Odds ratios

From MedSchoolWiki

Background

When a binary outcome variable is modeled using logistic regression, it is assumed that the logit transformation of the outcome variable has a linear relationship with the predictor variables. This makes the interpretation of the regression coefficients somewhat tricky. In this page, we will walk through the concept of odds ratio and try to interpret the logistic regression results using the concept of odds ratio in a couple of examples.

From probability to odds to log of odds: Everything starts with the concept of probability. Let's say that the probability of success of some event is .8. Then the probability of failure is 1- .8 = .2. The odds of success are defined as the ratio of the probability of success over the probability of failure. In our example, the odds of success are .8/.2 = 4. That is to say that the odds of success are 4 to 1. If the probability of success is .5, i.e., 50-50 percent chance, then the odds of success is 1 to 1.

The transformation from probability to odds is a monotonic transformation, meaning the odds increase as the probability increases or vice versa. Probability ranges from 0 and 1. Odds range from 0 and positive infinity. Below is a table of the transformation from probability to odds and we have also plotted for the range of p less than or equal to 0.9

Examples

Let's begin with probability. Let's say that the probability of success is .8, thus [p = 0.8] Then the probability of failure is [q = 1 - p = 0.2]

The odds of success are defined as [odds(success) = p/q = .8/.2 = 4]

that is, the odds of success are 4 to 1. The odds of failure would be [odds(failure) = q/p = .2/.8 = 0.25]

This looks a little strange but it is really saying that the odds of failure are 1 to 4. The odds of success and the odds of failure are just reciprocals of one another, i.e., 1/4 = .25 and 1/.25 = 4. Next, we will add another variable to the equation so that we can compute an odds ratio.

Another example

This example is adapted from Pedhazur (1997). Suppose that seven out of 10 males are admitted to an engineering school while three of 10 females are admitted. The probabilities for admitting a male are,

p = 7/10 = .7 q = 1 - .7 = .3

Here are the same probabilities for females,

p = 3/10 = .3 q = 1 - .3 = .7

Now we can use the probabilities to compute the admission odds for both males and females,

odds(male) = .7/.3 = 2.33333; odds(female) = .3/.7 = .42857

Next, we compute the odds ratio for admission,

OR = 2.3333/.42857 = 5.44

Thus, for a male, the odds of being admitted are 5.44 times as large than the odds for a female being admitted.

Relationship to relative risk

In clinical studies, as well as in some other settings, the parameter of greatest interest is often the relative risk rather than the odds ratio. The relative risk is best estimated using a population sample, but if the rare disease assumption holds, the odds ratio is a good approximation to the relative risk — the odds is p / (1 − p), so when p moves towards zero, 1 − p moves towards 1, meaning that the odds approaches the risk, and the odds ratio approaches the relative risk. When the rare disease assumption does not hold, the odds ratio can overestimate the relative risk.

Notes & References

[1] http://www.ats.ucla.edu/stat/Stata/faq/oratio.htm.

[2] http://en.wikipedia.org/wiki/Odds_ratio

Credits & Notices

Authors-contributors to this page (listed alphabetically, last name, first & middle initial only, no institutional affiliations, no scientific titles):

Stawicki SP

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