R\'egularisation dans les Mod\`eles Lin\'eaires G\'en\'eralis\'es Mixtes avec effet al\'eatoire autor\'egressif

Catherine Trottier (IMAG; Jocelyn Chauvet (IMAG); UM3); Xavier Bry (IMAG)

REVIEW 5 major objections 5 minor 9 references

R\'egularisation dans les Mod\`eles Lin\'eaires G\'en\'eralis\'es Mixtes avec effet al\'eatoire autor\'egressif

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proposes regularized EM algorithms that estimate a generalized linear mixed model with individual and AR(1) time random effects on panel data.

desk verdict Plausible AR(1) extension of regularized EM for panel GLMMs, but the algorithmic core is left too implicit and there are no numerical results to back it yet. read the letter →

arxiv 1908.07478 v1 pith:JUSJBW3K submitted 2019-08-09 stat.ME

Jocelyn Chauvet (IMAG) , Catherine Trottier (IMAG , UM3) , Xavier Bry (IMAG) This is my paper · ORCID

classification stat.ME MSC 62J1262F1062M10

keywords GeneralizedlinearmixedmodelspaneldataEMalgorithmridgepenalizationautoregressiverandomeffectsupervisedcomponentscross-validation

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

On balanced panel data, with repeated observations on many individuals over time, this paper proposes regularized EM algorithms for a generalized linear mixed model: one random effect captures individual heterogeneity, and a second random effect follows a first-order autoregressive process shared by all individuals. For low-dimensional predictors the algorithm adds an L2 (ridge) penalty to the complete-data log-likelihood; for high-dimensional predictors it replaces that penalty with a supervised-component criterion. If the algorithms work, a single estimation routine would produce the fixed coefficients $\beta$, the two variance components $\sigma_1^2$ and $\sigma_2^2$, and the autoregressive parameter $\rho$ while coping with many redundant explanatory variables. The paper states a simulation agenda—Poisson-log panels, convergence checks, mean-squared-error decay, and stability across $\rho$—but reports no numerical results in this text.

What carries the argument

The load-bearing objects are the linearized working model $M: z = X\beta + U\xi + e$, refreshed at each iteration, and the regularized complete-data objective that the EM maximizes. In low dimension that objective is $L_{\mathrm{pen}} = L(\theta; z,\xi) - \frac{\lambda}{2}\beta^\top\beta$, with $\lambda$ selected by a generalized cross-validation criterion adapted to unequal error variances and applied to the working model's hat matrix $S_\lambda$. In high dimension the objective is $L_{\mathrm{reg}} = (1-s)L(\theta; z,\xi) + s\,\varphi(w)$, where $\varphi(w) = \bigl(\sum_{j=1}^{p} \operatorname{cor}^2(x_j, f)^\ell\bigr)^{1/\ell}$ measures how well the component $f = Cw$ summarizes the predictor block, $C$ being the principal-component scores of $X$ with nonzero eigenvalues. The E-step computes conditional expectations of the penalized complete-data log-likelihood given the working response, and the M-step updates $\theta$; the AR(1) time effect enters through the $U_2$ design matrix and through the conditional moments of $\xi_2$.

What would settle it

Simulate balanced Poisson-log panel data with known $\beta$, $\sigma_1^2$, $\sigma_2^2$, and $\rho$, run the L2-penalized EM, and compare the penalty selected by generalized cross-validation with the penalty that minimizes the true predictive mean-squared error of the working model: if the two diverge systematically as $\rho$ approaches $\pm 1$, or if the estimated $\sigma_2^2$ and $\rho$ are badly biased, the cross-validation transfer or the AR(1) conditional moments would be refuted.

Watch

Extended reading notes

Core claim

Each iteration of a standard linearization scheme for generalized linear mixed models produces a working model $M: z = X\beta + U\xi + e$, where $z$ is a first-order Taylor approximation of the transformed response and $\operatorname{Var}(e\mid\xi)=\Gamma$ is diagonal. The paper's central claim is that this working model can be estimated by a regularized EM: in the low-dimensional case, maximize the penalized complete log-likelihood $L_{\mathrm{pen}} = L(\theta; z, \xi) - \frac{\lambda}{2}\beta^\top\beta$, with $\lambda$ chosen by a generalized cross-validation criterion designed for unequal error variances; in the high-dimensional case, maximize a regularized objective built from $L_{\mathrm{reg}} = (1-s)L(\theta; z,\xi) + s\,\varphi(w)$, where $\varphi(w)$ aggregates squared correlations between predictors and a component $f = Cw$ built from the nonzero-eigenvalue principal components of $X$. Both routes are claimed to estimate $\theta = (\beta, \sigma_1^2, \sigma_2^2, \rho)$ for a GLMM with an iid individual random effect and an AR(1) time random effect shared by all individuals.

