Time-Varying Functional Cox Model

Andrew Leroux; Hongyu Du

REVIEW 4 major objections 8 minor 12 references

Time-Varying Functional Cox Model

T0 review · 4 major / 8 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a functional predictor's effect on survival can be modeled as a smooth bivariate surface over the functional domain and follow-up time, estimated either by a Cox-Poisson transformation or a faster landmark…

desk verdict Useful extension of the functional Cox model with a practical landmark estimator, but inference validity rests on an unproven Poisson equivalence and the paper ships no code. read the letter →

arxiv 2412.14478 v1 pith:ZVRDBEL2 submitted 2024-12-19 stat.ME

Hongyu Du , Andrew Leroux This is my paper · ORCID

classification stat.ME MSC 62N0162G0862R10

keywords functionalregressionCoxmodeltime-varyingeffectslandmarkapproachpenalizedsplinesdataanalysisaccelerometrysurvival

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 Cox survival model in which a functional predictor measured at baseline—such as a full day of physical activity—is allowed to affect the hazard through a coefficient surface that varies smoothly in both the predictor's domain and follow-up time. The paper develops two estimation routes: a Cox-Poisson transformation that yields accurate estimates with roughly nominal pointwise coverage, and a landmark approximation that partitions follow-up into windows and is 10 to 42 times faster with only a small accuracy loss. Both are fit with penalized tensor product splines inside standard penalized regression software. Applied to NHANES accelerometry data, the model finds that the association between diurnal activity patterns and mortality attenuates over an eight-year follow-up.

What carries the argument

The object that carries the argument is the bivariate coefficient surface $\gamma(u,t)$, which multiplies the functional predictor $Z(u)$ inside the log hazard as $\int_U Z(u)\gamma(u,t)\,du$. Estimation rests on two pieces of machinery: the Cox-Poisson likelihood transformation, in which each subject contributes one row per event time while at risk so the partial likelihood becomes a Poisson likelihood, and the landmark decomposition, which replaces continuous time $t$ by landmark times $s_l$ with prediction windows and stratifies the baseline hazard. Both routes expand $\gamma$ in penalized tensor product splines—cyclic cubic splines over the functional domain and cubic splines over time—with smoothing parameters selected by restricted maximum likelihood.

What would settle it

Simulate a dataset with a known coefficient surface that changes sharply in time (for instance, a step at $t=0.5$), fit the Cox-Poisson TV-FLCM and an exact partial-likelihood fit with the same spline basis, and compare pointwise coverage: systematic divergence or coverage well below 95% near the change point would disprove the claimed validity of the Poisson approximation for general smooth surfaces.

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Extended reading notes

Core claim

The central claim is that a functional linear Cox model with time-varying coefficients—$\log \lambda_i(t) = \log \lambda_0(t) + \int_U Z_i(u)\gamma(u,t)\,du$—can be estimated and used for inference when $\gamma(u,t)$ is a smooth bivariate surface. Estimation is carried out by expanding $\gamma(u,t)$ in penalized tensor product splines and exploiting the Cox-Poisson likelihood connection, or by replacing $t$ with discrete landmark times $s_l$ and fitting a stratified landmark model. Simulations across four surfaces and sample sizes up to 4000 show the Cox-Poisson estimator recovers the surface with approximately nominal pointwise coverage, and the landmark estimator with short windows ($w=0.04$) has AMSE only 5.6% to 91.3% higher while reducing computation time by a factor of 10 to 42. The authors use the landmark model on the NHANES dataset (4445 subjects, 1440 minute-level activity values per subject) and report that the mortality association with diurnal activity attenuates over the follow-up.

Load-bearing premise

The paper assumes the Cox-Poisson likelihood transformation remains valid when the functional coefficient is a smoothly varying bivariate function of time and functional domain, but it gives no derivation or regularity conditions for the continuous-time setting, and the nominal coverage claim rests on simulations of four smooth surfaces.

Editorial extensions

If this is right

Because the full model can be expressed as a Poisson regression, standard penalized regression software can fit it, making the method immediately usable for small-to-medium datasets.
The landmark version scales to high-dimensional functional predictors (e.g., 1440 minute-level activity values) on a laptop, which was previously infeasible.
Pointwise confidence intervals from the Cox-Poisson route let researchers test where and when a functional predictor's effect is non-zero rather than only whether it is zero.
The NHANES finding that diurnal activity effects attenuate over eight years suggests future studies should model effect decay rather than assume a constant association.
The modeling framework extends naturally to multiple scalar and functional predictors, as the authors note.

