanis, & N. Myers (Eds.), An Introduction to Intermediate and Advanced Statistical Analyses for Sport and Exercise Scientists (pp.233-253). Hoboken, NJ: John Wiley and Sons, Ltd.

Wei, Y. & Higgins, J. (2013). Estimating within study covariances in multivariate meta-analysis with multiple outcomes. Statistics in Medicine, 32(7), 119-1205.

Olkin, I. & Gleser, L. (2009). Stochastically dependent effect sizes. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (pp. 357-376). New York: Russel Sage Foundation.

Examples

######################################################
# Example: Geeganage2010 data
# Preparing covarianceS for multivariate meta-analysis
######################################################
data(Geeganage2010)
## set the correlation coefficients list r
r12 <- 0.71
r.Gee <- lapply(1:nrow(Geeganage2010), function(i){matrix(c(1, r12, r12, 1), 2, 2)})

computvcov <- smd.vcov(nt = subset(Geeganage2010, select = c(nt_SBP, nt_DBP)),
               nc = subset(Geeganage2010, select = c(nc_SBP, nc_DBP)),
               d = subset(Geeganage2010, select = c(SMD_SBP, SMD_DBP)), r = r.Gee,
               name = c("SMD_SBP", "SMD_DBP"))
# name variance-covariance matrix as S
S <- computvcov$matrix.vcov
## fixed-effect model
y <- computvcov$ef
MMA_FE <- summary(metafixed(y = y, Slist = computvcov$list.vcov))
#######################################################################
# Running random-effects model using package "mvmeta" or "metaSEM"
#######################################################################
# Restricted maximum likelihood (REML) estimator from the mvmeta package
#library(mvmeta)
#mvmeta_RE <- summary(mvmeta(cbind(SMD_SBP, SMD_DBP),
#                     S = S,
#                     data = y,
#                     method = "reml"))
#mvmeta_RE

# maximum likelihood estimators from the metaSEM package
# library(metaSEM)
# metaSEM_RE <- summary(meta(y = y, v = S))
# metaSEM_RE
##############################################################
# Plotting the result:
##############################################################
obj <- MMA_FE
# obj <- mvmeta_RE
# obj <- metaSEM_RE
# pdf("CI.pdf", width = 4, height = 7)
plotCI(y = computvcov$ef, v = computvcov$list.vcov,
        name.y = NULL, name.study = Geeganage2010$studyID,
        y.all = obj$coefficients[,1],
        y.all.se = obj$coefficients[,2])
#> $`Plotting SMD_SBP`
#> 
#> $`Plotting SMD_DBP`
#> 
# dev.off()