Load-bearing premise

The method stands or falls on the assumption that the criterion used to choose the ridge penalty remains valid when the working model contains a shared first-order autoregressive time effect that correlates all observations, and that the conditional averages needed in the E-step for that effect can actually be computed; the paper does not prove either.

Editorial extensions

If this is right

The low-dimensional algorithm would let a practitioner fit a panel GLMM with a persistent shared time shock, estimating the autoregressive persistence $\rho$ together with the fixed coefficients and variance components.
The high-dimensional algorithm would extend supervised-component regression to panel GLMMs, producing a small number of interpretable linear components rather than sparse zero coefficients when $p$ exceeds $n$.
For Poisson-log count panels, the methods would give regularized estimates of log-rate coefficients while respecting both within-individual dependence and serial time dependence.
The paper's announced simulation checks—convergence speed, mean-squared error decreasing with sample size, and stable behavior for $\rho \in (-1,1)$—are the concrete criteria by which the claim would be verified.

Reading between the lines

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

The paper does not report the numerical experiments it announces, so the convergence and mean-squared-error claims remain programmatic rather than demonstrated.
A fair test of the method's weakest link would compare the penalty selected by generalized cross-validation with an oracle penalty on simulated panels with strong AR(1) dependence; if they diverge as $\rho$ approaches $\pm 1$, the penalty-selection step does not transfer to the mixed-effects working model.
Because the EM machinery only needs conditional moments of the Gaussian random-effect vector, the AR(1) structure could be replaced by other stationary time or spatial structures without changing the algorithm's skeleton.
In high dimension, comparing the supervised-component objective against lasso-type GLMM penalties would show whether the regularization buys interpretability, predictive accuracy, or both; the paper does not make that comparison.

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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

Colleague, this is a short French methods note that proposes two regularized EM algorithms for GLMMs with panel data and an AR(1) time random effect. That combination is genuinely not in the earlier papers they cite: Eliot et al. cover ridge EM for LMMs with independent random effects, Schelldorfer et al. do lasso GLMMs, and Chauvet et al. extend SCGLR to GLMMs, again with independent effects. So the AR(1) extension is a real gap, and the paper is clear about what it adds.

What I like: the model setup is careful (two random effects, one individual, one shared AR(1) process), and the algorithmic skeleton follows a standard Schall linearization plus penalized EM. The component-based variant for high-dimensional data is a reasonable adaptation of SCGLR, and the structural relevance criterion is well motivated. The writing, even in French, is straightforward and the bibliography is appropriate; self-citation in Chauvet et al. is not lazy here because the extension really does build on that prior work.

Now the soft spots, and they are substantial. Section 3.1 defines the GCV step in Algorithm 1, but the hat matrix Sλ is never explicitly defined. The notation suggests a mixed-model smoother, but the reader has to guess how it depends on θ and λ. The E-step says 'calculate Qpen' but gives no expression for the conditional moments of ξ1 and ξ2 under the AR(1) process; that is exactly the step where the autocorrelation enters, so it is not a minor omitted detail. And the GCV criterion is taken from Andrews (1991), which is derived for independent heteroskedastic errors. The working model z = Xβ + Uξ + e has a shared AR(1) time component, so the covariance matrix is non-diagonal with an N-block structure. Andrews' conditions do not hold, and no argument or simulation is given to show GCV still performs. Because λ[t] is picked inside the same iteration that updates β, σ1², σ2² and ρ, a biased λ could feed through all four. Section 4 lists three planned simulation objectives but reports no results, so the paper is a proposal, not yet a validated method.

Overall: the central idea is plausible and the gap is real, but the manuscript as submitted does not support the algorithms' claim. I would not cite it yet, but I would send it to a knowledgeable referee rather than desk reject. The referee should ask for explicit EM updates themselves the full derivation of Qpen and the conditional moments, a definition of Sλ, and either a justification of GCV in this mixed-model AR(1) setting or a simulation study showing GCV tracks oracle risk. If those are delivered, the paper would be a useful applied-methods contribution.

