arxiv: 2605.11768 · v1 · submitted 2026-05-12 · 📊 stat.ME

Recognition: 2 theorem links

· Lean Theorem

Using NonTargeted HPV Infections in Studies with Risk Compensation

Lola Etievant

Pith reviewed 2026-05-13 05:30 UTC · model grok-4.3

classification 📊 stat.ME

keywords HPV vaccinerisk compensationcausal inferenceconfoundingmediationdirect effectobservational studies

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The pith

Using nontargeted HPV infections removes both confounding bias and behavioral mediation to isolate the vaccine's direct immunological effect on targeted strains.

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

Observational studies of HPV vaccine effectiveness face bias from unmeasured differences in sexual behavior between vaccinated and unvaccinated women, and also from possible risk compensation where vaccination leads to riskier behavior. The paper demonstrates that infections with nontargeted HPV strains can be used to adjust for these issues. This adjustment removes both the confounding bias and the indirect effect mediated by behavior changes, leaving an estimate of the vaccine's direct immunological protection against targeted strains. Readers would care because this provides a causal interpretation for the estimates commonly produced in practice, though it may not reflect the total protection women experience if they alter their behavior after vaccination.

Core claim

When unmeasured sexual behavior acts as both a confounder and a mediator between vaccination and targeted HPV infections, the quantity estimated using nontargeted HPV infections represents the direct causal effect of the vaccine on targeted strains through immune mechanisms alone, excluding any behavioral component.

What carries the argument

Nontargeted HPV infections used as a reference outcome to isolate the direct effect in a causal model with unmeasured confounding and mediation by sexual behavior.

If this is right

The method produces an estimate with a clear causal meaning as the direct immunological effect.
In the presence of risk compensation the estimate suggests higher protection than women actually experience in total.
Unblinded randomized trials would allow nontargeted infections to isolate the indirect behavioral effect separately from the direct effect.
Public health evaluations must distinguish the direct immunological quantity from the total effect that includes behavior changes.

Where Pith is reading between the lines

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

The same reference-outcome approach could apply to other multi-strain vaccines where behavior change might confound efficacy estimates.
Routine collection of nontargeted infection data in observational vaccine studies would support better causal claims about direct protection.
Estimating the full total effect including behavioral responses likely requires different study designs such as blinded trials.

Load-bearing premise

The vaccine has no direct effect on nontargeted HPV strains and the causal structure contains no additional unmeasured paths that would prevent the nontargeted infections from correctly isolating the direct effect.

What would settle it

A controlled study showing that the vaccine directly affects nontargeted strains, or a dataset with measured behavior where the adjusted estimate fails to match the known immunological protection, would disprove the direct-effect interpretation.

Figures

Figures reproduced from arXiv: 2605.11768 by Lola Etievant.

**Figure 2.** Figure 2: Graph depicting the relationships among the variables in - [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** Figure 3: Boxplots of estimates with Joint-MH (red), Joint-Reg (green), MH (blue), and [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

**Figure 4.** Figure 4: Graph depicting the relationships among the variables in the unblinded random [PITH_FULL_IMAGE:figures/full_fig_p046_4.png] view at source ↗

**Figure 5.** Figure 5: Boxplots of Joint-NC estimates from 5,000 simulated studies under the un [PITH_FULL_IMAGE:figures/full_fig_p048_5.png] view at source ↗

**Figure 6.** Figure 6: Graph depicting the relationships among the variables in the observational [PITH_FULL_IMAGE:figures/full_fig_p051_6.png] view at source ↗

