RMS Parametric Survival Models

data analysis models
1

Regression Modeling Strategies: Parametric Survival Models

This is the 18th of several connected topics organized around chapters in Regression Modeling Strategies. The purposes of these topics are to introduce key concepts in the chapter and to provide a place for questions, answers, and discussion around the chapter’s topics.

Overview | Course Notes

RMS18

2

I have a basic question. On the book section 18.1.2 an example is given for the exponential distribution. The expected life length \mu is 1/\lambda and the MLE of \mu is then w/n_u. However, I have a problem understanding is why the estimated variance of \mu is w^2/n_u^3?

4

Here’s Claude AI’s answer.

Variance of MLE of Mean Survival Time (Exponential with Right Censoring)

Setup

Let the data be t_1, \ldots, t_n with censoring indicators \delta_i \in \{0,1\} (1 = event). Define:

  • d = \sum_i \delta_i — total number of events
  • T = \sum_i t_i — total exposure time (events + censored)

MLE of the rate parameter

The log-likelihood is \ell(\lambda) = d \log\lambda - \lambda T, giving:

\hat\lambda = \frac{d}{T}

Variance of \hat\lambda

The Fisher information is \mathcal{I}(\lambda) = d/\lambda^2, so:

\widehat{\text{Var}}(\hat\lambda) = \frac{\hat\lambda^2}{d} = \frac{d}{T^2}

Variance of \hat\mu = 1/\hat\lambda

By the delta method, with g(\lambda) = 1/\lambda and g'(\lambda) = -1/\lambda^2:

\widehat{\text{Var}}(\hat\mu) = \frac{\hat\mu^2}{d} = \frac{T^2}{d^3}

Key insight

Precision depends only on the number of events d, not sample size or censoring pattern.
The coefficient of variation of \hat\mu is 1/\sqrt{d}.

A 95% CI is best formed on the log scale:

\exp\!\left(\log\hat\mu \pm \frac{1.96}{\sqrt{d}}\right)
1 Like
5

How do you deal with the challenges of the proportional hazards assumption that Hernan raised? The HR is a conditional-on-survival quantity that selects on the outcome over time, by time t, the surviving treated and control populations are no longer comparable even under baseline randomization. (b) The HR depends on follow-up duration; the same trial reported at 1 year, 3 years, and 5 years will produce different HRs even when the underlying biological effect is unchanged. Thank you.

6

Under proportional hazards, there is a 1-1 mapping between the HR and difference in cumulative incidence so HR is a fully intent-to-treat quantity. When PH is not satisfied you can include treatment \times \log(t) interaction and base inference on any intent-to-treat transformation of those parameters and the underlying survival curve, i.e., cumulative incidence or restricted mean survival time.

1 Like