# Linear Regression for a discrete count dependent variable?

I want to model the number of trips taken by households to investigate the effects of income, number of cars available, etc. on the number of trips.

One potential issue is that the probability distribution for Y_i at X_i won't be normally distributed. The number of trips are whole integers in a small range, which I believe violates an assumption of general linear models. How does this translate to models where the values the dependent variable can take on is also limited, but over a much larger range (for instance, regressions with income as the dependent variable)?

• A Poisson regression model would make sense here. A household can take any number of trip, so theoretically, there is not a real upper bound. With a Poisson regression model you don't assume normality of the responses anymore. – Greenparker Mar 7 '16 at 21:07
• When you say "GLM", what do you mean? I'd normally take it to be short for Generalized Linear Model (which as you can see in the first line of that wikipedia link, is commonly abbreviated to GLM. If that is what you mean, then this class of models exponential family includes models for count data, the simplest and most widely used of which would be Poisson regression). Do you mean general linear model instead? – Glen_b Mar 7 '16 at 21:17
• What does the distribution of the outcome look like? What is "small range?" I agree with @Greenparker to use Poisson, but if you have a "large number of zeroes," you should consider a zero-inflated model as well. There are statistical tests that will tell you which model is better. – Marquis de Carabas Mar 7 '16 at 21:51
• I suspect that a zero inflated Poisson or hurdle model might be more appropriate, depending on your definition of "trip" (and I'd encourage you to be very specific about your definitions of things like "trips", "income", and "cars available." For example, if by "trips" you mean trips by car, then you would probably not want to use a Poisson model since there is zero chance of taking a car trip if no car was available. Instead, you'd want to use something like a hurdle model that boils down to saying if there is no car available the probability is zero. Otherwise, model the probability. – StatsStudent May 14 '17 at 1:09