# Interpreting logistic regression coefficient of a ratio predictor

I'm fitting a logistic regression model in which my predictor of interest is a ratio of measurements in millimeters. Possible values for this ratio range from 0 to ~2.0, with typical values around 0.9-1.2. I want to measure the association between this variable and a binary outcome. I fit a model and obtained the following results:

Coefficients:
Estimate  Std.err  Wald Pr(>|W|)
(Intercept)   -6.52673  1.09162 35.75  2.2e-09 ***
x              0.91680  0.36187  6.42    0.011 *  # OR = 2.5


I know that the generic way to interpret these results would be: "For a one unit increase in X, the odds of Y=1 increase by a factor of 2.5..." I keep getting hung up on the interpretation of this result, as a 1 unit change in x (e.g., from 0.6 to 1.6) would be a very extreme/physically impossible change for this particular ratio (which represents an index). Is there a way I should transform this variable or the results so that I can describe changes of 0.10 rather than 1.00?

If a one-unit increase in $$X$$ results in an increase in log-odds of $$2.5$$, then a $$0.1$$-unit increase in $$X$$ results in an increase in log-odds of $$0.25$$.

If you drive a tenth as long, you go a tenth as far. That $$2.5$$ is analogous to your speed.

• Great, thank you. So I'm wondering how I would report the OR, given that OR=0.25 would represent a reduction in odds. The log-odds I was looking at was 0.91680 [exp(0.91680)=2.5]. When I move the decimal point over by 1, [exp(0.091680)], I get an OR of 1.1. Or would I report a log odds of 1.25 (corresponding to a 25% increase)? Would it be incorrect to report that OR? Apologies if I'm missing something obvious or overthinking this. Mar 22, 2020 at 22:08
• Unless I’ve forgotten how logarithm math works, it’s not an odds ratio.
– Dave
Mar 25, 2020 at 0:56
• I was trying to make sure the relationship was the same after exponentiating the log-odds*0.10, i.e., converting it to odds. Apr 4, 2020 at 0:36

The coefficient of X is 0.91680, so OR for X is calculated as exp(0.91680 )=2.501273. The interpretation of the OR is that for every increase of 1-unit in X, the estimated odds of the event are multiplied by 2.501273. However, If your data is not increased by 1-unit, for instance, falls between 0 and 2, it may make more sense to say that for every 0.1 unit increase in x, the estimated odds of the event are multiplied by exp(0.91680×0.1)=1.096014.

You can find more explanations in PennState Eberly College of Science, STAT 463, Lesson 12: Logistic, Poisson & Nonlinear Regression.

• please summarize the content of your link, and provide a reference if possible, in case it dies in the future Oct 1, 2020 at 9:18
• Thanks. I have edited the hyperlink in my post. Oct 1, 2020 at 12:14