Reading between the lines

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

The landmark window length is an implicit bias-variance dial: shorter windows reduce the bias from assuming a constant effect within the window but inflate variance, so an automated or cross-validated choice of $w$ could replace the manual settings used here.
The undercoverage near $t=0$ for the rapidly oscillating surface suggests that inference may degrade when many events occur very early; boundary corrections or a different parameterization of the time basis could be tested.
Because landmarking conditions on survival to each landmark time, the same machinery could be used for dynamic prediction, updating risk estimates as new covariate or follow-up information accumulates.
The Poisson equivalence might be extended to surfaces with limited smoothness or interactions between time and the functional domain, but the current simulation evidence does not support that generalization.

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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

Quick take: this is a solid applied-methods paper with a practical contribution. The TV-FLCM lets the effect of a functional predictor vary smoothly in both the functional domain and event time, which addresses PH violations for functional predictors. The key novelty relative to Leroux (2020) is dropping the u <= t constraint, giving a rectangular coefficient surface. The landmark estimator, implemented through mgcv's cox.ph family, is a genuinely useful computational workaround for large datasets; the 10-42x speedup over the Poisson approach with only modest accuracy loss is credible from the simulations.

What the paper does well: the simulation study is careful. Four functional forms, sample sizes 2000-4000, 500 replicates, and both AMSE and pointwise coverage are reported. The Poisson method hits around 95% average coverage for the smooth surfaces, and even for the oscillatory 10cos(4pi(t-u)) surface average coverage stays in the 94-95% range except near t=0, which they acknowledge. The NHANES application shows the landmark method scales to 4445 subjects with 1440-dimensional predictors, which is the kind of real problem that motivates the method.

The soft spots are real but mostly fixable. The biggest one: Section 2.3 asserts that tensor product splines turn the Cox model into a Poisson regression, citing Whitehead (1980), but gives no derivation, no offset construction, and no regularity conditions. This is a standard equivalence, but for a bivariate smooth coefficient it deserves at least a careful statement of the data-expansion steps, and the paper's own Section 6 says rigorous asymptotic theory is future work. That means the nominal coverage results are evidence only for the simulated scenarios, not a general guarantee. I'd call this a major gap but not a fatal one—the simulations are encouraging, and the mgcv implementation likely does the right thing.

Minor issues: the NHANES results have no confidence intervals, so the headline 'attenuation over time' is unquantified. No code or data are publicly available, which makes the R snippets hard to trust as a full reproducibility package. The writing has some rough edges (typos, the phrase 'these novels').

Who this is for: applied survival analysts working with accelerometry or other functional predictors, and functional data methodologists. It deserves a serious referee; the derivation gap should be a major revision, not a desk reject.

Referee Report

4 major / 8 minor

Summary. The paper proposes two approaches for estimating a time-varying functional linear Cox model, in which the effect of a baseline functional predictor on the hazard is a bivariate smooth coefficient gamma(u,t) of the functional domain u and follow-up time t. The first approach, TV-FLCM, is said to exploit the Cox-Poisson likelihood connection and is intended for small-to-medium datasets; the second, TV-FLCM-L, uses landmark times and short prediction windows to reduce the computational burden for larger datasets. Estimation is carried out with penalized tensor-product splines via the mgcv package. The simulation study with four gamma surfaces and sample sizes 2000-4000 reports lower AMSE for TV-FLCM than for TV-FLCM-L, nominal or near-nominal average coverage for TV-FLCM in several scenarios, and 10-42 times faster computation for the landmark approach with w=0.04. The NHANES application estimates how the association between diurnal motor activity and mortality attenuates over an eight-year follow-up and illustrates the scalability of the landmark method.

