Weibull Models As Proportional Hazards And AFT

One Weibull Model, Two Interpretations

A Weibull duration model can describe how covariates change event risk over time, or how they stretch and shrink the event-time scale. These are not two unrelated models: when the Weibull shape parameter is common across covariate values, they are two parameterizations of the same distribution. The translation matters because a positive coefficient has opposite directional interpretations in the two forms: a higher hazard means a shorter event time, while a larger time scale means longer event times.

Let $T$ be a positive event time and let $x$ be a vector of covariates. We use a Weibull model with shape $k>0$, scale $\lambda_0>0$, and covariate vector $\gamma$. The reference covariate value is $x=0$; the scale $\lambda_0$ has the same units as time, while $k$ is dimensionless. The central condition for the equivalence developed here is that $k$ does not vary with $x$.

Definition: The Weibull accelerated-failure-time (AFT) parameterization is $$ S(t\mid x)=\exp\left\{-\left(\frac{t}{\lambda_0\exp(x'\gamma)}\right)^k\right\},\qquad t>0. $$ Here $S(t\mid x)=P(T>t\mid x)$ is the conditional survival function. The conditional Weibull scale is $\lambda(x)=\lambda_0\exp(x'\gamma)$. The common-shape condition means the same $k$ applies for every $x$.

The scale $\lambda(x)$ is a time-scale parameter, not generally the mean or median. For example, the survival probability at $t=\lambda(x)$ is $\exp(-1)$, regardless of $k$. The AFT interpretation follows directly from the scale: changing $x$ multiplies the time scale by $\exp(x'\gamma)$. In particular, increasing a single covariate by one unit while holding the others fixed multiplies the scale by $\exp(\gamma_j)$.

Deriving the Proportional-Hazards Form

The hazard is the instantaneous event rate conditional on having survived up to time $t$. For a continuous event time with density $f(t\mid x)$, it is $h(t\mid x)=f(t\mid x)/S(t\mid x)$. Equivalently, if $H(t\mid x)=-\log S(t\mid x)$ is the cumulative hazard, then $h(t\mid x)$ is its derivative with respect to $t$.

Starting from the Weibull survival function, the cumulative hazard is

H(t\mid x)=-\log S(t\mid x) =\left(\frac{t}{\lambda_0\exp(x'\gamma)}\right)^k =\left(\frac{t}{\lambda_0}\right)^k\exp(-k x'\gamma).

Define the reference-group cumulative hazard as $H_0(t)=(t/\lambda_0)^k$. Comparing the last expression with the proportional-hazards form $H(t\mid x)=H_0(t)\exp(x'\beta)$ gives the coefficient translation below.

Theorem: A Weibull model with a common shape $k$ is both an AFT model and a proportional-hazards model. Its coefficients satisfy $$ \beta=-k\gamma,\qquad \gamma=-\frac{\beta}{k}. $$ The baseline cumulative hazard and hazard are $$ H_0(t)=\left(\frac{t}{\lambda_0}\right)^k,\qquad h_0(t)=\frac{k}{\lambda_0}\left(\frac{t}{\lambda_0}\right)^{k-1}. $$ Thus $H(t\mid x)=H_0(t)\exp(x'\beta)$ and $h(t\mid x)=h_0(t)\exp(x'\beta)$.

The hazard expression follows by differentiating $H(t\mid x)=H_0(t)\exp(x'\beta)$ with respect to $t$; the covariate term is constant in $t$. Consequently, the hazard ratio for two covariate values $x_a$ and $x_b$, at the same time $t$, is $\exp((x_a-x_b)'\beta)$. This ratio does not depend on $t$, which is the proportional-hazards property.

Start with the Weibull cumulative hazard.
From the survival function, $H(t\mid x)=(t/\lambda_0)^k\exp(-k x'\gamma)$.

Separate baseline time dependence from covariates.
Set $H_0(t)=(t/\lambda_0)^k$, leaving the covariate multiplier $\exp(-k x'\gamma)$.

Match the proportional-hazards coefficient.
Since $\exp(-k x'\gamma)=\exp(x'\beta)$ for every $x$, the coefficients obey $\beta=-k\gamma$.

