What is the Odds Ratio in Logistic Regression?

Most logistic regression software outputs (or can be asked to output) odds ratios along with the regression coefficients. These odds ratios are the exponential of the corresponding regression coefficient:

    \[\text{odds ratio} = e^{\hat\beta}\]

For example, if the logistic regression coefficient is \hat\beta=0.25 the odds ratio is e^{0.25} = 1.28.

Interpretation

The odds ratio is the multiplier that shows how the odds change for a one-unit increase in the value of the X. Continuing the example above, if the odds are 1 to 4 or 0.25, then increasing the X variable by 1 unit will change the odds to 0.25\times 1.28 = 0.32 or pretty close to 1 to 3.

Note: Do not confuse the odds of 0.25 as a probability — the corresponding probability is 0.20. Similarly, the odds of 0.32 corresponds to a probability of 0.242.

Another way to try to interpret the odds ratio is to look at the fractional part and interpret it as a percentage change. For example, the odds ratio of 1.28 corresponds to a 28% increase in the odds for a 1-unit increase in the corresponding X.

The formula is:

    \[\text{Percent Change in the Odds} = \left( \text{Odds Ratio} - 1 \right) \times 100\]

As a final example, if the odds ratio is 0.94, then there is a 6% decrease in the odds for a 1-unit increase in the corresponding X.

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