Calculation of Mortality Probability from the AFT Model
Unlike proportional hazard models that multiply the risk, the Accelerated Failure Time (AFT) model acts multiplicatively on time itself. On Cross Validated, this is often described as the "dog year" effect: if a covariate has an acceleration factor of 7, one year of chronological time for a subject is equivalent to seven years of "baseline" time.
1. The Log-Linear Foundation
The AFT model is typically expressed in a log-linear form, making it look similar to a standard linear regression:
$$\log(T) = \mu + \beta_1 X_1 + \beta_2 X_2 + \dots + \sigma \epsilon$$Where:
- $T$: The random variable representing time to death.
- $\mu$: The intercept (baseline log-time).
- $\beta$: Regression coefficients.
- $\sigma$: A scale parameter related to the distribution's variance.
- $\epsilon$: The error term following a specific distribution (e.g., Gumbel for Weibull, Normal for Log-Normal).
2. Calculating the Survival Probability $S(t)$
To find the probability of mortality, we first calculate the Survival Probability ($S(t)$), which is the probability of being alive at time $t$. The general formula for a subject with covariates $X$ is:
$$S(t|X) = S_0\left( \frac{t}{\exp(\mu + \sum \beta X)} \right)$$Where $S_0$ is the survival function of the baseline distribution. The term $\exp(\mu + \sum \beta X)$ is the Acceleration Factor (AF). If $AF > 1$, time is "stretched," and survival is prolonged.
3. From Survival to Mortality Probability
The Mortality Probability (also known as the Cumulative Distribution Function or $F(t)$) is simply the complement of the survival function:
$$P(\text{Death} \leq t) = F(t) = 1 - S(t)$$Example: The Log-Normal AFT Model
If we assume a Log-Normal distribution for $T$, the mortality probability is calculated using the standard normal CDF ($\Phi$):
$$P(\text{Death} \leq t) = \Phi\left( \frac{\log(t) - (\mu + \beta X)}{\sigma} \right)$$4. Comparison of Distributional Forms
| Distribution | Error Term ($\epsilon$) | Mortality Probability $F(t)$ |
|---|---|---|
| Weibull | Gumbel (Smallest Extreme Value) | $1 - \exp(-\exp(\frac{\log(t) - \eta}{\sigma}))$ |
| Log-Normal | Normal (Gaussian) | $\Phi(\frac{\log(t) - \eta}{\sigma})$ |
| Log-Logistic | Logistic | $\frac{1}{1 + \exp(-\frac{\log(t) - \eta}{\sigma})}$ |
Note: $\eta = \mu + \sum \beta X$ represents the linear predictor.
5. Implementation in R (using survival package)
In practice, "Super Users" calculate these probabilities by extracting the linear predictor and scale from a model fit:
- Fit the model:
fit <- survreg(Surv(time, status) ~ age + treatment, dist="weibull") - Predict linear predictors:
lp <- predict(fit, type="linear") - Calculate $S(t)$:
1 - pweibull(t, shape=1/fit$scale, scale=exp(lp))
Conclusion
Calculating mortality probability from an AFT model provides a more intuitive "time-based" perspective than the "risk-based" perspective of Cox models. By 2026 standards, AFT models are increasingly preferred in clinical trials because they allow researchers to state exactly how many months or years of life are gained or lost due to a specific treatment. Understanding the link between the log-linear predictor and the baseline CDF is the key to accurate mortality forecasting.
Keywords
calculate mortality probability AFT model, survival analysis acceleration factor, log-normal AFT formula, Weibull AFT mortality risk, Cross Validated survival analysis, survival probability calculation 2026, accelerated failure time probability density.
