# Poisson Regression

### Member Training: Types of Regression Models and When to Use Them

February 1st, 2013 by

Linear, Logistic, Tobit, Cox, Poisson, Zero Inflated… The list of regression models goes on and on before you even get to things like ANCOVA or Linear Mixed Models. In this webinar, we will explore types of regression models, how they differ, how they’re the same, and most importantly, when to use each one.

Note: This training is an exclusive benefit to members of the Statistically Speaking Membership Program and part of the Stat’s Amore Trainings Series. Each Stat’s Amore Training is approximately 90 minutes long.

Not a Member? Join! Karen Grace-Martin helps statistics practitioners gain an intuitive understanding of how statistics is applied to real data in research studies.

She has guided and trained researchers through their statistical analysis for over 15 years as a statistical consultant at Cornell University and through The Analysis Factor. She has master’s degrees in both applied statistics and social psychology and is an expert in SPSS and SAS.

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### A Few Resources on Zero-Inflated Poisson Models

February 15th, 2010 by

1. For a general overview of modeling count variables, you can get free access to the video recording of one of my The Craft of Statistical Analysis Webinars:

Poisson and Negative Binomial for Count Outcomes

2. One of my favorite books on Categorical Data Analysis is:

Long, J. Scott. (1997).  Regression models for Categorical and Limited Dependent Variables.  Sage Publications.

It’s moderately technical, but written with social science researchers in mind.  It’s so well written, it’s worth it.  It has a section specifically about Zero Inflated Poisson and Zero Inflated Negative Binomial regression models.

3. Slightly less technical, but most useful only if you use Stata is Regression Models for Categorical Dependent Variables Using Stata, by J. Scott Long and Jeremy Freese.

4. UCLA’s ATS Statistical Software Consulting Group has some nice examples of Zero-Inflated Poisson and other models in various software packages.

### Zero-Inflated Poisson Models for Count Outcomes

February 12th, 2010 by

There are quite a few types of outcome variables that will never meet ordinary linear model’s assumption of normally distributed residuals.  A non-normal outcome variable can have normally distribued residuals, but it does need to be continuous, unbounded, and measured on an interval or ratio scale.   Categorical outcome variables clearly don’t fit this requirement, so it’s easy to see that an ordinary linear model is not appropriate.  Neither do count variables.  It’s less obvious, because they are measured on a ratio scale, so it’s easier to think of them as continuous, or close to it.  But they’re neither continuous or unbounded, and this really affects assumptions.

Continuous variables measure how much.  Count variables measure how many.  Count variables can’t be negative—0 is the lowest possible value, and they’re often skewed–so severly that 0 is by far the most common value.  And they’re discrete, not continuous.  All those jokes about the average family having 1.3 children have a ring of truth in this context.

Count variables often follow a Poisson or one of its related distributions.  The Poisson distribution assumes that each count is the result of the same Poisson process—a random process that says each counted event is independent and equally likely.  If this count variable is used as the outcome of a regression model, we can use Poisson regression to estimate how predictors affect the number of times the event occurred.

But the Poisson model has very strict assumptions.  One that is often violated is that the mean equals the variance.  When the variance is too large because there are many 0s as well as a few very high values, the negative binomial model is an extension that can handle the extra variance.

But sometimes it’s just a matter of having too many zeros than a Poisson would predict.  In this case, a better solution is often the Zero-Inflated Poisson (ZIP) model.  (And when extra variation occurs too, its close relative is the Zero-Inflated Negative Binomial model).

ZIP models assume that some zeros occurred by a Poisson process, but others were not even eligible to have the event occur.  So there are two processes at work—one that determines if the individual is even eligible for a non-zero response, and the other that determines the count of that response for eligible individuals.

The tricky part is either process can result in a 0 count.   Since you can’t tell which 0s were eligible for a non-zero count, you can’t tell which zeros were results of which process.  The ZIP model fits, simultaneously, two separate regression models.  One is a logistic or probit model that models the probability of being eligible for a non-zero count.  The other models the size of that count.

Both models use the same predictor variables, but estimate their coefficients separately.  So the predictors can have vastly different effects on the two processes.

But a ZIP model requires it be theoretically plausible that some individuals are ineligible for a count.  For example, consider a count of the number of disciplinary incidents in a day in a youth detention center.  True, there may be some youth who would never instigate an incident, but the unit of observation in this case is the center.  It is hard to imagine a situation in which a detention center would have no possibility of any incidents, even if they didn’t occur on some days.

