Run meta-analysis across per-study DE results

Integrates per-study differential expression results from mx_de_all into a single gene-level meta-analysis. Two families of methods are available: effect-size models, which pool the per-study log2 fold changes weighted by their sampling variance, and p-value combination methods, which combine per-study evidence without pooling effects.

Usage

mx_meta(
  de_results,
  method = c("random_effects", "fixed_effects", "fisher", "stouffer", "inverse_normal",
    "minp", "maxp", "rop", "awmeta"),
  min_studies = 2,
  tau_method = c("REML", "PM", "DL"),
  hksj = FALSE,
  directional = TRUE,
  weights = NULL,
  r = NULL,
  alpha = 0.05,
  n_samples = NULL,
  ...
)

Arguments

de_results

A named list of data.frame objects (the de_result slots produced by mx_de), or a named list of metaXpressStudy objects after mx_de_all. Required columns: gene_id, log2FC, pvalue, padj; effect-size methods also need lfcSE.

method

Character scalar. One of:

"random_effects": (default) Inverse-variance random-effects model; between-study variance \(\tau^2\) estimated by tau_method. Estimates the average effect across populations and reports a prediction interval.
"fixed_effects": Inverse-variance common-effect model. Appropriate only when studies are near-replicates (low \(I^2\)).
"fisher": Fisher's sum of \(-2\log p\). With directional = TRUE (default) the two one-sided Fisher tests are computed and combined as in Owen (2009), so evidence in opposite directions does not accumulate. Like all Fisher-type tests it can still be driven by one extreme study; check direction_consistency.
"stouffer": Unweighted Stouffer Z on signed z-scores.
"inverse_normal": Weighted inverse-normal (Stouffer) combination with weights \(\sqrt{n_s}\), as in Marot et al. (2009) and Rau et al. (2014).
"minp": Tippett's minimum p: DE in at least one study. Non-directional.
"maxp": Wilkinson's maximum p: DE in all studies. Non-directional.
"rop": r-th ordered p-value (Song & Tseng 2014): DE in at least r studies. Non-directional.
"awmeta": AWmeta (Hu et al. 2026). The p-value is the adaptively weighted Fisher statistic (AW-Fisher; Li & Tseng 2011), calibrated with the AWFisher package (Huo et al. 2020), which also returns binary per-study weights indicating which studies contribute. The effect size is an REML random-effects estimate over the selected studies only (AW-REM). Requires AWFisher and lfcSE. Non-directional, as in the original method; check direction_consistency. The selected studies are reported in aw_studies.

min_studies

Integer. Minimum number of studies in which a gene must be measured. Default: 2.

tau_method

Estimator of the between-study variance for "random_effects": "REML" (default; restricted maximum likelihood), "PM" (Paule-Mandel) or "DL" (DerSimonian-Laird). REML and PM are preferred over DL, which underestimates \(\tau^2\) (Veroniki et al. 2016; Langan et al. 2019).

hksj

Logical. Apply the Hartung-Knapp-Sidik-Jonkman adjustment to the random-effects standard error and use a \(t_{k-1}\) reference distribution. Recommended when few studies are pooled because the standard Wald test is anti-conservative (IntHout et al. 2014). Default: FALSE.

directional

Logical. For "fisher", combine one-sided p-values in each direction (default TRUE). FALSE gives the classical Fisher test on two-sided p-values, which ignores sign.

weights

For "stouffer"-type methods and the effect estimate reported by p-value methods: NULL (default) uses per-study sample sizes from the n_case/n_control columns written by mx_de for "inverse_normal" and equal weights for "stouffer". A numeric vector (one per study) overrides this.

r

Integer. Order statistic for "rop". Default: ceiling(k / 2) for a gene measured in k studies.

alpha

Numeric. Confidence level is 1 - alpha for confidence and prediction intervals. Default: 0.05.

n_samples

Deprecated alias for weights supplied as total sample sizes; its square root is used.

...

Unused.

