AFT model: by which factor does the time to the event change? (time ratio)

In brief

The AFT model (accelerated failure time) says by which factor the time to the event changes: "twice as long" instead of "half the risk". It is in the card "How long does it take until an event?" under "More options", next to the Cox regression, with the same columns. The time ratio is exp(b) with a Wald interval; the output also shows the scale σ, the log-likelihood and the AIC of all four distributions (Weibull, exponential, log-normal, log-logistic). The default is Weibull.

Trap 1: the sign

A positive b in the AFT model means a longer time (time ratio above 1), in the Cox regression a higher risk (hazard ratio above 1). In the example lung (227 complete cases, 164 events) women have b = +0.401 in the AFT model and the time ratio 1.49 (95% interval [1.17, 1.90], p = .001): the survival time is about 1.5 times as long. In the Cox regression the same women have b = −0.553 and the hazard ratio 0.575. Same statement, opposite sign.

More traps

• The time ratio is not a hazard. For a desirable event (recovery) a time ratio above 1 is bad, because it takes longer.
• The times must be greater than 0, because the model works with the logarithm of the time. The Cox regression also accepts 0. Statable says so in one sentence before calculating.
• R does not report perfect separation (a group without events) in the AFT model: it returns numbers (in the test σ ≈ 0.41 and b ≈ 9.8) that mean nothing. Statable checks beforehand for a group without events and warns.
• The distribution cannot be "tested". The AIC gives an order: for lung Weibull 2274.9, log-logistic 2285.0, exponential 2295.1, log-normal 2303.8 (smaller = better fit). The person chooses the distribution according to the subject matter; the AIC is not a test.
• A characteristic in small units, for example age in years, gives a time ratio near 1 (0.993 per year). Better read it per 10 years: about 0.93.
• Few events: the rule of thumb of at least 10 events per coefficient applies here too. There are no time-dependent covariates, no left truncation and no interval censoring.

Weibull and hazard ratio

Only the Weibull model is also a proportional hazards model. There HR = exp(−b/σ); Statable shows it in a table. For lung: age 1.010, woman 0.578, ECOG 1.591 (Cox: 1.011, 0.575, 1.590). The interval of the hazard ratio treats σ as fixed and is therefore an approximation. The exponential model is the Weibull model with σ = 1.

Reporting the result

AFT model (Weibull): TR = 1.49, 95% CI [1.17, 1.90], p = .001 (sex); σ = 0.73; AIC = 2274.9; 227 people, 164 events

For specialists

survival::survreg with dist = "weibull", "exponential", "lognormal" or "loglogistic"; time ratio exp(b), Wald interval with qnorm(.975); AIC = −2·logLik + 2·(number of coefficients + scale).

library(survival)
m <- survreg(Surv(time, status == 2) ~ age + sex + ph.ecog, data = lung, dist = "weibull")
summary(m)   # exp(Koeffizient) = Zeitverhältnis
AIC(m)