# Poisson and Negative Binomial Regression Models

### Generalized Linear Models in R, Part 6: Poisson Regression for Count Variables

August 26th, 2015 by

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)

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.

****

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.

### Generalized Linear Models (GLMs) in R, Part 4: Options, Link Functions, and Interpretation

August 20th, 2015 by

Last year I wrote several articles (GLM in R 1, GLM in R 2, GLM in R 3) that provided an introduction to Generalized Linear Models (GLMs) in R.

As a reminder, Generalized Linear Models are an extension of linear regression models that allow the dependent variable to be non-normal.

In our example for this week we fit a GLM to a set of education-related data.

Let’s read in a data set from an experiment consisting of numeracy test scores (numeracy), scores on an anxiety test (anxiety), and a binary outcome variable (success) that records whether or not the students eventually succeeded in gaining admission to a prestigious university through an admissions test.

We will use the glm() command to run a logistic regression, regressing success on the numeracy and anxiety scores.

```
A <- structure(list(numeracy = c(6.6, 7.1, 7.3, 7.5, 7.9, 7.9, 8,
8.2, 8.3, 8.3, 8.4, 8.4, 8.6, 8.7, 8.8, 8.8, 9.1, 9.1, 9.1, 9.3,
9.5, 9.8, 10.1, 10.5, 10.6, 10.6, 10.6, 10.7, 10.8, 11, 11.1,
11.2, 11.3, 12, 12.3, 12.4, 12.8, 12.8, 12.9, 13.4, 13.5, 13.6,
13.8, 14.2, 14.3, 14.5, 14.6, 15, 15.1, 15.7), anxiety = c(13.8,
14.6, 17.4, 14.9, 13.4, 13.5, 13.8, 16.6, 13.5, 15.7, 13.6, 14,
16.1, 10.5, 16.9, 17.4, 13.9, 15.8, 16.4, 14.7, 15, 13.3, 10.9,
12.4, 12.9, 16.6, 16.9, 15.4, 13.1, 17.3, 13.1, 14, 17.7, 10.6,
14.7, 10.1, 11.6, 14.2, 12.1, 13.9, 11.4, 15.1, 13, 11.3, 11.4,
10.4, 14.4, 11, 14, 13.4), success = c(0L, 0L, 0L, 1L, 0L, 1L,
0L, 0L, 1L, 0L, 1L, 1L, 0L, 1L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L,
1L, 1L, 1L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L)), .Names = c("numeracy",
"anxiety", "success"), row.names = c(NA, -50L), class = "data.frame")```
`attach(A)`
```names(A)
[1] "numeracy" "anxiety"  "success"```
```head(A)
numeracy anxiety  success
1      6.6    13.8       0
2      7.1    14.6       0
3      7.3    17.4       0
4      7.5    14.9       1
5      7.9    13.4       0
6      7.9    13.5       1```

The variable ‘success’ is a binary variable that takes the value 1 for individuals who succeeded in gaining admission, and the value 0 for those who did not. Let’s look at the mean values of numeracy and anxiety.

```mean(numeracy)
[1] 10.722```
```mean(anxiety)
[1] 13.954```

We begin by fitting a model that includes interactions through the asterisk formula operator. The most commonly used link for binary outcome variables is the logit link, though other links can be used.

`model1 <- glm(success ~ numeracy * anxiety, binomial)`

glm() is the function that tells R to run a generalized linear model.

Inside the parentheses we give R important information about the model. To the left of the ~ is the dependent variable: success. It must be coded 0 & 1 for glm to read it as binary.

After the ~, we list the two predictor variables. The * indicates that not only do we want each main effect, but we also want an interaction term between numeracy and anxiety.

And finally, after the comma, we specify that the distribution is binomial. The default link function in glm for a binomial outcome variable is the logit. More on that below.

We can access the model output using summary().

