I am trying to understand some formulas in a paper, explaining how to re-calculate a population life table by eliminating a certain cause of death. I have a problem understanding a particular step in their calculations.
First, let's denote the probability of surviving in a certain interval as $$p_x = 1- q_x$$, where $q_x$ is the probability of dying in a certain interval and let $$r_x= (q_x^{attrb})/(q_x)$$ be the fraction of age-specific deaths that are attributable to the risk factor of interest. Here, $q_x^{attrb}$ denotes the age-specific death rate for the risk factor of interest and $q_x$ denotes the age-specific all-cause death rate. Then they estimate the probability of surviving in the age interval if the risk factor of interest was deleted with an exponential formula: $$p_x^′= p_x^{(1-r_x)}$$ , where $p_x^′$ represents the counterfactual probability of survival in a given age interval after eliminating the risk factor.
What I do not understand is - where is this formula for probability of survival with the exponential function coming from? I guess it must be something standard because it is not explained in the paper, but I can't get my head around it. I was thinking first they might assume a Cox-proportional hazard model of survival, but I couldn't derive the formula from there. Any help will be much appreciated!