# RMS Parametric Survival Models

**URL:** <https://discourse.datamethods.org/t/rms-parametric-survival-models/28766>\
**Category:** models\
**Tags:** survival-analysis\
**Created:** [May 30, 2026, 1:05pm UTC](https://discourse.datamethods.org/t/rms-parametric-survival-models/28766 "2026-05-30T13:05:11Z")\
**Posts on this page:** 5\
**Page:** 1

<div class="post-metadata">

**Author:** ![f2harrell](https://discourse.datamethods.org/user_avatar/discourse.datamethods.org/f2harrell/32/165_2.png) [@f2harrell](https://discourse.datamethods.org/u/f2harrell)\
**Post date:** [May 30, 2026, 1:05pm UTC](https://discourse.datamethods.org/t/rms-parametric-survival-models/28766/1 "2026-05-30T13:05:12Z")

</div>

# Regression Modeling Strategies: Parametric Survival Models

This is the 18th of several connected topics organized around chapters in [Regression Modeling Strategies](https://hbiostat.org/rms). 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](https://hbiostat.org/rmsc/parsurv)

RMS18

---

<div class="post-metadata">

**Author:** ![hgomez](https://discourse.datamethods.org/letter_avatar_proxy/v4/letter/h/c77e96/32.png) [@hgomez](https://discourse.datamethods.org/u/hgomez)\
**Post date:** [May 30, 2026, 4:57pm UTC](https://discourse.datamethods.org/t/rms-parametric-survival-models/28766/2 "2026-05-30T16:57:20Z")

</div>

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?

---

<div class="post-metadata">

**Author:** ![f2harrell](https://discourse.datamethods.org/user_avatar/discourse.datamethods.org/f2harrell/32/165_2.png) [@f2harrell](https://discourse.datamethods.org/u/f2harrell)\
**Post date:** [May 30, 2026, 8:19pm UTC](https://discourse.datamethods.org/t/rms-parametric-survival-models/28766/4 "2026-05-30T20:19:55Z")

</div>

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)

---

<div class="post-metadata">

**Author:** ![trumanfrancis](https://discourse.datamethods.org/user_avatar/discourse.datamethods.org/trumanfrancis/32/47_2.png) [@trumanfrancis](https://discourse.datamethods.org/u/trumanfrancis)\
**Post date:** [May 31, 2026, 11:52am UTC](https://discourse.datamethods.org/t/rms-parametric-survival-models/28766/5 "2026-05-31T11:52:35Z")

</div>

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.

---

<div class="post-metadata">

**Author:** ![f2harrell](https://discourse.datamethods.org/user_avatar/discourse.datamethods.org/f2harrell/32/165_2.png) [@f2harrell](https://discourse.datamethods.org/u/f2harrell)\
**Post date:** [June 2, 2026, 7:13am UTC](https://discourse.datamethods.org/t/rms-parametric-survival-models/28766/6 "2026-06-02T07:13:54Z")

</div>

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.