Weibull Models as PH and AFT Models

One Weibull Model, Two Regression Interpretations

The Weibull distribution is unusual among common survival distributions because a regression model using it can be written both as a proportional-hazards (PH) model and as an accelerated-failure-time (AFT) model. These are not two separate fits that happen to agree: under a particular parameterization, the same conditional event-time distribution has both representations. The conversion matters because the two coefficients answer different questions. A PH coefficient describes a relative hazard; an AFT coefficient describes a relative event-time scale.

Let \(T\) be a positive event time and let \(x\) be a vector of fixed covariates. Use a Weibull model with common shape \(k>0\), baseline scale \(\lambda_0>0\), and PH coefficient vector \(\beta\):

S(t\mid x)=\exp\left[-\left(\frac{t}{\lambda_0}\right)^k\exp(\beta^\mathsf{T}x)\right],\qquad t>0.

Here \(S(t\mid x)=P(T>t\mid x)\) is the conditional survival function. The scale \(\lambda_0\) has units of time, while \(k\) is dimensionless. The covariates and coefficients are paired so that \(\beta^\mathsf{T}x\) is dimensionless. When \(x=0\), the survival distribution has scale \(\lambda_0\); it is not generally the median event time.

Definition: In the Weibull PH parameterization, the conditional cumulative hazard is \(H(t\mid x)=(t/\lambda_0)^k\exp(\beta^\mathsf{T}x)\), and the hazard is \(h(t\mid x)=k t^{k-1}\lambda_0^{-k}\exp(\beta^\mathsf{T}x)\). This is a PH model because, for two covariate vectors \(x_1\) and \(x_0\), the hazard ratio is constant in time: \(h(t\mid x_1)/h(t\mid x_0)=\exp\{\beta^\mathsf{T}(x_1-x_0)\}\).

Deriving the AFT Representation

To expose the time-scale interpretation, first rearrange the survival function. For a positive \(t\), take negative logarithms and then logarithms again:

-\log S(t\mid x)=\left(\frac{t}{\lambda_0}\right)^k\exp(\beta^\mathsf{T}x), \qquad \log\{-\log S(t\mid x)\}=k\log t-k\log\lambda_0+\beta^\mathsf{T}x.

Equivalently, the random event time can be generated by taking \(E\sim\operatorname{Exp}(1)\), an exponential random variable with mean 1. Equating \(E\) to the cumulative hazard at the event time gives \(E=(T/\lambda_0)^k\exp(\beta^\mathsf{T}x)\). Solving for \(T\) and taking logarithms yields:

\log T=\log\lambda_0-\frac{\beta^\mathsf{T}x}{k}+\frac{\log E}{k}.

Thus the AFT coefficient vector is \(\gamma=-\beta/k\), and the AFT error scale is \(\sigma=1/k\). To state the error distribution explicitly, write \(W=\log E\). Its cumulative distribution function is \(P(W\leq w)=1-\exp\{-\exp(w)\}\), so the AFT form is \(\log T=\log\lambda_0+\gamma^\mathsf{T}x+\sigma W\), with \(\gamma=-\beta/k\) and \(\sigma=1/k\). This specifies the Weibull AFT model without requiring a separate distributional assumption for the error.

Key result: For a Weibull model with common shape \(k\), PH and AFT coefficients are related by \(\gamma=-\beta/k\), or, in the reverse direction, \(\beta=-k\gamma\). For a covariate contrast \(x_1-x_0\), the hazard ratio is \(\exp\{\beta^\mathsf{T}(x_1-x_0)\}\), while the event-time acceleration factor is \(\exp\{\gamma^\mathsf{T}(x_1-x_0)\}\). The two factors are related by raising the hazard ratio to the power \(-1/k\).

The acceleration factor compares corresponding event-time quantiles. If it is 1.25, then each quantile for the covariate profile in the numerator is 1.25 times the corresponding quantile for the reference profile. This is a 25% increase in the modeled time scale, not a statement that every individual’s event time is deterministically multiplied by 1.25. A positive PH coefficient means a higher hazard and a shorter modeled time scale; its corresponding AFT coefficient is negative.

