Modeling Opioid Mortality Rates using Poisson Regression

This is a general statistics question about Poisson Regression. I have age-adjusted and crude rates for opioid mortality for the period 2014-2016. I want to use Poisson regression, but I am not sure if this is the right statistical analysis for the type of data that I have. Now, I am not that well verse when it comes to statistics, so I was hoping someone could give me some advice on how to proceed. I know that Poisson regression use models count, but I believe you can model mortality rates too. My dependent variable would be the age-adjusted or crude rates, while my independent variables would consist of socio-economic indicators such as low educational attainment, poverty, unemployment, occupation, median household income as percentages pulled from the American Community Survey 2013-2017.

After doing a little bit of research, I realized that I need to use an offset if I am trying to model mortality rates. An example I have crude rates for opioid mortality for 100 counties in North Carolina. My dependent variable would be my death counts for each county for 2014-2016, while my offset would be the total population at risk for the study period (2014-2016).

I note your additional information in the comments stating that you do have access to the underlying numerator and denominator values that give you the estimated rates. As others have pointed out, this is necessary for models of this kind.

In any case, what I would like to suggest here is that Poisson regression is (almost) never a good model; you should always use a negative binomial model (or some other two-parameter model) instead. Because the Poisson distribution has only a single parameter, the variance of the response in the Poisson model is not "free" and so it is not estimated properly from the data. Models with two parameters allow us to estimate the mean and variance from the data and so they are preferable in almost all cases. (And of course, models with even more parameters allow estimation of higher order moments, but estimating the mean and variance well is the most important part.)

Do you have numerator and denominator values for those mortality rates? To fit a Poisson model using rates, you need to have the original counts (the numerator and the denominator of the rate). The Poisson model is fit to the counts and uses the log of the denominator as an offset (exposure) variable.

Now if the denominators are all the same for these rates, you actually don't need them, but you still need the numerators, so you could create a "count" model. Modeling rates, rather than counts, is important when values of numerators are affected by values of denominators. If the denominators are all the same and you know what they are, you can obtain the numerators by simple arithmetic.

For your crude rate variable, that should be easy, as it should be defined as something like the total number of deaths per year per 1,000 people. But for age-adjusted rates, that may not be possible, as they are computed using death counts for different age groups and counts of individuals in those age groups, so you would need those too, and even then you would not be able to arrive at one number that you could model in count form.

If you are unable to obtain numerator and/or denominator counts for one of your dependent variables, then Poisson is not an appropriate approach. Your alternative could potentially be a basic linear regression (OLS). You would need to of course check the assumptions of OLS estimation and perhaps do something like a log-transformation of your dependent variable to fix heteroscedasticity.

• I do have the original numerator and denominator that was calculated for the crude rates. Furthermore, I was thinking of applying Poisson Regression for white and nonwhite crude rates. I have the numerators and denominators for those as well. May 20 '19 at 15:28
• I also do not have a rate for each year, but rather a average rate for all three years for the total population, white, and nonwhite (2014-2016). May 20 '19 at 15:34
• The general stance here is contentious, although quite what is defined as Poisson regression is also an issue. The most important ingredient in Poisson regression is arguably a logarithmic link function. What the conditional distribution is can be regarded as secondary. Existence proof: there are implementations of Poisson regression that don't require two kinds of input, or even integer input at all. Naturally none of that denies a need to be careful about what you're doing. For more from this point of view, see blog.stata.com/2011/08/22/… May 21 '19 at 18:05
• Further, plain vanilla regression is often a poor alternative, which can be inappropriate for rates, as they can't be negative and are often heteroscedastic. May 21 '19 at 18:07
• It's a naturally a really good idea to use numerator and denominator information together if that is what you have. May 21 '19 at 18:11