Value

A metaXpressResult. Its meta_table has one row per gene with columns:

gene_id: Gene identifier.
meta_log2FC, meta_se: Pooled log2 fold change and its standard error. For p-value methods this is the inverse-variance common-effect estimate, reported as a descriptive effect size.
ci_lower, ci_upper: 1 - alpha confidence interval.
pi_lower, pi_upper: Random-effects prediction interval for the effect in a new study (Higgins et al. 2009); requires \(k \ge 3\).
meta_stat: Test statistic (Z, t or \(\chi^2\) as appropriate).
meta_pvalue, meta_padj: Combined p-value and its Benjamini-Hochberg adjustment across genes.
tau2, i_squared, q_stat, q_pvalue: Between-study variance, \(I^2\) (%), Cochran's Q and its p-value.
n_studies, n_up, n_down: Studies measuring the gene, and those with positive / negative log2FC.
direction_consistency: Fraction of studies whose log2FC sign agrees with meta_log2FC.

Details

Use meta_padj (BH-adjusted) for significance. For a typical disease-signature meta-analysis we recommend "random_effects" with tau_method = "REML", and reporting \(I^2\) and the prediction interval alongside the pooled effect. P-value combination methods answer different questions (any vs. all vs. most studies) and should be chosen a priori, not by comparing numbers of significant genes.

A study contributes to a gene only when it reports log2FC and pvalue (plus lfcSE for effect-size methods). DESeq2 sets pvalue to NA for count outliers flagged by Cook's distance, so such genes are treated as not measured in that study. (DESeq2's independent filtering affects only padj, which is not used.)

All methods assume studies are independent. Fold-change direction must be harmonised across studies (same case/control orientation); check with mx_study_concordance.

References

DerSimonian, R. & Laird, N. (1986) Meta-analysis in clinical trials. Controlled Clinical Trials, 7(3), 177–188. doi:10.1016/0197-2456(86)90046-2

Viechtbauer, W. (2005) Bias and efficiency of meta-analytic variance estimators in the random-effects model. Journal of Educational and Behavioral Statistics, 30(3), 261–293. doi:10.3102/10769986030003261

Paule, R.C. & Mandel, J. (1982) Consensus values and weighting factors. Journal of Research of the National Bureau of Standards, 87(5), 377–385. doi:10.6028/jres.087.022

Hartung, J. & Knapp, G. (2001) A refined method for the meta-analysis of controlled clinical trials with binary outcome. Statistics in Medicine, 20(24), 3875–3889. doi:10.1002/sim.1009

IntHout, J., Ioannidis, J.P.A. & Borm, G.F. (2014) The Hartung-Knapp-Sidik-Jonkman method for random effects meta-analysis is straightforward and considerably outperforms the standard DerSimonian-Laird method. BMC Medical Research Methodology, 14, 25. doi:10.1186/1471-2288-14-25

Higgins, J.P.T., Thompson, S.G. & Spiegelhalter, D.J. (2009) A re-evaluation of random-effects meta-analysis. Journal of the Royal Statistical Society: Series A, 172(1), 137–159. doi:10.1111/j.1467-985X.2008.00552.x

Higgins, J.P.T. & Thompson, S.G. (2002) Quantifying heterogeneity in a meta-analysis. Statistics in Medicine, 21(11), 1539–1558. doi:10.1002/sim.1186

Veroniki, A.A. et al. (2016) Methods to estimate the between-study variance and its uncertainty in meta-analysis. Research Synthesis Methods, 7(1), 55–79. doi:10.1002/jrsm.1164

Langan, D. et al. (2019) A comparison of heterogeneity variance estimators in simulated random-effects meta-analyses. Research Synthesis Methods, 10(1), 83–98. doi:10.1002/jrsm.1316

Owen, A.B. (2009) Karl Pearson's meta-analysis revisited. Annals of Statistics, 37(6B), 3867–3892. doi:10.1214/09-AOS697

