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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.

mzp
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Emily
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2 Answers2

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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?

Henry
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  • That does seem much more reasonable. Sorry, how are you deriving probabilities of mortality (0.503, etc.)? – Emily Apr 03 '18 at 17:04
  • @Emily odds of $o$ translate into probabilities of $p=\dfrac{o}{1+o}$ as the inverse function of translating probabilities into odds $o=\dfrac{p}{1-p}$ – Henry Apr 03 '18 at 17:22
  • @Henry, the 2.6 odds ration must be the odds of survival, I think, the OP has made a mistake (I think). After treatment, the survival probability should go up and not lowered unless the treatment is trying to kill you. In which case Odds of survival in controls would be $\frac{0.72}{0.28} = 2.5714$ Multiply this by 2.6 and you get $6.68571$ and now use probability of survival for the treatment = $\frac{6.68571}{7.68571} = 0.8698$ – Satish Ramanathan Apr 04 '18 at 05:49
  • @SatishRamanathan - you may be correct that treatment is supposed to improve outcomes but that is not how I read "an of odds ratio =2.6 mortality in treatment" – Henry Apr 04 '18 at 07:46
  • @Henry, Certainly, you have answered correctly to the question, but I suspect that the question is ill posed and I wonder if the OP understands the implication if it is real research in physical sciences. – Satish Ramanathan Apr 04 '18 at 08:43
  • @SatishRamanathan sorry, just saw this comment. The way I posed the question is correct but I understand your concern. The treatment was intended to help but likely does more harm than good, surprisingly – Emily May 16 '18 at 19:50
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$Odds_Ratio = \frac{P_{11}.P_{00}}{P_{10}.P_{01}} = 2.6$

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

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

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

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

Satish Ramanathan
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