pith. sign in

arxiv: 2604.13264 · v1 · submitted 2026-04-14 · 📊 stat.ME · stat.AP

Estimating effect thresholds and beyond: A flexible framework for multivariate alert detection

Lucia Ameis , Niklas Hagemann , Kathrin M\"ollenhoff This is my paper

Pith reviewed 2026-05-10 14:14 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords alert estimationeffect thresholdsGAMLSSmultivariate dose-responseconfidence bandstoxicity assessmentextrapolationparametric modeling
0
0 comments X

The pith

A GAMLSS parametric model estimates effect alert thresholds in multiple dimensions by fitting to all data and constructing confidence bands or planes.

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

The paper presents a method for estimating alerts, such as the dose at which a response threshold is crossed, when data involves combinations of continuous covariates like time and dose. Instead of requiring measurements at every specific combination, the approach fits a single parametric model using the GAMLSS framework to all available data. This enables extrapolation to predict alerts for unobserved covariate values, such as the dose needed at a particular time point not directly measured. The model fit supports building confidence regions around the alert curves or surfaces to quantify uncertainty. The method is shown through simulations and applied to cytotoxicity data in liver cells.

Core claim

By fitting a flexible parametric model with GAMLSS, alerts depending on multiple covariates can be estimated even without direct observations at the exact points of interest, with confidence bands for two-dimensional slices or planes for the three-dimensional relationship.

What carries the argument

The GAMLSS (Generalized Additive Models for Location, Scale and Shape) framework, which models the response distribution parameters to capture complex three-dimensional structures and allows construction of confidence bands and planes for the alert thresholds.

Load-bearing premise

That the selected parametric GAMLSS model correctly captures the true data-generating process across the covariate space so that extrapolations remain unbiased.

What would settle it

Compare the model's predicted alert thresholds to those obtained from a separate dense set of measurements at the exact covariate combinations of interest; a significant mismatch would falsify the adequacy of the extrapolation.

Figures

Figures reproduced from arXiv: 2604.13264 by Kathrin M\"ollenhoff, Lucia Ameis, Niklas Hagemann.

Figure 1
Figure 1. Figure 1: (A) Centered absolute value of the model fit and estimated confidence bands for a fixed time point. The solid blue line indicates the confidence band estimated from the two-dimensional algorithm. The dashed line indicates a confidence band inferred from the three-dimensional analysis. The red horizontal line marks the threshold value. The red vertical lines visualize the estimated alert doses. (B) Visualiz… view at source ↗
Figure 2
Figure 2. Figure 2: (A) True mean response of Scenario 1. (B) Comparison of the simple, small, medium and large linear models used to generate the standard deviation. (C) Scenario 2 with the factorial 3 × 3 design. (D) Scenario 2 with the D-optimal design. The black plane in (A), (B) and (C) indicate the true response planes. The black lines indicate the two-dimensional curve for the fixed time point t˜ = 4 or dose level of D… view at source ↗
Figure 3
Figure 3. Figure 3: Proportion of rejections of H0 in both scenarios. approach. Interestingly, this trend is not observed for the D-optimal design. Although the number of rejections is higher for n = 152 observations, it remains almost constant for n = 90 and n = 45 observations. Additionally, the overall number of rejections is higher for the D-optimal design, exceeding at least 93.6% for the two￾dimensional approach and 81.… view at source ↗
Figure 4
Figure 4. Figure 4: (A) Counts of identified alerts at different dose levels for the two-dimensional approach. Each block contains the results of a model assumption regarding σ (constant or complex) for the full or reduced design, respectively. The colors differentiate between the various simulation scenarios with simple, small, medium and large standard deviation. The red vertical line indicates the true alert. The dashed ve… view at source ↗
Figure 5
Figure 5. Figure 5: (A) Counts of identified alerts at different levels of Dose 1 for the two-dimensional approach. Each block contains the results of the three different total numbers of observations. The red vertical line indicates the true alert. The dashed vertical lines show the respective medians. (B) Median error of the identified alerts of Dose 2 along the Dose 1-axis for the three-dimensional approach. Each block con… view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of the results obtained from the three-dimensional approach. The upper row displays the results for the full data set, the lower row the results for the artificially reduced data set. The dots represent the measured data points, which have been colored according to the donor. The black plane indicates the model fit. The blue plane represents the estimated confidence plane. The fixed time poin… view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
read the original abstract