Referee Report

5 major / 5 minor

Summary. The paper proposes regularized EM estimation of generalized linear mixed models (GLMMs) for balanced panel data. The model has two random effects: an individual-specific i.i.d. effect ξ1 ~ N(0, σ1² I_N) and a time-specific AR(1) effect ξ2 shared by all individuals, with ξ2,t+1 = ρ ξ2,t + ν_t, ν_t iid N(0, σ2²). Section 2 defines the model and the linear predictor η = Xβ + Uξ. Section 3 introduces two estimation strategies: for p < n, an L2-penalized EM algorithm built on the Schall (1991) linearization and a GCV-based choice of the ridge parameter λ (Algorithm 1); for p >> n, a supervised component-based regularized EM maximizing an objective Qreg with a structural relevance criterion φ(w). Section 4 describes a planned simulation study but contains no numerical results.

Significance. If fully developed and validated, the contribution would address a real gap: extending ridge and component-based regularized estimation to GLMMs with an autoregressive time random effect. The model setup is clearly stated, and the combination of Schall linearization with penalized complete-data likelihood is a plausible route. The paper also correctly identifies that existing mixed-model regularized methods typically assume independent random effects. However, the significance is currently conditional: the algorithms are not fully specified (the hat matrix Sλ, the E-step moments, and the M-step updates are missing), and there is no simulation or application demonstrating that the methods work. The supervised component-based algorithm is only sketched. As a methods contribution, the paper needs both complete derivations and empirical evidence before its claims can be assessed.

major comments (5)

[Section 3.1, Algorithme 1, step (2.b)] The hat matrix S[t]_λ is never defined. The GCV criterion depends on S[t]_λ as the smoother matrix satisfying zhat[t] = S[t]_λ z[t]. In the mixed model M, z = Xβ + Uξ + e, the natural penalized estimator of β depends on θ[t] through Γ[t] and through the AR(1) covariance of Uξ, so S[t]_λ is not the simple ridge smoother used by Golub et al. (1979) or Andrews (1991). The paper must give the explicit expression for S[t]_λ, or the penalized mixed-model equations from which it is derived; otherwise the GCV objective in step (2.b) cannot be evaluated.
[Section 3.1, Algorithme 1, steps (2.c) and (2.d)] The E-step and M-step are described only symbolically. The conditional distribution of ξ = (ξ1, ξ2) given z[t] and θ[t] is not derived, the complete-data log-likelihood L(θ; z, ξ) is not written explicitly in terms of σ1², σ2² and ρ, and the maximizing updates for β and the variance components are not given. Without these computations, Algorithm 1 is not implementable and the 'EM penalisé' claim lacks a derivation. In particular, the conditional moments for the AR(1) random effect shared by all individuals are nontrivial and need to be exhibited.
[Section 3.1, step (2.b), GCV criterion] The GCV criterion is taken from Andrews (1991), which concerns regression with independent heteroskedastic errors. In the working model M, the error term Uξ + e has covariance V = UDU^T + Γ, which is non-diagonal and has a block structure induced by the AR(1) time effect shared across individuals. No argument is given for extending GCV to this dependent-error setting. Since λ[t] is selected inside the EM loop and then used in the E-step and M-step, a systematically biased λ would propagate into β, σ1², σ2² and ρ. The authors should either adapt GCV to the AR(1) mixed model with a derivation, or supply simulations showing that GCV-selected λ tracks the oracle prediction-risk minimizer.
[Section 3.2] The high-dimensional algorithm is described only through the objective Qreg. No algorithmic details are given for maximizing Qreg with respect to the component weights w, for extracting multiple components under orthogonality constraints, for performing the E-step and M-step in the presence of the AR(1) random effect, or for calibrating s and l by cross-validation. The sentence 'les composantes de rangs supérieurs sont calculées comme celle de rang 1, après l'ajout de contraintes d'orthogonalité' is not a sufficient algorithmic specification. This is a major gap because the supervised component-based regularized EM is one of the two central contributions.
[Section 4] Section 4 contains only a plan: 'nous présenterons des études sur données simulées' with three objectives. There are no simulation results, no tables, no figures, no comparison with existing methods, and no code or data. For a methods paper, the absence of any numerical evidence makes it impossible to verify the convergence behavior, the MSE properties, or the effect of the AR(1) parameter ρ ∈ (−1,1) that the authors themselves list as objectives. The manuscript should report the actual simulation study in this section.

minor comments (5)