Worked Coefficient Conversions

Worked Example: Converting a PH Coefficient to a Time Ratio

Suppose a hypothetical Weibull model has shape \(k=2\), and a one-unit increase in a treatment indicator has PH coefficient \(\beta=0.80\). Assume the treatment and reference groups share the same shape, as required by this model. Find the hazard ratio and the AFT coefficient and time ratio.

The hazard ratio for a one-unit increase is:

\mathrm{HR}=\exp(0.80)=2.226.

The treatment group therefore has 2.226 times the modeled hazard of the reference group at each time, under the PH model. Convert the coefficient using \(\gamma=-\beta/k\):

\gamma=-\frac{0.80}{2}=-0.40,\qquad \mathrm{AF}=\exp(-0.40)=0.6703.

The modeled event-time scale, and hence each corresponding event-time quantile, is 0.6703 times as large for the treatment profile. That is a 32.97% shorter time scale. The interpretation is consistent: the positive PH coefficient corresponds to a hazard increase and a time-scale decrease.

Worked Example: Checking the Conversion With a Median

Consider another hypothetical model with baseline scale \(\lambda_0=10\) months, common shape \(k=2\), and binary covariate coefficient \(\beta=\log 2\). Calculate the reference and covariate-positive medians, then verify their ratio using the AFT form. The Weibull assumption is that both profiles share shape \(2\), with the covariate changing only the scale as specified.

For a survival probability of \(0.5\), solve \(S(t\mid x)=0.5\). The median is:

t_{0.5}(x)=\lambda_0 \left\{\frac{\log 2}{\exp(\beta x)}\right\}^{1/k}.

For the reference profile \(x=0\), the median is \(10\sqrt{\log 2}=10(0.8326)=8.326\) months. For \(x=1\), \(\exp(\beta x)=2\), so the median is \(10\sqrt{\log 2/2}=10(0.5887)=5.887\) months. The ratio is \(5.887/8.326=0.7071\).

The AFT conversion gives \(\gamma=-\log(2)/2=-0.3466\), so \(\exp(\gamma)=\exp(-0.3466)=0.7071\). The calculated medians and the AFT time ratio agree. This agreement holds for every corresponding quantile, not just the median, because the profiles share a common shape and differ by a scale factor.

Worked Example: Converting From AFT to PH

Suppose a hypothetical Weibull AFT model reports \(\gamma=0.25\) per unit increase in a continuous covariate and error scale \(\sigma=0.50\). The model assumes a common error scale across covariate values. Find the shape, the PH coefficient, and the hazard ratio for a one-unit increase.

The Weibull shape is the inverse of the AFT error scale:

k=\frac{1}{\sigma}=\frac{1}{0.50}=2.

Using \(\beta=-k\gamma\), the PH coefficient is \(\beta=-2(0.25)=-0.50\). Therefore, the one-unit hazard ratio is \(\exp(-0.50)=0.6065\). The AFT time ratio is \(\exp(0.25)=1.284\), meaning that the modeled event-time quantiles are 1.284 times as large per one-unit covariate increase. A lower hazard and a longer event-time scale are consistent descriptions of this association.

Assumptions, Diagnostics, and Failure Modes

The dual interpretation depends on a specific model structure, not merely on using software that labels a distribution “Weibull.” The conditional distribution must have a common shape \(k\) across covariate profiles, while covariates act through the PH linear predictor or, equivalently, the AFT location term. Covariates are treated as fixed in the formulas above. For inference with right-censored observations, censoring must also be noninformative conditional on the modeled covariates for the usual survival likelihood interpretation to apply.

One useful model check follows directly from the derivation: \(\log\{-\log S(t\mid x)\}\) is linear in \(\log t\), with slope \(k\) shared across profiles. In a population-level comparison, the transformed curves should be approximately straight and parallel. In practice, estimated curves can be noisy, especially with heavy censoring, so this is a diagnostic rather than a proof of the Weibull assumption. A visibly changing slope across time or markedly nonparallel profiles can signal that the common-shape Weibull model is inadequate.

