Accelerated Failure-Time Models › Formula

Positive event times allow us to model $\log T$. The basic AFT model then assumes a linear location on this log scale and a common error distribution and scale across covariate profiles. The log transformation does not guarantee symmetry: normal errors are symmetric, while the minimum Gumbel errors used for Weibull lifetimes are not. A one-unit increase in a linearly entered covariate changes log time by its coefficient, giving a multiplicative factor on the time scale. Doubling a raw covariate is not generally a fixed one-unit change.

We begin with notation. For subject $i$, let $T_i>0$ be the true event time, let $C_i$ be the censoring time, and let

$$Y_i = \min(T_i, C_i)$$

be the observed time. Define the event indicator

$$\delta_i = \begin{cases} 1, & \text{if } T_i \leq C_i, \\ 0, & \text{if } T_i > C_i. \end{cases}$$

Thus $\delta_i=1$ means we observed the event, and $\delta_i=0$ means the observation was right-censored. Let $\mathbf{x}_i=(1,x_{i1},\ldots,x_{ip})^\top$ include an intercept and $p$ covariates, and let $\boldsymbol{\beta}=(\beta_0,\beta_1,\ldots,\beta_p)^\top$ be the corresponding coefficient vector.

What we ultimately want is the conditional distribution of $T_i$ given $\mathbf{x}_i$, but the observed data give only partial information: for events we see $T_i$, for censored subjects we only know that $T_i > y_i$. The rest of this section does three things in order. We express the covariate effect as a location shift on the log scale and quantify it as a time ratio. We assemble a likelihood that handles events and censored observations in the same expression. We examine the two distributional choices that determine the shape of the resulting survival curves.

The AFT model writes the log event time as

$$\log T_i = \beta_0 + \beta_1 x_{i1} + \cdots + \beta_p x_{ip} + \sigma W_i,$$

where $W_i$ is a standardized error term with a chosen distribution, and $\sigma>0$ is a scale parameter. In matrix notation,

$$\log T_i = \mathbf{x}_i^\top \boldsymbol{\beta} + \sigma W_i.$$

Equivalently,

$$T_i = \exp(\mathbf{x}_i^\top \boldsymbol{\beta}) \exp(\sigma W_i).$$

The two factors on the right have clearly separated roles. The first, $\exp(\mathbf{x}_i^\top \boldsymbol{\beta})$, depends only on covariates and shifts the entire distribution of event times. The second, $\exp(\sigma W_i)$, is subject-level noise that has the same distribution for every subject. This separation is why AFT coefficients are interpretable: all of the covariate effect is carried by a single location shift on the log scale, and everything else is the shape of the error distribution.

To see what the shift means on the time scale, compare two subjects with covariate vectors that differ only in the $j$-th component by one unit, and imagine that the individual noise $W_i$ is identical for both. Then

$$\frac{T_i(x_j+1)}{T_i(x_j)} = \frac{\exp(\mathbf{x}_i^\top \boldsymbol{\beta}+\beta_j)\exp(\sigma W_i)} {\exp(\mathbf{x}_i^\top \boldsymbol{\beta})\exp(\sigma W_i)} = \exp(\beta_j).$$

This ratio is the time ratio, also called the acceleration factor. If $\beta_j=0.2$, then $\exp(0.2)=1.2214$, so the event time is multiplied by about 1.22, or prolonged by 22.1 percent. If $\beta_j=-0.2$, then $\exp(-0.2)=0.8187$, so the event time is multiplied by about 0.82, or shortened by 18.1 percent. The same factor applies to every quantile of the event-time distribution, because the $p$-th quantile of $T_i$ depends on covariates only through the common factor $\exp(\mathbf{x}_i^\top \boldsymbol{\beta})$. That is the precise sense in which a covariate "stretches the clock."

For likelihood work it is more convenient to translate the model into a survival function. If $S_W(w)=P(W>w)$ is the survival function of the standardized error, then

$$S_T(t\mid \mathbf{x}) = P(T>t\mid \mathbf{x}) = P\left(W>\frac{\log t-\mathbf{x}^\top \boldsymbol{\beta}}{\sigma}\right) = S_W\left(\frac{\log t-\mathbf{x}^\top \boldsymbol{\beta}}{\sigma}\right).$$

Once $S_W$, $\boldsymbol{\beta}$, and $\sigma$ are fixed, the survival curve can be evaluated from the assumed error survival function. Cox regression can also produce survival curves after baseline estimation; the difference here is the explicit parametric event-time distribution. Different choices of $S_W$ give different tail shapes and different hazard patterns, which is the source of the modeling decision discussed later.

For a censored subject, the observed record does not reveal $T_i$; it gives a lower bound on that time. Evaluating the event density at the censoring time would invent an observed event. The right-censoring decomposition resolves this by assigning the two kinds of observation two structurally different likelihood pieces:

• If subject $i$ has an observed event at $y_i$, we have observed the value of $T_i$, so its contribution is the density $f_T(y_i\mid \mathbf{x}_i)$.
• If subject $i$ is censored at $y_i$, the only information available is that $T_i>y_i$, so its contribution is the probability of that event, the survival function $S_T(y_i\mid \mathbf{x}_i)$.

