Improved Risk Ratio Approximation by Complementary Log-Log Models: A Comparison with Logistic Models

Kenji Beppu; Yuji Tsubota

REVIEW 2 major objections 4 minor 16 references

Improved Risk Ratio Approximation by Complementary Log-Log Models: A Comparison with Logistic Models

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For any two outcome probabilities, the complementary log ratio is always closer to the risk ratio than the odds ratio is.

desk verdict Sound, genuinely new estimand-level result; the practical recommendation needs more support. read the letter →

arxiv 2506.00889 v1 pith:U7NZRMCM submitted 2025-06-01 stat.ME stat.AP

Yuji Tsubota , Kenji Beppu This is my paper

classification stat.MEstat.AP MSC 62J12

keywords riskratioapproximationcomplementarylog-logmodeloddsAranda-Ordazfamilybinaryregressionlinkfunctionlogmaximumrelativediscrepancy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that the effect measure of a complementary log-log model, which it names the complementary log ratio, always tracks the risk ratio more closely than the odds ratio does. The comparison is exact: for any two distinct outcome probabilities in the unit interval, the maximum relative discrepancy between the risk ratio and the complementary log ratio is strictly smaller than the same discrepancy for the odds ratio. The proof works inside the Aranda-Ordaz family of link transformations, a one-parameter family that contains the complementary log-log link at one end and the logit link at the other. Because the complementary log-log model is a standard GLM already available in R and SAS, the practical payoff is that practitioners can get closer risk-ratio approximations simply by changing the link function. The paper does not argue against direct risk-ratio estimation; it offers the complementary log-log model as an accessible alternative when direct methods are unstable.

What carries the argument

The central object is the Aranda-Ordaz family of transformations $W_\lambda(\theta)=\{(1-\theta)^{-\lambda}-1\}/\lambda$ for $0<\lambda\le1$ and $W_0(\theta)=-\log(1-\theta)$, which interpolates between the complementary log-log link ($\lambda=0$) and the logit link ($\lambda=1$). The proof works by writing the generalized ratio as $WR(\lambda)=(e^{\lambda b}-1)/(e^{\lambda a}-1)$ with $a=-\log(1-p_0)$ and $b=-\log(1-p_1)$, then showing that the derivative of $\log WR(\lambda)$ is positive because the function $h(x)=x e^x/(e^x-1)$ is strictly increasing for $x>0$. Monotonicity of $WR(\lambda)$ plus convexity of $W_\lambda$ in $\theta$ yields monotonicity of the discrepancy $B(\lambda)$ and the direction of the over- or underestimation.

What would settle it

Fix any $0<p_0\ne p_1<1$ and compute $RR=p_1/p_0$, $CLR=\log(1-p_1)/\log(1-p_0)$, $OR=\frac{p_1/(1-p_1)}{p_0/(1-p_0)}$, and the two maxima $\max(RR/CLR,CLR/RR)$ and $\max(RR/OR,OR/RR)$; a single pair with the former not smaller than the latter would refute Corollary 1. Equivalently, check whether $\frac{d}{d\lambda}\log WR(\lambda)$ ever becomes negative for some $\lambda\in(0,1)$ and $p_0\ne p_1$, which would overturn Theorem 1.

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Extended reading notes

Core claim

The paper's central result is that, for every pair of distinct outcome probabilities $p_0,p_1\in(0,1)$, the complementary log ratio $CLR=\log(1-p_1)/\log(1-p_0)$ is a better approximation to the risk ratio $RR=p_1/p_0$ than the odds ratio $OR$ is, in the sense of the maximum relative discrepancy $B=\max(RR/\text{measure},\text{measure}/RR)$. This is Corollary 1, and it follows from the stronger Theorem 1: within the Aranda-Ordaz family $W_\lambda(\theta)$, the function $B(\lambda)=\max(RR/WR(\lambda),WR(\lambda)/RR)$ is strictly increasing in $\lambda$ over $[0,1]$. Since $WR(1)=OR$ and $WR(0)=CLR$, the complementary log-log link minimizes approximation error inside the family while the logit link maximizes it. Lemma 1 supplies the directional part: $WR(\lambda)$ always overestimates the risk ratio when $RR>1$ and underestimates it when $RR<1$, so the discrepancy never flips sign.

Load-bearing premise

The proof compares the complementary log ratio and the odds ratio as exact functions of the two true outcome probabilities, so it does not account for estimation error, convergence problems, or the standard errors of fitted models; if those sources of variability are large enough, the practical advantage of the complementary log-log model could shrink or disappear.

