How should I interpret Exp (B) = 3.661E-11 in negative binomial regression? I am stumped when it comes to these E-11, E-10, and E-12 values. I got these values on interaction terms consisting of categorical variables. Can anyone assist?

I think I am clear on how to interpret it when Exp (B) is greater than 1. For Exp (B) values greater than one, the criterion variable (Y) is expected to be the value of the Exp (B) times higher as a result of a one unit change in the (X) predictor. One would apply the following formula: (Exp (B) – 1.00) * 100. If Exp(B) = 1.5 then: (1.5 – 1.00) * 100 = 0.5 * 100 = 50%. One could then say that for every one standard deviation increase in the predictor (X) one would expect the mean criterion variable (Y) to increase by 50%.

For Exp (B) values that are less than one, the same approach is used. If the Exp (B) value of a predictor (X) were 0.75 it would indicate that with one unit increase in the predictor (X) the predicted score of the criterion variable would be expected to be a factor of 0.75 times the mean criterion variable (Y). To determine the percentage of decrease in the mean rate of the criterion variable, one would use the same formula: (Exp (B) – 1.00) * 100 such that (0.75 – 1) * 100 = – 0.25 * 100 = – 0.25%. One could say that the mean criterion variable (Y) would be expected to decrease by 25% for every one standard deviation increase in the predictor variable (X).

When the Exp(B) value is so small like 3.661E-11 then applying the formula results in 1 – 0.00000000003661 = .9999 * 100 = 99.99% so the mean criterion variable would be 99.99% lower? I am just having a difficult time wrapping my brain around that. Could it be that although these Exp(B) values are statistically significant, that they really could be indicating a problem with the data? The frequency counts of the interaction terms that generated these values are very low (between 1 and 4).

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#### Best Answer

I think your last sentence hits the nail on the head here. What is happening is that for some combination of your predictor variables there is a zero count. So the software is trying to generate a coefficient which will send the intercept (which is presumably positive) to zero. But there is no multiplier which does that so it stops when it reaches a very small predicted value. If you look at the coefficients on the log scale it is about -24 which is trying to go to $-infty$ but stopping there. For advice on what to do you could try browsing the tag separation or looking at How to deal with perfect separation in logistic regression? Note that most of those are about logistic regression but similar issues arise for you here.

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