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Lesson 5 of 7

Parametric and AFT Models

For four lessons, every model Dr. Rao fit has refused to say one thing. Kaplan-Meier drew a staircase but stopped dead at the last observed death. Cox handed her a hazard ratio but shrugged when she asked what the baseline hazard actually looked like. Both were deliberately silent about the shape of survival time, and that silence has a cost: neither can tell a patient "you have about 26 months," and neither can say anything at all about the world past the end of the study.

This lesson breaks the silence. You will commit to a shape for the survival time itself, a smooth curve with a formula, and in return you get to predict real survival times, read a treatment's effect as "the drug multiplies your time by 1.65," and extend the survival curve confidently past the last follow-up.

By the end of this lesson you will be able to:

  • Fit exponential and Weibull survival models with survreg, and read the Weibull shape to say whether risk rises, falls, or holds steady over time
  • Interpret an accelerated failure time (AFT) coefficient as a time ratio that stretches the whole survival curve by one constant factor
  • Predict a median survival time and a survival probability beyond the follow-up window, and choose a distribution (and know when a parametric fit beats Cox, and when it is risky)

Prerequisites: Lesson 1 (the survival function \(S(t)\), the hazard \(h(t)\), right-censoring, Surv()), Lesson 2 (the empirical staircase and its median), Lesson 3 (the hazard ratio \(e^{\beta}\), and that Cox never names the baseline hazard), and Lesson 4 (the proportional-hazards promise \(S_1=S_0^{\text{HR}}\)). You can run R and read a coefficient table.

Two questions Cox cannot answer

Where the Cox model stops

Dr. Rao's new drug works. Lesson 3 proved it: her Cox model put the hazard ratio well below 1, protective even after adjusting for age. But two clinical questions keep coming back, and the Cox model cannot answer either one.

First, a patient wants a number, not a ratio. "Doctor, how long do I have?" A hazard ratio of 0.47 says nothing about months. To answer, you need an actual survival time, and for that you need the whole survival curve, not just a comparison of two hazards.

Second, the trial closed at 36 months, but people live longer than that. Roughly what fraction of drug patients are still alive at 5 years? Cox's baseline survival is a nonparametric staircase that simply ends at the last observed event; ask it about month 60 and it has nothing to say. There is no step out there to read.

Both gaps have the same fix: instead of leaving the survival time's shape unspecified, we will assume it follows a specific distribution. First, meet the data. We build Dr. Rao's 400-patient trial right here, because each lesson runs in its own fresh R session:

RInteractive R
library(survival) set.seed(2024) n <- 400 trt <- rbinom(n, 1, 0.5) # 1 = new drug, 0 = standard care age <- round(rnorm(n, 60, 9)) # the truth we simulate: the drug and youth MULTIPLY survival time scale_i <- exp(3.0 + 0.5 * trt - 0.02 * (age - 60)) t_event <- rweibull(n, shape = 1.5, scale = scale_i) # each patient's true event time # follow-up: the study closes at 36 months, with some earlier dropout dropout <- runif(n, 8, 60) seen <- pmin(dropout, 36) # last month we could still observe this patient time <- pmin(t_event, seen) # what we record: event or last contact status <- as.integer(t_event <= seen) # 1 = died, 0 = censored (still alive) trial <- data.frame( time = round(time, 1), status = status, arm = factor(ifelse(trt == 1, "drug", "standard"), levels = c("standard", "drug")), age = age ) head(trial, 5) #> time status arm age #> 1 30.2 0 drug 60 #> 2 36.0 0 standard 64 #> 3 8.4 1 drug 53 #> 4 16.0 1 drug 62 #> 5 7.8 1 standard 69 table(trial$arm, trial$status) # censored (0) and deaths (1) per arm #> #> 0 1 #> standard 46 152 #> drug 99 103

  

  





Read a couple of rows so the table is concrete. Row 3 is a 53-year-old drug patient who died at month 8.4 (status 1). Row 2 is a 64-year-old on standard care who was still alive when the study closed at month 36, so we censor them (status 0): we know only that they lived at least 36 months. The table shows the drug arm is much more heavily censored (99 of 202 still alive at the end) precisely because those patients survive longer, exactly the good problem a parametric model helps us describe.



Key Insight
Kaplan-Meier and Cox are nonparametric about the baseline: they never write down a formula for how survival time is distributed. That makes them robust, but it is also why they cannot extrapolate or report a survival time. A parametric model trades a distributional assumption for the power to do both.