Imagine to have two groups of people, the first one more strongly exposed to a pollutant than the second one, and the first one developing a certain disease more often. Having measurements of the pollutant one can apply logistic regression to estimate the significance of influence of the pollutant on developing the disease. One can also calculate an odds ratio of this scenario.
Often, odds ratios are based on one unit change of the independent variable, e.g. changing the pollutant concentration for 1 mg/ml yields an odds ratio of 4 to 1 to develop the disease.
My question is now, how can I recalculate an odds ratio based on a change for several unit changes?
My first guess was the OR of the new range is OR of one unit change to the power of range size in units. Where range is, for example, 10 unit changes of the independent variable. So 4 to the power of 10, in my example.
Or must it be multiplied?
And what is the mathematical basis to prove the calculation?
Best Answer
I Think you're right!
If the independent variable is continuous, logistic regression will give you OR for a one-unit change. For a n-unit change you can use the power of the OR, i.e. OR^n. Multiplication is valid for log(odds).
Look at this page for some mathematical explanations (chapter "Logistic regression with a single continuous predictor variable"). FAQ: How do I interpret odds ratios in logistic regression? Some calculations are presented for a one-unit increase in math variable, you can easily replicate it for a 10 unit increase.
Look also at page 156 of this book: Multivariable Analysis: A Practical Guide for Clinicians and Public Health