# AFT model interpretation

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!

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