Significance. If the Cox-Poisson equivalence and the associated coverage guarantees can be established rigorously, the paper offers a practical and potentially widely used extension of functional Cox regression to time-varying coefficients, building on the stable and popular mgcv infrastructure. The landmark variant addresses a real computational bottleneck for high-dimensional functional predictors such as 1440-minute accelerometry profiles, and the simulation study covers several nontrivial coefficient shapes, including an interaction-like surface. The NHANES application is directly relevant to a large public-health literature. The paper also deserves credit for including the stacked-landmark data construction, explicit model formulas, and a sensitivity-style comparison of landmark window choices. However, the main inference claim depends on an unproven Poisson-likelihood transformation for a continuously varying bivariate coefficient, and the computational pathway for the TV-FLCM fits is not fully documented in the main text.

major comments (4)

[Section 2.4 (with Section 2.3 and Section 6)] The central inference step is asserted rather than derived. Section 2.4 states that 'By extending the functional term using tensor product splines, the entire Cox model is transformed into a Poisson regression model,' citing Whitehead (1980). The paper does not give the stacked Poisson likelihood, the offset or stratum construction, or a proof that the score equations coincide with those of the Cox partial likelihood for the bivariate smooth coefficient surface gamma(u,t) entering through the integral term. The standard Whitehead construction is a profile-likelihood/Poisson equivalence that requires a specific treatment of the baseline hazard and risk sets; it is not immediate for the present continuous-time setting. Please add a derivation or state a theorem with sufficient conditions (e.g., no tied event times, independent censoring, compactly supported spline bases), and reconcile the abstract's unconditional 'valid estimation and inference' claim with Section 6's statement that developing rigorous asymptotic theory is future work.
[Section 5.1 and Tables 2-5] The simulation data-generating mechanism is incompletely specified. The hazard is written as log lambda_i(t|X_i, Z_i(u), s_l) = log lambda_0(t|s_l) + integral Z_i(u) gamma(u,s_l) du, but the baseline hazard lambda_0 is never defined, and the survival curve S_i(t|Z_i) that is inverted to generate event times is not given in equation form. In addition, the text introduces a noisy version Z_{i,real}(u)=Z_i(u)+epsilon_i(u) after saying the predictor is 'measured with bias,' but it is not stated whether the fitted landmark and Poisson models use Z_{i,real} or the true Z_i. Consequently, the AMSE and coverage numbers in Tables 2-5 are not reproducible from the information provided.
[Section 3.2] The R code for the separate landmark model uses `ti(umat, by=Zlmat, bs=c("cc","cr"), k=c(5,5), mc=c(T,F))` with a single variable, while the proposed model of Eq. (3) involves a bivariate smooth in (u,s_l); as written, the code does not implement the tensor-product smooth described in Section 2.5. Furthermore, no R code or dataset construction is shown for the TV-FLCM (Poisson) estimator whose results appear in Tables 2-5. Since the paper advertises stable software implementation, please correct the code snippets and provide the full estimation pathway for TV-FLCM, including the family, offset, stacking rule, or a pointer to an online supplement with complete code.
[Section 5.3 and Table 4] The abstract's claim that 'The Cox-Poisson method provides nominal coverage probabilities' is too strong for the reported results. Table 4 shows average coverages of 95.1%, 94.3%, and 94.0% for N=2000, 3000, and 4000 for gamma(u,t)=10cos(4*pi*(t-u)), with the text acknowledging 'substantial undercoverage for t close to 0.' The stated explanation (many events before t=0.05) describes the event process but does not explain why the Wald intervals are invalid in that region; if the Poisson equivalence is the basis for the intervals, this is a known failure regime for the inference claim. Please qualify the abstract statement to 'empirically nominal in the scenarios considered' and either correct the intervals in regions with heavy early events or provide a theoretical explanation for the undercoverage.

minor comments (8)

