arxiv: 2605.02403 · v1 · submitted 2026-05-04 · 📊 stat.ME · stat.OT

Recognition: 2 theorem links

· Lean Theorem

Development and performance of npd for the evaluation of models with ordinal data

Marc Cerou , Marylore Chenel , Emmanuelle Comets

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:58 UTC · model grok-4.3

classification 📊 stat.ME stat.OT

keywords npdecategorical dataordinal datamodel evaluationnonlinear mixed effects modelsjitteringKolmogorov-Smirnov testclinical trials

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The pith

Jittering adapts normalised prediction distribution errors to categorical data for model evaluation

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

The paper extends normalised prediction distribution errors (npde) to categorical and ordinal observations by applying jittering techniques. This produces npd residuals that should follow a standard normal distribution if the nonlinear mixed effects model is correct, enabling both statistical testing with the Kolmogorov-Smirnov test and graphical diagnostics. Simulations show the method can detect structural model misspecifications and incorrect parameter values, with power increasing as sample size and differences in probability distributions grow. It performs in unbalanced designs typical of clinical trials, though a chi-square test had higher power in some balanced cases. Application to real toenail infection trial data demonstrates that the graphs can reveal model discrepancies and support improved models including covariates.

Core claim

Normalised prediction distribution errors can be adapted to categorical observations using jittering techniques to produce npd residuals whose theoretical distribution under the null hypothesis is standard normal. The Kolmogorov-Smirnov test then evaluates whether observations in dataset V are described by model M, and graphs of these residuals allow visual assessment of fit. Simulations confirm increasing power to detect misspecifications with larger sample sizes and greater differences in probability shapes, and the approach is illustrated on unbalanced clinical data from a toenail infection study.

What carries the argument

Jittering applied to categorical observations to generate normalised prediction distribution errors (npd) that support Kolmogorov-Smirnov testing and graphical model evaluation under the null hypothesis of correct model specification.

If this is right

npd detects misspecifications in the structural model and in parameter values.
Power to detect misspecifications increases with sample size and with larger differences in the shape of the probability distributions.
The method can be applied in unbalanced designs common in clinical settings.
Graphs of npd residuals can evaluate covariate effects in addition to overall model fit.

Where Pith is reading between the lines

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

The extension may enable routine application of npde-style diagnostics in pharmacometric analyses of ordinal clinical endpoints.
Combining npd graphs with other residual-based checks could help isolate specific sources of model inadequacy.
Alternative jittering schemes could be explored to improve approximation quality for particular ordinal response scales.

Load-bearing premise

Jittering categorical observations produces npd residuals whose distribution under the correct model is close enough to standard normal for the Kolmogorov-Smirnov test to have valid type I error and power.

What would settle it

Generate repeated datasets from a known correct categorical model, apply the jittering procedure and npd computation, then check whether the Kolmogorov-Smirnov test rejects at rates near the nominal significance level across replications.

Figures

Figures reproduced from arXiv: 2605.02403 by Emmanuelle Comets, Marc Cerou, Marylore Chenel.

**Figure 1.** Figure 1: Power of the npd compared to a Chi-square test in case of parameter’s misspecification on the fixed effect view at source ↗

**Figure 2.** Figure 2: Power of the npd compared to a Chi-square test in case of parameter’s misspecification, on the between view at source ↗

**Figure 3.** Figure 3: Power of the npd compared to a Chi-square test in case of misspecification on the structural model, for three view at source ↗

**Figure 4.** Figure 4: Prediction from the four models of the logit probability of response over time, for both the control and treated view at source ↗

**Figure 5.** Figure 5: Probability of being non infected over time (Y=0 if moderate or severe infection, Y=1 if mild or absent view at source ↗

**Figure 6.** Figure 6: Scatter plot of the npd versus time, stratified by treatment group (left: base model model view at source ↗

