Statistical modeling of groundwater quality assessment in Iran using a flexible Poisson likelihood

Afshin Fallah; Havard Rue; Hossein Baghishani; Mahsa Nadifar

REVIEW 4 major objections 6 minor 49 references

Statistical modeling of groundwater quality assessment in Iran using a flexible Poisson likelihood

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A Bayesian gamma-count model with the SPDE approximation improves spatial count predictions over Poisson and negative binomial alternatives.

desk verdict Spatial gamma-count regression is a natural and useful extension; the evidence for its superiority is currently undercut by a misread test statistic and a broken negative binomial baseline. read the letter →

arxiv 1908.02344 v1 pith:6LQPLYUP submitted 2019-08-06 stat.ME stat.AP

Mahsa Nadifar , Hossein Baghishani , Afshin Fallah , Havard Rue This is my paper · ORCID

classification stat.MEstat.AP MSC 62F1562M3062J1262P12

keywords gamma-countdistributionspatialcountdataoverdispersionunderdispersionrenewaltheoryINLASPDEgroundwaterquality

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 aims to give analysts of spatial count data a likelihood that does not force variance to equal the mean. It does this by embedding the gamma-count distribution—obtained when event waiting times are gamma rather than exponential—in a Bayesian spatial hierarchical model, with the spatial field handled through the SPDE approximation and inference by INLA. On groundwater-quality counts from Golestan province, Iran, and in a simulation study, the paper reports that this model fits and predicts better than Poisson and negative binomial models whenever counts are over- or under-dispersed. If correct, practitioners gain a fast, widely usable tool for spatial counts that relaxes equidispersion at the cost of a single extra parameter.

What carries the argument

The central object is the gamma-count distribution from renewal theory: replace the exponential waiting times that generate the Poisson distribution by gamma waiting times $\tau_k\sim\operatorname{Gamma}(\alpha,\gamma)$. The count in $(0,T)$ then has probability mass function as the difference of two incomplete gamma CDFs, and the gamma's reproductive property keeps this a stable numerical expression. The dispersion parameter $\alpha$ is the single additional ingredient: it collapses the model to Poisson at $\alpha=1$, and indexes over- and under-dispersion through negative and positive duration dependence. The spatial machinery is the SPDE representation of a Matern Gaussian field as a Gaussian Markov random field, which gives a sparse precision matrix; combined with the latent-Gaussian structure, this is what makes INLA fast enough for routine use.

What would settle it

Take spatial count data with strong over-dispersion in one half of the domain and near-Poisson behavior in the other, fit the proposed constant-$\alpha$ gamma-count model, and compare its coverage and WAIC with a model that lets $\alpha$ vary spatially. If the flexible-$\alpha$ model is clearly better, the central practical claim fails for spatially heterogeneous dispersion.

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Extended reading notes

Core claim

The paper's central claim is that a gamma-count likelihood coupled with a Gaussian spatial random field is a better sampling model for spatially referenced dispersed counts than the usual Poisson and negative binomial alternatives. In the gamma-count construction, waiting times are iid $\operatorname{Gamma}(\alpha,\gamma)$, and the count pmf is $P(Y=y)=G(y\alpha,\gamma T)-G((y+1)\alpha,\gamma T)$, where $G$ is the incomplete gamma ratio; $\alpha=1$ gives Poisson, $\alpha<1$ over-dispersion, and $\alpha>1$ under-dispersion. Writing $\gamma_i=\alpha\exp(x_i'\beta+\varphi(s_i))$ keeps the model inside the latent-Gaussian class, so INLA computes posterior marginals without MCMC. For the groundwater data the estimated dispersion is $\hat{\alpha}=0.309$, and WAIC, DIC, and cross-validated logarithmic score all prefer the gamma-count model over Poisson and negative binomial.

Load-bearing premise

The load-bearing premise is that a single dispersion parameter $\alpha$ describes every location and observation; if dispersion varies spatially, the gamma-count likelihood is misspecified and the reported gains may not transfer.