[Equation (2)] Equation (2) writes the scalar covariate term as X_i beta, whereas Section 2.1 and the landmark model in Eq. (3) allow beta to depend on time; please make the notation consistent, for example by writing X_i beta(t).
[Section 2.5, displayed partial-likelihood formulas] In the two displayed partial-likelihood expressions, the quantity inside the logarithm is written as e^{eta_i} in the sum over risk-set members; it should be e^{eta_j}, with the sum indexed by j.
[Section 2.1] The citation list contains 'Bender et al. 2018, ?'; the missing reference or citation key should be completed.
[Section 4.2] The window specification '(0.5, 0.25, 1.2, 0.8, 1.25, 1, 0.5, 0.75, 1, 0.65, 0.1, 0.5) + 0.3' is ambiguous; please clarify whether 0.3 is added to each window length or to each interval endpoint.
[Tables 2-5] The reported AMSE and coverage values are averaged over 500 simulations but do not include Monte Carlo standard errors; adding these would help readers judge whether differences across methods and sample sizes are meaningful.
[Section 4.3] The conclusion that diurnal effects attenuate over the eight-year follow-up is drawn from a visual comparison of landmark curves; a quantitative summary (e.g., the integrated negative-area per landmark time, or a formal trend test) would strengthen the claim.
[Section 2.7] When discussing REML smoothing-parameter selection for the stacked landmark data, the paper notes that observations are treated as independent; because each subject contributes multiple landmark rows, this is a pseudo-likelihood feature, and the possible effect on smoothing-parameter or interval estimates should be stated as a limitation.
[Section 5.3 and Tables 2-5] The 'small loss of accuracy' conclusion applies only to the short window w=0.04; the w=Inf results are dramatically worse (e.g., AMSE 44.11 vs 2.51 in Table 4 at N=4000). The paper should state explicitly that w=Inf is not a generally advisable choice and that the simulation-based advice concerns short windows only.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the empirical claims are evaluated against externally simulated true surfaces, and the self-citations are minor and non-load-bearing.

full rationale

The paper's derivation chain does not reduce to its own inputs. The TV-FLCM is defined as a functional Cox model with a bivariate coefficient surface, estimated through penalized tensor-product splines, and the Cox-Poisson connection is attributed to the external result of Whitehead (1980) rather than derived from the paper's own fitted values. The landmark approach is imported from Van Houwelingen (2007) and Rizopoulos et al. (2017), and the penalized-spline machinery is from Wood (2006, 2017) and mgcv. Simulation results are compared against independently generated true functional coefficient surfaces, so the AMSE and coverage claims are external benchmarks, not fitted parameters renamed as predictions. The NHANES application is descriptive and makes no out-of-sample predictive claim. The self-citations to Leroux (2020) and Cui et al. (2021) are used for model setup and a standard sum-to-zero identifiability constraint; they do not carry the burden of the paper's central empirical or methodological claims. The paper's own admission in Section 6 that rigorous asymptotic theory remains future work is a limitation on inference support, not evidence of circularity. Overall, no specific circular step can be exhibited from the paper's equations, and the low score reflects only the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim depends on several modeling choices: the spline basis dimensions and landmark design are user-specified free parameters; the Cox-Poisson equivalence and the landmark approximation are domain assumptions imported from prior literature; the mean centering identifiability trick is ad hoc to this paper. No new physical or statistical entities are introduced.

free parameters (4)

Ks (landmark/time basis dimension) = 5 or 10 in examples
Number of marginal spline basis functions for the time/landmark direction; chosen by hand and stated to be important (Section 2.5).
Ku (functional domain basis dimension) = 5 or 10 in examples
Number of marginal spline basis functions for the functional domain u; chosen by hand (Section 2.5).
K1 (scalar covariate effect basis dimension) = not specified precisely
Number of basis functions for the time-varying scalar effect beta(s_l); chosen by hand (Section 2.5).
Landmark time set and window lengths = S={0,0.04,...,0.96}, w=0.04 or infinity in simulations; varied in application
User-specified design choices that determine the approximation; sensitivity assessed (Sections 2.4, 5.1).

assumptions (5)

domain assumption Cox-Poisson equivalence for the time-varying functional coefficient
Section 2.3 asserts the Cox model is transformed into a Poisson regression model without derivation; inference relies on this.
domain assumption Landmark piecewise-constant approximation with landmark-specific baseline hazards
Section 2.4 assumes gamma is constant within each window and conditions on survival to s_l; this is the defining approximation of TV-FLCM-L.
domain assumption Independence of stacked landmark rows for REML smoothing parameter selection
Section 2.7 states REML 'implicitly assumes the independence of the observations across strata', which is violated because subjects appear in multiple landmark datasets.
ad hoc to paper Identifiability via mean centering of the functional predictor by landmark
Section 2.6 introduces mean centering and sum-to-zero constraints to separate the coefficient surface from the baseline hazard; this is a modeling choice specific to this paper.
domain assumption Dense functional observations allow accurate Riemann sum approximation
Section 2.5 replaces the integral over u with a numerical sum using integration weights; accuracy depends on grid fineness.