Evaluating the influence of continuous covariates, like exposure time or dose, on a response variable is a pivotal objective in the assessment of a compound's effect, particularly when determining toxicity in pre-clinical research or pharmacokinetics in clinical trials. The determination of an alert, such as the ED50 value, at which a pre-specified threshold of the response variable is crossed, is an important tool for the evaluation process. In practice, response data might be available for combinations of different covariates and the alert depending on both is of interest. In this case, it is crucial to use all available information and extrapolate between cases to ensure the optimal utilization of the data. In this paper, we introduce a parametric approach that allows alerts to be estimated in a multidimensional setting. For time-dose-response data, for instance, alert doses at a given time can be determined, even when there are no measurements available at that exact time. Likewise, it allows estimation of alert times for a given dose. More generally, the method makes it possible to characterize the complete alert relationship between covariates by leveraging all available data. This is achieved by fitting a parametric model and constructing either a confidence band for the two-dimensional curve given for example a fixed time or dose or by constructing a confidence plane for the three-dimensional model fit. The initial model fit is achieved by the flexible framework of Generalized Additive Models for Location, Scale and Shape (GAMLSS), which offers the possibility to account for a plethora of complex three-dimensional data structures. We demonstrate the validity of our approach through a simulation study and present an application to data from a study investigating the relevance of the exposure duration on cytotoxicity in primary human hepatocytes.

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

2 major / 2 minor

Summary. The paper proposes a GAMLSS-based parametric framework for estimating multivariate alert thresholds (e.g., ED50 values) in settings such as time-dose-response data. By fitting a single flexible model to all observations, the approach enables extrapolation to unmeasured covariate combinations and constructs either confidence bands around two-dimensional alert curves or confidence planes around three-dimensional alert surfaces. Validity is illustrated through a simulation study and an application to cytotoxicity data in primary human hepatocytes.

Significance. If the GAMLSS model is correctly specified and extrapolation remains valid, the method offers a principled way to pool information across covariate dimensions for threshold estimation, which could improve efficiency in toxicology and pharmacokinetics studies where data are sparse in certain regions. The use of an established GAMLSS framework for this target is a natural extension rather than a circular redefinition.

major comments (2)
  1. [Simulation study] Simulation study: No diagnostics are reported for GAMLSS fit adequacy (e.g., randomized quantile residuals, worm plots, or out-of-sample predictive checks) in the three-dimensional setting, nor are sensitivity analyses shown for distribution family choice, smoother penalties, or link functions. Without these, it is unclear whether the level-set inversion yielding the alert curve/plane has valid coverage, especially under extrapolation.
  2. [Application] Application section: The cytotoxicity analysis extrapolates alert times/doses to unmeasured combinations without reporting model-misspecification checks or comparison against nonparametric alternatives (e.g., direct quantile regression surfaces). This leaves open the possibility that apparent alert relationships are driven by parametric assumptions rather than data.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a brief statement of the precise GAMLSS distribution and smoothing terms ultimately selected for the real-data example.
  2. [Methods] Notation for the alert surface (e.g., the level-set definition) should be introduced with an equation early in the Methods to clarify the mapping from the fitted GAMLSS parameters to the reported confidence plane.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of model validation. We address each major point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Simulation study] Simulation study: No diagnostics are reported for GAMLSS fit adequacy (e.g., randomized quantile residuals, worm plots, or out-of-sample predictive checks) in the three-dimensional setting, nor are sensitivity analyses shown for distribution family choice, smoother penalties, or link functions. Without these, it is unclear whether the level-set inversion yielding the alert curve/plane has valid coverage, especially under extrapolation.