The PH and AFT implications are linked: with this model, a covariate effect cannot be time-constant on the hazard scale but have a different, unrelated effect on quantile ratios. If the data show a time-varying hazard ratio, the standard Weibull PH representation is unsuitable. If profiles do not behave like scaled versions of one common event-time distribution, the standard Weibull AFT representation is also unsuitable. A model that allows covariate-specific shapes may describe some data better, but it generally loses these simple, simultaneous PH and AFT interpretations.

State the parameterization.
Confirm whether coefficients are reported as PH coefficients \(\beta\), or as AFT coefficients \(\gamma\), and identify the shape \(k\) or error scale \(\sigma\).

Check the common-shape condition.
For the standard dual interpretation, use one \(k\) across covariate profiles; in AFT notation this is one common \(\sigma=1/k\).

Convert and interpret in context.
Use \(\gamma=-\beta/k\) or \(\beta=-k\gamma\), then exponentiate the appropriate coefficient contrast to obtain a hazard ratio or time ratio.

How the Weibull Model Compares With Alternatives

The Cox proportional-hazards model also represents constant hazard ratios, but it does not require a Weibull baseline hazard and does not, in general, imply a common AFT time ratio. The Weibull model is more restrictive about the baseline hazard, but its parametric form supplies the AFT interpretation as well as the PH one. A general AFT model can describe time-scale effects using other event-time distributions; only the Weibull parameterization gives this exact PH correspondence. The exponential model is the special Weibull case \(k=1\), where the hazard is constant over time and the conversion simplifies to \(\gamma=-\beta\).

This flexibility of interpretation is not a guarantee of fit. The common shape constrains both how the baseline hazard changes over time and how covariate profiles relate. When that constraint is implausible, reporting both a hazard ratio and a time ratio from the same fitted Weibull model does not resolve the problem; both summaries inherit the same model assumptions.

Common Pitfalls

Using the wrong conversion. The relation is \(\gamma=-\beta/k\), not \(\gamma=-k\beta\). The shape is in the denominator when converting from PH to AFT.
Dropping the minus sign. A larger hazard corresponds to a shorter time scale. A positive PH coefficient must therefore translate to a negative AFT coefficient.
Confusing scale with median. In the reference group, \(\lambda_0\) is the Weibull scale; the median is \(\lambda_0(\log 2)^{1/k}\).
Assuming all Weibull regressions have both interpretations. The standard result requires a common shape. If shape varies by covariate profile, the simple coefficient conversion need not hold.
Calling an acceleration factor a hazard ratio. A time ratio compares event-time quantiles; a hazard ratio compares instantaneous event rates among those still event-free.

Key takeaway: With a common Weibull shape \(k\), PH and AFT are two parameterizations of the same survival model. Translate coefficients with \(\gamma=-\beta/k\), interpret exponentiated PH contrasts as hazard ratios and exponentiated AFT contrasts as time ratios, and check whether the shared-shape model is credible before relying on either summary.

Check Your Understanding

Answer each question, then compare your work with the solutions below.

A Weibull PH model has \(k=1.5\) and \(\beta=0.60\) for a one-unit covariate increase. Find the hazard ratio, AFT coefficient, and acceleration factor.
An AFT model has error scale \(\sigma=0.25\) and coefficient \(\gamma=-0.10\). Find \(k\), the PH coefficient, and the one-unit hazard ratio.
Why is the Weibull baseline scale \(\lambda_0\) not generally the baseline median? Give the median formula.
What does it imply if the PH coefficient is positive but the converted AFT coefficient is also positive under the stated convention?

Solutions. (1) \(\mathrm{HR}=\exp(0.60)=1.822\); \(\gamma=-0.60/1.5=-0.40\); and \(\mathrm{AF}=\exp(-0.40)=0.6703\). (2) \(k=1/0.25=4\); \(\beta=-k\gamma=-4(-0.10)=0.40\); and \(\mathrm{HR}=\exp(0.40)=1.492\). (3) The baseline median is \(\lambda_0(\log 2)^{1/k}\), which generally differs from \(\lambda_0\). (4) The signs conflict with the required relation \(\gamma=-\beta/k\) for \(k>0\); check the coefficient definitions, covariate contrast, and parameterization before interpreting the output.