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Cite this review

Pith. "Pith review of Time-Varying Functional Cox Model." pith.science (2026). https://pith.science/paper/ZVRDBEL2

@misc{pith2026241214478,
  author       = {Pith},
  title        = {Pith review of: Time-Varying Functional Cox Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZVRDBEL2}},
  note         = {Machine review of arXiv:2412.14478}
}

read the original abstract

We propose two novel approaches for estimating time-varying effects of functional predictors within a linear functional Cox model framework. This model allows for time-varying associations of a functional predictor observed at baseline, estimated using penalized regression splines for smoothness across the functional domain and event time. The first approach, suitable for small-to-medium datasets, uses the Cox-Poisson likelihood connection for valid estimation and inference. The second, a landmark approach, significantly reduces computational burden for large datasets and high-dimensional functional predictors. Both methods address proportional hazards violations for functional predictors and model associations as a bivariate smooth coefficient. Motivated by analyzing diurnal motor activity patterns and all-cause mortality in NHANES (N=4445, functional predictor dimension=1440), we demonstrate the first method's computational limitations and the landmark approach's efficiency. These methods are implemented in stable, high-quality software using the mgcv package for penalized spline regression with automated smoothing parameter selection. Simulations show both methods achieve high accuracy in estimating functional coefficients, with the landmark approach being computationally faster but slightly less accurate. The Cox-Poisson method provides nominal coverage probabilities, while landmark inference was not assessed due to inherent bias. Sensitivity to landmark modeling choices was evaluated. Application to NHANES reveals an attenuation of diurnal effects on mortality over an 8-year follow-up.

Figures

Figures reproduced from arXiv: 2412.14478 by the authors.

**Figure 1.** (a) Functional effect γ(u, t) = sin(2 ∗ π ∗ u)/(t + 0.5). (b) Functional predictor Z(u) = sin(5 ∗ u). (c)The integral of functional effect times functional predictor when t = 0.1. (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗

**Figure 2.** Estimated functional effects in NHANES dataset application. In panel (a), the win [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** (a) Heatmap showing the average MIMS over 24 hours for subjects experiencing events [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 11 canonical work pages

[1]

Andersen, P. K. & Gill, R. D. (1982), ‘Cox’s regression model for counting processes: a large sample study’, The annals of statisticspp. 1100–1120. Andrinopoulou, E.-R., Eilers, P. H., Takkenberg, J. J. & Rizopoulos, D. (2018), ‘Improved dynamic predictions from joint models of longitudinal and survival data with time-varying effects using p- splines’, Bi...

work page 1982
[5]

(b) Heatmap showing the average MIMS over 24 hours for subjects who survived beyond each time point

25 Table 1: Landmark Dataset ID T d X svec umat zmat smat lmat zlmat 1 1 0 7 0 0 2 4 6 1 0.3 0.7 1.1 0 0 0 0 2 2 2 2 2 0.6 1.4 2.2 1 2 0 7 1 0 2 4 6 1 0.3 0.7 1.1 1 1 1 1 2 2 2 2 2 0.6 1.4 2.2 1 3 0 7 2 0 2 4 6 1 0.3 0.7 1.1 2 2 2 2 2 2 2 2 2 0.6 1.4 2.2 1 4 0 7 3 0 2 4 6 1 0.3 0.7 1.1 3 3 3 3 2 2 2 2 2 0.6 1.4 2.2 1 4.5 1 7 4 0 2 4 6 1 0.3 0.7 1.1 4 4 4 ...

work page 2000
[8]

The simulations are performed on different sample sizes (N=2000, 3000 or

True Effect Methods Sample Sizes N=2000 N=3000 N=4000 w=0.04 w=∞ Poisson CI Tests Methods N=2000 N=3000 N=4000 AMSE w=0.04 0.037 0.026 0.019 w=∞ 0.087 0.055 0.038 Poisson 0.034 0.024 0.018 Coverage Rate CI 95.6% 95.4% 95.3% Computation Time w=0.04 9 seconds 16 seconds 25 seconds Poisson 374 seconds 378 seconds 329 seconds 28 Table 4: The results of the es...

work page 2000
[9]

In this complex case, TV-FLCM-L demonstrates its ability to capture the underlying structure and critical features of the estimated functional effects

