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Variable selection via fused sparse-group lasso penalized multi-state models incorporating molecular data

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A fused sparse-group lasso penalty for multi-state Cox models recovers weak but real transition effects that a plain lasso shrinks to zero.

desk verdict A useful but under-specified extension of fused sparse-group lasso to multi-state Cox models; the algorithm as written is dimensionally inconsistent, so the simulations need to be re-anchored to a precise estimator before the method can be trusted. read the letter →

arxiv 2411.17394 v1 pith:ETYWTK7T submitted 2024-11-26 stat.ME stat.AP

Kaya Miah , Jelle J. Goeman , Hein Putter , Annette Kopp-Schneider , Axel Benner This is my paper · ORCID
classification stat.MEstat.AP MSC 62N0162J07
keywords multi-statemodelsCoxregressionfusedsparse-grouplassoADMMvariableselectionhigh-dimensionaldatatransition-specifichazardsregularization
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

This paper proposes a fused sparse-group lasso (FSGL) penalty for multi-state Cox models that combines overall sparsity, fusion of covariate effects across user-specified similar transition pairs, and group lasso over transitions. The authors claim that this penalty, fitted with an alternating direction method of multipliers (ADMM) algorithm, selects a sparser and more accurate model than a global lasso when the analyst can say which transitions behave alike. In a proof-of-concept simulation mirroring a nine-state acute myeloid leukemia pathway, FSGLmstate kept the true negative effect $\beta_{1,8} = -0.8$ that the lasso shrank to zero on average, while correctly zeroing all true null effects. The method also produced higher true positive rates, lower false discovery rates, and lower mean squared error for non-zero coefficients, and in the AML application it selected DNMT3A-R882 mutations for both relapse transitions. A sympathetic reader would care because late, rarely observed transitions are exactly where plain lasso regularization tends to drop weak real effects.

What carries the argument

The engine is the fused sparse-group lasso penalty $p_{\lambda,\mathrm{FSGL}}(\boldsymbol\beta) = \lambda[\alpha\gamma\sum_{q,p}|\beta_{p,q}| + (1-\gamma)\sum_{(q,q')\in S}\sum_p |\beta_{p,q}-\beta_{p,q'}| + (1-\alpha)\gamma\sum_q \|\boldsymbol\beta_q\|_2]$, which lets the analyst specify a set $S$ of similar transition pairs whose coefficients are fused. Estimation is done by an ADMM algorithm that introduces an auxiliary variable $\boldsymbol\theta = \boldsymbol K\boldsymbol\beta$, where the penalty structure matrix $\boldsymbol K$ stacks identity rows (lasso), contrast rows (fusion), and group-membership rows (group lasso), separating the smooth Cox partial likelihood from the non-smooth penalty. The $\boldsymbol\theta$-update is a closed-form vector soft-thresholding operation $S_\kappa(\boldsymbol a) = (1 - \kappa/\|\boldsymbol a\|_2)_+\boldsymbol a$, and an adaptive step size accelerates convergence. Clinical covariates can be left unpenalized through individual penalty scaling factors.

What would settle it

Simulate data where two transitions declared similar have true effects of opposite sign, for example $\beta_{1,3} = 1.2$ and $\beta_{1,7} = -0.8$, and check whether FSGLmstate shrinks both toward their average so that the fused estimates sit near zero and TPR/FDR are no better than the lasso's. A parallel check would apply the method to the AML data with a deliberately misspecified similarity set and observe whether the selected effects shift or vanish.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that FSGL penalized multi-state Cox regression recognizes the similarity structure of covariate effect pairs $\beta_{1,3} = \beta_{1,7} = 1.2$ and $\beta_{1,4} = \beta_{1,8} = -0.8$ while setting all other true negative covariate effects to zero, whereas LASSOmstate sets the late-transition effect $\beta_{1,8} = -0.8$ to zero on average. In the simulation of the 9-state AML model with 1000 observations and 16 regression parameters, FSGLmstate detects all non-zero effects more often, achieves a median FDR of 0.29 versus 0.38 for the lasso, and gives smaller mean squared errors for the non-zero coefficients. In the application to the randomized phase III AML trial, the method selects DNMT3A-R882 mutations for transitions 3 and 7 (CR1 to first relapse and CR2 to second relapse) and a negative treatment effect on first relapse, consistent with the trial's cumulative incidence of relapse analysis.

