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arxiv: 2605.06237 · v1 · submitted 2026-05-07 · 📊 stat.ME · stat.AP

Recognition: unknown

Bayesian Fractional Polynomials for Optimal Dosage Estimation with Fish Nutrition Applications

{\AA}shild Krogdahl, Aliaksandr Hubin, Guro L{\o}kka, Trond M. Kortner

Pith reviewed 2026-05-08 07:29 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords Bayesian fractional polynomialsdose-response modelingoptimal dosage estimationBayesian model averagingnonlinear regressionfish nutrition experiments
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0 comments X

The pith

Bayesian averaging over fractional polynomials recovers optimal dose levels more accurately than standard approaches by accounting for uncertainty in the dose-response shape.

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

The paper develops a Bayesian fractional polynomial framework to model nonlinear dose-response curves and estimate the dosage that produces the best outcome. It places priors over a range of fractional polynomial degrees and powers, then averages across the resulting models to quantify uncertainty in both the curve and the location of its optimum. Simulations across varied true response shapes show the method recovers the true optimal dose with lower error than fixed-model or nonparametric benchmarks. The same procedure is applied to real aquaculture data on fish nutrient needs, producing dose recommendations together with credible intervals that reflect model choice uncertainty.

Core claim

The central claim is that a Bayesian model average over a library of fractional polynomial regressions yields accurate point estimates and uncertainty statements for the dose that maximises or minimises a biological response, outperforming fixed functional forms in simulation recovery of known optima and providing practical dose recommendations on fish nutrition trials.

What carries the argument

Bayesian model averaging across a discrete space of fractional polynomial models, which supplies both a flexible nonlinear basis and posterior probabilities that automatically down-weight misspecified forms when locating the response optimum.

If this is right

  • Dose recommendations in aquaculture and pharmacology can be reported with explicit uncertainty intervals that incorporate doubt about the correct curve shape.
  • Model averaging reduces the risk that an arbitrary choice of polynomial degree or power produces a misleading location for the optimum.
  • The same workflow applies directly to any continuous dose variable whose effect on a measured response is expected to be smooth but of unknown exact shape.

Where Pith is reading between the lines

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

  • The framework could be extended to jointly model multiple responses or multiple nutrients by expanding the model space to include multivariate fractional polynomials.
  • When the response is known to be monotonic, restricting the polynomial space to monotonic members would shrink uncertainty around the optimum without changing the core averaging procedure.

Load-bearing premise

Fractional polynomials of the degrees and powers considered can approximate the true underlying dose-response curve closely enough that the location of its optimum is not systematically shifted.

What would settle it

A simulation in which the true response is generated from a functional family far from any fractional polynomial (for example a high-frequency oscillation or a sharp threshold) and the method's recovered optimum deviates substantially from the known true value.

read the original abstract

The problem of optimal dosage estimation arises in diverse scientific domains, from pharmacology and toxicology to aquaculture and environmental studies. Statistical modeling of nonlinear dose-response relationships is essential to quantify biological effects and determine response-optimal levels. This paper introduces a flexible Bayesian fractional polynomial (BFP) framework for modeling such relationships, allowing for model uncertainty quantification and robust prediction through Bayesian model averaging. Extensive simulation results demonstrate that the proposed BFP approach yields accurate estimation of optimal dose levels, outperforming benchmarks significantly. The approach is demonstrated on real data from fish nutrient requirement experiments.

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.

Referee Report

3 major / 2 minor

Summary. The paper introduces a Bayesian fractional polynomial (BFP) framework for modeling nonlinear dose-response relationships to estimate optimal dosage levels. It employs Bayesian model averaging over a family of fractional polynomials to quantify model uncertainty and robustly predict the argmax, with claims of accurate optimal dose recovery in extensive simulations (outperforming benchmarks) and an application to real fish nutrient requirement data.

