Proportional hazards is one way to compare event-time distributions, not the definition of survival regression. Accelerated failure-time models act directly on the time scale, while transformation models provide a common framework containing proportional hazards and proportional odds.
Accelerated failure-time structure
Definition. An accelerated failure-time (AFT) model is \[ \log T=\beta^\top Z+\varepsilon, \qquad \varepsilon\perp Z, \] where the error distribution \(F_\varepsilon\) may be parametric or unspecified. Equivalently, \[ T=e^{\beta^\top Z}T_0, \qquad T_0=e^\varepsilon. \]
Proposition. If \(S_0\), \(f_0\), and \(\lambda_0\) are the survival, density, and hazard of \(T_0\), then \[ S(t\mid z)=S_0(te^{-\beta^\top z}), \] \[ f(t\mid z)=e^{-\beta^\top z} f_0(te^{-\beta^\top z}), \] \[ \lambda(t\mid z)=e^{-\beta^\top z} \lambda_0(te^{-\beta^\top z}). \]
Proof. Since \(T=e^{\beta^\top z}T_0\), \[ P(T>t\mid z)=P(T_0>te^{-\beta^\top z}). \] Differentiate to obtain the density, then divide density by survival for the hazard.
Example (acceleration factor). For a binary treatment \(A\), \[ \frac{Q_{T\mid A=1}(q)} {Q_{T\mid A=0}(q)} =e^\beta \] for every quantile \(q\in(0,1)\). If \(e^\beta=1.4\), every conditional survival quantile is stretched by \(40\%\). This interpretation is global on the time scale, unlike a hazard ratio.
Parametric censored likelihood
Theorem. Let \[ e_i(\beta)=\log X_i-\beta^\top Z_i. \] If \(\varepsilon\) has density \(f_\varepsilon(\cdot;\eta)\) and survival \(S_\varepsilon(\cdot;\eta)\), the event-time log-likelihood under independent right censoring is \[ \ell(\beta,\eta) =\sum_i\left[ \Delta_i\{\log f_\varepsilon(e_i;\eta)-\log X_i\} +(1-\Delta_i)\log S_\varepsilon(e_i;\eta) \right]. \]
Proof. The transformation \(\varepsilon=\log T-\beta^\top Z\) has Jacobian \(1/t\), so \[ f_T(t\mid Z)=\frac1t f_\varepsilon(\log t-\beta^\top Z). \] The event-time survival is \(S_\varepsilon(\log t-\beta^\top Z)\). Substitute these into the right-censoring likelihood.
Normal, logistic, and extreme-value errors yield lognormal, log-logistic, and Weibull survival models. Parametric extrapolation beyond observed follow-up is possible only because the chosen error tail supplies it.
Weibull duality
Theorem. The Weibull family is both AFT and proportional hazards. If \[ S_0(t)=e^{-t^\alpha}, \] then the AFT model \(T=e^{\gamma^\top Z}T_0\) satisfies \[ S(t\mid Z) =\exp\{-t^\alpha e^{-\alpha\gamma^\top Z}\} \] and therefore is a Cox model with \[ \beta=-\alpha\gamma,\qquad \lambda_0(t)=\alpha t^{\alpha-1}. \]
Proof. Apply the AFT survival identity: \[ S(t\mid Z)=S_0(te^{-\gamma^\top Z}) =\exp\{-(te^{-\gamma^\top Z})^\alpha\}. \] The cumulative hazard is \(t^\alpha e^{-\alpha\gamma^\top Z}\), which factors into a baseline term and \(e^{\beta^\top Z}\) with \(\beta=-\alpha\gamma\).
Outside the Weibull family, an AFT model does not generally have proportional hazards. Agreement of Weibull AFT and PH coefficient signs reflects the minus sign in \(\beta=-\alpha\gamma\): a covariate extending survival decreases hazard.
Rank-based semiparametric estimation
Definition. Let residuals \[ e_i(\beta)=\log X_i-\beta^\top Z_i. \] The Gehan estimating function is \[ U_G(\beta) =\sum_{i=1}^n\sum_{j=1}^n \Delta_i(Z_i-Z_j) 1\{e_j(\beta)\ge e_i(\beta)\}. \] It is the subgradient of a convex rank-loss function, up to sign.
Proposition (unbiasedness at the AFT parameter). Under independent censoring after the required conditioning and the AFT model, \[ E\{U_G(\beta_0)\}=0. \]
Proof outline. At \(\beta_0\), uncensored residual event times share the common error law. For each ordered pair \((i,j)\), exchangeability makes the expected contribution \[ \Delta_i(Z_i-Z_j)1\{e_j\ge e_i\} \] cancel with its reversed-pair counterpart after accounting for independent censoring through the observed risk indicator. A counting-process derivation writes the rank score as an integral of centered covariates against residual-time martingales, whose expectation is zero.
Solving \(U_G(\beta)=0\) gives a root-\(n\) estimator under regularity. Its nonsmooth objective can be handled by linear programming, induced smoothing, or convex optimization. Other weights generate log-rank-type rank estimators with different efficiency across error laws.
Buckley-James imputation
Definition. Let \[ Y_i=\log X_i,\qquad e_i(\beta)=Y_i-\beta^\top Z_i. \] Estimate the residual survival distribution by Kaplan-Meier, and replace a censored log time by \[ \widetilde Y_i(\beta) =\beta^\top Z_i+ E_{\widehat F_\beta} \{\varepsilon\mid\varepsilon>e_i(\beta)\}. \] The Buckley-James estimator solves least-squares normal equations using \(\Delta_iY_i+(1-\Delta_i)\widetilde Y_i(\beta)\).
This is a self-consistency algorithm: impute conditional residual means under the current coefficient, then refit. It can be unstable near the largest censored residual because the residual tail mean is weakly identified.
General transformation models
Definition. A semiparametric transformation model has \[ G\{S(t\mid Z)\}=H(t)+\beta^\top Z, \] where \(G:(0,1)\to\mathbb R\) is known and monotone, while \(H\) is an unknown monotone baseline transformation.
Example.
- With \(G(s)=\log\{-\log s\}\), \[ \log\{-\log S(t\mid Z)\} =\log\Lambda_0(t)+\beta^\top Z, \] which is proportional hazards.
- With \(G(s)=\log\{(1-s)/s\}\), \[ \operatorname{logit}F(t\mid Z)=H(t)+\beta^\top Z, \] which is proportional odds.
Proposition (proportional odds). In the proportional-odds model, \[ \frac{F(t\mid z_1)/S(t\mid z_1)} {F(t\mid z_0)/S(t\mid z_0)} =e^{\beta^\top(z_1-z_0)} \] for every \(t\).
Proof. Subtract the two transformation equations and exponentiate.
Choosing a regression scale
PH, AFT, proportional odds, restricted mean, and quantile models encode different invariances:
- PH keeps instantaneous hazard ratios constant;
- AFT keeps ratios of survival quantiles constant;
- proportional odds keeps cumulative event odds ratios constant;
- restricted-mean regression targets \(\int_0^\tau S(t\mid Z)\,\mathrm{d}t\);
- quantile regression targets selected conditional event-time quantiles.
The model should follow the scientific contrast and observed shape, not a default software routine. Comparing residual processes, survival-curve crossings, and time-scale interpretations is more informative than choosing solely by one global likelihood value.