Odds Ratio

The odds ratio (OR) is a nonparametric, inferential test of association. It evaluates whether the odds of a certain event (or outcome) happening is the same for two groups. Specifically, the OR measures the ratio of the odds that an event or result will occur to the odds of the event not happening, given one group’s exposure to some factor to which the other group has not been exposed. This entry discusses the use of OR, its assumptions, and its calculation, significance and strength testing for the OR, and interpretation of the OR, giving an example of OR use.

The OR answers the following questions: “What are the odds that a child who has not been vaccinated will contract chicken pox as compared with the odds of a vaccinated child contracting chicken pox?”

The OR is the ratio of the following two:

• 1.
  the ratio between the number in the control group with outcome 1 and the number in the control group with outcome 2 and
• 2.
  the ratio between the number in the experimental group with outcome 1 and the number in the experimental group with outcome 2.

The OR is a robust and fairly versatile statistic used in many clinical studies and educational research designs with two study groups. The OR provides an effect size like other correlational statistics, but in a form very different from other effect size statistics. The OR provides information on how high the odds are for one condition versus the other. Therefore, the interpretation of the OR result is very different from the interpretation of other effect size statistics.

The OR assumes subjects were randomly and independently sampled from the population of interest. That is, selection of one subject is unrelated to the selection of any other subject. For example, in drug treatment studies, patients are assigned randomly to either the experimental or control group such that there is no bias in group assignment. The variables are the counts of subjects in each condition, not ratios or proportions.

To calculate the OR, the data must be in a 2 × 2 table, as shown in Table 1.

The formula to calculate the OR is as follows:

$$OR=\frac{A/B}{C/D}\text{or}\frac{A\times D}{B\times C}.$$

The formula AD ÷ BC is called the cross product and is mathematically equivalent to the original formula (A ÷ B) ÷ (C ÷ D). That is, the results for the two formulas are the same.

For interpretation, it is very important that the table is set correctly and the numbers correctly entered into the formula’s numerator and denominator for the independent and dependent variables. The odds of the experimental group experiencing the outcome must be placed in the numerator, while the odds of the control (untreated) group experiencing the outcome must be placed in the denominator. Reversing the placement will lead to an uninterpretable result.

Several significance tests may be used for the OR, but the Fisher exact test is typically used for a 2 × 2 table. The researcher may also use a chi-square test or a maximum likelihood ratio chi-square test. Strength testing can also be done using the φ coefficient.

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