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I am interested in performing a regression on data on a population. This population causes events X and Y. I have monthly data for population, event X, and event Y. Do I need to change my variables X and Y into rates (X divided by population) before running the regression?

Some details about the model: the population are students at a university. That's one IV. Another IV will be a yet to be determined demographic characteristic. As output variables, I have cheating instances and plagiarism instances. I'm going to combine these into one DV "academic dishonesty" and run three models, one with cheating as a DV, the other with plagiarism, and the last with academic dishonesty as the DV. In all cases university population and demographics will be the two IVs.

Should I change these cheating and plagiarism instances into rates? Thanks for the advice.

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    $\begingroup$ It depends on the model and on what these events are. Could you edit this question to include some more information about that? $\endgroup$
    – whuber
    Aug 6 '14 at 21:37
  • $\begingroup$ Usually events occur within a population, and are not caused by it. I feel there's lots more required before we can answer this question. $\endgroup$ Aug 7 '14 at 0:21
  • $\begingroup$ Fair enough. Added details. At first I did not think rates were necessary but a colleague recommended switching to rates to control for the effect of population (i.e. more students = more cheating) as a way to control for population size. But I feel as though since I'm already including population in my model, changing to rates will "double control" for population. Thoughts? $\endgroup$
    – digits
    Aug 7 '14 at 14:50
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    $\begingroup$ You can model the rate of, say, cheating, without transforming anything by adding log(population) as an offset to the regression model. In R: model <- glm(cheating ~ demog1 + demog2 + offset(log(population)), family=poisson) where all but the demographic variables are numbers of students. $\endgroup$ Aug 7 '14 at 15:53
  • $\begingroup$ Thanks folks. I liked going through your suggestions and I've worked it out. Enjoy your weekends $\endgroup$
    – digits
    Aug 8 '14 at 23:43
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I would consider a log transformation, which imposes a constant elasticity of substitution on your model.

If you estimate

$$ (NoCases) = \alpha + \beta (Characteristics) $$

In this case, $\beta$ is the increase on the number of cases resulting from a marginal increase in the characteristics.

You will of course see that larger universities have a higher number of cases. You could control for university size by including that as a characteristic, but there may still be some effects that you'll want to control for in another way. Your colleague is correct that you should do something to control for this.

Using rates is one way, but I would suggest you estimate

$$ log(NoCases) = \alpha + \beta(log(Characteristics))$$

In this case, $\beta$ represents the percent increase in cases due to a percent increase in your characteristics. This solves your rates issue, and will likely correct some outlier problems, too!

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A feature of models in which the dependent variable is a rate per unit of population is that they are likely to be subject to heteroscedasticity. Other things being equal, smaller populations are likely to be associated with greater variability in rates. An easy way to see this is to compare the rates for one student and for a large population of students, making the simplifying assumption that no student has more than one instance of academic dishonesty in any one month. The rate for the one student must be either 0 or 1. Within the population, however, it is likely that some students are honest and some dishonest, so that the rate will be between 0 and 1. A larger population allows for more averaging out than a smaller one.

Although heteroscedasticity does not lead to bias in regression coefficients estimated by OLS, it does imply that the conventionally calculated standard errors of those coefficients are unreliable. This would mean, in particular, that a test of significance of the estimated coefficient of your demographic characteristic would be unreliable.

The above is not intended to suggest that you should avoid using rates. However, if you do use rates and you intend to apply hypothesis tests or calculate confidence intervals, then you need to consider how you will address heteroscedasticity arising from different population sizes. Three approaches you might consider are:

  1. OLS with robust or heteroscedasticity-consistent standard errors.
  2. Weighted least squares, giving a higher weighting to larger populations since they are expected to be associated with lower rate variability.
  3. Calculating your regression on a sub-sample of your data selected so that the populations are fairly similar. This is unlikely to be helpful unless you have data for a very large number of universities.

All these approaches have their pros and cons. One of many sources which consider in general terms the relative merits of 1 and 2 above is here.

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