Editorial extensions

If this is right

Any count dataset currently forced into a Poisson or negative binomial model can be re-fit with the gamma-count likelihood through INLA, with the Poisson result recovered at $\alpha=1$.
Spatial predictions of the response field inherit the improved dispersion modeling; in the Golestan example the estimated range is smoother, about 64 km, and identifies low-quality water in the central region.
Simulation results in the paper indicate the gamma-count model recovers regression and Matern covariance parameters with lower RMSE than Poisson and negative binomial under both over- and under-dispersion, so inference about covariate effects is more reliable in those settings.
Because the gamma-count family nests the Poisson model, model comparison via WAIC and DIC gives a direct check for whether the extra dispersion parameter is needed.

Reading between the lines

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

Editorial inference: letting $\log\alpha$ depend on covariates or on a second spatial field is the natural next step; if dispersion varies by region, the constant-$\alpha$ model would understate uncertainty in high-dispersion areas.
The same renewal-theory construction with Weibull or lognormal waiting times should transfer directly to the INLA-SPDE machinery, giving a family of flexible spatial count models rather than a single one.
A likely unstated payoff is that the gamma-count likelihood should also work for spatio-temporal counts and point-pattern aggregation, where over- and under-dispersion are common; the paper's arguments do not depend on the groundwater setting.

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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

Here's the short version: the paper does something real — it puts gamma-count regression in a spatial setting with SPDE/INLA, which no one seems to have done before. But the evidence section has two problems that need fixing before the headline claim is credible.

First, the Dean-Lawless test. They report T = -5.4466. That statistic is negative, so the one-sided overdispersion test does not reject; if anything it points to underdispersion. The text says the null is rejected and concludes overdispersion. That's a sign error. It matters because the estimated alpha is 0.309, which in the GC model means overdispersion. So either the test is computed wrong or the interpretation is; either way, it's an internal contradiction.

Second, the simulation's NB baseline is broken. At alpha=1, where the true model is Poisson, the NB column shows RMSE 20.15 for the intercept. That's not a model underperforming; that's a fit that didn't converge or an unidentifiable parameterization. A correctly specified NB nests Poisson and should give RMSE close to the Poisson's 0.22. With that baseline, the reported PR(GC/NB)=0.944 at alpha=1 is meaningless. The GC model may well be better than NB in the real-data comparison, but the simulation doesn't establish it.

There's also a mild circularity: they fit once, take the estimated sigma and range, then use those to set the PC priors and refit the same data. That should at least be flagged as a sensitivity analysis rather than presented as a neutral prior choice.

On the plus side, the model itself is clearly stated, the link to Zeviani et al. and Winkelmann is honest, and the use of the existing 'gammacount' family in R-INLA is a practical plus. The constant-dispersion limitation is acknowledged. The paper is well structured and the math checks out as far as I can tell.

No code or data accompanies the paper, which makes the simulation issues harder to diagnose. If it were mine, I'd want the NB fits rerun with convergence checks, the Dean-Lawless interpretation corrected, and the prior choice handled properly.

Who's this for? Spatial statisticians who work with INLA and count data, and environmental health researchers modeling water quality. They'll get a usable template even if the comparison claims need revision.

For peer review: yes, this deserves a serious referee. The idea is sound and the flaws are fixable. I'd send it out, though I'd expect major revisions before it's reliable. Engage with it, but don't take the abstract's 'significant improvement over negative binomial' at face value.

Referee Report

4 major / 6 minor

Summary. The paper proposes a Bayesian hierarchical spatial gamma-count (GC) regression model for count data, built on the renewal-theoretic idea that nonexponential waiting times induce a flexible count distribution capable of capturing both over- and under-dispersion. The authors embed this GC likelihood in a latent Gaussian model and fit it with INLA and the SPDE approach for geostatistical data. They apply the model to a groundwater-quality dataset from Golestan Province, Iran, where the response is the number of months (out of the study period) in which electrical conductivity indicates drinkable water. The paper claims, on the basis of a simulation study and the real-data example, that the GC model significantly outperforms both Poisson and negative binomial alternatives. The manuscript also reports model-selection criteria (WAIC, DIC, log score) and posterior summaries for regression and spatial parameters.

