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This paper proves that raw cognitive-screening scores can be guaranteed to outperform demographically-corrected scores in classification accuracy, under three explicit conditions, and that corrections fail to guarantee fairness.

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2026-08-02 09:21 UTC pith:EY5QA3SH

load-bearing objection Genuinely useful framework and a real step forward for continuous scores, but the main theorem's proof breaks down for discrete scores — and the application is discrete. the 3 major comments →

arxiv 2606.31418 v2 pith:EY5QA3SH submitted 2026-06-30 stat.ME

On the choice of using raw or demographically-corrected scores

Ignacio Gonzalez-Perez , Mats Julius Stensrud , Marco Piccininni This is my paper
classification stat.ME
keywords demographic correctioncognitive screeningROC curveclassification accuracyco-monotonicitydemographic insensitivitycausal inferencefairness
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper takes a long-standing dispute in neuropsychology — whether screening tests for cognitive impairment should be scored raw or adjusted for age and education — and gives a nonparametric answer. It proves a formal inequality: if the test is well-designed (monotone likelihood ratio), the correction is well-designed (corrected score independent of demographics among healthy people), and a co-monotonicity condition links demographics to both raw scores and impairment, then at any fixed false-positive rate the raw score has at least as high a true-positive rate as the corrected score. The co-monotonicity condition is the substantive premise, and it has a clear causal reading: demographic correction hurts when demographics affect both test performance and the risk of impairment in the same direction. A parallel theorem shows the opposite when the condition reverses. Practitioners who defend corrections as fair also lose: the paper shows corrections equalize false-positive rates but do not, in general, make decisions insensitive to demographics.

Core claim

The central claim is Theorem 1. For a fixed false-positive rate α, the difference between the raw and corrected ROC curves is bounded below by an expectation involving the joint likelihood ratio of (X,V) and the covariate-specific false-positive rate; when the co-monotonicity assumption holds, the weighted Chebyshev inequality forces that lower bound nonnegative. Thus raw scores dominate corrected scores pointwise in the ROC plane. The proof works for discrete and continuous scores and for a general class of corrections (Definition 1), and a multivariate extension is given in Appendix C. Under the symmetric counter-monotonicity assumption, corrected scores dominate instead, and if both hold

What carries the argument

The machinery is the comparison of marginal ROC curves through covariate-specific ROC curves, whose slopes are likelihood ratios. Because a correction is an increasing transformation within each demographic stratum, covariate-specific ROC curves of raw and corrected scores coincide (Lemma 1); the marginal comparison therefore reduces to a covariance between the likelihood ratio and the deviation of the stratum-specific false-positive rate from the marginal rate. The decisive tool is the weighted Chebyshev inequality, applied under the probabilistic co-monotonicity condition (Assumption 3), which says the demographic gradient in raw scores among healthy people and the demographic gradient in

Load-bearing premise

The load-bearing premise is Assumption 3, probabilistic co-monotonicity: among healthy people the demographic group with lower raw scores must also be the group in which impairment is more likely at a fixed raw score; if those two gradients ever point in opposite directions, the proof's Chebyshev step changes sign and corrected scores can be the better classifier.

