Question

1. Our main concrete example of a proportional hazards regression model is Weibull regression. (a) What is the baseline hazard function for Weibull regression? Assume eo is part of the baseline hazard function (b) Suppose that the Weibul regression mode is the true model for a set of data. When we fit a proportional hazards regression model by maximum partial likeli- hood and estimate B1, what function of the Weibull regression model parameters are we estimating?

Answer 1

Weibull regression model is one of the most popular forms of parametric regression model that it provides estimate of baseline hazard function, as well as coefficients for covariates. Because of technical difficulties, Weibull regression model is seldom used in medical literature as compared to the semi-parametric proportional hazard model. To make clinical investigators familiar with Weibull regression model, this article introduces some basic knowledge on Weibull regression model and then illustrates how to fit the model with R software. The SurvRegCensCov package is useful in converting estimated coefficients to clinical relevant statistics such as hazard ratio (HR) and event time ratio (ETR). Model adequacy can be assessed by inspecting Kaplan-Meier curves stratified by categorical variable. The eha package provides an alternative method to model Weibull regression model. The check.dist() function helps to assess goodness-of-fit of the model. Variable selection is based on the importance of a covariate, which can be tested using anova() function. Alternatively, backward elimination starting from a full model is an efficient way for model development. Visualization of Weibull regression model after model development is interesting that it provides another way to report your findings.

(a)

Before exploring R for Weibull model fit, we first need to review the basic structure of the Weibull regression model. The distribution of time to event, T, as a function of single covariate is written as (1):

In(T) = β₀ + β₁x + σε

where β₁ is the coefficient for corresponding covariate, ε follows extreme minimum value distribution G(0, σ)and σ is the shape parameter. This is also called the accelerated failure-time model because the effect of the covariate is multiplicative on time scale and it is said to “accelerate” survival time. In contrast, the effect of covariate is multiplicative on hazard scale in the proportional hazard model. The hazard function of Weibull regression model in proportional hazards form is:

h(t, x, β, λ) = λt^λ−1e^{−1(β₀+β₁x)} = λt^λ−1e^−λβ₀e^−λβ₁x = λγt^λ−1e^−λβ₁x = h₀(t)e^θ₁x

where γ=e−β0σ=eθ0, θ₁ = −β₁/σ, and the baseline hazard function is h₀(t) = λγt^λ−1. σ is a variance-like parameter on log-time scale. γ = 1/σ is usually called a scale parameter. Parameter λ is a shape parameter. Parameter θ₁ has a hazard ratio (HR) interpretation for subject-matter audience.

The accelerated failure-time form of the hazard function can be written as:

h(t, x, β, λ) = λt^λ−1e^{−λ(β₀+β₁x)} = λγ(te^−β₁x)^λ−1e^−β₁x

Weibull regression model can be written in both accelerated and proportional forms, allowing for simultaneous description of treatment effect in terms of HR and relative change in survival time [event time ratio (ETR)]