Meta-analysis

A single study often finds differences that others do not reproduce: of the genes called differentially expressed in Alzheimer’s disease in one dataset, over 85% were not reproduced in any of 16 others Nakatsuka et al., 2025. A meta-analysis combines the estimates of several studies into one, weighs each by its precision, and measures how much the studies disagree. meta_analysis does it for every gene at once, as metafor’s rma.uni does for one effect without moderators Viechtbauer, 2010, and differential_expression_meta_analysis does it for the results of limma_voom or limma_trend on several cubes, matched by cell type, contrast and gene id. meta_analysis returns a MetaAnalysisFit, and differential_expression_meta_analysis a MetaAnalysisResult, with a table for every cell type and contrast (MetaAnalysisResult.table).

The model¶

Study $i$ estimates a gene’s effect by $y_i$, with a standard error $s_i$ whose square, $v_i$, is taken as known. In the random-effects model the true effects of the studies vary around a mean $\mu$ with the between-study variance $\tau^2$ DerSimonian & Laird, 1986:

With the weights $w_i = 1 / (v_i + \tau^2)$, $\hat\mu = \sum w_i y_i / \sum w_i$ with standard error $1 / \sqrt{\sum w_i}$. The fixed-effect model, one true effect for every study, is the case $\tau^2 = 0$: the inverse-variance weighted mean.

The between-study variance¶

• "reml", metafor’s default: the restricted maximum likelihood estimate Viechtbauer, 2005. Where the restricted likelihood has two maxima, as when an imprecise study lies far from precise ones that agree, CytePool takes the larger one; metafor’s Fisher scoring can stop at the smaller one.

• "dl": DerSimonian and Laird’s method of moments, $\max(0, (Q - (k - 1)) / (\sum w_i - \sum w_i^2 / \sum w_i))$ with $w_i = 1 / v_i$.

• "pm": Paule and Mandel’s estimate Paule & Mandel, 1982, at which the generalized Q statistic equals its expectation $k - 1$.

• "fixed": $\tau^2 = 0$.

Tests and heterogeneity¶

The default z-test refers $\hat\mu / \mathrm{SE}$ to the normal distribution. test="knapp_hartung" multiplies the standard error by $\sqrt{q}$, with $q = \sum w_i (y_i - \hat\mu)^2 / (k - 1)$, and uses a t distribution on $k - 1$ degrees of freedom Hartung, 1999Sidik & Jonkman, 2002Knapp & Hartung, 2003, which metafor’s documentation recommends for few studies.

Cochran’s $Q = \sum (y_i - \hat\mu_{\text{fixed}})^2 / v_i$ tests whether the studies agree Cochran, 1954. $I^2$ is the share of the variation that comes from heterogeneity rather than sampling error, and $H^2$ the ratio of the total to the sampling variance Higgins & Thompson, 2002; CytePool reports $I^2$ as a proportion, where metafor reports a percentage.

Genes that some studies did not measure¶

NaN marks a study that did not measure or test a gene. Each gene is analysed over the studies that measured it; differential_expression_meta_analysis analyses a gene tested by at least min_studies studies (two by default, since one study says nothing about $\tau^2$). It takes limma’s moderated standard errors as the known sampling errors.

Limits¶

The studies are taken as independent, with known sampling variances. Meta-regression on study-level covariates, multivariate meta-analysis of correlated effects, and shrinkage across conditions such as mash Urbut et al., 2019 are not implemented.