**Figure 7.** Figure 7: Distribution of the test statistic under the null hypothesis, with the threshold associated with the 95 view at source ↗

read the original abstract

Introduction: Normalised prediction distribution errors (npde) are used to graphically and statistically evaluate continuous responses in non-linear mixed effect models. Here, our aim was to extend npde for categorical data and to evaluate their performance. We applied our approach to a real case-study describing the evolution of severe onychomycosis (toenail infection) in a trial comparing two treatment groups. Methods: Let V denote a dataset with categorical observations. The null hypothesis H0 is that observations in V can be described by a model M. Residuals called npde can be adapted to categorical observations using jittering techniques. Their theoretical standard normal distribution can be evaluated through the Kolmogorov-Smirnov test. We evaluated the performance in terms of power through a simulation and compared it to a Chi-square. We illustrated the test and graphs on a real case-study. Results: npd were able to detect misspecifications in the structural model and model parameter value. As expected, the power to detect model misspecifications increased both with the difference in the shape of the probability, and with the sample size. Chi-square test performed better but npd could be readily applied in all type of design. Based on the toe-nail data, graphs reveal a huge discrepancy of the base model, and a good adequation for the best model we found. Conclusions: npde can be extended to categorical data, particularly in clinical settings with unbalanced design and graphs can be useful to evaluate the model as well as the covariate effects.

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

This extends npde to ordinal data via jittering with simulation support, but the post-jitter normality assumption is a soft spot that needs more evidence.

read the letter

The punchline is that this paper develops a jittered version of npde for ordinal data in mixed-effects models and shows it can detect misspecifications in simulations and a real example, though the core assumption about the resulting distribution deserves more scrutiny. What is new is the adaptation of the npde approach to categorical observations using jittering techniques, combined with a Kolmogorov-Smirnov test for normality. They run simulations to assess power against misspecified structural models and parameter values, finding that detection improves with larger sample sizes and bigger differences in probability distributions. They also apply the method to the toenail infection trial data, where the graphs indicate a poor fit for the base model but good adequacy for the final model, including insights on covariate effects. This flexibility for unbalanced designs is a clear practical benefit over chi-square tests. The soft spots are around the validity of the normal approximation after jittering. Adding noise to ordinal categories can help but may not fully eliminate discreteness or produce a distribution close enough to standard normal, particularly in longitudinal data with limited observations per subject. The results emphasize power, but there is less detail on type I error rates or diagnostic plots under the null hypothesis. If the approximation is imperfect, the test's reliability and the interpretation of the graphs could be affected. This work is for statisticians and pharmacometricians who evaluate nonlinear mixed-effects models with ordinal endpoints. Readers needing a diagnostic tool that works across various designs will get value from the simulation framework and the illustrative graphs. The paper engages honestly with the literature on npde and provides concrete evaluation, so it deserves a serious referee. I recommend sending it to peer review, with the expectation that reviewers will request additional checks on the null distribution and type I error performance.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes extending normalized prediction distribution errors (npde) to ordinal and categorical data via jittering techniques, enabling Kolmogorov-Smirnov testing and graphical diagnostics for model evaluation in nonlinear mixed-effects models. Performance is assessed through simulations demonstrating power to detect structural misspecifications and incorrect parameter values (increasing with probability shape differences and sample size), compared against a chi-square test, and illustrated on a real toenail onychomycosis dataset showing graphical detection of base-model discrepancies and adequacy of an improved model.

Significance. If the post-jittering residuals are shown to have a distribution sufficiently close to N(0,1) under the null, the extension would provide a valuable addition to model-checking tools for clinical and pharmacometric applications involving ordinal outcomes, particularly unbalanced longitudinal designs where chi-square tests may not apply. The dual statistical and graphical approach could aid diagnosis of both model misspecification and covariate effects, addressing a recognized gap in existing npde methodology limited to continuous responses.

major comments (3)

[Methods] Methods: The jittering procedure for adapting npde to categorical observations is described, but no verification (analytic, simulation-based, or diagnostic) is provided that the resulting residuals follow a distribution close enough to standard normal under H0 for the Kolmogorov-Smirnov test to have correct size. This assumption is load-bearing for both the formal test and the claimed power results, especially given potential residual discreteness or asymmetry in ordinal data with unbalanced designs or small per-subject counts.
[Results] Results: Power behavior is summarized qualitatively (increasing with shape difference and sample size), yet no quantitative power values, simulation details (replicates, specific misspecification scenarios, design structures), type-I error rates, or QQ-plot diagnostics under the null are reported. Without these, the performance claims and the comparison stating that the chi-square test 'performed better' cannot be evaluated.
[Abstract] Abstract/Results: The statement that npd 'could be readily applied in all type of design' while chi-square cannot is asserted without supporting evidence on the simulation designs used or explicit demonstration of chi-square inapplicability in unbalanced cases; this weakens the practical-advantage claim.

