Odds Ratio

An Example: Odds Ratios for Two Dichotomous Outcomes

David P. Strachan, Barbara K. Butland, and H. Ross Anderson report on the occurrence of hay fever for 11-year-old children with and without eczema and present the results in a contingency table (Table 1).

First, the probability of hay fever for those with eczema (the top row) is calculated. This probability is 141/561 = .251. Thus, the odds of hay fever for those with eczema are

about 1 to 3. Analogously, the probability of hay fever for those without eczema is

From the example, one can infer that the odds of hay fever for eczema sufferers are 4.89 times the odds for noneczema patients. Thus, having eczema almost quintuples the odds of getting hay fever.

The ratio of the two odds can, as shown in Equation (1), also be computed as a ratio of the products of the diagonally opposite cells and is also referred to as the cross-product ratio.

The OR is bounded by zero and positive infinity. Values above and below 1 indicate that the occurrence of an event is more likely for one or the other group, respectively. An OR of exactly 1 means that the two odds are exactly equal, implying complete independence between the variables.

Does the Odds Ratio Imply a Significant Relationship?

To determine whether or not this OR is significantly different from 1.0 (implying the observed relationship or effect is most likely not due to chance), one can perform a null hypothesis significance test. Usually, the OR is first transformed into the log odds ratio [log(OR)] by taking its natural logarithm. Then, this value is divided by its standard error and the result compared to a test value. In this example, the OR of 4.89 is transformed to the log(OR) of 1.59. A log(OR) of 0 implies independence, whereas values further away from 0 signify relationships in which the probability or odds are different for the two groups. Note that log(OR) is symmetrical and is bounded by negative and positive infinity.

...