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Regularized Bidimensional Estimation of the Hazard Rate

Vivien Goepp, Jean-Christophe Thalabard, Grégory Nuel, Olivier Bouaziz

Abstract

In epidemiological or demographic studies, with variable age at onset, a typical quantity of interest is the incidence of a disease (for example the cancer incidence). In these studies, the individuals are usually highly heterogeneous in terms of dates of birth (the cohort) and with respect to the calendar time (the period) and appropriate estimation methods are needed. In this article a new estimation method is presented which extends classical age-period-cohort analysis by allowing interactions between age, period and cohort effects. This paper introduces a bidimensional regularized estimate of the hazard rate where a penalty is introduced on the likelihood of the model. This penalty can be designed either to smooth the hazard rate or to enforce consecutive values of the hazard to be equal, leading to a parsimonious representation of the hazard rate. In the latter case, we make use of an iterative penalized likelihood scheme to approximate the L0 norm, which makes the computation tractable. The method is evaluated on simulated data and applied on breast cancer survival data from the SEER program.

Regularized Bidimensional Estimation of the Hazard Rate

Abstract

In epidemiological or demographic studies, with variable age at onset, a typical quantity of interest is the incidence of a disease (for example the cancer incidence). In these studies, the individuals are usually highly heterogeneous in terms of dates of birth (the cohort) and with respect to the calendar time (the period) and appropriate estimation methods are needed. In this article a new estimation method is presented which extends classical age-period-cohort analysis by allowing interactions between age, period and cohort effects. This paper introduces a bidimensional regularized estimate of the hazard rate where a penalty is introduced on the likelihood of the model. This penalty can be designed either to smooth the hazard rate or to enforce consecutive values of the hazard to be equal, leading to a parsimonious representation of the hazard rate. In the latter case, we make use of an iterative penalized likelihood scheme to approximate the L0 norm, which makes the computation tractable. The method is evaluated on simulated data and applied on breast cancer survival data from the SEER program.

Paper Structure

This paper contains 14 sections, 18 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Fused Regularized Estimation
  2. Modelization
  3. Penalized Likelihood
  4. Numerical Optimization
  5. Fused L$_{2}$ Penalty Estimate
  6. Fused Adaptive Ridge Estimate
  7. Choice of the Penalty Parameter $\kappa$
  8. Simulation Study
  9. Simulation Designs
  10. Smooth true hazard
  11. Piecewise constant true hazard
  12. Performance of the Estimation Methods in Terms of MSE
  13. Perspective Plots of the Estimation Methods
  14. Real Data Application

Figures (5)

  • Figure 1: Diagrams representing the lives of individuals: in the age-period plane (a) -- called Lexis diagram -- and in the age-cohort plane (b). Solid lines represent lives of individuals until occurrence of the event of interest. The same age, cohort, and period intervals are displayed in gray.
  • Figure 2: True hazard of the two simulation designs: smooth hazard in heatmap (a) and perspective plot (b) and piecewise constant hazard in heatmap (c) and perspective plot (d).
  • Figure 3: Smooth true hazard and corresponding estimates. The sample size is $4000$ and the hazard estimates are medians taken over $500$ simulations. The estimations are performed in the age-cohort plane and with different methods. Panel (a) represents the true hazard used to generate the data, Panel (b) represents the hazard estimated using the age-cohort model, Panel (c) represents the smoothed estimate, and Panel (d) represents the segmented estimate with the EBIC criterion.
  • Figure 4: Piecewise constant true hazard and corresponding estimates. The sample size is $4000$ and the hazard estimates are medians taken over $500$ simulations. The estimations are performed in the age-cohort plane and with different methods. Panel (a) represents the true hazard used to generate the data, Panel (b) represents the hazard estimated using the age-cohort model, Panel (c) represents the smoothed estimate, and Panel (d) represents the segmented estimate with the EBIC criterion.
  • Figure 5: Estimated hazard of death after diagnosis of breast cancer for different stages of cancer. The estimate is obtained with the L$_{0}$ regularization. The upper right corner of every graph corresponds to the region where no data are available. Note that the grey-color scales are different between panels.