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I am slightly confused about the way accelerated failure time models work in comparison to proportional hazard models.

  1. Is it a correct statement that while the PH model assumes covariates multiply the hazard function (hence proportional hazards), the covariates multiply the survival function in an AFT model?

  2. Is $S(t,x)=S_0(te^{x'\beta})$ the general expression for all AFT models? If $e^{x'\beta}=2$ (i.e. the acceleration factor), doesn't that mean the function is stretched relative to the baseline, rather than "accelerated"? If $e^{x'\beta}$ is the acceleration factor then shouldn't the function divide by this term instead of multiplying by it?

  3. I believe a Weibull model acts as both an AFT and PH model simultaneously. Does this mean that if $e^{x'\beta}=2$, the hazard for somebody with that set of covariates is multiplied by 2, AND the survival function is compressed by a factor of 2?

Any help is appreciated!

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1 Answer 1

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  1. For the PH model, the covariates appear in the exponential that multiplies the baseline hazard:

$$ h_\text{PH}(t|\mathbf x_i, \boldsymbol\theta,\boldsymbol\beta)=h_0(t|\boldsymbol\theta)\exp\left\{\mathbf x_i^\mathsf T\boldsymbol\beta\right\}.$$

In an AFT model, they multiply the time and the baseline hazard:

$$h_\text{AFT}(t|\mathbf x_i, \boldsymbol\theta,\boldsymbol\beta)=h_0\left(t\{\exp\left\{\mathbf x_i^\mathsf T\boldsymbol\beta\right\}|\boldsymbol\theta\right) \exp\left\{\mathbf x_i^\mathsf T\boldsymbol\beta\right\} .$$

  1. As you can see from the previous formulation, the exponential of the linear predictor affects both the time level and baseline hazard level. The AFT model can be reformulated as a log linear regression model. So, the covariates have an interpretation in terms of how the affect the survival time (see: Survival Analysis: Parametric Models).

  2. If the baseline hazard is Weibull (i.e. $\propto t^{\nu-1}$), then the AFT = PH, as the linear predictor affects simultaneously the baseline hazard and the time scale. However, this happens at different scales (see Survival Analysis: Parametric Models). To get the interpretation in terms of the AFT model, you need to re-parameterize the model and interpret in the corresponding scale. Note that $e^{x^\mathsf T\beta}$ affects both scales in the AFT model. You can find information about how to fit PH and AFT models in R at Example: Regression .

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