# Multinomial logistic regression: Interpretation of odds ratios as relative risks

In the context of an epidemiological study, a multinomial regression analysis was used to obtain odds ratios for an outcome variable with four different categories.

proc logistic data=ha8  descending;
class gp    (ref='1')/ param=ref;
run;


For example we got results as follows (the values are fictional):

• 1.4 is the multinomial logit estimate for a one unit increase in BMI (continuous) for group 4 relative to group 1. If a subject were to increase his BMI by one point, the multinomial log-odds for group 4 relative to group 1 would be expected to increase by 40%.
• 1.3 is the multinomial logit estimate for a one unit increase in BMI (continuous) for group 3 relative to group 1. If a subject were to increase his BMI by one point, the multinomial log-odds for group 3 relative to group 1 would be expected to increase by 30%.
• 1.2 is the multinomial logit estimate for a one unit increase in BMI (continuous) for group 2 relative to group 1. If a subject were to increase his BMI by one point, the multinomial log-odds for group 2 relative to group 1 would be expected to increase by 20%.

Which conditions are necessary in order to interpret odds ratios obtained from this multinomial logistic regression, as relative risks?

Many thanks!

The safe thing is to never interpret odds ratios as risk ratios. If you want risk ratios use a log link function and check if that models is reasonable. I don't know how to extend that to more than two outcome categories.

It depends on the context of your problem. Per the rare disease assumption, the relative risk will approximately be equal to the odds ratio when the prevalence of the disease (or whatever outcome you are measuring) is sufficiently rare. There isn't a specific number that delineates "sufficiently rare," but generally a disease can be considered rare if the proportion of individuals afflicted with the disease is 5% or lower.

You might want to be careful if you're working within the context of genetic epidemiology, however. No shameless plug intended, but my thesis explored the rare disease assumption as applied to genetic epidemiology. My advisor and I discovered that the rare disease assumption is necessary but not sufficient to assume the relative risk and the odds ratio are approximately equal. In these cases, we saw through simulations that relying on the rare disease assumption (with no additional assumptions) might lead to inflated Type I error rates, misrepresented Type II error rates, and biased estimates of the relative risk. These generally depended on factors like minor allele frequency, but we could not conclusively establish or quantify an explicit relationship. Just something to consider depending on the context of your problem!