16 💻 Poisson regr exer
This is a brief into with a collection of exercises concerning Poisson regression stuff! Some of them are actually (noised changing data) samples from the exam, so dig into it! Some of them will be done in class during lecture, some others are left to you as a to-do-at-home exercise.
16.1 What about Poisson regression
Poisson Regression involves regression models in which the response variable is in the form of counts and not fractional numbers. For example, the count of number of births or number of wins in a football match series. Also the values of the response variables follow a Poisson distribution.
The general mathematical equation for Poisson regression is:
$$
log(y) = _0 + _1x_1 + _2x_2 … _Nx_N
$$
The basic syntax for glm() function in Poisson regression is :
glm(formula, data, family)16.2 Walkthrough 🚶
We have the in-built data set warpbreaks which describes the effect of wool type (A or B) and tension (low, medium or high) on the number of warp breaks per loom. Let’s consider breaks as the response variable which is a count of number of breaks. The wool type and tension are taken as predictor variables.
poisson_regression <-glm(formula = breaks ~ wool+tension, data = warpbreaks,family = "poisson")
summary(poisson_regression)In the summary we look for the p-value in the last column to be less than 0.05 to consider an impact of the predictor variable on the response variable. As seen the wooltype B having tension type M and H have impact on the count of breaks.
16.3 mixed Exercises 👨💻
In class if you are stuck either ask to the teacher or assemble a group to work with! Collaboration in key 👯♂️!
Let’s go 🎬
Exercise 16.1 In Sweden all motor insurance companies apply identical risk arguments to classify customers, and thus their portfolios and their claims statistics can be combined. The Committee was asked to look into the problem of analyzing the real influence on claims of the risk arguments and to compare this structure with the actual tariff.
The dataset motorins is contained into the package faraway. Fit a poisson regression with Claims being the dependent and all the others being the independent variables
As you do with linear regression and logistic regression you just specify first the model formula, then data and the end the link. The link is actually the poisson. The patter should look familiar to you. This is happening because each of the models we have been doing so far fall under the umbrella of Generalized Linear Models, i.e. GLM which share a common worflow.
Answer to Question 16.1:
library(faraway)
data(motorins)
poisson_model <- glm(Claims ~ Payment, family = poisson, data = motorins)
summary(poisson_model)Exercise 16.2 Within the library faraway, the gala dataset records the counts of the numbers of species of tortoise found on 30 Galapagos Islands. The relationship between the number of plant species and several geographic variables is of interest.
- check teh distribution of the variablw
Species - fit a poisson model on
Speciesgiven all the others. - Which of them are statistically significant with
Answer to Question 16.2:
you need to load the same library as before and data.
library(faraway)
data("gala")then check the distribution of Species
hist(gala$Species)in the end fit the model:
poisson_regression_2 = glm(Species~., data = gala, family = "poisson")
summary(poisson_regression_2)by looking at the pvalues you see that: Endemics, Area and Nearest are the most significant ones.
Exercise 16.3 the fishing data set in the stats4nr package contains data on the number of fish caught by visitors to a state park. It includes the following variables:
-
livebait: whether or not the group used live bait (0/1), -
camper: whether or not the group brought a camper on their visit (0/1), -
persons: the number of people in the group, -
child: the number of children in the group, and -
count: number of fish caugh
do:
- plot the histogram, the mean and the variance of
count, is it normally distributed? - fit a poisson regression on
countwithperson,childandcamper, are those coefficients significant?
Answer to Question 16.3:
install.packages(“stats4nr”) data(“fishing”)
then check the variable count:
hist(fishing$count)
mean(fishing$count)
var(fishing$count)then fit the model:
fish_model = glm(count ~ persons + child + camper, data = fishing, family = "poisson")
summary(fish_model)Exercise 16.4 You hypothesize that as forests become older, their canopies become dominated by only a few species, which limits light availability for other species growing in the understory. Data to examine this hypothesis were acquired from the Hubachek Wilderness Research Center, an experimental forest located in northern Minnesota (Gill et al. 2019). Data were collected from 36 forest inventory plots with at least one tree larger than 12.7 cm in diameter. The number of species (num_spp), number of standing dead trees (num_dead), and average height of the trees in the plot (ht_m, meters) are the variables of interest.
Run the code below to create the data set sp_ht to use in this analysis
sp_ht <- tibble(
num_spp = c(4, 3, 2, 4, 2, 4, 2, 3, 2, 2,
5, 1, 7, 4, 4, 7, 4, 6, 4, 5,
3, 5, 5, 3, 7, 2, 5, 5, 3, 5,
3, 5, 2, 4, 5, 4),
num_dead= c(1, 1, 0, 0, 1, 0, 1, 1, 0, 0,
2, 0, 0, 3, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 1, 0, 0, 0,
0, 2, 0, 0, 0, 0),
ht_m = c(17.6, 19.8, 26.0, 17.4, 19.8, 20.7,
14.7, 17.6, 11.6, 19.4, 12.7, 16.4,
12.6, 17.9, 23.2, 14.5, 18.6, 15.0,
11.5, 15.1, 13.8, 9.9, 20.2, 7.3,
6.2, 11.4, 16.5, 8.7, 9.8, 6.9,
18.0, 10.0, 12.8, 13.7, 12.8, 16.7)
)- Does the count variable of interest
num_sppdisplay characteristics? Comment on the usefulness of using the Poisson distribution for this variable? - Create a histogram for the
num_sppvariable and comment on its distribution? - Fit two count regression models predicting the number of species based average height. One model should assume the Poisson and the other a negative model distribution. Based on their AIC values, which model would you prefer and why?
Answer to Question 16.4: