It is defined as the ratio of the odds of an event occurring in one group to the odds of it occurring in another group, or to a sample-based estimate of that ratio.

The odds ratio compares the relative odds in each group

One advantage of the odds ratio is that it is not dependent on whether we focus on the event's occurrence or its failure to occur. If the odds ratio for an event deviates substantially from 1.0, the odds ratio for the event's failure to occur will also deviate substantially from 1.0, though in the opposite direction.

Shown below is the typical 2 by 2 table.

You can understand the odds ratio by first noticing what the odds are in each row of the table. The odds for row Y- are a/b. The odds for row Y+ are c/d. The odds ratio (OR) is simply the ratio of the two odds.

which can be simplified to

Notice that if the odds are the same in each row, then the odds ratio is 1.

An odds ratio of 1 indicates that the condition or event under study is equally likely in both groups. An odds ratio greater than 1 indicates that the condition or event is more likely in the first group. And an odds ratio less than 1 indicates that the condition or event is less likely in the first group.

Consider the following data on survival of passengers on the Titanic. There were 462 female passengers: 308 survived and 154 died. There were 851 male passengers: 142 survived and 709 died (see table below).

Clearly, a male passenger on the Titanic was more likely to die than a female passenger.

DEAD ALIVE TOTAL

MALE 709 142 851

FEMALE 154 308 462

TOTAL 863 450 1,313

The odds ratio compares the relative odds of death in each group.

For males, the odds were 5:1 in favor of death (709X308= 218,372).

For females, the odds were exactly 2:1 against dying (154X142= 21,868)

The odds ratio (218,372/21,868= 9.986 ). There is a ten fold greater odds of death for males than for females.

EXAMPLE suppose that in a sample of 100 men, 60 have

drunk wine in the previous week, while in a sample of 100 women only 20 have drunk wine in the same period.

WHO DRUNK WINE

WHO DIDN’T DRUNK WINE

TOTAL

MEN 60 40 100

WOMEN 30 70 100

TOTAL 90 110 200

The odds of a man drinking wine are 60 to 40, or 2:1

the odds of a woman drinking wine are only 20 to 80, or 1:2

Now, (60X70 =4200) for men and (30X40 = 1200 ) for women (4200/1200=3.5 ) so the odds ratio is 4, showing that men are

much more likely to drink wine than women.

the odds ratio is commonly used as a means of expressing the results in some forms of clinical trials, in survey research, and in epidemiology, such as in case-control studies.