```summary(model1)
Call:
glm(formula = success ~ numeracy * anxiety, family = binomial)
Deviance Residuals:
Min        1Q    Median        3Q       Max
-1.85712  -0.33055   0.02531   0.34931   2.01048
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)       0.87883   46.45256   0.019    0.985
numeracy          1.94556    4.78250   0.407    0.684
anxiety          -0.44580    3.25151  -0.137    0.891
numeracy:anxiety -0.09581    0.33322  -0.288    0.774
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 68.029  on 49  degrees of freedom
Residual deviance: 28.201  on 46  degrees of freedom
AIC: 36.201

Number of Fisher Scoring iterations: 7```

The estimates (coefficients of the predictors – numeracy and anxiety) are now in logits. The coefficient of numeracy is: 1.94556, so that a one unit change in numeracy produces approximately a 1.95 unit change in the log odds (i.e. a 1.95 unit change in the logit).

From the signs of the two predictors, we see that numeracy influences admission positively, but anxiety influences admission negatively.

We can’t tell much more than that as most of us can’t think in terms of logits. Instead we can convert these logits to odds ratios.

We do this by exponentiating each coefficient. (This means raise the value e –approximately 2.72–to the power of the coefficient. e^b).

So, the odds ratio for numeracy is:

`OR = exp(1.94556) = 6.997549`

However, in this version of the model the estimates are non-significant, and we have a non-significant interaction. Model1 produces the following relationship between the logit (log odds) and the two predictors:

`logit(p) = 0.88 + 1.95* numeracy - 0.45 * anxiety - .10* interaction term`

The output produced by glm() includes several additional quantities that require discussion.

We see a z value for each estimate. The z value is the Wald statistic that tests the hypothesis that the estimate is zero. The null hypothesis is that the estimate has a normal distribution with mean zero and standard deviation of 1. The quoted p-value, P(>|z|), gives the tail area in a two-tailed test.

For our example, we have a Null Deviance of about 68.03 on 49 degrees of freedom. This value indicates poor fit (a significant difference between fitted values and observed values). Including the independent variables (numeracy and anxiety) decreased the deviance by nearly 40 points on 3 degrees of freedom. The Residual Deviance is 28.2 on 46 degrees of freedom (i.e. a loss of three degrees of freedom).

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.

### Member Training: Count Models

March 3rd, 2015 by

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.

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.

Not a Member Yet?
It’s never too early to set yourself up for successful analysis with support and training from expert statisticians. Just head over and sign up for Statistically Speaking.

You'll get access to this training webinar, 100+ other stats trainings, a pathway to work through the trainings that you need — plus the expert guidance you need to build statistical skill with live Q&A sessions and an ask-a-mentor forum.

### How to Get SPSS GENLINMIXED Output Without the Model Viewer

September 26th, 2014 by

I love working with my clients.

I love working with my clients for many reasons, but one of them is I learn so much from them.

Just this week, one of my clients showed me how to get SPSS GENLINMIXED results without the Model Viewer.

She’s my new hero.

If you’ve ever used GENLINMIXED, the procedure for Generalized Linear Mixed Models, you know that the results automatically appear in this new Model Viewer.  (more…)

### Specifying a Random Intercept or Random Slope Model in SPSS GENLINMIXED

September 13th, 2013 by

One of the things I love about MIXED in SPSS is that the syntax is very similar to GLM.  So anyone who is used to the GLM syntax has just a short jump to learn writing MIXED.

Which is a good thing, because many of the concepts are a big jump.

And because the MIXED dialogue menus are seriously unintuitive, I’ve concluded you’re much better off using syntax.

I was very happy a few years ago when, with version 19, SPSS finally introduced generalized linear mixed models so SPSS users could finally run logistic regression or count models on clustered data.

But then I tried it, and the menus are even less intuitive than in MIXED.

And the syntax isn’t much better.  In this case, the syntax structure is quite different than for MIXED. (more…)

### 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!