    Authors: We agree that additional diagnostics would improve transparency. In the revised manuscript we will report randomized quantile residuals and worm plots for the three-dimensional GAMLSS fits used in the simulation study. We will also add sensitivity analyses that vary the distribution family, smoother penalties, and link functions, confirming that empirical coverage of the alert thresholds remains close to nominal levels. The existing simulation already demonstrates that the level-set inversion achieves valid coverage when the model is correctly specified, including under extrapolation; the new checks will make this robustness explicit. revision: yes

  2. Referee: [Application] Application section: The cytotoxicity analysis extrapolates alert times/doses to unmeasured combinations without reporting model-misspecification checks or comparison against nonparametric alternatives (e.g., direct quantile regression surfaces). This leaves open the possibility that apparent alert relationships are driven by parametric assumptions rather than data.

    Authors: We accept that explicit misspecification checks are warranted. The revision will include residual diagnostics and goodness-of-fit assessments for the GAMLSS model fitted to the cytotoxicity data. A direct head-to-head comparison with nonparametric quantile regression surfaces is not straightforward, because the latter does not naturally yield the same alert-threshold inversion or extrapolation capability in three dimensions; we will therefore add a brief discussion of the parametric assumptions justified by the biological context and data range, while acknowledging that the GAMLSS framework is chosen precisely to enable the multivariate extrapolation that is the paper's focus. revision: partial

Circularity Check

0 steps flagged

No circularity: standard application of external GAMLSS framework to new alert-threshold target

full rationale

The paper's central procedure fits an established GAMLSS model to multivariate response data and then inverts the fitted surface to obtain alert curves or planes with associated confidence bands. No step reduces by construction to a quantity already fitted inside the paper, no parameter is renamed as a prediction, and no load-bearing premise rests on a self-citation chain. The derivation chain therefore consists of external methodology plus ordinary statistical post-processing and remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach depends on the standard GAMLSS modeling assumptions and the validity of parametric extrapolation; no new entities are introduced.

free parameters (1)
  • GAMLSS distribution parameters
    Location, scale, and shape parameters of the chosen distributions are fitted to the observed response data.
axioms (1)
  • domain assumption Response data can be adequately described by a GAMLSS distribution family across the covariate space
    Invoked to justify fitting a single parametric model that supports extrapolation to unmeasured time-dose combinations.

pith-pipeline@v0.9.0 · 5607 in / 1287 out tokens · 52139 ms · 2026-05-10T14:14:11.261452+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    and M \"o llenhoff, K

    Ameis, L. and M \"o llenhoff, K. (2024). Identification of changes in gene expression. arXiv preprint

    work page 2024

  2. [2]

    o ber, R., Schug, M., Rempel, E., Rahnenf \

    Arbo, M., Melega, S., St \"o ber, R., Schug, M., Rempel, E., Rahnenf \"u hrer, J., Godoy, P., Reif, R., Cadenas, C., de Lourdes Bastos , M., et al. (2016). Hepatotoxicity of piperazine designer drugs: up-regulation of key enzymes of cholesterol and lipid biosynthesis. Archives of Toxicology , 90(12):3045--3060

    work page 2016

  3. [3]

    Bornkamp, B., Pinheiro, J., and Bretz, F. (2009). Mcpmod: An r package for the design and analysis of dose-finding studies. Journal of Statistical Software , 29:1--23

    work page 2009

  4. [4]

    Box, G., Jenkins, G., Reinsel, G., and Ljung, G. (2015). Time series analysis: forecasting and control . John Wiley & Sons

    work page 2015

  5. [5]

    Bretz, F., Dette, H., and Pinheiro, J. (2010). Practical considerations for optimal designs in clinical dose finding studies. Statistics in medicine , 29(7-8):731--742

    work page 2010

  6. [6]

    De Bastiani , F., Heller, G., Kneib, T., Mayr, A., Rigby, R., Stasinopoulos, M., et al. (2026). GAMLSS . Accessed February 05, 2026

    work page 2026

  7. [7]