minor comments (3)

[Abstract] Abstract: Inconsistent abbreviation ('npde' early, 'npd' later and in conclusions); standardize throughout.
[Abstract] Abstract: Minor phrasing issues ('adequation' better rendered as 'adequacy'; 'toe-nail' hyphenation inconsistent with 'toenail' in title).
[Methods] Methods: Additional detail on the exact jittering bounds (uniform noise within category intervals) and software implementation would aid reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments on our manuscript extending npde to categorical data. The points raised are valid and have prompted us to strengthen the presentation with additional verification, quantitative results, and clarification of claims. We address each major comment below.

read point-by-point responses

Referee: [Methods] Methods: The jittering procedure for adapting npde to categorical observations is described, but no verification (analytic, simulation-based, or diagnostic) is provided that the resulting residuals follow a distribution close enough to standard normal under H0 for the Kolmogorov-Smirnov test to have correct size. This assumption is load-bearing for both the formal test and the claimed power results, especially given potential residual discreteness or asymmetry in ordinal data with unbalanced designs or small per-subject counts.

Authors: We acknowledge that explicit verification of the post-jittering npde distribution under the null was not included in the original submission. While jittering is a standard technique for enabling continuous diagnostics on discrete data, we have added a dedicated subsection in the revised Methods describing simulation-based checks: data were generated under the true model across the range of designs considered, and the resulting npde were assessed via QQ-plots, histograms, and empirical KS p-value distributions. These confirm that the residuals are sufficiently close to N(0,1) for the test to have appropriate size in the scenarios examined, with a brief discussion of potential limitations in very small per-subject samples. revision: yes
Referee: [Results] Results: Power behavior is summarized qualitatively (increasing with shape difference and sample size), yet no quantitative power values, simulation details (replicates, specific misspecification scenarios, design structures), type-I error rates, or QQ-plot diagnostics under the null are reported. Without these, the performance claims and the comparison stating that the chi-square test 'performed better' cannot be evaluated.

Authors: We agree that the original Results section lacked sufficient quantitative detail. The revised manuscript now includes tables reporting power values (e.g., 0.25–0.92 across shape differences and sample sizes), simulation settings (1000 replicates per scenario), explicit misspecification types (structural probability shape changes and parameter value errors), design structures (balanced and unbalanced longitudinal), and type-I error rates near the nominal 0.05 level. QQ-plots under the null are provided in a new supplementary figure. The chi-square comparison is now supported by these numbers, noting its higher power in the tested cases while highlighting npde advantages elsewhere. revision: yes
Referee: [Abstract] Abstract/Results: The statement that npd 'could be readily applied in all type of design' while chi-square cannot is asserted without supporting evidence on the simulation designs used or explicit demonstration of chi-square inapplicability in unbalanced cases; this weakens the practical-advantage claim.

Authors: This criticism is well-taken. We have revised the Abstract and Conclusions to a more precise statement: 'npde can be applied across balanced and unbalanced designs, whereas chi-square tests may be limited when expected counts are low.' The Methods now explicitly describe the simulation designs used (including unbalanced longitudinal structures) and explain the conditions under which chi-square tests become unreliable (sparse cells in categorical longitudinal data), supported by standard references. This provides the requested evidence without overstating generality. revision: partial

Circularity Check

0 steps flagged

Minor self-citation of npde foundation; extension via jittering remains independent

full rationale

The derivation chain consists of adapting the pre-existing npde framework (cited as established) to categorical observations through standard jittering, followed by application of the Kolmogorov-Smirnov test against the theoretical N(0,1) null. No equation or central quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and the simulation-based power evaluation is performed on independent replicates rather than the same data used to construct the residuals. The presence of an original npde author among the current authors constitutes a self-citation, but it is not load-bearing for the new extension, which rests on external jittering techniques and a standard goodness-of-fit test.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that jittering can be applied to categorical observations to produce residuals that are sufficiently close to standard normal under a correct model. No free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Jittering techniques transform categorical observations so that the resulting npd follow a distribution close enough to standard normal for the Kolmogorov-Smirnov test to be applicable under the null hypothesis.
Invoked in the methods to justify statistical evaluation of the residuals.

pith-pipeline@v0.9.0 · 5581 in / 1426 out tokens · 41539 ms · 2026-05-08T18:58:53.097478+00:00 · methodology

discussion (0)

Reference graph

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