Download Exact Confidence Intervals For Odds Ratio

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.