What would settle it

Find a large cohort in which, among cognitively healthy people, older age is associated with lower raw scores, but conditional on a fixed raw score older age is associated with lower (not higher) dementia prevalence — then Assumption 3 fails at that threshold and the predicted raw-score edge need not appear. Equivalently, a precise estimate of the lower bound τ(α) that is significantly negative at any false-positive rate α would directly contradict Theorem 1's sufficient condition applied to that setting.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If Assumption 3 holds at every false-positive rate, the area under the ROC curve of the raw score is at least that of the corrected score (Corollary 3).
  • If Assumption 4 (counter-monotonicity) holds instead, the corrected score wins; when both hold, raw and corrected scores have identical marginal ROC curves.
  • A well-designed correction equalizes false-positive rates across demographic groups but need not equalize true-positive rates; demographic corrections therefore do not by themselves deliver demographic insensitivity of decisions.
  • The results give a nonparametric explanation for the repeated empirical finding that age and education corrections do not improve dementia screening accuracy.
  • In the open neuroimaging dataset analyzed, point estimates of the lower bound favor raw scores at most thresholds, though the nonparametric confidence intervals include zero; a parametric model gives significant evidence for raw-score superiority and for Assumption 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural next step is a pre-registered accuracy study that estimates the lower bound τ(α) with enough precision to settle the sign; the paper's inequality turns that sign into a decision rule.
  • The same inequality could be used to decide when to keep or drop demographic corrections in other fields — kidney-function, osteoporosis, lung-function, and cardiovascular risk scores all have analogous demographic adjustments.
  • The demographic-insensitivity framework suggests a testable criterion: a corrected score is fair in the paper's sense only if intervention on the sensitive attribute changes neither true- nor false-positive classification probabilities; the analysis finds the age-corrected score remains age-sensitive among impaired individuals.
  • The nonparametric part of the data analysis is inconclusive; the empirical conclusion that raw scores win in this cohort depends on a correctly specified parametric model, so the applied claim is weaker than the theoretical one.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies whether raw cognitive-screening scores or demographically corrected scores have better classification accuracy. It defines a general class of corrections, states sufficient conditions for raw scores to dominate corrected scores at a fixed FPR (Theorem 1), gives a converse condition for corrected scores (Theorem 2), adds a causal interpretation of the key co-monotonicity assumption, and proposes formal notions of demographic insensitivity. The theory is applied to MMSE data from OASIS-3, with a nonparametric analysis of the lower-bound quantity τ(t) and a parametric analysis supporting raw-score superiority. The paper is transparent about assumptions and includes substantial appendix proofs.

Significance. If the central theorem were correct, the paper would provide a useful, largely assumption-light explanation for the recurring empirical finding that demographic corrections do not improve, and can worsen, classification accuracy in cognitive screening. The multivariate extension in Appendix C, the causal interpretation in Section 2.3, and the intervening-variable formalization of demographic insensitivity in Appendix D are thoughtful and go beyond existing parametric analyses. The authors are also commendably explicit about the inconclusiveness of their nonparametric estimates. However, the main theorem is not established for discrete scores, and the paper's primary application (the MMSE) is discrete; the stress-test concern is real. A concrete counterexample falsifies Theorem 1 as stated under the paper's own ROC definition.

major comments (3)
  1. [§2.1–2.2 and Appendix B.1, Theorem 1] The proof fails at atoms of the score distribution. It defines t_S(α)=F^{-1}_{S,0}(α) and uses the identity E[α^X_V−α|D=0]=0, where α^S_v=P(S≤t_S(α)|D=0,V=v). For discrete S, F_{S,0}(t_S(α))=c_S≥α, generally strictly, so this identity is false; under Assumption 2, α^Z_v=c_Z, not α. Thus the comparison is not at the claimed FPR α and the Chebyshev step collapses. Remark 1's interpolation convention is not used in the proof; if it is used, the proof's first equality ROC_X(α)−ROC_Z(α)=P(X≤t_X(α)|D=1)−P(Z≤t_Z(α)|D=1) is also false. Concretely: let V∈{0,1} be equiprobable, P(D=1|V=0)=0.2, P(D=1|V=1)=0.8; X|D=0,V=0 uniform on {0,1}, X|D=0,V=1 uniform on {−1,0}, X|D=1,V=0 supported on {0,1} with masses .8,.2, and X|D=1,V=1 supported on {−1,0} with masses .8,.2. The z-score correction satisfies Assumptions 1–3. For α=0.6, F^{-1}_{X,0}(0.6)=0 and F^{-1}_{Z,0}(0.6)=1; the displayed ROC definition
  2. [§4.2, Table 1] The empirical implementation inherits the discrete-score problem. The reparametrized quantity τ(t)=E[LR_{X,V}(t,V)(α^X_V(t)−α(t))|D=0] is the theorem's lower bound only if t=F^{-1}_{X,0}(F_{X,0}(t)); for discrete X this is generally false, and the theorem's inequality is false at atoms. Therefore the claim that rejecting H0:τ(t)<0 provides evidence for raw superiority is not supported by Theorem 1. In addition, all reported right-sided 95% confidence intervals in Table 1 contain zero, so the nonparametric analysis is inconclusive; the parametric normal model is the only analysis that yields significance, and it assumes continuous scores rather than the observed discrete MMSE.
  3. [Assumption 2 and discrete corrections] Assumption 2 is much more restrictive for discrete scores than the text suggests. For a deterministic strictly increasing correction, Z⊥V|D=0 requires that, after alignment of the transformed support points, the conditional probability masses of X|D=0,V=v are identical across strata; location-scale shifts are a special case. For age-group-specific MMSE distributions, the usual z-score correction will generally violate Assumption 2. Since Theorem 1 and the Section 4 application rely on Assumption 2, the paper should either characterize when Assumption 2 holds for discrete tests or explicitly use the Appendix C.1 route without it.
minor comments (3)
  1. [Throughout] Typos: Definition 2 says 'neither or Type I nor II'; Appendix B.1 says 'Regrading' instead of 'Regarding'; Appendix D.2 says 'at leats' instead of 'at least'. These should be corrected.
  2. [§4.2] The bootstrap p-values are only described as 'considerably below 5%'. Please report the actual p-values or the bootstrap confidence intervals.
  3. [Appendix D.3] After defining bν_weighted,D(s), the next paragraph states that bν_weighted,S(s) is consistent; this appears to be a typo for bν_weighted,D(s). Also, Tables 2 and 3 are large; a compact summary of which intermediate nulls are rejected would improve readability.

