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.

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**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.*

Deidre Coghlan says

I am struggling to understand what an appropriate goodness-of-fit test would be if you used Gaussian? Would I need to look at the AIC values for the best fit ? Thanks!

Atik says

model1 <- glm(Students ~ Days, poisson)

In this comment is Students variable follow IID Poisson random variables??

Karen Grace-Martin says

Atik, tecnhically it’s Students|Days is assumed to follow iid Poisson. That assumption would have to be checked.

vikas says

sir

i want to know the code for poisson regression for count data set without using inbuild function i.e glm() function.

can you help me.

Thanks

Fran says

I would like to know that, too. I’m with Robert, why is it greater?

Thank’s

Daz says

Hi Fran,

In my R Companion textbook, I found another way to test for overdispersion. If you divide Residual deviance / residual df and the value is above 1.5 (which is generally considered the threshold), you should use a quasi model.

In this case, it would be 0.945.

The site for R-Companion (online) is http://rcompanion.org/rcompanion/e_07.html

Please know that I’m looking at similar models, but using logistic regressions (for which the answer may be slightly different). Best!

Robert says

I don’t have understand what you said.

You said “the residual deviance is greater than the degrees of freedom”, but df are 107 and residual deviance is 101.17. Could you open my mind ? Thank you