Editorial extensions

If this is right

In any binary comparison with distinct outcome probabilities, no pair $(p_0,p_1)$ can make the odds ratio a better risk-ratio approximation than the complementary log ratio under the maximum relative discrepancy measure.
Inside the Aranda-Ordaz family, approximation error is ordered by the parameter: $\lambda=0$ (complementary log-log) is best, $\lambda=1$ (logit) is worst, and intermediate links fall between.
When $RR>1$, both the odds ratio and the complementary log ratio overestimate the risk ratio, and when $RR<1$ both underestimate it; the complementary log ratio's deviation is always the smaller one.
Because the complementary log-log model is a standard GLM available in R and SAS, switching from logistic regression to a complementary log-log model gives a closer risk-ratio approximation with essentially no extra implementation cost.
The paper's recommendation is scope-limited: direct risk-ratio estimation remains preferable when it is stable, and the complementary log-log model is proposed as a fallback, not a replacement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the proof is deterministic and says nothing about estimation uncertainty; in finite samples, the complementary log ratio estimator's advantage in bias could be outweighed by larger variance or worse confidence-interval coverage, which the paper does not address.
Inference: the monotonicity result suggests the Aranda-Ordaz parameter could be tuned, for example choosing a small positive $\lambda$ to retain some logistic-like inferential behavior while keeping most of the risk-ratio bias reduction, although the paper only advocates the $\lambda=0$ endpoint.
Inference: because $CLR=\log(1-p_1)/\log(1-p_0)$ is a ratio of log-survival probabilities, the result connects naturally to proportional-hazards thinking, hinting that complementary log-log regression may be especially appropriate in settings where a hazard-ratio-style interpretation is already intended.
Inference: a testable extension is to compare, in simulation, the relative bias and coverage of cloglog versus logistic links for estimating risk ratios under covariate adjustment; the paper's theory predicts smaller relative bias for cloglog, but the sampling behavior is an open question.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline: the math is sound and the result is genuinely new — complementary log ratio is provably closer to risk ratio than odds ratio for any two outcome probabilities — but the paper's practical pitch outruns what the proof alone supports.

What it does well: Corollary 1 is not stated in the cited literature. The proof is clean: convexity of the Aranda-Ordaz transform gives the inequality, and the monotonicity argument with h(x)=x e^x/(e^x−1) is solid. Presenting logit and cloglog as special cases of the Aranda-Ordaz family is a genuinely nice unifying move, and the paper is short and readable. The derivation is parameter-free, so there is no circularity.

The soft spots: first, the abstract promises 'simulation studies further reinforce our theoretical findings,' but no simulations appear in the supplied manuscript. If that is an artifact of the version, fine; as presented, the empirical support cannot be checked. Second, and more substantive, the theorem is about estimands, not estimators. For fixed p0 and p1, CLR is closer to RR than OR. But in a fitted GLM, both the estimated OR and the estimated CLR are random. The proof says nothing about their variances, standard errors, or mean squared error around the true RR. So the recommendation to prefer cloglog in routine regression practice is a step beyond the theorem. The estimand-level result is useful and may well carry over, but that needs simulation or at least an explicit caveat. There is a minor typo in the appendix: the second derivative should be (λ+1)(1−θ)^{−λ−2}, but convexity still holds, so the proof is unaffected.

Who it's for: biostatisticians and epidemiologists who routinely report odds ratios for common outcomes and want a simple way to get closer to risk ratios. The theoretical result is a nice, citable fact. A serious referee should engage with it: the proof deserves checking and the practical claims need to be brought in line with the evidence. My recommendation: send it to peer review, and ask for either the simulations or a rewritten abstract and discussion that limit the claim to the estimand comparison.

Referee Report

2 major / 4 minor

Summary. The paper defines the complementary log ratio (CLR) as log(1−p1)/log(1−p0), the estimand of a complementary log-log regression, and compares it with the odds ratio (OR) as an approximation to the risk ratio RR = p1/p0. Using the Aranda-Ordaz family Wλ(θ) with λ=0 giving the cloglog transformation and λ=1 giving the logit, the paper proves Lemma 1 (WR(λ) always lies on the same side of RR as the inequality between p0 and p1), Theorem 1 (the maximum relative discrepancy B(λ) is strictly increasing on [0,1]), and Corollary 1 (CLR is closer to RR than OR under the maximum relative discrepancy measure). The paper then recommends complementary log-log models as a practical alternative to logistic models for risk ratio approximation.

