# Is there a difference between controlling for population size directly vs. putting variables in per capita terms?

I want to look at the impact of a number of factors (e.g nutrition, alcohol sales, GDP, etc.) on mortality. I have a dataset with 10-30 years of data on these variables aggregated at the state level. (i.e. I'm looking at the impact of total state-wide alcohol sales in a given year on total deaths in that state in that year). Obviously, population size is a likely omitted variable if I don't include it. However, I am wondering whether I should include it by transforming variables to their per-capita form or whether I should control for population directly.

It seems that creating per capita variables is the conventional thing to do, but it seems to me that just adding in population size as a control is simpler and it allows for population size to have its own unique effect on the dependent variable. For example, in this case, we might think that states with larger populations, all else being equal, have more deaths because it is harder to organize the healthcare system efficiently to meet the demands of a large population.

What should I do here? Are these two approaches functionally equivalent?

## 2 Answers

Its not that simple, at least not in a linear additive model. It also fundamentally depends on your dependent variable.

Take a simple example where the true Data Generating Process (DGP) a increase from 1 in a 1000 individuals with a certain condition to 2 in a 1000 is associated with a 10% increase in some variable of interest and there is no association between population size proper and your dependent variable. Take two states with resp. 10,000 and 100,000 inhabitants and resp. 20 and 40 individuals with a certain treatment. You propose adding as regressors [20, 10,000] and [40, 100,000] whereas per capita regressors would have been 2 and 0.4. Your model will not accurately reflect the association you are looking for because the per capita ratio between the two observations is decreasing (2 versus 0.4) whereas in absolute terms it is increasing (20 versus 40). Therefore, the two approaches are clearly not equivalent. Which you prefer depends on your dependent variable and the relationship you are testing, but in an OLS framework variables are independent so adding the total population as one regressor doesn't mean the other regressor is scaled by such a variable - they are approached as independent numbers by the model. Note that if your dependent variable is a rate or per capita value, you generally would want your regressors to be the same (just think about the above situation).

There is no problem, however, adding both per capita ratios AND some population size indicator (probably best to take the natural log of the true population) - that seems to be the best approach in your case?

Hope this helps

• "Your model will not accurately reflect the association you are looking for because the per capita ratio between the two observations is decreasing (2 versus 0.4) whereas in absolute terms it is increasing (20 versus 40)." But wouldn't the model with population size as a control take into account the fact that the population sizes are different? – leecarvallo Aug 8 '19 at 16:10
• Again it depends on what you are looking for. In a model with just the per capita variable as a regressor, the interpretation of your coefficient will be the association between an increase of one per thousand on your dependent variable. In a model with population size and for example the actual number of people with some condition, the interpretation for the two coefficients is the association between a. one more person with a certain condition and b. between one more person living in your state and your dependent variable, conditional on one another. Clearly not the same, right? – Mark Verhagen Aug 8 '19 at 16:31
• I suppose they're not literally the same, but with the latter model, you're getting at the impact of one more person with a certain condition holding population constant, which is functionally very close, right? Like other the math of figuring out state-wide averages for variables, wouldn't we expect that the relationship between, say, per capita mortality and per capita alcohol sales would be very similar to the relationship between mortality and alcohol sales while controlling for population size? – leecarvallo Aug 8 '19 at 17:12
• Yes, exactly. In fact, you'd just be dividing the left and right hand side by a constant (e.g. 1,000). It just depends on what you're testing, and it would be a bad idea if you would e.g. have an absolute number as the dependent variable and a ratio as the independent! Given that it is 'just' adding a constant on both sides, adding the total population in the first or second versions, it is simply adding an additional regressor for the total state size into the equation. It doesn't functionally change the interpretation of the other regressor, except it being conditional on something else then – Mark Verhagen Aug 9 '19 at 7:15

Let $$Y_{ij}$$ be the number of death at state $$i$$ in year $$j$$, $$P_{ij}$$ be the corresponding # of people, $$X_{ij}$$ be a vector of other covariates, such as GDP.

Then fit a Poisson regression model with random state specific intercept: $$Y_{ij} \sim Poisson(\lambda_{ij}P_{ij})$$ $$\lambda_{ij} = exp(X_{ij}\beta + \epsilon_i)$$ $$\epsilon_{ij}\sim N(0,\sigma)$$ In most software, $$\log(P_{ij})$$ is called as offset.