Load-bearing premise

The benefit of the method rests on the analyst correctly specifying which pairs of transitions truly have similar covariate effects; if the true effects on a specified pair differ in sign or magnitude, the fusion penalty pulls them together and biases the estimates.

Editorial extensions

If this is right

  • Analysts can build sparse multi-state models that keep weak true effects on late, rarely observed transitions instead of shrinking them to zero.
  • Expert knowledge about which transitions behave alike becomes part of the estimation, not just the interpretation.
  • Clinical predictors can remain unpenalized while molecular covariates are regularized, so the final model mixes both data types.
  • The ADMM framework separates the likelihood from the non-smooth penalty, which can be reused for other structured penalties.

Reading between the lines

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

  • The simulation validates the method on data generated to match the assumed similarity structure; sensitivity to misspecified pairs (sign flips or magnitude differences) is a natural next check the paper does not run.
  • The same fusion idea could be applied to other structured survival settings, such as competing risks with known cause similarities or transitions ordered along a disease ladder.
  • Since late transitions have few events, the gain over lasso may be larger when event counts are unbalanced; a directed simulation varying event counts per transition would test this.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes a fused sparse-group lasso (FSGL) penalty for multi-state Cox-type regression models, combining sparsity, fusion of coefficients across user-specified pairs of similar transitions, and group-wise selection of transition-specific coefficient vectors. An ADMM algorithm is presented, tuned by an approximate generalized cross-validation criterion. The method is evaluated in a proof-of-concept simulation with two binary covariates and eight transitions, and applied to an AML study with 24 mutation covariates and four clinical predictors. The central claim is that the FSGL penalty improves variable selection over a global lasso in settings where some covariate effects are equal across transitions.

Significance. If the method is correctly specified and the reported simulation reflects the actual estimator, the paper would provide a practically relevant extension of structured regularization to multi-state event-history models, complementing existing work such as the structured fusion lasso of Sennhenn-Reulen and Kneib. The authors give credit for providing R code in the supporting information and for structuring the simulation study according to ADEMP/ADEMP-PreReg criteria. However, the manuscript's central algorithmic derivation contains internal inconsistencies that make it unclear whether the evaluated estimator is the one defined in Section 3. These issues must be resolved before the empirical comparisons in Sections 5 and 6 can be interpreted as evidence for the stated method.