Significance. If the central claim is correct, the paper offers practitioners a useful and computationally tractable extension of spatial count models: a single additional dispersion parameter that nests the Poisson model, implemented through R-INLA. This would be a practical contribution to spatial epidemiology and environmental statistics. The paper also demonstrates the use of the already-implemented 'gammacount' family in R-INLA, and the reliance on the SPDE approach is standard. However, the current manuscript contains several internal errors and an apparently broken negative binomial baseline in the simulations, so the strength of the evidence for the headline claim is not yet established.

major comments (4)

[Section 1.2, Eq. (1)] The Dean-Lawless test statistic is reported as T = -5.4466. Because the test in Eq. (1) is a one-sided test of H0: tau = 0 versus H1: tau > 0, a large negative value indicates underdispersion, not overdispersion. The text then states that H0 is rejected at level 0.05 and concludes that the counts are overdispersed. This is logically inconsistent. Moreover, Section 5 reports an estimated dispersion parameter alpha = 0.309 with 95% credible interval (0.203, 0.434), which by the paper's own convention (alpha < 1) implies overdispersion, contradicting the implication of the test statistic. Please correct the sign interpretation or the test computation, and reconcile the contradictory conclusions.
[Section 4, Table 1] The negative binomial baseline in the simulation appears to be non-converged or incorrectly implemented. When alpha = 1 (the true model is Poisson), the NB model yields RMSEs of 20.1498 for the intercept and 5.6965 for the slope, while the Poisson and GC models have RMSEs around 0.22 and 0.06. Since the NB model nests the Poisson model, a correctly specified and well-fitted NB should have RMSEs comparable to those of the Poisson model. The large RMSEs indicate a convergence failure or a faulty parameterization of the NB model in R-INLA. Consequently, the claim of a 'significant improvement over the negative binomial' model based on this simulation study is not supported by the presented evidence. The simulations should be rerun with a reliably fitted NB baseline, and the results should be reported alongside a discussion of any encountered convergence issues.
[Section 5, after Table 5] The PC priors for the Matérn field parameters are set using estimates obtained from the same dataset. The authors first fit the selected model and obtain estimates sigma = 0.74 and r = 49.18, and then use these estimates to specify P(sigma > 0.74) = 0.05 and P(r < 49) = 0.05. This is a double use of the data: the data are used to choose the prior hyperparameters and then used again to obtain the posterior. This empirical-Bayes-like procedure should be explicitly acknowledged and justified, or the priors should be fixed a priori. As written, the posterior credible intervals in Table 6 do not account for this prior adaptation and may be overconfident.
[Section 5, Table 5] The WAIC values in Table 5 range from 652.540 to 18336.51, while the DIC values range from 221.052 to 236.823. WAIC and DIC are both information criteria typically reported on the same deviance scale, and such a discrepancy of two orders of magnitude is implausible. This suggests that the WAIC (or DIC) computation is incorrect or that the criteria are defined on different scales without clarification. Since the model-selection conclusions rely on these criteria, please verify the computational definition, check the implementation, and report the criteria on a consistent scale.

minor comments (6)

