Following formula is used to calculate the odds ratio (O.R.) and its confidence interval (C.I.).
OR = a*d / b*c, where:
- a is the number of times both A and B are present,
- b is the number of times A is present, but B is absent,
- c is the number of times A is absent, but B is present, and
- d is the number of times both A and B are negative.
To calculate the confidence interval, we use the log odds ratio, log(OR) = log(a*d/b*c), and calculate its standard error:
se(log(OR)) = √1/a + 1/b + 1/c +1/d
The confidence interval, C.I., is calculated as:
CI = exp(log(OR) ± Zα/2*√1/a + 1/b + 1/c + 1/d),
where Zα/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96).
Note: The logarithms included in the formulae above are natural logarithms, i.e., log base e, sometimes denoted ln().
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