# Logistic regression: change in the ratio by a percentage

In this article in the section of "Logistic regression with multiple predictor variables and no interaction terms" is stated:

This fitted model says that, holding math and reading at a fixed value, the odds of getting into an honors class for females (female = 1)over the odds of getting into an honors class for males (female = 0) is exp(.979948) = 2.66. In terms of percent change, we can say that the odds for females are 166% higher than the odds for males.

How was the value of 166% calculated? It seems that the author just subtracted 1 from 2.66 and multiplied by 100% but according to which formula?

When the odds are 1, this means males and females are just as likely of getting into an honor class. So when the odds are 1+1 = 2, this means the odds are now twice as likely or a 100% increase in odds (from 1 to 2). So, you'll need to subtract 1 from the odds to obtain the percent increase. So in your example the odds are 2.66 (going from 1 to 2.66 means the odds increased by 2.66-1 = 1.66. This means that there is a 166% increase in the odds ($2.66-1)\times100$%). If the odds were less than 1, it would not be a percent increase, but a decrease. For example if the odds were .97, then that would mean there was a 1-.97 = .03 or 3% reduction in the odds.
To look at this another way, think of it like this. If I have 100 cupcakes and then make 25 more, than I have increased the the number of cupcakes by 25%, or I have produced $\frac{{125\mbox{ cupcakes}}}{{100\mbox{ cupcakes}}} = 1.25$ times as many cupcakes ($1.25 \times 100$ original cupcakes = 125 cupcakes). The "1" in front of the ".25" implies an increase.