[Throughout] The manuscript contains many typesetting errors, including garbled characters (e.g., 'ﬂexible' for 'flexible', 'shuch' for 'such', 'T able' for 'Table'), inconsistent spacing, and missing article template formatting. A careful editorial pass is needed.
[References, item [28]] The reference to Pearson (1894) is listed with the year '1994' in the reference list. The correct year is 1894.
[Section 2, Eq. (4)] The infinite sum in Eq. (4) is stated to have no closed form; please clarify what numerical method is used to evaluate it in the likelihood computations, as this affects the reproducibility of the results.
[Section 3, Eq. (9) and following text] The definitions of the SPDE precision-matrix components ilde{C} and G are given in prose; it would be clearer to define them with explicit equations, especially the diagonal matrix ilde{C}.
[Section 4, Table 3] The text says MSPE is computed for 'one simulated GRF' but Table 3 appears to report averages over replications. Please clarify how MSPE is aggregated across the simulation replications.
[Section 5, Figure 6] The prediction maps in Figure 6 are described only briefly; a sentence explaining what the color scale represents and how the standard deviation maps should be interpreted would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the gamma-count likelihood and the INLA/SPDE implementation are not defined in terms of the quantities the paper claims to predict.

full rationale

The paper's central claim is that a spatial gamma-count model improves on Poisson and negative binomial alternatives. The gamma-count pmf (Eq. 2) is derived from gamma waiting times via renewal theory, independent of the groundwater data; the regression model (Eq. 8) links the predictor to E(τ) and nests Poisson at α=1. The simulation study draws data from the assumed GC model, so it is a generative consistency check rather than a tautological proof. The real-data analysis uses fixed likelihoods and standard SPDE/INLA machinery; no fitted parameter is renamed as a prediction. The empirical-Bayes step in Section 5 — using first-stage estimates σ̂=0.74 and r̂=49.18 to set PC-prior quantiles (P(σ>0.74)=0.05, P(r<49)=0.05) and then refitting the same data — double-uses the data and is a legitimate statistical concern, but it does not make the WAIC/DIC comparison equivalent to those estimates by construction; the posterior range actually moves away from the prior constraint (r posterior mean 64.46 km vs prior r<49 km). Self-citations to INLA/SPDE/PC-prior literature are methodological and not load-bearing in the model derivation. The reported Dean-Lawless statistic T=-5.45 appears inconsistent with the claimed overdispersion conclusion, but that is a correctness/evidence issue, not circularity. Under the stated rules, no specific equation-level or definitional reduction is exhibited, so the appropriate circularity score is 0.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The model rests on the gamma-count renewal assumption, conditional independence, the Matérn/SPDE spatial field, and the availability of a numerically accurate R-INLA implementation. The main uncharged costs are the data-dependent PC prior hyperparameters and the constant dispersion assumption.

free parameters (5)

alpha (gamma shape / dispersion) = 0.309 posterior mean in application; simulation settings 0.1, 1, 1.5, 3
Central dispersion parameter of the gamma-count model, estimated from data. In the application the posterior mean is 0.309, indicating the model's inferred dispersion direction.
sigma (marginal SD of Matérn field) = Posterior mean about 0.74 in application
SPDE spatial field scale, estimated from data.
r (spatial range) = Posterior mean 49 km in preliminary fit; 64.46 km in final GC model
SPDE spatial correlation range, estimated from data.
PC prior hyperparameters sigma0, r0 = sigma0=0.74, r0=49
Set using posterior estimates from the same data before the final model fit, a double use of data noted in Section 5.
Prior hyperparameters a, b, Sigma_beta, c, d, e, f = a=0.01, b=0.01, Sigma_beta=1000I, c=log(0.02), d=10, e=log(14), f=10
Weakly informative priors chosen by hand and used in both simulation and application; standard for INLA models.

