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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
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
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.