Significance. The theoretical result is clean, elementary, and parameter-free: for any fixed p0 and p1, CLR is a better approximation to RR than OR in the maximum-relative-discrepancy sense, and λ=0 is the minimizer within the restricted Aranda-Ordaz family. This is a useful reference result for categorical data analysis and is proved with a short convexity and monotonicity argument. The strength of the manuscript lies in this deterministic, estimand-level ordering. Its limitation is that the practical recommendation in Section 4 and the abstract's claim of simulation support go beyond what the proof establishes, because nothing is said about sampling variability or the behavior of estimated CLR in finite samples.

major comments (2)

[Section 4 (Discussion) and Corollary 1] The paper's practical recommendation—that practitioners should prefer complementary log-log models to logistic models when the goal is to approximate risk ratios—is based on Corollary 1, which is a deterministic statement about the true estimands CLR and OR for fixed p0 and p1. In any fitted GLM, the estimated OR and estimated CLR are random variables, and the proof says nothing about their sampling variability, standard errors, or mean squared error around the true RR. A lower-bias estimand can still yield a worse practical approximation if its estimator has larger variance, so the transition from Corollary 1 to the practical claim in Section 4 requires additional support, e.g., asymptotic variance calculations, a finite-sample simulation study, or a substantially qualified conclusion.
[Abstract and full text] The abstract states that 'Simulation studies further reinforce our theoretical findings,' but the supplied manuscript contains no simulation section, table, or code. Since the practical claim rests on finite-sample behavior, the absence of the promised simulations makes that support uncheckable. The authors should either add a simulation study that compares estimated CLR and estimated OR in fitted regressions, or remove and explicitly qualify the simulation claim.

minor comments (4)

[eAppendix A.1 (proof of Lemma 1)] The displayed second derivative of Wλ(θ) is incorrect: it should be (λ+1)(1−θ)^{−λ−2}, not λ(1−θ)^{−λ}. The convexity conclusion is unaffected because (λ+1)>0, but the formula should be corrected.
[eAppendix A.1 (proof of Theorem 1)] The proof invokes 'without loss of generality' that 0<p0<p1<1, but it does not explicitly justify why this is WLOG. The justification is that B(λ) is symmetric under exchanging p0 and p1, so adding one sentence about this symmetry would make the argument complete.
[References and in-text citations] The citation 'Zhang and Kai (1998)' is incorrect: the author is Kai F. Yu (JAMA 280(19), 1690–1691, 1998), so the in-text citations and the reference entry 'Zhang, J. and F. Y. Kai (1998)' should be corrected to 'Zhang, J. and Yu, K. F. (1998)'.
[Section 3, definition of B(λ)] The displayed definition of B(λ) has a line-break artifact that makes it appear as max(RR WR(λ), WR(λ) RR); it should be typeset clearly as max(RR/WR(λ), WR(λ)/RR). The surrounding text makes the intent clear, but the equation should be reformatted.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Corollary 1 is a parameter-free analytic inequality proved by calculus; no fitted inputs or load-bearing self-citations.

full rationale

The load-bearing claim (Corollary 1) is derived from Theorem 1, whose proof in eAppendix A.1 differentiates log WR(λ) and shows that h(x)=x e^x/(e^x-1) is strictly increasing for x>0 because e^x>x+1. This is ordinary calculus applied directly to the definitions OR=p1(1-p0)/(p0(1-p1)) and CLR=log(1-p1)/log(1-p0). No parameter is fitted to any data, no external benchmark is required, and no author self-citation is load-bearing. The Aranda-Ordaz parameterization is cited for its mathematical form, but the paper proves the inequality itself rather than importing it from the citation. Thus the comparison does not reduce to its inputs by construction. The abstract's statement that 'Simulation studies further reinforce our theoretical findings' is unsupported in the visible manuscript, and the extrapolation from estimand-level comparison to finite-sample regression practice is a real gap, but neither is a circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. The transformation parameter lambda is a known Aranda-Ordaz quantity and the proof does not fit any values to data. The axioms are standard calculus facts used inside the proof. The only non-visible supporting evidence is the claimed simulation study, which would be an empirical check rather than a load-bearing axiom.

assumptions (3)

standard math The Aranda-Ordaz family W_lambda(theta) is well-defined and continuous at lambda = 0 with the limiting definition.
Used in Section 2.2 to define WR(lambda) and to justify continuity in the proof of Lemma 1. This is a standard mathematical definition, not an ad hoc assumption.
standard math The functions W_lambda are convex in theta for 0 < lambda <= 1, as shown by the second derivative being positive.
The second derivative calculation in eAppendix A.1 justifies the convexity used in the proof of Lemma 1. It is a direct calculus result.
standard math For x > 0, e^x > x + 1, which is used to show h(x) = x e^x / (e^x - 1) is strictly increasing.
Used in the proof of Theorem 1 to establish monotonicity of WR(lambda). This is a standard inequality.