major comments (4)
  1. [§4.2, Eq. (4), Algorithm 1] The penalty structure matrix K and the dimensions of the auxiliary variables are internally inconsistent with the penalty in Eq. (4). Equation (4) contains P·|S| fusion terms of the form |β_{p,q} − β_{p,q'}|, which require a fusion matrix D with P·|S| rows. The text instead defines D ∈ R^{s×PQ} for s pairs of similar transitions, so with s=2 and P=24 (as in the AML application) only two contrast rows are described, not the 48 needed to fuse all covariates. Relatedly, the group blocks G_m are defined as P×PQ matrices, each contributing P rows, so the total row count of K is 2PQ + sP, not M = PQ + s + PQ as stated. Since θ ∈ R^M and β ∈ R^{PQ}, the Algorithm 1 final step 'obtain β̂ = θ̂' is dimensionally undefined. The manuscript never specifies which coordinates of θ correspond to the regression coefficients, nor how the vector soft-thresholding in step 5 acts on the group-level components. This is not a cosmetic issue: without a precise statement of K and the final estimator, the TPR/FDR comparisons in Figures 3–5 cannot be attributed to the FSGL estimator defined by Eq. (4).
  2. [§4.2, Algorithm 1, step 5] The θ-update is not consistently defined for the group penalty. The text states θ_m = S_{λ_m w_m / ρ}(K_m β^{r+1} + ν_m^r / ρ) for m=1,…,M with scalar θ_m, but for the group blocks K_m = G_{m-PQ-s} is a P×PQ matrix, so θ_m should be a vector of length P, not a scalar. The proximal operator for the group-lasso term ‖β_q‖_2 requires application to the whole group vector, with threshold λ(1−α)γ√P/ρ, and no such group-wise operation is described. The paper should state explicitly whether the reported simulations use scalar-wise soft-thresholding or group-wise soft-thresholding; this distinction changes the estimator that is being evaluated.
  3. [§5.1–5.2] The simulation evidence is confined to a single favorable setting: two binary covariates, N=1000, 16 parameters, and a true coefficient vector that is exactly constant on the user-specified similar pairs S={(3,7),(4,8)}. The paper provides no sensitivity analysis for misspecified similarity pairs, where the fusion penalty would bias estimates by forcing unequal effects together, nor does it compare against the existing structured fusion lasso of Sennhenn-Reulen and Kneib, which is the closest competitor. As written, the claim that the fusion penalty 'improves' selection over the lasso is supported only for the narrow scenario in which the method's structural assumptions are exactly correct. The authors should add a misspecification analysis (e.g., true effects of opposite sign or different magnitudes on a fused pair) and, ideally, a comparison with a relevant existing structured penalty.
  4. [§4.3, GCV criterion] The approximate GCV effective degrees of freedom uses Σ_λ = diag{p'(β̂_{p,q}) / |β̂_{p,q}|}, which is derived from a locally quadratic approximation of a separable penalty. For the group-lasso and fusion components of Eq. (4), the penalty is not separable and is non-differentiable at zero, so the diagonal approximation is not justified as stated. The manuscript should either derive the appropriate effective degrees-of-freedom expression for the FSGL penalty or provide empirical evidence (e.g., comparison with a validation-based tuning criterion) that the approximation yields reasonable tuning in the settings considered.
minor comments (6)
  1. [§4.2] The displayed K matrix omits the exact row counts and labels for the group blocks; the figure is schematic but the text should state that the displayed rows are only illustrative, not a full representation for general P and Q.
  2. [§4.2, Algorithm 1] The line 'obtain β̂ = θ̂' should be replaced with a formula that selects the relevant coordinates of θ (typically the first PQ entries if K begins with the identity block), and the dimensions of all variables should be restated consistently.
  3. [§5.2] The statement that LASSOmstate 'sets to zero on average' the true effect β_{1,8} = −0.8 is vague; it would be helpful to report the proportion of simulation runs in which each true nonzero coefficient is selected, not only the aggregate TPR/FDR.
  4. [§2.2] The notation switches between N (individuals) and n (long-format rows); a sentence clarifying the relationship between N, n, and the risk set formulation would improve readability.
  5. [§6] The application panel for transition 3 in Figure 8 labels variables as 'FLG.1' etc.; for clarity, the figure should use the same variable ordering and naming across panels, and the caption should state that crosses denote nonzero estimated effects.
  6. [Abstract] The abstract describes the method as targeting 'high-dimensional data', but the simulation uses P=2 covariates and the application uses only 24 mutation variables; consider softening 'high-dimensional' or explicitly stating the dimension range covered.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the simulation benchmarks against independently generated true coefficients, and the fusion penalty is the method under test rather than an input disguised as a prediction.

full rationale

The paper proposes a fused sparse-group lasso penalty for multi-state Cox models and derives an ADMM algorithm to minimize the penalized negative partial log-likelihood. The derivation chain is self-contained: the objective function in Section 3 (Eq. 4) defines the penalty, and Section 4.2 constructs the ADMM updates from that objective via the constraint θ = Kβ. No equation in the derivation assumes the conclusion; the estimator is not defined in terms of the simulation outcomes or the application results. The simulation study generates data from transition-specific Cox hazards with pre-specified true coefficients (e.g., β1,3 = β1,7 = 1.2 and β1,4 = β1,8 = −0.8) independently of the fitting algorithm, then evaluates TPR, FDR, bias, and MSE against those known values. This is a standard empirical validation, not a circular reduction. The simulation does place true equal effects on the same transition pairs S used by the fusion penalty, but that is a deliberate proof-of-concept design to test whether the method can exploit the assumed structure; the improvement over LASSOmstate is not algebraically forced, since fusion could in principle overshrink or fail to select, so the reported benefit is an empirical finding. GCV tuning is a conventional model-selection device and does not define the estimand or the evaluation metric. The paper cites the authors' own R packages (mstate, penalized, penMSM) and prior work, but these are used as computational tools or as background, not as load-bearing evidence for the central claim; no uniqueness theorem or unverified result from the authors' prior work is invoked to forbid alternatives. A separate internal consistency issue exists in the dimensional specification of the penalty structure matrix K and the fusion matrix D (Eq. 4 implies P·|S| fusion terms while K is described with only s fusion rows), but this is a correctness or reproducibility concern, not a circularity: no equation reduces to its own input. Overall, the central claim has independent content and the evaluation is not circular.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The method rests on standard multi-state survival assumptions plus a domain-specific similarity prior that is user-specified and not validated in the paper. No new scientific entities are postulated. The tuning parameters and ADMM step size are selected by the algorithm or set by the user, so they are listed as free parameters.