Circularity Check

0 steps flagged

No significant circularity: Theorem 1 is derived from explicit Assumptions 1–3 using standard inequalities; the OASIS-3 analysis checks the assumptions rather than fitting the conclusion; self-citations are background only.

full rationale

Walking the derivation chain, Definition 1 does define corrected scores as per-stratum increasing transforms of X, and Lemma 1's equality of covariate-specific ROC curves is a direct consequence of that definition. The paper does not present this as a prediction; the claimed result is the marginal ROC comparison, which is not encoded in Definition 1. Theorem 1 is proved in Appendix B.1 from Assumptions 1–3: Assumption 1 gives concavity, Assumption 2 gives α^Z_v = α by stipulation (a well-designed correction), and Assumption 3 is a substantive co-monotonicity condition on observable quantities, used only to apply Chebyshev's weighted inequality. No fitted constants enter the proof, and the lower bound in Eq. (2) is a genuine sufficient-statistic expression rather than a restatement of ROC_X ≥ ROC_Z. The OASIS-3 application is an empirical check: Section 4.2 estimates τ(t) nonparametrically and fits parametric models only to translate Assumption 3 into coefficient conditions; it does not fit a parameter to the raw-minus-corrected ROC difference and call it a prediction. The demographic-insensitivity analysis (Section 4.3, Appendices D/F) gives identification theorems and plug-in estimators, again without fitting the target conclusion. References [21,22] include the authors' prior work, but they are cited for background, motivation, and causal interpretation; the main theorem does not rely on them, and no uniqueness or ansatz result is imported from those papers. The reviewer's discrete-score objection is a correctness concern about whether the quantile identity E[α^X_V−α|D=0]=0 holds for MMSE-like discrete scores under the generalized-inverse definition; that is not a circular reduction of the conclusion to the assumptions, so it is outside the circularity score.

Axiom & Free-Parameter Ledger

5 free parameters · 8 axioms · 1 invented entities

The central theorem is conditional on Definition 1 and Assumptions 1-3; no fitted parameters enter the proof. Free parameters appear only in the OASIS-3 parametric models used to test Assumption 3 and in the estimated z-score correction. V^S is a conceptual entity introduced to formalize demographic insensitivity.