True Effect Methods Sample Sizes N=2000 N=3000 N=4000 w=0.04 w=∞ Poisson CI Tests Methods N=2000 N=3000 N=4000 AMSE w=0.04 5.563 5.051 4.782 w=∞ 44.28 44.17 44.11 Poisson 3.533 2.900 2.510 Coverage Rate CI 95.1% 94.3% 94.0% Computation Time w=0.04 17 seconds 25 seconds 31 seconds Poisson 168 seconds 290 seconds 379 seconds 29 7 Supplementary material 7.1 ...

work page 2000
[10]

Furthermore, the Poisson regression method achieves accurate coverage rates, further enhancing its robustness

Increasing the sample size in such intricate functions improves the accuracy of the overall shape and reduces the averaged mean squared error (AMSE) for both the landmark and Poisson methods. Furthermore, the Poisson regression method achieves accurate coverage rates, further enhancing its robustness. In summary, regardless of the functional effect’s comp...

work page 2000
[12]

True Effect Methods Sample Sizes N=2000 N=3000 N=4000 w=0.04 w=∞ Poisson CI Tests Methods N=2000 N=3000 N=4000 AMSE w=0.04 0.169 0.137 0.133 w=∞ 0.884 0.772 0.768 Poisson 0.159 0.126 0.119 Coverage Rate CI 92.0% 93.0% 94.0% Computation Time w=0.04 40 seconds 42 seconds 47 seconds Poisson 720 seconds 924 seconds 930 seconds 31 Table 6: The table of surviva...

work page 2000
[61]

& Reyes, E

Thomas, L. & Reyes, E. M. (2014), ‘Tutorial: Survival estimation for cox regression models with time- varying coefficients using sas and r’, Journal of Statistical Software, Code Snippets61(1), 1–23. URL: https://www.jstatsoft.org/index.php/jss/article/view/v061c01 Tian, L., Zucker, D. & Wei, L. (2005), ‘On the cox model with time-varying regression coeff...

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Cox, D. R. (1975), ‘Partial likelihood’, Biometrika 62(2), 269–276. Crainiceanu, C. M., Goldsmith, J., Leroux, A. & Cui, E. (2024), Functional Data Analysis with R, 1st edn, Chapman and Hall/CRC. URL: https://doi.org/10.1201/9781003278726 Crowther, M. J., Abrams, K. R. & Lambert, P. C. (2013), ‘Joint modeling of longitudinal and survival data’, The Stata ...

work page doi:10.1201/9781003278726 1975

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Ruppert, D., Wand, M. P. & Carroll, R. J. (2003), Semiparametric regression, number 12, Cambridge university press. Stadtm¨ uller, U. & Zampiceni, M. (2014), An introduction to functional data analysis, in ‘Stochastic Geometry, Spatial Statistics and Random Fields: Models and ...

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(2020), Statistical methods for the analysis of functional data under models with complex association structure, PhD thesis, Johns Hopkins University

Leroux, A. (2020), Statistical methods for the analysis of functional data under models with complex association structure, PhD thesis, Johns Hopkins University. Leroux, A., Crainiceanu, C., Smirnova, E. & Cao, Q. (2018), ‘rnhanesdata: Nhanes accelerometry data pipeline’, R pa...

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[3000]

The survival curves for the first, second, and fourth functions are remarkably similar

All four functions demonstrate well-defined survival curves. The survival curves for the first, second, and fourth functions are remarkably similar. In contrast, the third survival curve reveals a notable divergence, indicating that while some subjects survived until the study...

2000
[4000]

The simulations are performed on different sample sizes (N=2000, 3000 or

True Effect Methods Sample Sizes N=2000 N=3000 N=4000 w=0.04 w=∞ Poisson CI Tests Methods N=2000 N=3000 N=4000 AMSE w=0.04 0.050 0.033 0.029 w=∞ 0.139 0.101 0.089 Poisson 0.045 0.031 0.026 Coverage Rate CI 95.1% 95.0% 95.0% Computation Time w=0.04 10 seconds 12 seconds 19 seco...