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Cite this review

Pith. "Pith review of Improved Risk Ratio Approximation by Complementary Log-Log Models: A Comparison with Logistic Models." pith.science (2026). https://pith.science/paper/U7NZRMCM

@misc{pith2026250600889,
  author       = {Pith},
  title        = {Pith review of: Improved Risk Ratio Approximation by Complementary Log-Log Models: A Comparison with Logistic Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7NZRMCM}},
  note         = {Machine review of arXiv:2506.00889}
}

read the original abstract

Odds ratios obtained from logistic models fail to approximate risk ratios with common outcomes, leading to potential misinterpretations about exposure effects by practitioners. This article investigates the complementary log-log models as a practical alternative to produce risk ratio approximation. We demonstrate that the corresponding effect measure of complementary log-log models, called the complementary log ratio in this article, consistently provides a closer approximation to risk ratios than odds ratios. To compare the approximation accuracy, we adopt the one-parameter Aranda-Ordaz family of link functions, which includes both the logit and complementary log-log link functions as special cases. Within this unified framework, we implement a theoretical comparison of approximation accuracy between the complementary log ratio and the odds ratio, showing that the former always produces smaller approximation bias. Simulation studies further reinforce our theoretical findings. Given that the complementary log-log model is easily implemented in standard statistical software such as R and SAS, we encourage more frequent use of this model as a simple and effective alternative to logistic models when the goal is to approximate risk ratios more accurately.

Figures

Figures reproduced from arXiv: 2506.00889 by the authors.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Agresti, A. (2010). Analysis of ordinal categorical data , Volume 656. John Wiley & Sons

work page 2010
[2]

Agresti, A. (2012). Categorical Data Analysis . John Wiley & Sons

work page 2012
[3]

Altman, D. G., J. J. Deeks, and D. L. Sackett (1998). Odds ratios should be avoided when events are common. BMJ: British Medical Journal\/ 317\/ (7168), 1318

work page 1998
[4]

Aranda-Ordaz, F. J. (1981). On two families of transformations to additivity for binary response data. Biometrika\/ 68\/ (2), 357--363

work page 1981
[5]

Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character\/ 222\/ (594-604), 309--368

work page 1922
[6]

Hosmer Jr, D. W., S. Lemeshow, and R. X. Sturdivant (2013). Applied logistic regression . John Wiley & Sons

work page 2013
[7]

Knol, M. J., R. G. Duijnhoven, D. E. Grobbee, K. G. Moons, and R. H. Groenwold (2011). Potential misinterpretation of treatment effects due to use of odds ratios and logistic regression in randomized controlled trials. PLoS One\/ 6\/ (6), e21248

work page 2011
[8]

Knol, M. J., S. Le Cessie, A. Algra, J. P. Vandenbroucke, and R. H. Groenwold (2012). Overestimation of risk ratios by odds ratios in trials and cohort studies: alternatives to logistic regression. Cmaj\/ 184\/ (8), 895--899

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[9]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Agresti, A. (2010). Analysis of ordinal categorical data , Volume 656. John Wiley & Sons

work page 2010

[2] [2]

Agresti, A. (2012). Categorical Data Analysis . John Wiley & Sons

work page 2012

[3] [3]

Altman, D. G., J. J. Deeks, and D. L. Sackett (1998). Odds ratios should be avoided when events are common. BMJ: British Medical Journal\/ 317\/ (7168), 1318

work page 1998

[4] [4]

Aranda-Ordaz, F. J. (1981). On two families of transformations to additivity for binary response data. Biometrika\/ 68\/ (2), 357--363

work page 1981

[5] [5]

Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character\/ 222\/ (594-604), 309--368

work page 1922

[6] [6]

Hosmer Jr, D. W., S. Lemeshow, and R. X. Sturdivant (2013). Applied logistic regression . John Wiley & Sons

work page 2013

[7] [7]

Knol, M. J., R. G. Duijnhoven, D. E. Grobbee, K. G. Moons, and R. H. Groenwold (2011). Potential misinterpretation of treatment effects due to use of odds ratios and logistic regression in randomized controlled trials. PLoS One\/ 6\/ (6), e21248

work page 2011

[8] [8]

Knol, M. J., S. Le Cessie, A. Algra, J. P. Vandenbroucke, and R. H. Groenwold (2012). Overestimation of risk ratios by odds ratios in trials and cohort studies: alternatives to logistic regression. Cmaj\/ 184\/ (8), 895--899

work page 2012