assumptions (6)

domain assumption Waiting times between events are i.i.d. Gamma(alpha, gamma).
Defines the gamma-count distribution in Section 2; if the waiting time distribution is misspecified, the likelihood is misspecified.
domain assumption Counts are conditionally independent given the latent spatial field and parameters.
Standard hierarchical model assumption introduced in Section 3 and used for the marginal likelihood in equation (10).
domain assumption The spatial effect follows a stationary isotropic Matérn GRF with smoothness nu=1, approximated by SPDE.
Required for the INLA-SPDE computational approach in Section 3; the paper fixes nu=1 as in Lindgren et al. (2011).
domain assumption The gamma-count likelihood can be treated as a latent Gaussian model family in R-INLA with linear predictor entering as gamma = alpha exp(eta).
The paper relies on the R-INLA 'gammacount' implementation without deriving its numerical accuracy for non-integer alpha (Section 3.1).
domain assumption Constant dispersion alpha across all observations.
Acknowledged in Section 6 as a restricting factor; if false, the model is misspecified.
domain assumption PC priors for Matérn parameters are appropriate.
Standard INLA prior framework, but the hyperparameters are data-dependent in the application because they are set using the same data's posterior estimates.

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Pith. "Pith review of Statistical modeling of groundwater quality assessment in Iran using a flexible Poisson likelihood." pith.science (2026). https://pith.science/paper/6LQPLYUP

@misc{pith2026190802344,
  author       = {Pith},
  title        = {Pith review of: Statistical modeling of groundwater quality assessment in Iran using a flexible Poisson likelihood},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LQPLYUP}},
  note         = {Machine review of arXiv:1908.02344}
}

read the original abstract

Assessing water quality and recognizing its associated risks to human health and the broader environment is undoubtedly essential. Groundwater is widely used to supply water for drinking, industry, and agriculture purposes. The groundwater quality measurements vary for different climates and various human behaviors, and consequently, their spatial variability can be substantial. In this paper, we aim to analyze a groundwater dataset from the Golestan province, Iran, for November 2003 to November 2013. Our target response variable to monitor the quality of groundwater is the number of counts that the quality of water is good for a drink. Hence, we are facing spatial count data. Due to the ubiquity of over or underdispersion in count data, we propose a Bayesian hierarchical modeling approach based on the renewal theory that relates nonexponential waiting times between events and the distribution of the counts, relaxing the assumption of equidispersion at the cost of an additional parameter. Particularly, we extend the methodology for the analysis of spatial count data based on the gamma distribution assumption for waiting times. The model can be formulated as a latent Gaussian model, and therefore, we can carry out the fast computation by using the integrated nested Laplace approximation method. The analysis of the groundwater dataset and a simulation study show a significant improvement over both Poisson and negative binomial models.

Figures

Figures reproduced from arXiv: 1908.02344 by the authors.

**Figure 1.** Locations of groundwater stations (blue stars) in Golestan province (left panel), a plot of the study area in Golestan Province (right panel) For assessing the quality of groundwater, we use Electrical Conductivity (EC) that is measured by passing an electric current between two metal plates (electrodes) in the water sample and measuring how readily current flows (i.e., conducted) between the plates. This variable i… view at source ↗

**Figure 2.** A simulated Mat´ern field (first column), and the prediction (posterior mean) of the field based on GC model (second column), Poisson model (third column), and NB model (forth column). The results are obtained for α = 0.1 (first row), α = 1 (second row), α = 1.5 (third row), α = 3 (fourth row) [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Histograms of PIT values for the GC model (first row), the Poisson model (second row), and the NB model (third row). The results are obtained for α = 0.1 (first column), α = 1 (second column), α = 1.5 (third column), α = 3 (fourth column). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The matrix of scatter plots of covariates for groundwater data Fitting models The first step required to fit the geostatistical model for groundwater data, by the SPDE approach, is the triangulation of the considered spatial domain depicted in [PITH_FULL_IMAGE:figures…

**Figure 1.** Figure 1: Figure 5 displays a non-convex triangulation for the area study inside the [PITH_FULL_IMAGE:figures/full_fig_p017_1.png]

**Figure 5.** Figure 5: The Golestan province triangulation. The red dots denote the observation locations of the groundwater stations. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: Prediction maps of the response variable (in the log scale) including posterior mean (first row) and posterior sd (second row), obtained from the GC model (first column), Poisson model (second column), and NB model (third column). computational time. To overcome these …

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