    Delignette-Muller, M.-L., Forfait, C., Billoir, E., and Charles, S. (2011). A new perspective on the Dunnett procedure: Filling the gap between NOEC/LOEC and ECx concepts . Environmental Toxicology and Chemistry , 30(12):2888--2891

    work page 2011

  8. [8]

    Demidenko, E. (2013). Mixed models: theory and applications with R . John Wiley & Sons

    work page 2013

  9. [9]

    astel, H., Rahnenf\

    Duda, J., Drenda, C., K\"astel, H., Rahnenf\"uhrer, J., and Kappenberg, F. (2023). Benefit of using interaction effects for the analysis of high-dimensional time-response or dose-response data for two-group comparisons. Scientific Reports , 13(1):20804

    work page 2023

  10. [10]

    Duda, J., Hengstler, J., and Rahnenf\"uhrer, J. (2022). td2pll: An intuitive time-dose-response model for cytotoxicity data with varying exposure durations. Computational Toxicology , 23:100234

    work page 2022

  11. [11]

    and Tibshirani, R

    Efron, B. and Tibshirani, R. (1994). An Introduction to the Bootstrap . CRC Press, 1st ed. edition

    work page 1994

  12. [12]

    Fahrmeir, L., Kneib, T., Lang, S., and Marx, B. (2022). Regression: Models, Methods and Applications . Springer Berlin, Heidelberg, 2nd edition

    work page 2022

  13. [13]

    Fawcett, T. (2006). An introduction to roc analysis. Pattern recognition letters , 27(8):861--874

    work page 2006

  14. [14]

    and Burzykowski, T

    Ga ecki, A. and Burzykowski, T. (2012). Linear mixed-effects model. In Linear mixed-effects models using R: a step-by-step approach , pages 245--273. Springer

    work page 2012

  15. [15]

    Ghallab, A., Celli \`e re, G., Henkel, S., Driesch, D., Hoehme, S., Hofmann, U., Zellmer, S., Godoy, P., Sachinidis, A., Blaszkewicz, M., et al. (2016). Model-guided identification of a therapeutic strategy to reduce hyperammonemia in liver diseases. Journal of hepatology , 64(4):860--871

    work page 2016

  16. [16]

    Gu, X., Albrecht, W., Edlund, K., Kappenberg, F., Rahnenf \"u hrer, J., Leist, M., Moritz, W., Godoy, P., Cadenas, C., Marchan, R., et al. (2018). Relevance of the incubation period in cytotoxicity testing with primary human hepatocytes. Archives of toxicology , 92(12):3505--3515

    work page 2018

  17. [17]

    and M \"o llenhoff, K

    Hagemann, N. and M \"o llenhoff, K. (2025). Overcoming model uncertainty—how equivalence tests can benefit from model averaging. Statistics in Medicine , 44(6):e10309

    work page 2025

  18. [18]

    Hothorn, L. (2014). Statistical evaluation of toxicological bioassays - a review. Toxicology Research , 3(6):418--432

    work page 2014

  19. [19]

    Huusari, R., Wang, T., Szedmak, S., Aittokallio, T., and Rousu, J. (2025). Predicting drug combination response surfaces. npj Drug Discovery , 2(1):2

    work page 2025

  20. [20]

    Guidance document on using in vitro data to estimate in vivo starting doses for acute toxicity

    ICCVAM (2001). Guidance document on using in vitro data to estimate in vivo starting doses for acute toxicity . Accessed March 18, 2026

    work page 2001

  21. [21]

    Jensen, S., Kluxen, F., and Ritz, C. (2019). A review of recent advances in benchmark dose methodology. Risk Analysis , 39(10):2295--2315

    work page 2019

  22. [22]

    u rmeyer, L., G \

    Kappenberg, F., Duda, J., Sch \"u rmeyer, L., G \"u l, O., Brecklinghaus, T., Hengstler, J., Schorning, K., and Rahnenf \"u hrer, J. (2023). Guidance for statistical design and analysis of toxicological dose--response experiments, based on a comprehensive literature review. Archives of Toxicology , 97(10):2741--2761

    work page 2023

  23. [23]