Significance. If the simulation results prove robust to misspecification and the method generalizes, the BFP approach could supply a practical Bayesian tool for dosage optimization in pharmacology, toxicology, and aquaculture by incorporating model uncertainty via averaging. The fish nutrition demonstration shows applied utility, though external validation would increase impact.

major comments (3)
  1. [Abstract] Abstract: the assertion that 'extensive simulation results demonstrate that the proposed BFP approach yields accurate estimation of optimal dose levels, outperforming benchmarks significantly' is unsupported by any description of simulation design, data-generating processes, benchmark methods, error metrics, or sensitivity analyses; this directly undermines the central performance claim.
  2. [Methods / Model Space] Model assumptions (implicit in methods): the chosen fractional polynomial basis (powers such as -2, -1, -0.5, 0.5, 1, 2, 3) may fail to approximate common nutrient dose-response shapes with thresholds, plateaus, or sigmoidal saturation. No misspecification experiments are described that generate data outside this span, so systematic bias in recovered argmax cannot be ruled out.
  3. [Application to Fish Data] Real data application: the fish nutrient experiments provide no ground-truth optimal dose, so reported performance rests on internal model fit rather than recovery of a known optimum; this weakens the empirical support for the claim of accurate estimation.
minor comments (2)
  1. [Methods] Clarify the precise enumeration of the model space, the prior on model indicators, and the exact definition of the optimal dose estimator (e.g., posterior mean of argmax versus argmax of posterior mean).
  2. [Simulation Results] Simulation figures should report variability (e.g., standard errors or boxplots across replicates) rather than point summaries alone.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight important aspects of clarity and robustness in our presentation. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that 'extensive simulation results demonstrate that the proposed BFP approach yields accurate estimation of optimal dose levels, outperforming benchmarks significantly' is unsupported by any description of simulation design, data-generating processes, benchmark methods, error metrics, or sensitivity analyses; this directly undermines the central performance claim.

    Authors: We agree that the abstract would be strengthened by a brief summary of the simulation study. In the revised manuscript we will expand the abstract to include a concise description of the simulation design, the range of data-generating processes considered, the benchmark methods, the error metrics (bias and MSE for argmax recovery), and the sensitivity checks performed. Full details remain in the Simulation Studies section. revision: yes

  2. Referee: [Methods / Model Space] Model assumptions (implicit in methods): the chosen fractional polynomial basis (powers such as -2, -1, -0.5, 0.5, 1, 2, 3) may fail to approximate common nutrient dose-response shapes with thresholds, plateaus, or sigmoidal saturation. No misspecification experiments are described that generate data outside this span, so systematic bias in recovered argmax cannot be ruled out.

    Authors: We acknowledge that the selected power set may not perfectly capture every possible shape, including sharp thresholds or flat plateaus. To address this concern directly, we will add a new set of misspecification experiments in the revised paper. These experiments will generate data from models outside the fractional-polynomial family (e.g., logistic saturation and piecewise threshold functions) and evaluate the robustness of the Bayesian model-averaged argmax estimator under such conditions. revision: yes

  3. Referee: [Application to Fish Data] Real data application: the fish nutrient experiments provide no ground-truth optimal dose, so reported performance rests on internal model fit rather than recovery of a known optimum; this weakens the empirical support for the claim of accurate estimation.

    Authors: We agree that the real fish-nutrient data lack a known ground-truth optimum, as is typical for applied problems. The simulations, where the true optimum is known, constitute the primary evidence for accurate recovery. The fish-data example is presented as an illustration of practical utility and uncertainty quantification rather than as independent validation. We will revise the text to make this distinction explicit. revision: partial

Circularity Check

0 steps flagged

No circularity: BFP is a new modeling framework whose performance claims rest on external simulation and data benchmarks

full rationale

The paper defines a Bayesian fractional polynomial model space, applies Bayesian model averaging to obtain a predictive dose-response surface, and then locates the optimal dose as the argmax of that surface. These steps are constructive definitions of an estimator; they do not reduce any claimed result back to a quantity that was already fitted from the same data by construction. Simulation studies and the fish-nutrient application supply independent empirical checks rather than algebraic identities. No self-citation is invoked as a uniqueness theorem, no ansatz is smuggled, and no known empirical pattern is merely renamed. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only, specific free parameters such as fractional powers or prior hyperparameters cannot be identified; the approach likely relies on standard Bayesian priors and a discrete model space over polynomial forms without introducing new invented entities.

pith-pipeline@v0.9.0 · 5398 in / 1062 out tokens · 73951 ms · 2026-05-08T07:29:58.246599+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

7 extracted references

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