Marot, G., Foulley, J.-L., Mayer, C.-D. & Jaffrezic, F. (2009) Moderated effect size and P-value combinations for microarray meta-analyses. Bioinformatics, 25(20), 2692–2699. doi:10.1093/bioinformatics/btp444

Rau, A., Marot, G. & Jaffrezic, F. (2014) Differential meta-analysis of RNA-seq data from multiple studies. BMC Bioinformatics, 15, 91. doi:10.1186/1471-2105-15-91

Song, C. & Tseng, G.C. (2014) Hypothesis setting and order statistic for robust genomic meta-analysis. Annals of Applied Statistics, 8(2), 777–800. doi:10.1214/13-AOAS683

Li, J. & Tseng, G.C. (2011) An adaptively weighted statistic for detecting differential gene expression when combining multiple transcriptomic studies. Annals of Applied Statistics, 5(2A), 994–1019. doi:10.1214/10-AOAS393

Huo, Z., Tang, S., Park, Y. & Tseng, G. (2020) P-value evaluation, variability index and biomarker categorization for adaptively weighted Fisher's meta-analysis method in omics applications. Bioinformatics, 36(2), 524–532. doi:10.1093/bioinformatics/btz589

Hu, Y. et al. (2026) AWmeta empowers adaptively weighted transcriptomic meta-analysis. Current Issues in Molecular Biology, 48(5), 530. doi:10.3390/cimb48050530

Examples

# Three per-study results in the format written by mx_de()
set.seed(1)
genes <- paste0("G", 1:200)
de <- lapply(1:3, function(s) {
  lfc <- rnorm(200, c(rep(1, 20), rep(0, 180)), 0.3)
  se  <- runif(200, 0.2, 0.4)
  p   <- 2 * pnorm(-abs(lfc / se))
  data.frame(gene_id = genes, log2FC = lfc, lfcSE = se, pvalue = p,
             padj = p.adjust(p, "BH"), n_case = 5, n_control = 5)
})
names(de) <- c("S1", "S2", "S3")
res <- mx_meta(de, method = "random_effects", tau_method = "REML")
#> Running meta-analysis (random_effects) on 3 studies...
head(res@meta_table[order(res@meta_table$meta_pvalue), ])
#>    gene_id meta_log2FC   meta_se  ci_lower ci_upper  pi_lower pi_upper
#> 5       G5   1.2844615 0.1879690 0.9160490 1.652874 -1.543210 4.112133
#> 12     G12   1.0837146 0.1735513 0.7435604 1.423869 -1.121463 3.288893
#> 10     G10   1.0446984 0.1680409 0.7153443 1.374053 -1.090464 3.179861
#> 8       G8   1.0759511 0.1752744 0.7324196 1.419483 -1.151122 3.303024
#> 1       G1   0.9789829 0.1600478 0.6652951 1.292671 -1.054617 3.012583
#> 11     G11   1.2198264 0.1994541 0.8289036 1.610749 -1.314478 3.754131
#>    meta_stat  meta_pvalue    meta_padj       tau2 i_squared     q_stat
#> 5   6.833369 8.294346e-12 1.658869e-09 0.01419286         0 1.89583209
#> 12  6.244349 4.255700e-10 3.077624e-08 0.00000000         0 0.04912692
#> 10  6.216929 5.069799e-10 3.077624e-08 0.00000000         0 0.15762653
#> 8   6.138665 8.321762e-10 3.077624e-08 0.00000000         0 0.55516997
#> 1   6.116817 9.546260e-10 3.077624e-08 0.00000000         0 1.27237086
#> 11  6.115825 9.605844e-10 3.077624e-08 0.00000000         0 0.75678390
#>     q_pvalue n_studies n_up n_down direction_consistency
#> 5  0.3875478         3    3      0                     1
#> 12 0.9757358         3    3      0                     1
#> 10 0.9242125         3    3      0                     1
#> 8  0.7576112         3    3      0                     1
#> 1  0.5293077         3    3      0                     1
#> 11 0.6849620         3    3      0                     1