How to apply odds ratio to calculate change?

Question

I did a control/treatment experiment and found an of odds ratio $= 2.6 (1.03-6.58)$ mortality in treatment. Thus, for every one death in control there are $2.6$ deaths in treatment. I believe the beta associated with mortality in treatment is $\ln(2.6) = 0.9555$.

Baseline survival for control conditions is $0.72$, thus baseline mortality is $1-0.72 = 0.28$. How can I calculate new values for survival and mortality based on the odds ratio from my experiment?

What follows is my best attempt, but using this method with odds ratio $= 1.03$ or $6.58$ results in unreasonable values so I’m worried I’m making errors:

$1-0.9555 = 0.045$ change in survival

$0.045*0.72=0.0324$

$0.72-0.0324=0.688 \Rightarrow$ baseline values change to survival $= 0.688$ and mortality $= 0.312$ in treatment conditions.

Answer 1

An odds ratio of $2.6$ does not mean for every one death in control there are $2.6$ deaths in treatment

You have said the control probability of survival is $0.72$ and so probability of mortality $0.28$, which makes the odds of mortality $\frac{0.28}{0.72} \approx 0.389$ in control conditions

Multiplying these odds by $2.6$, $1.06$ and $6.58$ would give about $1.011$, $0.412$ and $2.559$ respectively, translating to probabilities of mortality of about $0.503,0.292, 0.719$ and so probabilities of survival of about $0.497, 0.708, 0.281$

Does an estimated probability of survival under treatment of $0.497$ ($0.281 - 0.708$) fit your intuition better?

Answer 2

$Odds_Ratio = \frac{P_{11}.P_{00}}{P_{10}.P_{01}} = 2.6$

$\frac{P_{11}\times 0.28}{0.72\times P_{01}} = 2.6$

gives you $P_{11} = 6.68571P_{01}$

You also know that $P_{11}+P_{01} = 1$

Thus gives you $P_{11} = .8698$ and $P_{01} = .13011$