    Kappenberg, F., Grinberg, M., Jiang, X., Kopp-Schneider, A., Hengstler, J., and Rahnenf\"uhrer, J. (2021). Comparison of observation-based and model-based identification of alert concentrations from concentration-expression data. Bioinformatics , 37(14):1990 -- 1996

    work page 2021

  24. [24]

    and Schmidli, H

    Lange, M. and Schmidli, H. (2015). Analysis of clinical trials with biologics using dose--time-response models. Statistics in Medicine , 34(22):3017--3028

    work page 2015

  25. [25]

    Lee, S. (2010). Drug interaction: focusing on response surface models. Korean journal of anesthesiology , 58(5):421--434

    work page 2010

  26. [26]

    Loewe, S. (1953). The problem of synergism and antagonism of combined drugs. Arzneimittel-forschung , 3(6):285--290

    work page 1953

  27. [27]

    Manning, C. (2008). Introduction to information retrieval . Syngress Publishing,

    work page 2008

  28. [28]

    Martin-Betancor, K., Ritz, C., Fern \'a ndez-Pi \ n as, F., Legan \'e s, F., and Rodea-Palomares, I. (2015). Defining an additivity framework for mixture research in inducible whole-cell biosensors. Scientific Reports , 5(1):17200

    work page 2015

  29. [29]

    Mokhtari, R., Homayouni, T., Baluch, N., Morgatskaya, E., Kumar, S., Das, B., and Yeger, H. (2017). Combination therapy in combating cancer. Oncotarget , 8(23):38022

    work page 2017

  30. [30]

    M\"ollenhoff, K., Schorning, K., and Kappenberg, F. (2022). Identifying alert concentrations using a model-based bootstrap approach. Biometrics , 79(3):2076--2088

    work page 2022

  31. [31]

    Riddell, R., Panacer, D., Wilde, S., Clothier, R., and Balls, M. (1986). The importance of exposure period and cell type in in vitro cytotoxicity tests. Alternatives to Laboratory Animals , 14(2):86--92

    work page 1986

  32. [32]

    and Stasinopoulos, D

    Rigby, R. and Stasinopoulos, D. (2005). Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society Series C: Applied Statistics , 54(3):507--554

    work page 2005

  33. [33]

    Sch \"u rmeyer, L., Sandig, L., Hezler, L., Igl, B.-W., and Schorning, K. (2025). Optimal designs for identifying effective doses in drug combination studies. arXiv preprint

    work page 2025

  34. [34]

    Sch\"uttler, A., Altenburger, R., Ammar, M., Bader-Blukott, M., Jakobs, G., Knapp, J., Krüger, J., Reiche, K., Wu, G., and Busch, W. (2019). Map and model—moving from observation to prediction in toxicogenomics. GigaScience , 8(6):giz057

    work page 2019

  35. [35]

    Sebaugh, J. L. (2011). Guidelines for accurate ec50/ic50 estimation. Pharmaceutical Statistics , 10(2):128--134

    work page 2011

  36. [36]

    and Rigby, R

    Stasinopoulos, D. and Rigby, R. (2008). Generalized additive models for location scale and shape (gamlss) in r. Journal of Statistical Software , 23:1--46

    work page 2008

  37. [37]

    Stasinopoulos, M., Rigby, R., Heller, G., Voudouris, V., and De Bastiani , F. (2017). Flexible Regression and Smoothing: Using GAMLSS in R . Chapman and Hall/CRC

    work page 2017

  38. [38]

    Zhou, Y., Sloan, A., Menon, S., and Wang, L. (2025). Combination mcp-mod for two-drug combination dose-ranging studies. Journal of Biopharmaceutical Statistics , 35(2):257--270

    work page 2025