free parameters (5)
  • β0, β1 (logistic model for D|V*) = not reported
    Fitted in the OASIS-3 parametric analysis (Eq. 3) used to test the sufficient-condition null in Eq. (4).
  • μ0, μ1, μ2, σ² (normal model for X|D,V*) = not reported
    Fitted in Eq. (3); μ1 and μ2 enter the test for raw-score superiority via the quantity μ1(β1 − μ1μ2/σ²).
  • η0, η1, δ0, δ1, δ2 (conditional model for Assumption 3) = η1δ1 product CI (−0.0025, −0.0012)
    Fitted to assess Assumption 3 through H0: η1δ1 > 0; bootstrap percentile interval reported in Section 4.2.
  • z-score correction mean/SD in normative group = not reported
    Estimated from the OASIS-3 non-impaired fold to construct age-corrected scores in Section 4.3.
  • age-quartile cutoffs (65, 70, 75) = 65/70/75
    Chosen as approximate age quartiles of OASIS-3; a modeling choice affecting all data analyses, not tuned to outcomes.
axioms (8)
  • domain assumption Definition 1: corrected score is a strictly increasing differentiable transform of the raw score within each V stratum
    Formalizes all corrections considered; used in Lemma 1 and throughout Section 2.
  • domain assumption Assumption 1: the conditional likelihood ratio LR_{X|V}(x|v) is decreasing in x
    Equivalent to concave covariate-specific ROC curves; needed for Lemma 3 and Theorem 1.
  • domain assumption Assumption 2: Z⊥V|D=0
    Assumes the correction removes demographic dependence among non-impaired; only guaranteed for location-scale families.
  • domain assumption Assumption 3: co-monotonicity of α^X_v and P(D=1|X=t,V=v)
    Key sufficient condition in Theorem 1; empirical and verifiable when D is known.
  • domain assumption Assumption 4: counter-monotonicity for corrected-score superiority
    Dual condition used in Theorem 2.
  • domain assumption Causal DAG V→D→X and V→X with no unmeasured common causes
    Used in Section 2.3 for interpretation and in Section 3/Appendix D for identifying demographic insensitivity.
  • domain assumption Identification conditions in Appendix D: positivity, consistency, dismissible component conditions, faithfulness, V^S partial isolation
    Needed to identify counterfactual score distributions in Theorems 5-9.
  • standard math Standard measure-theoretic tools: Chebyshev's weighted inequality, Bayes rule, δ-method, multinomial CLT
    Used in proofs of Lemma 3, Theorem 1, and Appendix F inference.
invented entities (1)
  • V^S (sensitive attributes / intervening variable) no independent evidence
    purpose: Formalize demographic insensitivity: a hypothetical variable representing sensory ability/background knowledge to which the test is sensitive, deterministically tied to observed demographics V.
    Introduced in Section 3.2; not directly measured and no falsifiable handle outside the paper. It is a conceptual device for defining counterfactual insensitivity.

pith-pipeline@v1.3.0-alltime-deepseek · 43020 in / 17292 out tokens · 155205 ms · 2026-08-02T09:21:22.701406+00:00 · methodology

0 comments
read the original abstract

Demographic corrections are routinely performed in many disciplines, including psychology. Yet, there are ongoing debates about whether these corrections are appropriate and improve classification accuracy. Here, we focus on cognitive screening tests, and show that common demographic corrections, like the z-score correction, can be detrimental for classification in some settings. Formally, we present sufficient conditions ensuring that raw scores outperform the demographically-corrected ones, and give a substantive interpretation of this result. We also investigate the claim that using demographically-corrected scores results in more fair decisions compared to using raw scores. We apply our results to the Mini-Mental State Examination in the OASIS-3 dataset.

Figures

Figures reproduced from arXiv: 2606.31418 by Ignacio Gonzalez-Perez, Marco Piccininni, Mats Julius Stensrud.

Figure 1
Figure 1. Figure 1: DAG representing the causal structure of the variables in our system. strong assumptions: it does not allow raw and corrected ROC curves to cross. For example, it is possible that ROC curves cross and still AUC(X) ≥ AUC(Z). This is considered a disadvantage of the AUC as a comparative tool [43]. However, our previous results like Theorems 1 and 2 are not affected by this caveat, as they perform local (at a… view at source ↗
Figure 2
Figure 2. Figure 2: Example of a causal DAG for the scenario introduced in Sec￾tion 3 (A) and its associated extended causal DAG (B). Definition 3 (Strong demographic insensitivity). We say that a score S is insen￾sitive to the demographics V in the strong sense if the distribution of S vS | {V = v, D = d} does not depend on vS for all (v, d) in V × {0, 1}. Yet, in our screening problem we are not necessarily interested in th… view at source ↗
Figure 3
Figure 3. Figure 3: Extended causal DAG introduced in [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

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