*by David Lillis, Ph.D.*

In my last couple of articles (Part 4, Part 5), I demonstrated a logistic regression model with binomial errors on binary data in R’s glm() function.

But one of wonderful things about glm() is that it is so flexible. It can run so much more than logistic regression models.

The flexibility, of course, also means that you have to tell it exactly which model you want to run, and how.

In fact, we can use generalized linear models to model count data as well.

In such data the errors may well be distributed non-normally and the variance usually increases with the mean values.

As with binary data, we use the glm() command, but this time we specify a Poisson error distribution and the logarithm as the link function.

The natural log is the default link function for the Poisson error distribution. It works well for count data as it forces all of the predicted values to be positive.

In the following example we fit a generalized linear model to count data using a Poisson error structure. The data set consists of counts of high school students diagnosed with an infectious disease within a period of days from an initial outbreak.

cases <-
structure(list(Days = c(1L, 2L, 3L, 3L, 4L, 4L, 4L, 6L, 7L, 8L,
8L, 8L, 8L, 12L, 14L, 15L, 17L, 17L, 17L, 18L, 19L, 19L, 20L,
23L, 23L, 23L, 24L, 24L, 25L, 26L, 27L, 28L, 29L, 34L, 36L, 36L,
42L, 42L, 43L, 43L, 44L, 44L, 44L, 44L, 45L, 46L, 48L, 48L, 49L,
49L, 53L, 53L, 53L, 54L, 55L, 56L, 56L, 58L, 60L, 63L, 65L, 67L,
67L, 68L, 71L, 71L, 72L, 72L, 72L, 73L, 74L, 74L, 74L, 75L, 75L,
80L, 81L, 81L, 81L, 81L, 88L, 88L, 90L, 93L, 93L, 94L, 95L, 95L,
95L, 96L, 96L, 97L, 98L, 100L, 101L, 102L, 103L, 104L, 105L,
106L, 107L, 108L, 109L, 110L, 111L, 112L, 113L, 114L, 115L),
Students = c(6L, 8L, 12L, 9L, 3L, 3L, 11L, 5L, 7L, 3L, 8L,
4L, 6L, 8L, 3L, 6L, 3L, 2L, 2L, 6L, 3L, 7L, 7L, 2L, 2L, 8L,
3L, 6L, 5L, 7L, 6L, 4L, 4L, 3L, 3L, 5L, 3L, 3L, 3L, 5L, 3L,
5L, 6L, 3L, 3L, 3L, 3L, 2L, 3L, 1L, 3L, 3L, 5L, 4L, 4L, 3L,
5L, 4L, 3L, 5L, 3L, 4L, 2L, 3L, 3L, 1L, 3L, 2L, 5L, 4L, 3L,
0L, 3L, 3L, 4L, 0L, 3L, 3L, 4L, 0L, 2L, 2L, 1L, 1L, 2L, 0L,
2L, 1L, 1L, 0L, 0L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 0L, 0L,
0L, 1L, 1L, 0L, 0L, 0L, 0L, 0L)), .Names = c("Days", "Students"
), class = "data.frame", row.names = c(NA, -109L))
attach(cases)
head(cases)
Days Students
1 1 6
2 2 8
3 3 12
4 3 9
5 4 3
6 4 3

The mean and variance are different (actually, the variance is greater). Now we plot the data.

plot(Days, Students, xlab = "DAYS", ylab = "STUDENTS", pch = 16)

Now we fit the glm, specifying the Poisson distribution by including it as the second argument.

model1 <- glm(Students ~ Days, poisson)
summary(model1)
Call:
glm(formula = Students ~ Days, family = poisson)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.00482 -0.85719 -0.09331 0.63969 1.73696
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.990235 0.083935 23.71 <2e-16 ***
Days -0.017463 0.001727 -10.11 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 215.36 on 108 degrees of freedom
Residual deviance: 101.17 on 107 degrees of freedom
AIC: 393.11
Number of Fisher Scoring iterations: 5

The negative coefficient for Days indicates that as days increase, the mean number of students with the disease is smaller.

This coefficient is highly significant (p < 2e-16).

We also see that the residual deviance is greater than the degrees of freedom, so that we have over-dispersion. This means that there is extra variance not accounted for by the model or by the error structure.

This is a very important model assumption, so in my next article we will re-fit the model using quasi poisson errors.

****

See our full R Tutorial Series and other blog posts regarding R programming.

**About the Author:** *David Lillis has taught R to many researchers and statisticians. His company, Sigma Statistics and Research Limited, provides both on-line instruction and face-to-face workshops on R, and coding services in R. David holds a doctorate in applied statistics.*

Count variables are common dependent variables in many fields. For example:

- Number of diseased trees
- Number of salamander eggs that hatch
- Number of crimes committed in a neighborhood

Although they are numerical and look like they should work in linear models, they often don’t.

Not only are they discrete instead of continuous (you can’t have 7.2 eggs hatching!), they can’t go below 0. And since 0 is often the most common value, they’re often highly skewed — so skewed, in fact, that transformations don’t work.

There are, however, generalized linear models that work well for count data. They take into account the specific issues inherent in count data. They should be accessible to anyone who is familiar with linear or logistic regression.

In this webinar, we’ll discuss the different model options for count data, including how to figure out which one works best. We’ll go into detail about how the models are set up, some key statistics, and how to interpret parameter estimates.

**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.**

### About the Instructor

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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