This derivation also explains the sign change. If $\gamma_j>0$, a one-unit increase in covariate $x_j$ lengthens the time scale by $\exp(\gamma_j)>1$; the corresponding hazard coefficient is $\beta_j=-k\gamma_j<0$, so the hazard ratio is below one. Conversely, $\beta_j>0$ corresponds to a shorter time scale.

Translating Effects and Quantiles

For a one-unit increase in $x_j$, holding the remaining covariates fixed, the AFT time ratio is $\exp(\gamma_j)$. It multiplies the model’s event-time quantiles by that factor. In the proportional-hazards form, the same increase has hazard ratio $\exp(\beta_j)=\exp(-k\gamma_j)$. These are different summaries: a time ratio is not the reciprocal of the hazard ratio except in special parameter settings.

The Weibull quantile at event probability $q$, where $0<q<1$, is obtained by solving $1-S(t\mid x)=q$. Therefore,

t_q(x)=\lambda_0\exp(x'\gamma)\left[-\log(1-q)\right]^{1/k}.

Every such quantile is multiplied by $\exp(\gamma_j)$ for a one-unit increase in $x_j$. For the median, set $q=0.5$, giving $t_{0.5}(x)=\lambda_0\exp(x'\gamma)(\log 2)^{1/k}$. This is useful for translating model parameters into an event-time summary with the original time units.

Worked Example: Translating a Coefficient and Comparing Medians

Suppose a hypothetical duration model has shape $k=1.5$, reference scale $\lambda_0=12$ months, and one covariate $x$ with AFT coefficient $\gamma=0.20$. Compare $x=0$ with $x=1$.

Coefficient translation. The proportional-hazards coefficient is $\beta=-k\gamma=-(1.5)(0.20)=-0.30$. Thus the hazard ratio for $x=1$ versus $x=0$ is $\exp(-0.30)=0.7408$. Under this model, the covariate is associated with a hazard about 25.92% lower at any given time.

Time scale and medians. The scale at $x=1$ is $12\exp(0.20)=14.6568$ months. The median factor is $(\log 2)^{1/1.5}\approx0.783219$, so the reference median is $12(0.783219)\approx9.399$ months. At $x=1$, it is $14.6568(0.783219)\approx11.48$ months.

The ratio of the medians is approximately $11.48/9.399\approx1.221$, matching the AFT time ratio $\exp(0.20)=1.2214$ up to rounding. The result describes a model-based association; it does not by itself establish a causal effect.

Worked Example: Converting a Group Time Ratio to a Hazard Ratio

Consider a hypothetical comparison of two groups with common shape $k=2.5$. Let the reference group have scale $20$ days, and suppose the comparison group’s scale is $20\exp(-0.162)$ days. The group indicator has AFT coefficient $\gamma=-0.162$.

Scales and time ratio. The comparison scale is $20\exp(-0.162)\approx20(0.85044)=17.0088$ days. Its time ratio relative to the reference group is $17.0088/20\approx0.85044$. Thus its model-based event-time quantiles are about 14.96% shorter.

Hazard ratio. The corresponding coefficient is $\beta=-k\gamma=-(2.5)(-0.162)=0.405$. Therefore, the comparison-to-reference hazard ratio is $\exp(0.405)\approx1.499$, constant over time under the model.

As a check using medians, $(\log 2)^{1/2.5}\approx0.86364$. The reference median is $20(0.86364)\approx17.27$ days, while the comparison median is $17.0088(0.86364)\approx14.69$ days. Their ratio is approximately $14.69/17.27\approx0.8504$, consistent with the time ratio. A higher hazard ratio and a time ratio below one point in opposite directions, as expected.

Survival Probabilities and the Shape Condition

The equivalence also allows direct calculation of conditional survival probabilities. For a covariate change, the proportional-hazards representation multiplies the cumulative hazard, while the AFT representation rescales time. Both yield the same survival probability when their parameters are translated consistently.

Worked Example: Checking Survival and Hazard at a Fixed Time

Suppose a hypothetical duration has $k=2$, reference scale $\lambda_0=10$ days, and a binary covariate with AFT coefficient $\gamma=-0.15$. Compare the reference value $x=0$ and the group $x=1$ at $t=10$ days.

