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Running a logistic regression, I have a dependent variable

 Loyal Customer can be 0 or 1

and an independent variable

 Return.rate is a continuous percent value from 0 to 1

If my odds ratio for Return.rate is 5, how do I interpret the result in words?

What is a unit? 1 which is 100%? Or 0.01 which is 1%?

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The interpretation of the odds ratio is that for each unitary increase in the value of the independent variable the chances that observation belongs to the modeled class increase by the value of the odds ratio. The interpretation applies to each unit increase in Return.rate, but it is dependent on the the actual variable scale. If the Return.rate data is in percent units (i.e. 1,2,3 etc.), then the odds ratio would apply to each increase in percent units. If the Return.rate data is given in fractions of 1 (i.e. 0.56, 0.57, 0.58, etc.) then the odds ratio would again apply to each unit increase, but interpretation would apply to the difference between a Return.rate equal to 0 and a Return.rate equal to 1 (i.e. 0% vs. 100%); everything in between or beyond those points would require additional calculations.

To use your example to further illustrate the point, assuming you obtained that OR value by modelling likelihood of belonging to Loyal Customer class 1, and that Return.rate is given in percent units (i.e. 1, 2, 3, etc.) then each percentage increase in Return.rate makes an observation 5 times more likely to belong to Loyal Customer class 1.

Put another way for two observations that are identical in all respects except for return rate, except that one observation has Return.rate 1% higher than the other (e.g. 2% vs. 1%), then the observation with the higher Return.rate will be five times more Likely to belong to Loyal Customer class 1. Let's call these observations A and B. A has a return rate of 1, while B has a return rate of 2.

The percentage probability difference can be calculated by going back from the odds ratio formula linking ORs and logistic regression coefficients:

                  OR = exp(coefficient)             
                  coefficient = ln(OR)

For an odds ratio of 5, the logistic regression coefficient is approximately 1.6. Please note that odds ratio will increase exponentially from (in our specific case) a base of 5 (i.e. a 2-unit difference will give an odds ratio of 25, 3-unit difference gives 125, etc.). In order to make use of a coefficient provided by logistic regression for such comparisons, the general form is:

                         OR = exp(d*coefficient)

Now, should the Return.rate have instead been expressed as a fraction of 1 (e.g. 0.57, 0.58, 0.59, etc.), most of the previous discussion still holds, however you have to perform extra calculations in order to be able to compare odds ratios between two observations. Let's now assume A has a Return.rate of 0.56 and B has a return.rate of 0.57.

Our marginal change in OR in this case would be:

                 Delta d*coeff = 0.57*1.6 - 0.56*1.6
                 Delta d*coeff = 0.016
                 Delta OR = exp(0.016) = 1.016

Also, should you wish to circumvent the calculation above to get OR, you can simply raise the OR to the power of the difference between the two observations. In our case that would mean raising 5 to the power 0.01.

Hope this helps.

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    $\begingroup$ +1 Also, I'd recommend rescaling the RetrunRate by multiplying it with 100, turning it into percent rather than a fraction would make the interpretation less clunky. Also, make sure to use "percent point" rather than "percent" change in return rate when interpreting the result, or the readers may think you have log-transformed return rate as well. $\endgroup$ – Penguin_Knight Apr 29 '15 at 12:29

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