Bayes factor functions for testing partial correlation coefficients
Pith reviewed 2026-05-22 23:54 UTC · model grok-4.3
The pith
Bayes factor functions assess partial correlations by varying priors on standardized effects and accumulating evidence across studies.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
BFFs for partial correlations are derived from test statistics and expressed as functions of a standardized effect size, providing summaries of hypothesis tests as alternative hypotheses vary over priors on standardized effects and enabling integration of evidence across studies.
What carries the argument
The Bayes Factor Function (BFF), which represents Bayes factors as functions of standardized effect size from the partial correlation test statistic.
If this is right
- Summaries of hypothesis tests can be obtained by varying the prior distributions on the standardized effect size.
- Evidence from multiple studies on partial correlations can be integrated using the BFFs.
- BFFs provide an alternative to p-values for evaluating the presence of partial correlation after controlling for other variables.
Where Pith is reading between the lines
- The method could facilitate sensitivity analyses for different prior choices in partial correlation studies.
- BFFs might be applicable to other correlation measures beyond partial correlations.
Load-bearing premise
That Bayes factors for partial correlations can be validly expressed as functions of standardized effect size derived from test statistics in a manner that supports cumulative evidence across studies.
What would settle it
Computing the Bayes factor directly for a specific prior on the standardized effect size for a given partial correlation dataset and comparing it to the value from the BFF at that effect size; mismatch would falsify the approach.
read the original abstract
Partial correlation coefficients are widely applied in the social sciences to evaluate the relationship between two variables after accounting for the influence of others. In this article, we present Bayes Factor Functions (BFFs) for assessing the presence of partial correlation. BFFs represent Bayes factors derived from test statistics and are expressed as functions of a standardized effect size. While traditional frequentist methods based on $p$-values have been criticized for their inability to provide cumulative evidence in favor of the true hypothesis, Bayesian approaches are often challenged due to their computational demands and sensitivity to prior distributions. BFFs overcome these limitations and offer summaries of hypothesis tests as alternative hypotheses are varied over a range of prior distributions on standardized effects. They also enable the integration of evidence across multiple studies.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces Bayes factor functions (BFFs) for partial correlation coefficients. BFFs are constructed from the partial-correlation t-statistic and expressed as functions of the standardized effect size (the partial correlation itself). The functions summarize evidence across a continuum of prior distributions on the effect size and permit direct multiplication of BFF values across independent studies at any fixed partial-correlation value.
Significance. If the derivations are correct, the BFFs supply a computationally light, prior-robust Bayesian summary for partial correlations that directly supports cumulative evidence across studies. This addresses documented shortcomings of p-values while avoiding the need to elicit a single prior, and extends the existing BFF framework from zero-order to partial correlations with only the appropriate degrees-of-freedom adjustment.
minor comments (1)
- [§3] The manuscript should explicitly state the mapping t = r_p * sqrt((n - k - 2)/(1 - r_p^2)) together with the adjusted degrees of freedom in the BFF formula so that readers can verify the extension from the zero-order case.
Simulated Author's Rebuttal
We thank the referee for the positive review, the accurate summary of the contribution, and the recommendation to accept the manuscript.
Circularity Check
No significant circularity detected
full rationale
The abstract and description frame BFFs for partial correlations as a direct functional extension of prior test-statistic Bayes factors, with the sole adjustment being the degrees of freedom in the t-to-r_p mapping. No quoted equations or steps reduce a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. The mechanism for multiplying BFFs across studies is supplied by the functional form itself rather than by re-deriving inputs from outputs. This is the normal case of an internally consistent methodological extension without load-bearing circular reductions.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
We define the non-centrality parameter under the alternative as λ = sqrt(n-p-1) ω, where ω = ρ* / sqrt(1-(ρ*)²). ... Lemma 1 ... BF10 = m1(t1|τ²,ν)/m0(t1)
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
BFFs ... expressed as functions of a standardized effect size ... enable the integration of evidence across multiple studies
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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