# Regression on fractional data

I'm writing a piece of research into the death rate of minorities on tv shows, as there's been a lot of outcry over the deaths of queer women in media but very little actual analysis on the topic. I've done multivariate data analysis in the past but nothing needing regression; I'm a second-year undergraduate statistics student.

In order to control for the correlation between total number of characters and number of deaths, I'm intending to express the deaths per year as a fractional value. Aside from a simple linear regression, are there any regression models that would be appropriate? I've looked into binomial regression (which might works, but I'm unsure) and Poisson regression already, which certainly wouldn't work as it's not integer data.

• Do you have the number of characters and the number of deaths, or just the fraction? – Ian_Fin Aug 30 '16 at 11:22
• You do in fact have integers - the number of deaths. Depending on what data you have exactly either logistic or Poisson regression might be appropriate so can you show us some more detail? – mdewey Aug 30 '16 at 12:12
• If you're directly comparing sets of proportions by category of person there's already tests for that -- I wouldn't use linear regression. If you have other covariates (besides the exposure variable, i.e. the number of characters) then I'd look at say logistic regression, but otherwise I'd probably look at something like a chi-squared test. – Glen_b Aug 31 '16 at 0:42
• I'm examining the rate of deaths, as a proportion of the total population, over time; the goal is to find out if 2016 is a significant outlier in terms of death and thus check the validity of all the claims that it is. My data is currently sorted by character, year they were introduced, year they left, and whether they died, so there's transformations to be done regardless. The issue is that there's likely an underlying relationship between the number of characters present in a year and the number that die, and that could skew the result of an analysis that just uses the number of deaths. – user129472 Sep 1 '16 at 4:25
• stats.stackexchange.com/questions/142338/… – kjetil b halvorsen Oct 27 '17 at 12:05

But I think that we can take care of that within the glm framework, I will first discuss a poisson regression (with log link). Let $Y_t$ be the number of deaths year $t$, the exposure $E_t$ the total number of characters, and possibly $x_t$ other covariates (in intensive form, see Goodness of fit and which model to choose linear regression or Poisson). Then $Y_t \sim \cal{P}(\lambda_t)$, $\lambda_t = \exp(\beta_0+1\cdot \log E_t + \beta^T x_t)$ where the log exposure $\log E_t$ is used as an offset (see Difference between offset and exposure in Poisson Regression). This is a baseline model which is not modelling any interaction between exposure and number of deaths. Then to see if there is any such interaction, extend the model as $\lambda_t = \exp(\beta_0+1\cdot \log E_t + s(\log E_t) + \beta^T x_t)$ where $s(\cdot)$ represents a smooth term, maybe a spline, or more simply a linear term. Then the estimate of the new term indicates if there is such an interaction. If there is overdispersion all this can be done within a negative binomial model.
One could argue (since the number of deaths $Y_t$ is bounded above by the exposure) that a binomial likelihood is more natural. That would have to be used with a log link, see for instance Log binomial regression with a case-control sample