[Section 3.2] The condition 'l /greaterorequalslant1' appears as a LaTeX artifact; it should read 'l ≥ 1'.
[Section 3.1, GCV formula] The notation ||·||²_{Γ[t]^{-1}} is not defined; it should be stated explicitly, for example as v^T Γ[t]^{-1} v for a vector v.
[Section 3, Algorithm 1, step (3)] The stopping rule 'stabilité conjointe' is vague; the authors should specify a convergence tolerance on the relative or absolute change of θ = (β, σ1², σ2², ρ).
[Section 3, step (3)] The update of the working variable z[t+1] and Γ[t+1] is stated only as 'mettre à jour'; writing the explicit formulas (e.g., z[t+1] = g(µ[t+1]) + (y − µ[t+1])g'(µ[t+1])) would make the algorithm reproducible.
[References] Reference [4] (Chauvet et al., 2016) lacks page numbers or a DOI; please complete the bibliographic information.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the L2-penalized EM and component-based EM are self-contained algorithmic constructions; the co-authored phi(w) regularizer is an explicit input, not a derived prediction.

full rationale

The derivation chain is not circular. In Algorithm 1, the working model M: z = X beta + U xi + e is obtained by first-order linearization, lambda is selected from the current working vector via GCV, and the E- and M-steps update theta = (beta, sigma1^2, sigma2^2, rho). The estimates are outputs of that iterative fitting scheme, not quantities predicted from the same fitted constants, so no fitted input is renamed as a prediction. The high-dimensional variant uses Qreg(theta, theta[t]) = E_xi|z[Lreg | theta[t]] with Lreg = (1-s)L + s phi(w), where phi(w) is the structural-relevance criterion of Bry and Verron (2015), a co-author citation. That citation is real but not circular: phi is an explicitly borrowed regularization term, and the paper's claimed contribution, an EM version adapted to panel GLMMs with an AR(1) random effect, is constructed from the linearized model and the AR(1) specification, not from Bry and Verron. Two genuine weaknesses exist but are not circularity: (a) the GCV criterion of Andrews (1991) is extended to a mixed model with individual and shared AR(1) effects without proof, and the hat matrix S_lambda is never defined; (b) Section 4 only promises future simulations, so numerical support is absent. Neither weakness makes the derivation equivalent to its own inputs. No uniqueness theorem, no ansatz smuggled via self-citation, and no renaming of a known result appear. Score 1 records the minor self-cited regularizer; the core derivation is self-contained.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central estimation rests on standard GLMM and AR(1) assumptions plus several ad hoc choices: the principal-component constraint f = Cw, the structural relevance criterion, and the automatically tuned penalty parameters λ, s, and l. No new entities (particles, forces, dimensions) are introduced.

free parameters (3)

λ (ridge penalty) = chosen via GCV at each EM iteration
Controls the L2 penalty in the low-dimensional algorithm; selected from the working data in Algorithm 1, step (2.b).
s (weight on structural relevance) = calibrated by cross-validation
Balances fit L and structural relevance φ(w) in Qreg for the component-based algorithm, Section 3.2.
l (exponent in φ(w)) = calibrated by cross-validation
Sets the aggregation order in the structural relevance criterion, Section 3.2.

assumptions (5)

domain assumption Y_i given ξ are independent and follow an exponential-family distribution with mean linked to η_i = x_i^T β + u_i^T ξ via a link function g.
Section 2, assumptions (i) and (ii), standard GLMM setup.
domain assumption The individual random effect ξ1 ~ N(0, σ1² I_N) and the time random effect ξ2 is an AR(1) process with innovations ν_t ~ N(0, σ2²), with ξ1 and ξ2 independent.
Section 2, part 2, defines the two random effects and their distributions.
domain assumption A first-order Taylor approximation of g(y) yields a working linearized model M: z = Xβ + Uξ + e with Var(e|ξ) = Γ.
Section 3, 'Étape de linéarisation'; this standard Schall-type approximation is essential to the EM derivation.
ad hoc to paper For identifiability, the component is constrained to f = Cw, where C = XU is the set of principal components of X with nonzero eigenvalues.
Section 3.2, 'Pour des raisons d'identifiabilité'; this restriction is introduced without external justification beyond identifiability.
ad hoc to paper The structural relevance criterion φ(w) = (Σ_j cor²(x_j, f)^l)^{1/l}, with parameters s and l calibrated by cross-validation, is a valid regularizer in the objective Qreg.
Section 3.2, defined via Bry and Verron (2015); its use as the regularizing criterion is assumed.