Assume independent subjects, conditionally independent censoring, and censoring-law parameters distinct from the event-time parameters. After removing factors that do not depend on the fitted parameters, the likelihood is the product:

$$L(\boldsymbol{\beta},\sigma) = \prod_{i=1}^{n} f_T(y_i\mid \mathbf{x}_i)^{\delta_i} S_T(y_i\mid \mathbf{x}_i)^{1-\delta_i}.$$

Taking logarithms turns the product into a sum:

$$\ell(\boldsymbol{\beta},\sigma) = \sum_{i=1}^{n} \left[ \delta_i \log f_T(y_i\mid \mathbf{x}_i) + (1-\delta_i)\log S_T(y_i\mid \mathbf{x}_i) \right].$$

To express each piece in terms of the standardized error, standardize each observation:

$$w_i = \frac{\log y_i-\mathbf{x}_i^\top \boldsymbol{\beta}}{\sigma}.$$

Because $T=\exp(\mathbf{x}^\top \boldsymbol{\beta}+\sigma W)$, the transformation from $W$ to $T$ has Jacobian

$$\frac{dt}{dw} = \sigma \exp(\mathbf{x}^\top \boldsymbol{\beta}+\sigma w) = \sigma t.$$

Therefore the density of $T$ is

$$f_T(t\mid \mathbf{x}) = \frac{f_W(w)}{\sigma t},$$

where $f_W$ is the density of the standardized error. Taking logs gives

$$\log f_T(t\mid \mathbf{x}) = -\log t-\log \sigma+\log f_W(w).$$

The Jacobian contributes $-\log t-\log\sigma$ to an event log density. For fixed observed times, the $-\log t$ term does not depend on the fitted parameters: omitting that term alone changes absolute time-scale likelihood values, but not the maximizer. It also cancels from AIC differences when both models omit the same term on the same event records. The $-\log\sigma$ term does depend on a fitted parameter and must remain in the objective. Retaining the complete density avoids mismatched scoring conventions. The survival function transforms without a Jacobian, because it is a probability:

$$S_T(t\mid \mathbf{x})=S_W(w).$$

Substituting these expressions into the log-likelihood, we obtain the general AFT censored log-likelihood:

$$\ell(\boldsymbol{\beta},\sigma) = \sum_{i=1}^{n} \left[ \delta_i\left(-\log y_i-\log \sigma+\log f_W(w_i)\right) + (1-\delta_i)\log S_W(w_i) \right].$$

Two features of this expression deserve emphasis. First, the two halves of the summand are structurally different: an event contributes a density with its Jacobian, while a censored observation contributes a survival probability. Each subject activates exactly one of the two. Second, if no observation is censored, the second term vanishes. The remaining likelihood gives the same parameter estimates as ordinary log-location-scale regression on $\log T_i$, differing from its log-time likelihood by the parameter-free sum $-\sum_i\log y_i$. Every parametric AFT model, including Weibull, log-normal, and log-logistic, is expressed simply by plugging in the corresponding $f_W$ and $S_W$ in this formula.

The general likelihood is now in hand; the remaining decision is what to plug in for $f_W$ and $S_W$. We compare two error distributions:

• The standard normal error is symmetric and produces a log-normal distribution for $T$. Its hazard can rise and then fall, which allows a non-monotone risk pattern.
• The minimum Gumbel error is less familiar but produces the Weibull distribution for $T$, which has a special algebraic connection to proportional hazards. That connection is what allows Weibull AFT coefficients to be translated into hazard ratios.

The distributions can give similar fits in part of the observed range while implying different tails. Their agreement in the middle is not guaranteed.

The survival curve comparison can be read alongside the corresponding hazard functions. Holding log-time location and scale fixed, the log-normal hazard is not forced to be monotone, while the Weibull hazard is monotone in this parameterization. The following curves use the same location and scale for both families, so the difference in shape comes from the error distribution rather than from a changed regression coefficient.

For the log-normal AFT model, $W\sim N(0,1)$. Then

$$S_T(t\mid \mathbf{x}) = 1-\Phi\left(\frac{\log t-\mathbf{x}^\top \boldsymbol{\beta}}{\sigma}\right) = \Phi\left(\frac{\mathbf{x}^\top \boldsymbol{\beta}-\log t}{\sigma}\right),$$

where:

• $\Phi$: the standard normal cumulative distribution function
• $t$: the event time
• $\mathbf{x}^\top \boldsymbol{\beta}$: the log-time location for the covariate profile
• $\sigma$: the log-time scale parameter

The density is

$$f_T(t\mid \mathbf{x}) = \frac{1}{t\sigma}\phi\left(\frac{\log t-\mathbf{x}^\top \boldsymbol{\beta}}{\sigma}\right),$$

where $\phi$ is the standard normal density. The hazard is the ratio

$$h_T(t\mid \mathbf{x}) = \frac{f_T(t\mid \mathbf{x})}{S_T(t\mid \mathbf{x})} = \frac{\phi(w)}{\sigma t\,\Phi(-w)}.$$

Here $w=(\log t-\mathbf{x}^\top\boldsymbol{\beta})/\sigma$. The log-normal hazard can rise and later fall; it tends to zero both as $t\downarrow0$ and as $t\to\infty$. This is different from a Weibull hazard, whose direction is controlled by one shape parameter.