Compare that to the number of alcoholic drinks consumed in a day, which could plausibly be fit with a ZIP model.  Some participants do drink alcohol, but will have consumed 0 that day, by chance.   But others just do not drink alcohol, so will never have a non-zero response.  The ZIP model can determine which predictors affect the probability of being an alcohol consumer and which predictors affect how many drinks the consumers consume.  They may not be the same predictors for the two models, or they could even have opposite effects on the two processes.

### The Exposure Variable in Poisson Regression Models

January 23rd, 2009 by

Poisson Regression Models and its extensions (Zero-Inflated Poisson, Negative Binomial Regression, etc.) are used to model counts and rates. A few examples of count variables include:

– Number of words an eighteen month old can say

– Number of aggressive incidents performed by patients in an impatient rehab center

Most count variables follow one of these distributions in the Poisson family. Poisson regression models allow researchers to examine the relationship between predictors and count outcome variables.

Using these regression models gives much more accurate parameter (more…)

### Poisson Regression Analysis for Count Data

December 31st, 2008 by

There are many dependent variables that no matter how many transformations you try, you cannot get to be normally distributed.  The most common culprits are count variables–the variable that measures the count or rate of some event in a sample.  Some examples I’ve seen from a variety of disciplines are:

Number of eggs in a clutch that hatch
Number of domestic violence incidents in a month
Number of times juveniles needed to be restrained during tenure at a correctional facility
Number of infected plants per transect

A common quality of these variables is that 0 is the mode–the most common value.  1 is the next most common, 2 the next, and so on.  In variables with low expected counts (number of cars in a household, number of degrees earned), (more…)

### Regression Models for Count Data

October 24th, 2008 by

One of the main assumptions of linear models such as linear regression and analysis of variance is that the residual errors follow a normal distribution. To meet this assumption when a continuous response variable is skewed, a transformation of the response variable can produce errors that are approximately normal. Often, however, the response variable of interest is categorical or discrete, not continuous. In this case, a simple transformation cannot produce normally distributed errors.

A common example is when the response variable is the counted number of occurrences of an event. The distribution of counts is discrete, not continuous, and is limited to non-negative values. There are two problems with applying an ordinary linear regression model to these data. First, many distributions of count data are positively skewed with many observations in the data set having a value of 0. The high number of 0’s in the data set prevents the transformation of a skewed distribution into a normal one. Second, it is quite likely that the regression model will produce negative predicted values, which are theoretically impossible.

An example of a regression model with a count response variable is the prediction of the number of times a person perpetrated domestic violence against his or her partner in the last year based on whether he or she had witnessed domestic violence as a child and who the perpetrator of that violence was. Because many individuals in the sample had not perpetrated violence at all, many observations had a value of 0, and any attempts to transform the data to a normal distribution failed.

An alternative is to use a Poisson regression model or one of its variants. These models have a number of advantages over an ordinary linear regression model, including a skew, discrete distribution, and the restriction of predicted values to non-negative numbers. A Poisson model is similar to an ordinary linear regression, with two exceptions. First, it assumes that the errors follow a Poisson, not a normal, distribution. Second, rather than modeling Y as a linear function of the regression coefficients, it models the natural log of the response variable, ln(Y), as a linear function of the coefficients.

The Poisson model assumes that the mean and variance of the errors are equal. But usually in practice the variance of the errors is larger than the mean (although it can also be smaller). When the variance is larger than the mean, there are two extensions of the Poisson model that work well. In the over-dispersed Poisson model, an extra parameter is included which estimates how much larger the variance is than the mean. This parameter estimate is then used to correct for the effects of the larger variance on the p-values. An alternative is a negative binomial model. The negative binomial distribution is a form of the Poisson distribution in which the distribution’s parameter is itself considered a random variable. The variation of this parameter can account for a variance of the data that is higher than the mean.

A negative binomial model proved to fit well for the domestic violence data described above. Because the majority of individuals in the data set perpetrated 0 times, but a few individuals perpetrated many times, the variance was over 6 times larger than the mean. Therefore, the negative binomial model was clearly more appropriate than the Poisson.

All three variations of the Poisson regression model are available in many general statistical packages, including SAS, Stata, and S-Plus.

References:

• Gardner, W., Mulvey, E.P., and Shaw, E.C (1995). “Regression Analyses of Counts and Rates: Poisson, Overdispersed Poisson, and Negative Binomial Models”, Psychological Bulletin, 118, 392-404.
• Long, J.S. (1997). Regression Models for Categorical and Limited Dependent Variables, Chapter 8. Thousand Oaks, CA: Sage Publications.