A Multivariate Meta-Analysis Model for High-Dimensional Data

Description

The MetaHD function performs a multivariate meta-analysis for high-dimensional data, combining summary estimates obtained from multiple studies by using restricted maximum likelihood estimation. In its default settings, the function fits the fastMetaHD model, which provides a memory-efficient and computationally faster implementation of the MetaHD methodology. Assuming a meta-analysis is based on N outcomes and K studies:

Usage

MetaHD(
  Y,
  Slist,
  Psi = NULL,
  method = c("multi","REM","FEM"),
  bscov = c("unstructured","diag","none"),
  useDivideConquer = FALSE,
  parallel = FALSE,
  est.wscor = FALSE,
  shrinkCor = TRUE,
  impute.na = FALSE,
  optim.algorithm = c("BOBYQA","hybrid","L-BFGS-B"),
  optim.maxiter = 2000,
  rigls.iter = 1,
  initPsi = NULL,
  impute.var = 10^4
)

Arguments

Details

If parallel = TRUE, the divide-and-conquer approach may be evaluated in parallel. Parallel execution is implemented using the future R package.

On Windows, users must set a future plan (e.g., future::plan(future::multisession, workers = ncores)) before calling MetaHD() in order to enable parallel computation.

On Linux and macOS, users may alternatively use future::plan(future::multicore, workers = ncores).

If no future plan is set, or if parallel = FALSE, computations are performed sequentially.

Value

A list of objects containing :

• estimate: An N-dimensional vector of the combined estimates.

• std.err: An N-dimensional vector of the associated standard errors.

• pVal: An N-dimensional vector of the p-values.

• I2.stat: I^2 statistics.

References

Liyanage JC, Prendergast L, Staudte R, De Livera AM (2024). MetaHD: a multivariate meta-analysis model for metabolomics data. Bioinformatics, 40(7), btae470. doi:10.1093/bioinformatics/btae470

Powell MJ (2009). The BOBYQA algorithm for bound constrained optimization without derivatives. Cambridge NA Report NA2009/06, University of Cambridge, 26, 26–46.

Sera F, Armstrong B, Blangiardo M, et al. (2019). An extended mixed-effects framework for meta-analysis. Statistics in Medicine, 38, 5429–5444.

Schäfer J, Strimmer K (2005). A shrinkage approach to large-scale covariance estimation and implications for functional genomics. Statistical Applications in Genetics and Molecular Biology, 4, 32.

Examples

# CREATE INPUT DATA
input_data <- MetaHDInput(realdata)
Y <- input_data$Y
Slist <- input_data$Slist

N <- ncol(Y)
K <- nrow(Y)

Smat <- matrix(0, nrow = K, ncol = N)
for (i in 1:K) {
 Smat[i, ] <- diag(Slist[[i]])
}

# MULTIVARIATE RANDOM-EFFECTS META-ANALYSIS 
model <- MetaHD(Y = Y, Slist = Slist, method = "multi")
model$estimate
model$pVal

# UNIVARIATE RANDOM-EFFECTS META-ANALYSIS
model <- MetaHD(Y = Y, Slist = Smat, method = "REM")
model$estimate
model$pVal

# UNIVARIATE FIXED-EFFECTS META-ANALYSIS
model <- MetaHD(Y = Y, Slist = Smat, method = "FEM")
model$estimate
model$pVal

Creating Input Data for MetaHD When Individual-Level Data are Available

Description

The MetaHDInput function creates input data Y (treatment effects) and Slist (within-study covariance matrices) for MetaHD when individual-level data are available. Assuming that the individual-level data are in the following format, with 'study' in column 1, 'group' in column 2 and outcomes in rest of the columns, with samples in rows.

Usage

MetaHDInput(data)

Arguments

Value

A list of objects containing :

• Y: A K \times N matrix of treatment effect sizes, where K is the number of studies and N is the number of outcomes.

• Slist: A list of length K containing N \times N within-study variance–covariance matrices of the treatment effects.

Examples

# CREATE INPUT DATA
input_data <- MetaHDInput(realdata)

## treatment effect-sizes
Y <- input_data$Y
head(Y)

## within-study variance–covariance matrices
Slist <- input_data$Slist
head(Slist[[1]])

P-value Combination Methods for High-Dimensional Data

Description

Combines individual p-values across multiple studies for each outcome using p-value combination methods applied independently per outcome. Includes traditional and weighted p-value combination approaches and a vote counting method.