For the Weibull AFT model, $W$ follows the standard minimum Gumbel distribution, whose survival function is

$$S_W(w)=\exp(-e^w),$$

where:

• $w$: the standardized log-time value $(\log t-\mathbf{x}^\top \boldsymbol{\beta})/\sigma$
• $e^w$: the exponential of $w$, which is always positive
• $\exp(-e^w)$: the Gumbel survival probability, decreasing from 1 toward 0 as $w$ increases

Substituting $w=(\log t-\mathbf{x}^\top \boldsymbol{\beta})/\sigma$ gives

$$S_T(t\mid \mathbf{x}) = \exp\left[-\exp\left(\frac{\log t-\mathbf{x}^\top \boldsymbol{\beta}}{\sigma}\right)\right].$$

If we set the shape parameter $k=1/\sigma$ and the scale parameter $\lambda(\mathbf{x})=\exp(\mathbf{x}^\top \boldsymbol{\beta})$, this becomes the usual Weibull survival function

$$S_T(t\mid \mathbf{x}) = \exp\left[-\left(\frac{t}{\lambda(\mathbf{x})}\right)^k\right].$$

The density is

$$f_T(t\mid \mathbf{x}) = \frac{k}{\lambda(\mathbf{x})} \left(\frac{t}{\lambda(\mathbf{x})}\right)^{k-1} \exp\left[-\left(\frac{t}{\lambda(\mathbf{x})}\right)^k\right].$$

To derive the hazard, we use the definition $h_T(t)=f_T(t)/S_T(t)$. Differentiating the survival function with the chain rule,

$$\frac{d}{dt}S_T(t\mid \mathbf{x}) = -S_T(t\mid \mathbf{x}) \exp\left(\frac{\log t-\mathbf{x}^\top \boldsymbol{\beta}}{\sigma}\right) \frac{1}{\sigma t}.$$

Since $f_T(t)=-dS_T(t)/dt$, dividing by $S_T(t)$ gives

$$h_T(t\mid \mathbf{x}) = \frac{1}{\sigma t} \exp\left(\frac{\log t-\mathbf{x}^\top \boldsymbol{\beta}}{\sigma}\right).$$

This factorization is why the Gumbel error is useful here. Write the hazard as

$$h_T(t\mid \mathbf{x}) = \left[ \frac{1}{\sigma t} \exp\left(\frac{\log t-\beta_0}{\sigma}\right) \right] \exp\left(-\frac{\sum_{j=1}^{p}x_j\beta_j}{\sigma}\right).$$

The sum in the second factor uses only the $p$ slope terms, excluding the intercept. The term in brackets depends only on time and the baseline parameters, and the remaining factor is constant over time. That is the proportional hazards structure, with hazard ratio

$$\text{HR}_j=\exp\left(-\frac{\beta_j}{\sigma}\right)$$

for a one-unit increase in covariate $j$. The sign flips relative to the AFT coefficient because a positive $\beta_j$ lengthens event times and therefore lowers the hazard. Among the two families studied here, the Weibull model has this proportional-hazards conversion; a log-normal AFT model does not generally have proportional hazards, because its hazard ratio changes with time. That distinction is the practical meaning of choosing one error distribution over the other.

The acceleration interpretation can be made exact through the quantile function. If $q_p$ is the $p$-th quantile of $W$, then the $p$-th quantile of $T$ given covariates is

$$t_p(\mathbf{x}) = \exp\left(\mathbf{x}^\top \boldsymbol{\beta}+\sigma q_p\right).$$

For a one-unit change in $x_j$, the ratio of quantiles is $\exp(\beta_j)$, independent of $p$. This is the constant time-ratio property, and it is what justifies calling $\exp(\beta_j)$ an acceleration factor rather than just a coefficient on the log scale.

The median depends on which $W$ we choose. For the log-normal model, the median is $\exp(\mathbf{x}^\top \boldsymbol{\beta})$ because the median of a standard normal is zero. For the Weibull model, the median is

$$\exp\left(\mathbf{x}^\top \boldsymbol{\beta}+\sigma \log(\log 2)\right),$$

because the minimum Gumbel median is $\log(\log2)\approx-0.3665$. Solving $\exp(-e^w)=1/2$ gives $e^w=\log2$, so the median shift is negative when multiplied by positive $\sigma$. The two models can therefore share the same $\mathbf{x}^\top \boldsymbol{\beta}$ and $\sigma$ while having different medians and different tails.

Finally, fitting requires numerical care. The log-likelihood is not guaranteed to be globally concave in all parameters. For example, joint concavity can fail in the log-normal location and log-scale parameterization even without censoring. In practice we optimize numerically, parameterize $\sigma$ as $\exp(\gamma)$ so that it stays positive, and use good starting values or multiple starts. Each parameterization needs its own existence and convergence checks; multiple starts do not establish a global optimum.