Statistics in Crosstabulations - Odds Ratio

An odds ratio is a statistical measure of dependence between 2 binary variables in a 2x2 table. An odds ratio of 1 indicates independence.  A typical example involves comparing two groups with respect to an event of interest occurring or not occurring.

The following table is from Agresti (1990) Categorical Data Analysis. It shows the breakdown of participants in a clinical trial testing whether aspirin taken regularly reduces the likelihood of heart attack.

 Heart Attack No Heart Attack Placebo 189 10,845 Aspirin 104 10,933

The odds of an individual in the placebo group experiencing heart attack is 189/10,845 = 0.01742739

The odds of an individual in the aspirin group experiencing heart attack is 104/10,933 = 0.009512485

Odds ratio = 0.01742739 / 0.009512485 = 1.832054, the odds of experiencing a heart attack is 1.83 times greater for the placebo group than for the aspirin group.

Asymptotic large sample confidence interval =  [1.440042 2.330780]

Formulas

For the following 2x2 table:

 a b c d

Odds ratio (or) = ad / bc

Large sample CI = exp ( ln (or) +/- zcrit * sqrt(1/a+1/b+1/c+1/d) )

zcrit = 1-alpha/2 quantile from the standard normal distribution where alpha = 1-confidence level