Abstract
Recent studies have demonstrated that conventional meta-analyses of partial correlation coefficients (PCC) are biased. Several adjustments have been shown in simulations to reduce these small-sample biases to negligibility. While many meta-analyses of partial correlation coefficients are conducted each year across several disciplines, the practical importance of these issues remains unknown. To address this question and to offer advice for applications, we survey 172 economic meta-analyses of PCCs. We find that small-sample biases are negligible in practice. However, some publication selection biases remain. Although random effects using Fisher’s z transformations have often been recommended, they reduce neither small-sample nor publication selection biases relative to conventional random effects. The family of unrestricted weighted least squares (UWLS) and the Hunter-Schmidt (HS) estimators produce smaller, arguably less biased, estimates of the mean PCC in these applications than either random effects with or without Fisher’s z transformations. These findings offer practical guidance for any discipline that meta-analyzes partial correlations.
The survey covers 172 meta-analyses containing 63,730 estimated partial correlations and their standard errors, extracted from 6,191 papers. That is about 371 estimates per meta-analysis, and about ten per primary study. The typical effect in this literature is small: the median partial correlation is around 0.046.
Status. Under review at Research Synthesis Methods; the second revision was submitted on 30 October 2026. The PDF here is the authors’ copy of that second revision, and there is no published version yet. The data and the code are public.
What changes if you switch estimator
Almost nothing, until you reach the margin. The corrections proposed for the small-sample bias barely move the mean away from the unadjusted estimator they correct. That holds for Fisher’s z applied to random effects or to UWLS, and for the degrees-of-freedom adjustment UWLS+3. Across 172 real meta-analyses, sample sizes are large and widely dispersed enough that any notable small-sample bias washes out. What little remains is likely publication selection bias, and that is a different problem with a different remedy.
Where the choice does bite is in the weighting rather than the transformation. The median UWLS mean is about 30% smaller than the corresponding random-effects mean, consistent with the ordering that simulations find when publication selection is present. Hunter-Schmidt is smaller still. And if 0.1 is taken as the smallest partial correlation of interest, roughly one in three of the “small” average effects that random effects reports becomes negligible under UWLS or Hunter-Schmidt. In its summary, the paper supports UWLS and Hunter-Schmidt over random effects, and it says that when in doubt one should report the more conservative estimate and emphasize the estimator with the best demonstrated performance under likely biases.
Ten ways to average the same 172 meta-analyses (Table 1 of the paper). Every literature is sign-aligned so that its median partial correlation is positive, so both small-sample bias and publication selection push a number up, and a smaller average is the one consistent with less of both. MSD vs PET-PEESE is the mean squared difference from the publication-bias-corrected estimate. Flipped is the same mean over the 50 meta-analyses whose estimates were mostly negative. The two Fisher’s z rows leave out 105 estimates with at most one degree of freedom, from two meta-analyses.
| Estimator of the mean PCC | Mean | Median | Std. dev. | MSD vs PET-PEESE | Flipped |
|---|---|---|---|---|---|
| Unweighted mean | 0.131 | 0.097 | 0.115 | 0.022 | 0.078 |
| Random effects (RE1) | 0.126 | 0.087 | 0.116 | 0.020 | 0.073 |
| Random effects, conventional SE (RE2) | 0.129 | 0.090 | 0.117 | 0.021 | 0.075 |
| UWLS (UWLS1) | 0.110 | 0.061 | 0.128 | 0.010 | 0.064 |
| UWLS, conventional SE (UWLS2) | 0.140 | 0.069 | 0.175 | 0.019 | 0.090 |
| UWLS+3 (degrees-of-freedom adjusted) | 0.109 | 0.061 | 0.127 | 0.009 | 0.063 |
| Random effects on Fisher’s z (REz) | 0.130 | 0.086 | 0.122 | 0.021 | 0.075 |
| UWLS on Fisher’s z (UWLSz) | 0.101 | 0.057 | 0.117 | 0.009 | 0.058 |
| Hunter-Schmidt (HS) | 0.096 | 0.056 | 0.109 | 0.010 | 0.055 |
| PET-PEESE, corrected for publication bias | 0.090 | 0.040 | 0.171 | — | 0.046 |
Read down the Mean column and the main pattern is visible. The small-sample corrections move little: REz gives 0.130 against random effects’ 0.126, and UWLS+3 gives 0.109 and UWLSz 0.101 against UWLS’s 0.110. The weighting moves it more: 0.126 for random effects against 0.096 for Hunter-Schmidt. And the estimators that come out smallest are also the ones closest to the publication-bias-corrected PET-PEESE, though PET-PEESE is itself only an estimate. The paper does not read this as proof: it declines to infer smaller bias from a smaller average, which would be affirming the consequent, and claims only a pattern consistent with the upward-bias hypothesis and with what simulations show when publication selection is present.
Three papers on partial correlations, and what each one does
Partial correlations are the workhorse effect size of economics meta-analysis, and three papers on this site are about them. They answer different questions and should not be confused:
- Meta-analyses of partial correlations are biased (2024) established the problem in theory and simulation, and proposed corrections.
- Reducing the biases of the conventional meta-analysis of correlations (2025) carried the same agenda to ordinary correlations, and brought in the Hunter-Schmidt approach.
- This paper asks whether any of it changes the answer in practice, by running the alternatives across 172 real meta-analyses rather than simulated data.
Data and code
The survey data and the analysis code are public at
github.com/PetrCala/pcc-survey: an R package holding the analysis dataset, the estimator and publication-selection routines, unit tests, a replication-guide vignette, and package versions locked with renv. The companion Stata and spreadsheet calculations sit under secondary-stats/.
Reference: T. D. Stanley, Petr Cala, Hristos Doucouliagos, Zuzana Irsova, and Tomas Havranek (2026), “Do methods matter in the meta-analysis of partial correlation coefficients?” Charles University, Prague. Available at meta-analysis.cz/pcc_survey.