**Figure 7.** Figure 7: Graph depicting the relationships among the variables in the observational [PITH_FULL_IMAGE:figures/full_fig_p055_7.png] view at source ↗

read the original abstract

Studies of HPV vaccine efficacy usually record infections with vaccine targeted and nontargeted strains. Contrary to blinded randomized controlled trials, confounding bias can be a threat and risk compensation may occur in observational studies. Etievant et al. (Biometrics, 2023) proposed to use cervical infections with nontargeted HPV strains to reduce or remove confounding bias of estimates of vaccine efficacy on targeted strains. However, they assumed that vaccinated women could not change their behavior after vaccination. We consider a more plausible setting where unmeasured sexual behavior acts as both a confounder and a mediator, and investigate if the quantity estimated in practice with their method has a clear causal meaning. We demonstrate that using nontargeted HPV infections can remove both confounding bias and the portion of the vaccine effect on the targeted HPV strains that is mediated through the change of behavior. In that case, the estimated quantity has a clear causal interpretation as it represents the direct immunological effect of the vaccine. However, it could be considered misleading from a public health perspective, as in the presence of risk compensation it would suggest higher protection than what women effectively experience. An unblinded randomized controlled trial would allow estimation of the total causal effect of the vaccine, and infections with nontargeted HPV strains could then be used to isolate the indirect behavioral effect of the vaccine.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows the 2023 nontargeted HPV estimator still isolates the direct immunological effect even with risk compensation, since behavior acts as both confounder and mediator.

read the letter

The main point is that the estimator from Etievant et al. (Biometrics, 2023) continues to recover the direct vaccine effect on targeted strains when unmeasured sexual behavior changes after vaccination. The paper models behavior as both a confounder and a mediator, then shows that nontargeted infections adjust for both the bias and the mediated portion, leaving only the immunological direct effect. This is the new piece: the prior work assumed no behavior change, and this relaxes that while keeping a clean causal reading. It does well by spelling out the distinction from the total effect and by noting that the direct-effect estimate could overstate real-world protection from a public-health standpoint. The suggestion to use an RCT for the total effect and then subtract to get the indirect behavioral part is also straightforward. The soft spots are limited. The no-direct-effect assumption on nontargeted strains is stated clearly but would benefit from a short sensitivity check or discussion of possible indirect paths. The abstract outlines the derivation, but explicit causal diagrams and any simulation results would make the identification steps easier to verify at a glance. The work stays theoretical, so an applied example would help readers see the practical difference. This is for statisticians and epidemiologists focused on observational vaccine studies or causal methods for infectious diseases. A reader already familiar with the 2023 paper will get the most out of the extension. The argument is coherent and directly engages the prior literature, so it deserves a serious referee rather than a desk reject. I would send it for peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript extends prior work on using nontargeted HPV infections to adjust for confounding in observational studies of HPV vaccine efficacy. It considers a setting where unmeasured sexual behavior acts as both a confounder and a mediator due to risk compensation after vaccination. The central claim is that the estimator based on nontargeted infections removes both the confounding bias and the mediated behavioral component, yielding a quantity with a causal interpretation as the direct immunological effect of the vaccine on targeted strains (distinct from the total effect that includes behavioral mediation).

Significance. If the identification assumptions hold, the result offers a practical method for obtaining a causally interpretable direct-effect estimate from observational HPV data that would otherwise be biased by unmeasured behavior. This is relevant for vaccine effectiveness studies where RCTs are not feasible, and the paper appropriately flags that the direct effect may overstate the protection experienced by women if risk compensation occurs. The work is grounded in explicit causal modeling rather than post-hoc parameter fitting.

major comments (1)

[Identification section / causal diagram] The identification result relies on the assumption of no direct vaccine effect on nontargeted strains and the absence of unmeasured paths from behavior to nontargeted infections that bypass the mediator structure. While the abstract and main argument state this, the manuscript would benefit from an explicit sensitivity analysis or bounding exercise showing how violations affect the estimator (e.g., via a simulation or analytic bound in the identification section).

minor comments (2)

[Abstract] The abstract states the result clearly but could briefly note the key identifying assumptions (no direct effect on nontargeted strains, behavior as sole unmeasured common cause) to help readers assess applicability.
[Methods / notation] Notation for the direct effect parameter and the adjusted estimator should be introduced with a short table or explicit mapping to the causal diagram for easier reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. We address the single major comment below and will incorporate the suggested addition in the revised manuscript.

read point-by-point responses

Referee: The identification result relies on the assumption of no direct vaccine effect on nontargeted strains and the absence of unmeasured paths from behavior to nontargeted infections that bypass the mediator structure. While the abstract and main argument state this, the manuscript would benefit from an explicit sensitivity analysis or bounding exercise showing how violations affect the estimator (e.g., via a simulation or analytic bound in the identification section).