The translated coefficient is $\beta=-2(-0.15)=0.30$, so the group hazard ratio is $\exp(0.30)=1.3499$. The reference cumulative hazard at 10 days is $H_0(10)=(10/10)^2=1$. Hence the group cumulative hazard is $1(1.3499)=1.3499$.

The reference survival probability is $\exp(-1)=0.3679$; the group survival probability is $\exp(-1.3499)\approx0.2592$. Directly from the AFT form, the group scale is $10\exp(-0.15)\approx8.6071$ days, and $S(10\mid x=1)=\exp[-(10/8.6071)^2]\approx\exp(-1.3499)\approx0.2592$, giving the same result.

As another check, the reference hazard at 10 days is $(2/10)(10/10)^{1}=0.2000$ per day. The group hazard is $0.2000(1.3499)=0.2700$ per day. The units are inverse days, and their ratio is the dimensionless hazard ratio.

A common shape is essential to the general equivalence. If covariates are allowed to change $k$, then the exponent of time in the cumulative hazard changes across covariate values. In general, the resulting hazard ratio will depend on time and cannot be represented by a standard proportional-hazards model with a fixed coefficient vector. A common-shape Weibull model may have an increasing hazard when $k>1$, a decreasing hazard when $k<1$, or a constant hazard when $k=1$; these time patterns are shared across covariate values, with covariates multiplying the hazard.

Assumptions and Practical Interpretation

The formulas specify a conditional distribution for positive event times. To use them for censored observations, the event-time model must be combined with an appropriate censoring assumption; in standard right-censored likelihood analysis, censoring is assumed independent of the event time conditional on the modeled covariates. The Weibull form also imposes a particular parametric shape on the event-time distribution, and the proportional-hazards interpretation requires a common $k$. These are modeling assumptions, not consequences of the parameter conversion.

In an econometric application, coefficient interpretation should identify both the covariate contrast and the scale being discussed. A hazard ratio compares instantaneous event rates among units still at risk at a given time. A time ratio compares corresponding model-based event-time quantiles. Neither alone is a statement that changing a covariate causally changes duration; causal conclusions require an appropriate design and additional assumptions.

Common Mistakes

Using the same sign in both parameterizations. The correct relation is $\beta=-k\gamma$, not $\beta=k\gamma$. A longer time scale implies a lower hazard, all else equal.
Calling a time ratio a hazard ratio. The AFT ratio is $\exp(\gamma_j)$; the hazard ratio is $\exp(-k\gamma_j)$. State which quantity is being interpreted.
Ignoring the shape parameter in the conversion. The multiplier $k$ is part of the coefficient mapping. A numerical AFT coefficient cannot be converted without knowing the Weibull shape.
Treating the scale as the median. The median is $\lambda(x)(\log 2)^{1/k}$. It equals the scale only for no general value of $k$; report the transformed value and its time units.
Assuming every Weibull regression is proportional hazards. This equivalence applies when the shape is common across covariates. Allowing covariate-specific shape generally removes the constant hazard-ratio property.

Key takeaway: With a common Weibull shape $k$, the AFT coefficient vector $\gamma$ and proportional-hazards coefficient vector $\beta$ describe the same covariate effects through $\beta=-k\gamma$. The AFT form gives time ratios $\exp(\gamma_j)$; the proportional-hazards form gives hazard ratios $\exp(\beta_j)$.

Check Your Understanding

Use the common-shape Weibull relationships to answer these questions.

If $k=2$ and an AFT coefficient is $\gamma_j=0.10$, what are the corresponding proportional-hazards coefficient and the time and hazard ratios for a one-unit increase?
Why does a positive AFT coefficient correspond to a hazard ratio below one?
For $k=1.5$, $\lambda_0=8$ weeks, and $x'\gamma=0$, write the formula for the median and calculate its approximate value in weeks.
What common-shape assumption is needed for a Weibull model to have a hazard ratio that is constant over time?
Explain why a hazard ratio of 1.5 is not a time ratio of $1/1.5$ in general.