Title: Probabilities, Odds & Odds Ratios
Course: UGQuants (SSPS) > Section: Logistic Regression

Probabilities, Odds & Odds Ratios

'Odds' express the probability of an event occurring in comparison to the probability of it not occurring.  Thus a probability of 0.5 would have odds of 1, because 0.5/0.5 = 1.

A probability of 0.25 would have odds of 0.25/0.75 = 0.33.

A probability of 0.75 would have odds of 0.75/0.25 = 3.

Table 1 compares odds and probabilities.  Any probability can be expressed as odds by dividing it by one minus the original probability.

Odds = probability/(1-probability)

Table 1: Comparison of odds and probabilities

P | 0.1 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 0.99
(1-p) | 0.99 | 0.9 | 0.7 | 0.5 | 0.3 | 0.1 | .01
Odds (p/(1-p)) | 0.01 | 0.111 | 0.429 | 1 | 233 | 9 | 99

If we express a probability as odds, we get rid of the upper bound of 1 to probabilities.  For example a probability of 0.99 becomes odds of 99 (to 1).  The lower bound, however, is still restricted to zero.

We can compare the odds for different values of an independent variable, such as whether and by how much men and women differ in their voting behaviour.  We do this by using the odds ratio, which simply expresses one odds compared to another odds.  With Yule’s Q the odds ratio can be used as a measure of association for two variables.  We can also use the odds ratio to compare the probability of a dependent variable taking a particular value, dependent upon the value taken by an independent variable.

Continuing with our example above, with the dependent variable being voting at last national election and the independent variable being the gender of respondent, suppose the probability of men having voted is 0.7 (i.e.  7 out of every 10 men reported this).  This could be expressed as odds of 2.33.  Suppose the probability of women having voted is 0.5 (i.e.  5 out of every 10 women reported this attitude).  This could be expressed as odds of 1.0.  The odds ratio for men to women would be 2.33/1 = 2.33.  The odds ratio for women to men would be 1/2.33= 0.429.  Note carefully how an odds ratio below one indicates a decrease in the odds of the dependent variable taking the value of interest (i.e. women are less likely than men to have voted), while an odds ratio above one indicates an increase (i.e. men are more likely than women to have voted).

(Since these are just made up numbers to simplify the example, you could check for yourself what the actual odds ratios were in the European Social Survey in 2010.)

What does an odds ratio of one indicate?  It shows that it doesn’t matter which gender is asked, as they have the same probability of holding a negative (or positive) opinion about gay male sex.  This is a feature known as statistical independence.