# Regression where dependent variable not normally distributed - logistic regression, negative binomial model, poisson model?

I'm conducting a study about social media posting with pets. My dependent variable is pet posting frequency (total pet pictures/total pictures posted), and a significant number of participants did not post pictures with pets in the 2-week time frame that I set, leaving a lot of zeros in the dataset. The independent variables are various psychological measures that are scored to indicate things like anxiety and depression.

Because of the zeroes, I'm told I won't be able to use multiple regression as I had planned, and that I will need to analyze the data using nonparametric statistics. Here were some of the suggestions:

1) split the data into those who post/ those who don't, then I can use logistic regression. There is also the potential to use an ordinal logistic regression model and to separate the dataset into three categories: those who do not post at all, those who post a little, and those who post a lot of pet pictures (though these parameters would be determined arbitrarily).

2) A negative binomial model and a poisson model

If anyone can explain the pros/cons, or what the most statistically sound approach would be to figuring out an appropriate regression to model this, I'd really appreciate it. Also, any additional information about the above 2 suggestions or others (examples, links) would be immensely helpful. I'm not a statistics person, so if you can explain in as basic terms as possible that reference my problem, that would be most helpful.

Thanks in advance for your time.

• The ratio of "total pet pictures/total pictures posted" is not a frequency, it's a rate. The count of total pet pictures is a frequency and, as such, is appropriate information for the models you are asking about. If you want to compare the performance of a ratio model with zero-inflated count or hurdle models then a tobit model approach isn't a bad idea. Truncated regression (at zero) would be another option. There's lots of literature about models for limited dependent variables out there, which is what you're dealing with. – DJohnson Mar 1 '17 at 18:45

## 1 Answer

Whoever told you you couldn't use multiple regression is wrong. To be sure, Negative Binomial and Poisson models are multiple regression models: they are a particular type of multiple regression. Multiple regression simply refers to the presence of multiple independent variables in your regression equation.

That being said, it's not clear from your posting if your dependent variable is pet posting frequency (i.e. a number) as you wrote or a proportion as you wrote parenthetically (i.e. "total pet pictures/total pictures posted"). I'll assume you meant frequency (i.e. an integer-valued count of the number of pet pictures posted).

The problem you describe can probably be best modeled by so called "hurdle models." These are models where there is some sort of sequential process that results in no events until some barrier or "hurdle" is overcome (in this case the decision to post or not to post a pet picture) and then, once the hurdle (decision) is overcome, the number of events is realized.

A zero-inflated poisson (ZIP) model might also be appropriate for your data. Search here on Cross Validated, and you'll see several discussions that explain the differences between the two types of models to get a feel for which one would best serve your needs. For example, see here, here, or here.

Also, you may find it helpful to examine the 1996 paper by Gurmu and Trivedi. In it they model the number of recreational boating trips by a family in one year and deal with the same "excess zero" problem and censoring problem you face in your own study. (Gurmu, S. and Trivedi, P. K. "Excess zeros in count models for recreational trips." Journal of Business and Economic Statistics (1996), 14, 469–477). I suspect you could model your two-week pet picture posting data the exact same way as they did recreational boating trips in a year.

• Why a hurdle model and not a (possibly zero-inflated) negative binomial model with the log(number of postings) as an offset (people with no postings at all, do not really contribute)? Negative binomial is typically preferable to Poisson, because it captures inter-individual variability that is not explained by included covariates. – Björn Jan 4 '17 at 8:42
• Absolutely @Björn. A ZINB model is well suited too. I suggested a ZIP model as it's a simpler model and certainly a stepping stone to other models that the OP will discover upon reading about the ZIP and/or hurdle models. Which model is right for the OP will depend on the intricacies of his study and data. – StatsStudent Jan 4 '17 at 20:43
• Thanks for the information so far. I found this article that I followed quite easily using R, and have what looks like a nicely fitting hurdle model using a negative binomial distirbution. Any thoughts on whether this is the right track: data.library.virginia.edu/getting-started-with-hurdle-models – sqlnewbie1979 Jan 4 '17 at 21:35
• Also, I was looking to do this on a proportion as I referenced. Does that change your answer? For example, first the hurdle is that someone posts in the first place, then perhaps how many are with pets. Thanks again! – sqlnewbie1979 Jan 4 '17 at 21:56
• You keep using the phrases that make me thing an actual count vs. a proportion (i.e. when I hear phrases like "how many are with pets, it makes me believe you mean an interger-valued response rather than a proprotion). The negative binomial a discrete probability distribution of the number of successes in N independent Bernoulli trial, so it is generally not appropriate for proportions. – StatsStudent Jan 5 '17 at 1:24