Authors: We agree that the identification result depends on these assumptions (no direct vaccine effect on nontargeted strains and no unmeasured paths from behavior to nontargeted infections that bypass the mediator). While the manuscript already states the assumptions explicitly, we concur that a sensitivity analysis would strengthen the presentation. In the revised version we will add a brief simulation study to the identification section that quantifies the bias in the estimator under controlled violations of each assumption, thereby providing readers with a practical assessment of robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends a prior causal identification strategy for HPV vaccine efficacy using nontargeted infections to a new setting that includes risk compensation as both confounder and mediator. The central result—that the estimator isolates the direct immunological effect—is obtained by applying standard potential-outcomes or DAG-based identification formulas under explicitly stated assumptions (no direct vaccine effect on nontargeted strains, unmeasured behavior as sole common cause/mediator). These formulas are derived from the model rather than fitted to the same data or defined in terms of the target quantity itself. The citation to Etievant et al. (2023) supplies the baseline method but does not carry the new claim about removal of the mediated behavioral component; that step is shown by direct manipulation of the identification expression. No self-definitional loops, fitted-input predictions, or ansatz smuggling appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard causal inference assumptions for confounding and mediation adjustment using a proxy variable (nontargeted infections); no free parameters or new entities are introduced.

axioms (1)

domain assumption Standard causal assumptions including consistency, positivity, and no unmeasured confounding for the paths involving vaccination, behavior, and targeted/nontargeted infections.
Invoked to allow identification of the direct effect via nontargeted infections as a proxy for behavior.

pith-pipeline@v0.9.0 · 5526 in / 1413 out tokens · 109968 ms · 2026-05-13T05:30:36.927963+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
AT E(Y1) = exp(β1) × EA[E{g1(Ã)|A,T=1}]/EA[E{g1(Ã)|A,T=0}], NDE(Y1,T=0)=exp(β1)

Reference graph

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As a reminder, this can be done by solving forθ= (α ∗ 1, β∗ 1, α∗ 2, β ∗

On the other hand, the approach proposed by [7] (the joint approach with no covariates, Joint- NC) uses data on{T, Y 1, Y2}and consists in jointly estimating E(Y1|T=1) E(Y1|T=0) and E(Y2|T=1) E(Y2|T=0) . As a reminder, this can be done by solving forθ= (α ∗ 1, β∗ 1, α∗ 2, β ∗

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E g1( ˜A)|W, T= 1 E g1( ˜A)|W, T= 0 # T + log h E g1( ˜A)|W, T= 0 i! Similarly, using Equation (6) in Section 2.5 in the Main Document, we have E(Y2 |W, T) = exp α2 +s 2(W) + log

the joint estimating equationPn i=1 U(Y 1,i, Y2,i, Ti;θ) = 0, with U(Y 1,i, Y2,i, Ti;θ) =   Y1,i−p∗ 1,i 1−p∗ 1,i Ti(Y1,i−p∗ 1,i) 1−p∗ 1,i Y2,i −p ∗ 2,i Ti(Y2,i −p ∗ 2,i)   , 26 and wherep ∗ 1,i = exp(α ∗ 1 +β ∗ 1Ti), andp ∗ 2,i = exp(α ∗ 2 +β ∗ 2Ti). This estimating equation is obtained from the log-likelihoods of the models in Equ...

