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The partly parametric and partly nonparametric additive risk model

Nils Lid Hjort, Emil Aas Stoltenberg

TL;DR

Methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified is developed, which also enables us to assess the goodness of fit of the model's parametric components.

Abstract

Aalen's linear hazard rate regression model is a useful and increasingly popular alternative to Cox' multiplicative hazard rate model. It postulates that an individual has hazard rate function $h(s)=z_1α_1(s)+\cdots+z_rα_r(s)$ in terms of his covariate values $z_1,\ldots,z_r$. These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions $α_j(s)$ are the parameters of the model and are estimated from data. This is traditionally accomplished in a fully nonparametric way. This paper develops methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified. Large-sample results are reached inside this partly parametric, partly nonparametric framework, which also enables us to assess the goodness of fit of the model's parametric components. In addition, these results are used to pinpoint how much precision is gained, using the parametric-nonparametric model, over the standard nonparametric method. A real-data application is included, along with a brief simulation study.

The partly parametric and partly nonparametric additive risk model

TL;DR

Methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified is developed, which also enables us to assess the goodness of fit of the model's parametric components.

Abstract

Aalen's linear hazard rate regression model is a useful and increasingly popular alternative to Cox' multiplicative hazard rate model. It postulates that an individual has hazard rate function in terms of his covariate values . These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions are the parameters of the model and are estimated from data. This is traditionally accomplished in a fully nonparametric way. This paper develops methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified. Large-sample results are reached inside this partly parametric, partly nonparametric framework, which also enables us to assess the goodness of fit of the model's parametric components. In addition, these results are used to pinpoint how much precision is gained, using the parametric-nonparametric model, over the standard nonparametric method. A real-data application is included, along with a brief simulation study.
Paper Structure (24 sections, 4 theorems, 95 equations, 5 figures)

This paper contains 24 sections, 4 theorems, 95 equations, 5 figures.

Table of Contents

  1. Introduction and summary
  2. The fully nonparametric and fully parametric cases
  3. The general integrated weighted least squares estimators
  4. Optimal nonparametric estimation
  5. The fully parametric model
  6. Estimation in the parametric and nonparametric model
  7. Estimating the parametric part
  8. Backfitting to re-estimate the nonparametric part
  9. Large-sample behaviour and optimality
  10. Large-sample theory for the parametric part
  11. Large-sample theory for the nonparametric part
  12. Assessing goodness of fit
  13. Goodness of fit processes
  14. Chi-squared tests
  15. Other tests
  16. ...and 9 more sections

Key Result

Proposition 2.1

Under standard regularity conditions, including those described in Assumptions assumptions:remark1, and supposing the model holds for a true parameter $\theta_0$, an inner point of the parameter space, $\sqrt{n}(\widehat{\theta}-\theta_0)$ tends to ${\rm N}_m(0,\Omega_0^{-1})$ in distribution.

Figures (5)

  • Figure 6.1: Histograms of $\sqrt{n}(\widehat{\theta}_k - \theta_k)/{\rm se}(\widehat{\theta}_k)$ and $\sqrt{n}\{\widehat{A}_j(t) - A_j(t)\}/{\rm se}(\widehat{A}_j(t))$ for $k = 1,2,3$, and $j = 3,4$. The cumulative regressors are evaluated at $t = 0.5$. The sample size was set to $n = 2000$, and the histograms are based on $200$ simulations. The green curves indicate the standard normal density.
  • Figure 6.2: Estimates of the cumulative regression functions in \ref{['eq::emp_model']}, fitted to the PBC-data set. The dashed lines indicate pointwise approximate $95$ percent confidence bands.
  • Figure 6.3: Estimated pointwise standard deviations of the estimators $\widetilde{A}_j(t)$ for $j =1,2,3$ of the Aalen model (in green), and $A_1(t,\widehat{\theta})$, $A_2(t,\widehat{\theta})$, and $\widehat{A}_3(t)$, of the partly parametric partly non-parametric model (in black).
  • Figure 6.4: The estimated survival curves corresponding to the estimated cumulative regression functions plotted in Figure \ref{['fig::empirical']}, for a non-treated individual with $\texttt{alb}_i$ equal to its mean. The dashed lines indicate approximate $95$ percent confidence bands.
  • Figure 6.5: The $R_{n,j}(t)$ functions of \ref{['eq:hereisRn']}, with weight functions $K_n(t) = 1$. The blue line shows $\sqrt{n}(\widetilde{A}_1(t) - \widehat{\theta}_1 t^{\widehat{\theta}_2})$, while the green line shows $\sqrt{n}(\widetilde{A}_2(t) - \widehat{\theta}_3 t)$.

Theorems & Definitions (8)

  • Proposition 2.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 5.1
  • proof