free parameters (4)
  • tuning parameter λ = 8.6 (LASSOmstate), 38.1 (FSGLmstate) in simulation; 20 in AML application
    Overall regularization strength selected by minimal GCV; central to the method's performance.
  • tuning parameter α = 1 (simulation optimum), 0.75 (application optimum)
    Balances lasso vs group lasso; selected by GCV grid.
  • tuning parameter γ = 0.25 (simulation optimum), 0.5 (application optimum)
    Balances sparse penalties vs fusion penalty; selected by GCV grid.
  • ADMM initial step size ρ0 = 1
    Initial augmented Lagrangian parameter; adjusted adaptively with τ=2, η=10 per He et al.
assumptions (6)
  • domain assumption Markov assumption: transition intensities depend only on the current state, not on the full event history.
    Stated in Section 2.1; the product-form multi-state partial likelihood relies on this.
  • domain assumption Transition-specific proportional hazards: covariate effects are constant over time on the hazard scale for each transition.
    The Cox-type multi-state model in Section 2.1 assumes time-fixed covariates and proportional hazards per transition.
  • ad hoc to paper The user-supplied set S of similar transition pairs and the transition grouping structure are correct.
    Section 3 assumes a subset S of pairs with similar effects; simulation and application both impose this prior. If wrong, the fusion penalty forces unequal effects together and biases estimates.
  • domain assumption The multi-state partial likelihood is valid given independent censoring, no tied event times, and correctly specified risk sets.
    The likelihood formulation in Section 2.2 is standard but inherits these usual survival analysis conditions.
  • standard math ADMM converges to the minimizer of the convex FSGL penalized partial likelihood for the chosen step sizes and tolerances.
    The paper invokes standard ADMM convergence theory (Boyd et al.) and stopping criteria adapted from Beer et al.
  • ad hoc to paper The GCV approximation with diagonal Σ_λ provides a valid estimate of predictive ability for tuning parameter selection in the presence of fusion penalties.
    Section 4.3; the effective degrees of freedom only account for element-wise penalty derivatives, not the full fusion penalty structure, and no justification is given for this approximation.

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Cite this review

Pith. "Pith review of Variable selection via fused sparse-group lasso penalized multi-state models incorporating molecular data." pith.science (2026). https://pith.science/paper/ETYWTK7T

@misc{pith2026241117394,
  author       = {Pith},
  title        = {Pith review of: Variable selection via fused sparse-group lasso penalized multi-state models incorporating molecular data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ETYWTK7T}},
  note         = {Machine review of arXiv:2411.17394}
}
read the original abstract

In multi-state models based on high-dimensional data, effective modeling strategies are required to determine an optimal, ideally parsimonious model. In particular, linking covariate effects across transitions is needed to conduct joint variable selection. A useful technique to reduce model complexity is to address homogeneous covariate effects for distinct transitions. We integrate this approach to data-driven variable selection by extended regularization methods within multi-state model building. We propose the fused sparse-group lasso (FSGL) penalized Cox-type regression in the framework of multi-state models combining the penalization concepts of pairwise differences of covariate effects along with transition grouping. For optimization, we adapt the alternating direction method of multipliers (ADMM) algorithm to transition-specific hazards regression in the multi-state setting. In a simulation study and application to acute myeloid leukemia (AML) data, we evaluate the algorithm's ability to select a sparse model incorporating relevant transition-specific effects and similar cross-transition effects. We investigate settings in which the combined penalty is beneficial compared to global lasso regularization.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.