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[32]

For example, assuming that the observed confounder is univariate and that s∗ 1 :w7→µ ∗ 1w+ν ∗ 1 w2 ands ∗ 2 :w7→µ ∗ 2w+ν ∗ 2 w2, we would solve forθ= (α ∗ 1, β∗ 1, µ∗ 1, ν∗ 1 , α∗ 2, β∗ 2, µ∗ 2, ν∗

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[33]

the joint estimating equation Pn i=1 U(Y 1,i, Y2,i, Wi, Ti;θ) = 0, with U(Y 1,i, Y2,i, Wi, Ti;θ) =   Y1,i−p∗ 1,i 1−p∗ 1,i Ti(Y1,i−p∗ 1,i) 1−p∗ 1,i Wi(Y1,i−p∗ 1,i) 1−p∗ 1,i W 2 i (Y1,i−p∗ 1,i) 1−p∗ 1,i Y2,i −p ∗ 2,i Ti(Y2,i −p ∗ 2,i) Wi(Y2,i −p ∗ 2,i) W 2 i (Y2,i −p ∗ 2,i)   , and wherep ∗ 1,i = exp(α...

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[34]

Again, following [7], this can be done by solving forθ= (β ∗ 1, β∗

On the other hand, the approach proposed by [7] consists in jointly estimatingβ ∗ 1 andβ ∗ 2 with the Mantel-Haenszel (MH) method, and then considering estimateβ ∗ 1 −β ∗ 2 (the joint Mantel-Haenszel approach, Joint-MH). Again, following [7], this can be done by solving forθ= (β ∗ 1, β∗

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[35]

the joint estimating equationPK k=1 ωk U X1,k, Z1,k, X2,k, Z 2,k;θ = 0 with U(X 1,k, Z1,k, X2,k, Z2,k;θ) =   X1,k n1,k −exp(β ∗

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[36]

Z1,k n0,k X2,k n1,k −exp(β ∗

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[37]

Z2,k n0,k   , andω k = n1,kn0,k nk , and whereX 1,k =Pn i=1 TiY1,iI(W i =w k),Z 1,k =Pn i=1(1−Ti)Y1,iI(W i = wk),X 2,k = Pn i=1 TiY2,iI(W i =w k),Z 2,k = Pn i=1(1−T i)Y2,iI(W i =w k),n 1,k = Pn i=1 TiI(W i =w k),n 0,k =Pn i=1(1−T i)I(W i =w k),n k =Pn i=1 I(W i =w k), withI() the Indicator function. This would lead to ˆβ∗ 1 = log PK k=1 n0,kX1,k nk PK...

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[38]

Z1,k n0,k 0 0−exp(β ∗

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[39]

Z2,k n0,k   . As a result, if Assumption A 1 and Assumption A 2,W hold under the setting of Figure 1(B)in the Main Document, ˆβ∗ 1 − ˆβ∗ 2 is an unbiased estimate of the log-natural direct effect. Again, the variance estimate can be obtained from adding the variances of ˆβ∗ 1 and ˆβ∗ 2 and subtracting two times their covariance. Finally, ifWonly has a...

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[40]

subtracted off

named the composite primary outcome. If a model of the form in Equation (1) or in Equation (5) holds separately forY (16) 1 andY (18) 1 but with a common functiong 1, then this is also approximately true forY 1 whenY (16) 1 andY (18) 1 are rare and the vaccine offers identical protection against infection by HPV 16 and infection by HPV 18. Indeed, if E(Y ...

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[41]

did not provide a rationale for their method. In particular, because the many articles discussing the use of negative control outcomes or proxies of the unmeasured confounders rely on more intricate methods (e.g., [7], and [29]), the approach proposed by [22] prob- ably does not allow for appropriate estimation the causal effect of the HPV vaccine on the ...

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[42]

As a preliminary step, we consider the setting without risk compensation of [7] (i.e., when ˜A=A) and when W=∅; see Supplementary Figure 7

under a setting such as the one in [7] or in the current work. As a preliminary step, we consider the setting without risk compensation of [7] (i.e., when ˜A=A) and when W=∅; see Supplementary Figure 7. More precisely, we assume E(Y1 |A, T) =g 1(A) exp(α1 +β 1T). As a reminder, under such a setting the average total effect ofTonY 1 equals exp(β1). But wit...

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