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Exact Confidence Intervals For Odds Ratio See text for formulas if you really want to study them. The principle of estimation and confidence intervals can be based on a likelihood approach. Let P(X=a | ψ ) be the probability of observing the value a in a two by two table. Also ψ represents the odds ratio. The maximum likelihood estimator of the odds ratio is the value that maximizes the probability of getting what we actually observed. So if we observe a=5 then find ψ so that P(X=a | ψ ) is a maximum. Computers can be programmed efficiently to do this. So the maximum likelihood estimator is this solution. We use the distribution based on the fixed row and columns in a table; so it is called the conditional maximum likelihood estimator. To get a confidence interval we choose all values of ψ that give a reasonable probability of being as extreme as a=5. So find the value of ψ P(X ≥ a | ψ ) =0.025 such that and the value such that P(X ≤ a | ψ ) =0.025. These two values will form the confidence interval for the odds ratio. A one sided test of hypothesis will reject the null hypothesis if P(X ≥ a | ψ ) =0.05. Two sided tests reject the null hypothesis if P(X ≥ a | ψ ) =0.025 or P(X ≤ a | ψ ) =0.025. Essentially a 95% confidence interval is the set of all odds ratios that would be accepted with a two sided test.