Simple version

Odds ratios represent the "multiplicative effect" that a given parameter has on the outcome. If a parameter has an odds ratio of 2, then an increase of 1 of that parameter's value will double the odds of the "success" being modeled with logistic regression.

Detailed version

Odds ratios are simply a transformation of the parameter estimates calculated for the logistic regression model. However, many find odds ratios to be much more useful when interpreting the results of logistic regression. The reason for this is that the parameter estimates tell you how much the "log odds" change when the independent (X) variable changes. But "log odds" are hard to explain. In contrast, odds ratios tell you how much the odds change when the independent (X) variable associated with that odds ratio changes. Before reading on, be sure you can tell the difference between probability and odds.

The standard form of the equation that multiple logistic regression fits is:

ln[P(Y=1)/P(Y=0)] = β0 + β1*X1 + β2*X2 + …, or

ln(Odds) = β0 + β1*X1 + β2*X2 + …

That is, the log odds can be expressed as a linear equation. If we exponentiate both sides, we get the following relationship:

eln(Odds) = eβ0 + β1*X1 + β2*X2 + …, or

Odds = eβ0 + β1*X1 + β2*X2 + …, or

Odds = (eβ0)*(eβ1*X1)*(eβ1*X2)*…

 

If we replace (eβ0) = β0, we get

Odds = (β0)*(β1X1)*(β1X2)*…

 

This illustrates the relationship of the parameter estimates and the odds ratios mentioned before: if you exponentiate β1 in the parameter estimate section, you’ll obtain the value reported for β1 in the odds ratio section of the results. Using this knowledge, it can be seen that the estimates for the odds ratios have the following interpretation:

For β2, we can say that a 1 unit increase in X2 has a multiplicative effect equal to the odds ratio estimate of β2 on the odds of Y when all the other X values are held constant.

As a simple example, let's say for a given set of values, you calculated the odds of a "success" to be 3 (sometimes called "3:1 odds" or "three to one odds"). If β2 was estimated to be 2, then an increase of X2 by 1 would result in an increase of the odds to 6 (or "6:1 odds").

Prism also provides confidence intervals for the odds ratio parameters. These are often misunderstood statistical concepts because they aren't quite what our intuition wants them to be. A proper interpretation of a confidence interval would read: We are 95% confident that the range between lowerVal and upperVal includes the true odds ratio for thisParameter

Note that calculated P values will be the same for both the parameter estimates and the odds ratios for a given logistic regression model.