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I am regressing a count variable (adverse consequences) against a continuous predictor variable between 0 and 1 where all values are beween 0 and 0.5 (blood alcohol.

I am using a hierarchical negative binomial regression and thus the regression estimates are log-count. I would like to express the relationship between the predictor and the outcome as a rate-ratio, and usually would exponentiate the estimate to say something like 'an increase of 1 unit of the predictor leads to an x-unit increase in the likelihood of an adverse event occurring'.

Here is the output of the glmmADMB model:

              Estimate Std. Error   z value      Pr(>|z|)
(Intercept)  -1.236837   0.092043 -13.43760  3.640743e-41
EstimatedBAC 10.318858   0.428410  24.08641 3.470139e-128

So for each one-unit increase in blood alcohol the expected increase in the log-count of the number of adverse consequences increase by ten.

However in this case because the predictor is blood-alcohol level it doesn't make sense to talk of a 1-unit increase in x because a blood-alcohol value of 1 is blood that is actually just ethanol!

The exponent of this log-count is 30,299. So does that mean that for every 1% increase in blood-alcohol concentration the chances of having an adverse events increase by 302.99 times?

I don't know what I am required to do so that the regression coefficient can be expressed in a more intuitive way, or am I expressing the statement in the previous paragraph correctly? I have looked around CV and the web and haven't been able to find an answer that matches my problem.

llewmills
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  • Why not use [beta regression](http://stats.stackexchange.com/questions/29038/regression-for-an-outcome-ratio-between-0-and-1?noredirect=1&lq=1)? – Tim Dec 04 '16 at 07:36
  • Would that change the way I interpret this result, or allow it to be framed in an incorrect way? And is there something wrong with negative binomial for overdispersed count data? – llewmills Dec 04 '16 at 10:41

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You could always enter blood alcohol in different units like 1 percent (by multiplying your current value by 100. The coefficient would then have the properties you desire of referring to the increment in log count of a 1% increase (or when exponentiated would be the percent increase per 1%). There is nothing wrong with your other approach except I think you have multiplied when you meant to divide. I usually find this very confusing so perhaps you are right and I am wrong but it does need double checking. Perhaps wise to have an extra coffee first.

mdewey
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