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Cite this review

Pith. "Pith review of R\'egularisation dans les Mod\`eles Lin\'eaires G\'en\'eralis\'es Mixtes avec effet al\'eatoire autor\'egressif." pith.science (2026). https://pith.science/paper/JUSJBW3K

@misc{pith2026190807478,
  author       = {Pith},
  title        = {Pith review of: R\'egularisation dans les Mod\`eles Lin\'eaires G\'en\'eralis\'es Mixtes avec effet al\'eatoire autor\'egressif},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JUSJBW3K}},
  note         = {Machine review of arXiv:1908.07478}
}

read the original abstract

We address regularised versions of the Expectation-Maximisation (EM) algorithm for Generalised Linear Mixed Models (GLMM) in the context of panel data (measured on several individuals at different time points). A random response y is modelled by a GLMM, using a set X of explanatory variables and two random effects. The first effect introduces the dependence within individuals on which data is repeatedly collected while the second embodies the serially correlated time-specific effect shared by all the individuals. Variables in X are assumed many and redundant, so that regression demands regularisation. In this context, we first propose a L2-penalised EM algorithm for low-dimensional data, and then a supervised component-based regularised EM algorithm for the high-dimensional case.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Andrews, D.W. (1991). Asymptotic optimality of generalized CL, c ross-validation, and generalized cross-validation in regression with heteroskedast ic errors. Journal of Econometrics, 47, 359 – 377

work page 1991
[2]

et Mortier, F

Bry, X., Trottier, C., Verron, T. et Mortier, F. (2013). Superv ised component gener- alized linear regression using a pls-extension of the ﬁsher scoring alg orithm. Journal of Multivariate Analysis , 119, 47 – 60

work page 2013
[3]

et Verron, T

Bry, X. et Verron, T. (2015). THEME: THEmatic model explorat ion through multiple co-structure maximization. Journal of Chemometrics , 29, 637 – 647

work page 2015
[4]

et Mortier, F

Chauvet, J., Trottier, C., Bry, X. et Mortier, F. (2016). Exten sion to mixed models of the Supervised Component-based Generalised Linear Reg ression. COMPSTAT: Proceedings in Computational Statistics

work page 2016
[5]

et Foulkes, A.S

Eliot, M., Ferguson, J., Reilly, M.P. et Foulkes, A.S. (2011). Ridge Re gression for Longitudinal Biomarker Data. The International Journal of Biostatistics , 7, 1, Article 37

work page 2011
[6]

et Wahba, G

Golub, G.H., Heath, M. et Wahba, G. (1979). Generalized cross - v alidation as a method for choosing a good ridge parameter. Technometrics, 21, 215 – 223

work page 1979
[7]

Green, P.J. (1990). On use of the EM for penalized likelihood estima tion. Journal of the Royal Statistical Society, Series B , 52, 443 – 452

work page 1990
[8]

Schall, R. (1991). Estimation in generalized linear models with rando m eﬀects. Biometrika, 78, 719 – 727

work page 1991

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[9]

et B¨ uhlmann, P

Schelldorfer, J., Meier, L. et B¨ uhlmann, P. (2014). Glmmlasso: a n algorithm for high-dimensional generalized linear mixed models using l1-penalization. Journal of Com- putational and Graphical Statistics , 23, 460 – 477. 6

2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Andrews, D.W. (1991). Asymptotic optimality of generalized CL, c ross-validation, and generalized cross-validation in regression with heteroskedast ic errors. Journal of Econometrics, 47, 359 – 377

work page 1991

[2] [2]

et Mortier, F

Bry, X., Trottier, C., Verron, T. et Mortier, F. (2013). Superv ised component gener- alized linear regression using a pls-extension of the ﬁsher scoring alg orithm. Journal of Multivariate Analysis , 119, 47 – 60

work page 2013

[3] [3]

et Verron, T

Bry, X. et Verron, T. (2015). THEME: THEmatic model explorat ion through multiple co-structure maximization. Journal of Chemometrics , 29, 637 – 647

work page 2015

[4] [4]

et Mortier, F

Chauvet, J., Trottier, C., Bry, X. et Mortier, F. (2016). Exten sion to mixed models of the Supervised Component-based Generalised Linear Reg ression. COMPSTAT: Proceedings in Computational Statistics

work page 2016

[5] [5]

et Foulkes, A.S

Eliot, M., Ferguson, J., Reilly, M.P. et Foulkes, A.S. (2011). Ridge Re gression for Longitudinal Biomarker Data. The International Journal of Biostatistics , 7, 1, Article 37

work page 2011

[6] [6]

et Wahba, G

Golub, G.H., Heath, M. et Wahba, G. (1979). Generalized cross - v alidation as a method for choosing a good ridge parameter. Technometrics, 21, 215 – 223

work page 1979

[7] [7]

Green, P.J. (1990). On use of the EM for penalized likelihood estima tion. Journal of the Royal Statistical Society, Series B , 52, 443 – 452

work page 1990

[8] [8]

Schall, R. (1991). Estimation in generalized linear models with rando m eﬀects. Biometrika, 78, 719 – 727

work page 1991