# Is it possible to convert the hazard difference of an Aalen additive model to the hazard ratios that are typically outputted by Cox PH models?

I am comparing the behavior of the survival and hazard function estimates of the two models using the same dataset and the same factors/covariates influencing survival time. I am not sure if it is proper to convert the hazard difference of Aalen into hazard ratios like in the context of Cox PH.

There is a limiting case in which you can get an approximate relationship between Cox and Aalen coefficients. The question is what you would want to accomplish with that, as the models are based on different assumptions about how covariates are associated with hazard.

With the hazard function written $$\lambda (t)$$, baseline hazard $$\lambda_0(t)$$, and covariate column vector $$x(t)$$ the Cox proportional hazards model is:

$$\lambda(t) = \lambda_0(t) e^{x(t)^T \beta(t)}$$ or $$\log\lambda(t) = \log\lambda_0(t) + x(t)^T \beta(t)$$

with potentially time-dependent regression coefficients $$\beta(t)$$, although initial models often assume time-invariant coefficients. The Aalen additive model is:

$$\lambda(t) = \lambda_0(t) + x(t)^T \alpha(t)$$

with regression coefficients $$\alpha(t)$$ explicitly modeled as functions of time.* The Aalen model can be rewritten:**

$$\lambda(t) = \lambda_0(t)\left(1 + x(t)^T \alpha(t)/\lambda_0(t)\right) .$$

With $$\log(1+z)\approx z$$ for small-magnitude $$z$$, the Aalen additive model can be put into a form similar to that for the Cox proportional hazards model:

$$\log \lambda(t) \approx \log\lambda_0(t)+ x(t)^T \alpha(t)/\lambda_0(t)$$

suggesting $$\beta(t) \approx \alpha(t)/\lambda_0(t)$$ for a small-magnitude second term in the approximation.

If there's a fixed baseline hazard $$\lambda_0$$ then $$\beta(t) \approx \alpha(t)/\lambda_0$$. But if there's a fixed baseline hazard, why not use a parametric exponential model for survival?

What's probably more useful is to examine how the Cox and Aalen models differ with your data, and whether the flexibility provided by the Aalen additive model adds anything useful to what can be learned from the Cox model.

*The standard summary of output from the aareg() function in R reports, for each covariate, the slope of the weighted regression of associated cumulative additive hazard versus time, a type of average over the separate coefficient estimates generated for each event time.

*This trick is based on Therneau and Grambsch, page 149, in a discussion of causes of nonproportionality in Cox models.