2000

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Andersen, P. K. & Gill, R. D. (1982), ‘Cox’s regression model for counting processes: a large sample study’, The annals of statisticspp. 1100–1120. Andrinopoulou, E.-R., Eilers, P. H., Takkenberg, J. J. & Rizopoulos, D. (2018), ‘Improved dynamic predictions from joint models of longitudinal and survival data with time-varying effects using p- splines’, Bi...

work page 1982

[2] [5]

(b) Heatmap showing the average MIMS over 24 hours for subjects who survived beyond each time point

25 Table 1: Landmark Dataset ID T d X svec umat zmat smat lmat zlmat 1 1 0 7 0 0 2 4 6 1 0.3 0.7 1.1 0 0 0 0 2 2 2 2 2 0.6 1.4 2.2 1 2 0 7 1 0 2 4 6 1 0.3 0.7 1.1 1 1 1 1 2 2 2 2 2 0.6 1.4 2.2 1 3 0 7 2 0 2 4 6 1 0.3 0.7 1.1 2 2 2 2 2 2 2 2 2 0.6 1.4 2.2 1 4 0 7 3 0 2 4 6 1 0.3 0.7 1.1 3 3 3 3 2 2 2 2 2 0.6 1.4 2.2 1 4.5 1 7 4 0 2 4 6 1 0.3 0.7 1.1 4 4 4 ...

work page 2000

[3] [8]

The simulations are performed on different sample sizes (N=2000, 3000 or

True Effect Methods Sample Sizes N=2000 N=3000 N=4000 w=0.04 w=∞ Poisson CI Tests Methods N=2000 N=3000 N=4000 AMSE w=0.04 0.037 0.026 0.019 w=∞ 0.087 0.055 0.038 Poisson 0.034 0.024 0.018 Coverage Rate CI 95.6% 95.4% 95.3% Computation Time w=0.04 9 seconds 16 seconds 25 seconds Poisson 374 seconds 378 seconds 329 seconds 28 Table 4: The results of the es...

work page 2000

[4] [9]

In this complex case, TV-FLCM-L demonstrates its ability to capture the underlying structure and critical features of the estimated functional effects

True Effect Methods Sample Sizes N=2000 N=3000 N=4000 w=0.04 w=∞ Poisson CI Tests Methods N=2000 N=3000 N=4000 AMSE w=0.04 5.563 5.051 4.782 w=∞ 44.28 44.17 44.11 Poisson 3.533 2.900 2.510 Coverage Rate CI 95.1% 94.3% 94.0% Computation Time w=0.04 17 seconds 25 seconds 31 seconds Poisson 168 seconds 290 seconds 379 seconds 29 7 Supplementary material 7.1 ...

work page 2000

[5] [10]

Furthermore, the Poisson regression method achieves accurate coverage rates, further enhancing its robustness

Increasing the sample size in such intricate functions improves the accuracy of the overall shape and reduces the averaged mean squared error (AMSE) for both the landmark and Poisson methods. Furthermore, the Poisson regression method achieves accurate coverage rates, further enhancing its robustness. In summary, regardless of the functional effect’s comp...

work page 2000

[6] [12]

True Effect Methods Sample Sizes N=2000 N=3000 N=4000 w=0.04 w=∞ Poisson CI Tests Methods N=2000 N=3000 N=4000 AMSE w=0.04 0.169 0.137 0.133 w=∞ 0.884 0.772 0.768 Poisson 0.159 0.126 0.119 Coverage Rate CI 92.0% 93.0% 94.0% Computation Time w=0.04 40 seconds 42 seconds 47 seconds Poisson 720 seconds 924 seconds 930 seconds 31 Table 6: The table of surviva...

work page 2000

[7] [61]

& Reyes, E

Thomas, L. & Reyes, E. M. (2014), ‘Tutorial: Survival estimation for cox regression models with time- varying coefficients using sas and r’, Journal of Statistical Software, Code Snippets61(1), 1–23. URL: https://www.jstatsoft.org/index.php/jss/article/view/v061c01 Tian, L., Zucker, D. & Wei, L. (2005), ‘On the cox model with time-varying regression coeff...

arXiv 2014

[8] [693]

Cox, D. R. (1975), ‘Partial likelihood’, Biometrika 62(2), 269–276. Crainiceanu, C. M., Goldsmith, J., Leroux, A. & Cui, E. (2024), Functional Data Analysis with R, 1st edn, Chapman and Hall/CRC. URL: https://doi.org/10.1201/9781003278726 Crowther, M. J., Abrams, K. R. & Lambert, P. C. (2013), ‘Joint modeling of longitudinal and survival data’, The Stata ...

work page doi:10.1201/9781003278726 1975