# Interpretation of incidence-rate ratios

So, I want to fit a random effects negativ-binomial model. For such a model STATA can produce exponentiated coefficients. According to the help file such coefficients can be interpreted as incidence-rate ratios. Unfortunately I am not a native English speaker and I do not really understand what incidence-rate ratios are or how I could translate them.

So my question is, how can I interpret incidence-rate ratios. E.g.:

If the model gives me an incidence-rate ratios of .7 for one var. would that mean that the number of expected observations (counts) on the dependent var. changes by .7 if the independent var changes by one unit?

Can anyone help?

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Ah, the incident rate ratio, my old friend.

You're correct. If we have a 0/1 variable, an IRR of 0.7 means that those with X = 1 will have 0.7 times the incident events as those with X = 0. If you want the actual number of predicted counts, you'll have to back-track to the unexponentiated model coefficients. Then your expected cases would be:

counts = exp(B0 + B1*X), where B0 is the intercept term, B1 is the coefficient for your variable (equal in this example to ~-0.3365) and X is the value of X for whatever group you're trying to calculate this for. I find that's occasionally a useful sanity check to make sure I haven't done something horribly wrong in the model itself.

If you're more familiar with Hazard Ratios from other areas of survival analysis, note that an incidence rate ratio is a hazard ratio, just with a very particular set of assumptions to it - that the hazard is both proportional and constant. It can be interpreted the same way.

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Thank you for your quick answer. The original coefficient is -.3365 but I think thats okay as exp(-.3365) roughly is .7 as well right?! –  Adrian Oct 14 '11 at 20:14
Heh - good job catching an error of mine. Protip: ln(7) = / = ln(0.7) –  Fomite Oct 14 '11 at 20:17

Yes, that sounds about right: to be precise, the expected count is multiplied by a factor of .7 when the independent variable increases by one unit.

The term 'incidence rate ratio" assumes that you're fitting a model with an exposure() (offset) term as well, typically specifying the time each unit was observed for, in which case instead of expected counts you have expected counts per unit time, i.e. rates. Calling them incidence rates is terminology from epidemiology.

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Great, thank you! But your answer leads me to a second question. I am fitting a model in which each unit is a number of events per month. So the exposure is the same for all units. So far I assumed that I don't have to define the exposure option in STATA if the exposure is the same for all units. Is that right or am I making a mistake here? –  Adrian Oct 14 '11 at 20:06
Yes, that's right. –  onestop Oct 15 '11 at 11:34