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Varying coefficient model for longitudinal data with informative observation times

Yu Gu, Yangjianchen Xu, Peijun Sang

TL;DR

This work addresses bias in varying-coefficient models for longitudinal data when observation times are informative. It introduces inverse-intensity weighting under a proportional-intensity model, coupled with a sieve estimation framework using a B-spline basis, to consistently estimate time-varying effects $\beta(t)$ and perform valid inference. The authors establish consistency, convergence rates, and asymptotic normality for $\widehat{\beta}(t)$, and implement a multiplier bootstrap for pointwise confidence intervals. Through simulations and an ADNI application, the weighted approach substantially reduces bias and yields appropriate coverage compared to unweighted analyses, with broad implications for studies where visit schedules depend on outcomes.

Abstract

Varying coefficient models are widely used to characterize dynamic associations between longitudinal outcomes and covariates. Existing work on varying coefficient models, however, all assumes that observation times are independent of the longitudinal outcomes, which is often violated in real-world studies with outcome-driven or otherwise informative visit schedules. Such informative observation times can lead to biased estimation and invalid inference using existing methods. In this article, we develop estimation and inference procedures for varying coefficient models that account for informative observation times. We model the observation time process as a general counting process under a proportional intensity model, with time-varying covariates summarizing the observed history. To address potential bias, we incorporate inverse intensity weighting into a sieve estimation framework, yielding closed-form coefficient function estimators via weighted least squares. We establish consistency, convergence rates, and asymptotic normality of the proposed estimators, and construct pointwise confidence intervals for the coefficient functions. Extensive simulation studies demonstrate that the proposed weighted method substantially outperforms the conventional unweighted method when observation times are informative. Finally, we provide an application of our method to the Alzheimer's Disease Neuroimaging Initiative study.

Varying coefficient model for longitudinal data with informative observation times

TL;DR

This work addresses bias in varying-coefficient models for longitudinal data when observation times are informative. It introduces inverse-intensity weighting under a proportional-intensity model, coupled with a sieve estimation framework using a B-spline basis, to consistently estimate time-varying effects and perform valid inference. The authors establish consistency, convergence rates, and asymptotic normality for , and implement a multiplier bootstrap for pointwise confidence intervals. Through simulations and an ADNI application, the weighted approach substantially reduces bias and yields appropriate coverage compared to unweighted analyses, with broad implications for studies where visit schedules depend on outcomes.

Abstract

Varying coefficient models are widely used to characterize dynamic associations between longitudinal outcomes and covariates. Existing work on varying coefficient models, however, all assumes that observation times are independent of the longitudinal outcomes, which is often violated in real-world studies with outcome-driven or otherwise informative visit schedules. Such informative observation times can lead to biased estimation and invalid inference using existing methods. In this article, we develop estimation and inference procedures for varying coefficient models that account for informative observation times. We model the observation time process as a general counting process under a proportional intensity model, with time-varying covariates summarizing the observed history. To address potential bias, we incorporate inverse intensity weighting into a sieve estimation framework, yielding closed-form coefficient function estimators via weighted least squares. We establish consistency, convergence rates, and asymptotic normality of the proposed estimators, and construct pointwise confidence intervals for the coefficient functions. Extensive simulation studies demonstrate that the proposed weighted method substantially outperforms the conventional unweighted method when observation times are informative. Finally, we provide an application of our method to the Alzheimer's Disease Neuroimaging Initiative study.
Paper Structure (17 sections, 2 theorems, 25 equations, 4 figures, 1 table)

This paper contains 17 sections, 2 theorems, 25 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary setup
  3. Data and outcome model
  4. Inverse intensity weighting
  5. Estimation procedure
  6. Estimation of weights
  7. Estimation of varying coefficients
  8. Functional principal component analysis
  9. Asymptotic properties and inference
  10. Regularity conditions and asymptotic properties
  11. Inference on varying coefficients
  12. Simulation studies
  13. Practical considerations
  14. Data generation process
  15. Simulation results
  16. ...and 2 more sections

Key Result

Theorem 4.1

Under Assumptions ass:ident--ass: intensity, the $j$th estimated coefficient function $\widehat{\beta}_j(t)$ satisfies for all $j \in [d]$, provided that $\log n/n = o(q_n^{-4})$.

Figures (4)

  • Figure 1: Observed ADAS-Cog 13 trajectories for five randomly selected subjects from the ADNI study.
  • Figure 2: Estimated varying coefficients. EW, the proposed weighted method with estimated weights; TW, the weighted method with true weights; UW, the unweighted method.
  • Figure 3: Empirical coverage percentage (CP) of the 95% pointwise confidence intervals for each varying coefficient. EW, the proposed weighted method with estimated weights; TW, the weighted method with true weights; UW, the unweighted method.
  • Figure 4: Estimated varying coefficients, eigenvalues, and first eigenfunction based on the ADNI data. Dashed curves show the 95% pointwise confidence intervals.

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 4.2