How to deal with non-proportional risk? Let's start with a toy example. We have an RCT with 2 treatment arms (Treatment A vs. Treatment B), and one outcome.
When we analyse survival curves according to the treatment arm, we found the following scenario:

It seems that the proportionality-hazard assumption is violated, and we therefore cannot use a Cox Regression Analysis to analyse the association between treatment arm and risk of outcome in this scenario.
How can we deal with this situation?
Is there any different model than Cox Regression analysis which are better suited to deal with such a situation?

I'm wondering if an accelerated-life model would be more appropriate here.
We would take the log life length to be a linear function of the explanatory variables,
$$ \mathrm{log}(Y_i) = X_i \beta + \varepsilon_i$$
or, equivalently,
$$ Y_i = \exp(X_i \beta) U_i\,
$$
where $U_i=\exp(\varepsilon_i)$. The conditional survival function for $Y_i$ given explanatory variables $X_i=x_i$ is then
$$
S_{Y_i|X_i}(y|x_i)=S_U(y/\exp(x_i \beta))\,.
$$
In other words, the survival function has a fixed form, but its argument is scaled by the values of the explanatory variables. For example, if $x_i \beta=0$ and $x_j \beta=\log2$, the effect is that individual $j$ ages twice as fast as individual $i$.
The downside is that this is usually a fully parametric model, so we need to choose a distribution for $U$ (typically log-logistic or Weibull).