Usage

MetaHDpval(
  pmat,
  method = c("Fisher", "Stouffer", "wZ", "Lancaster", "wFisher", "Vote counting"),
  weight = NULL,
  is.onetail = TRUE,
  eff.sign = NULL,
  alpha = 0.5
)

Arguments

Details

The MetaHDpval function offers five traditional and more recent p-value combination methods implemented using the metapro R package, as well as a vote counting method implemented using the metap R package:

• Fisher's method (Fisher, 1932), which combines logarithmically transformed p-values from individual studies for each outcome using Fisher’s statistic.

• Stouffer's method (Stouffer et al., 1949), which combines inverse normal–transformed p-values derived from individual study test statistics for each outcome.

• Weighted Z-method (wZ) (Mosteller and Bush, 1954), an extension of Stouffer’s method that incorporates study-specific weights, resulting in a weighted inverse normal combination.

• Lancaster's method (Lancaster, 1961), which generalizes Fisher’s method by introducing weights and exploits the additive property of the \chi^2-distribution.

• Weighted Fisher's method (wFisher) (Yoon et al., 2021), which extends Fisher’s method by allowing non-integer weights reflecting study-specific information (e.g., sample sizes). This approach replaces the \chi^2-distribution with the gamma distribution to accommodate non-integer degrees of freedom.

• Vote counting method (Becker, 1994), that classifies a study as positive if its p-value is less than alpha and as negative if it exceeds 1 - alpha, with studies falling in between treated as neutral and excluded. The number of positive studies is then counted, and a one-sided binomial test is applied to the non-neutral studies to obtain a combined p-value for each outcome.

Value

A numeric vector of length N containing the combined p-values for each outcome.

References

Yoon, S., Baik, B., Park, T., et al. (2021). Powerful p-value combination methods to detect incomplete association. Scientific Reports, 11, 6980. doi:10.1038/s41598-021-86465-y

Yoon, S. (2023). metapro: Robust P-Value Combination Methods (R package version 1.5.11). Comprehensive R Archive Network (CRAN). doi:10.32614/CRAN.package.metapro

Becker, B.J. (1994). Combining significance levels. In Cooper H, Hedges LV (eds.), A handbook of research synthesis, 215–230. Russell Sage, New York.

Dewey, M. (2025). metap: Meta-Analysis of Significance Values (R package version 1.13). Comprehensive R Archive Network (CRAN). doi:10.32614/CRAN.package.metap

Examples

## Example with 5 studies and 12 outcomes
set.seed(123)
pmat <- matrix(runif(15), nrow = 5, ncol = 12)
eff.sign <- matrix(sample(c(-1, 1), 60, replace = TRUE), nrow = 5, ncol = 12)
wmat <- matrix(sample(50:200, 60, replace = TRUE), nrow = 5, ncol = 12)

## Fisher's method
MetaHDpval(pmat, method = "Fisher", is.onetail = FALSE, eff.sign = eff.sign)

## Weighted Z method
MetaHDpval(pmat, method = "wZ", weight = wmat, is.onetail = FALSE, eff.sign = eff.sign)

## Vote counting
MetaHDpval(pmat, method = "Vote counting", alpha = 0.4)

An Individual-Level Metabolomics Dataset

Description

This is a subset of data, publicly available on MetaboAnalyst example datasets.

Usage

realdata

Format

A data frame with 172 observations on 14 metabolites.

Examples

head(realdata)

Simulated Dataset 1 : With Complete Data

Description

This dataset consists of a list of two data frames containing treatment effect-sizes and within-study covariance matrices

Usage

simdata.1

Format

A list of data frames as follows:

Y

treatment effect sizes of the metabolites in the form of a 12 x 30 matrix, where 12 is the number of studies and 30 is the number of metabolites.

Slist

12-dimensional list of 30 x 30 matrices representing within-study variances and covariances of the treatment effects

Examples

Y <- simdata.1$Y
Slist <- simdata.1$Slist

head(Y)
head(Slist[[1]])
head(Slist[[12]])

Simulated Dataset 2 : With Data Missing-At-Random

Description

This dataset consists of a list of two data frames containing treatment effect-sizes and within-study covariance matrices with missing values

Usage

simdata.2

Format

A list of data frames as follows:

Y

treatment effect sizes of the metabolites in the form of a 12 x 30 matrix, where 12 is the number of studies and 30 is the number of metabolites.

Slist

12-dimensional list of 30 x 30 matrices representing within-study variances and covariances of the treatment effects

Examples

Y <- simdata.2$Y
Slist <- simdata.2$Slist

head(Y)
head(Slist[[1]])
head(Slist[[12]])