A course in statistical modelling. session 06b: Modelling count data

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1 A Course in Statistical Modelling University of Glasgow 29 and 30 January, 2015 session 06b: Modelling count data Graeme Hutcheson 1 Luiz Moutinho 2 1 Manchester Institute of Education Manchester university 2 Adam Smith Business School Glasgow University Count data is common in the social sciences and is often the variable that is being modelled (the response). Examples of count variables include the number of children in a family, number of credit cards, number of applications, number of parking tickets or the number of excluded pupils. It is important to note that counts are different to continuous data (for example, they can t assume negative values) and need to be analysed using a different technique. The technique we use to analyse count data is Poisson regression, which is a GLM with a log link (the random and systematic components of the model are linked with a log function - giving a log-linear model). In other words, it is the log of the response variable that is linearly related to the explanatory variables.

2 To illustrate the need for the Poisson regression, the following data are analysed using OLS regression (for continuous data) and compared to an analysis using Poisson regression (for count data). It should be obvious from the output which of these models is more appropriate for our data. count data set The example we are going to use here is a simple made up one investigating the relationship between the number of children (0 to 4) and the salary earned by female employees in a particular company. Our model is... children salary which also shows how the dataset needs to be structured... number of children salary ( 1000) The dataset poissonexample.csv is available from the course web-site.

3 An OLS regression model of number of children... children OLS regression The OLS regression model shows a significant negative relationship between children and salary. This model is not particularly accurate, however, as the predicted number of children assume negative values (for salries above 125); something that is not possible salary OLS regression is NOT a good technique to use to model count data... A Poisson regression model of number of children... Poisson regression: default plot children The top graphic shows the default effect plot (Y is on a log scale) and shows clearly that the Poisson model is linear. Of particular note is the non linear scaling of the Y-axis. children salary Poisson regression: rescale.axis salary The lower graphic shows the effect plot drawn with the addition of the command rescale.axis=false and clearly shows that the predicted number of children is non-linear over the salary range and never goes below zero. The Poisson model is a more accurate model of count data than the OLS model...

4 The example above demonstrated that count data behave differently to continuous data making an OLS regression model inappropriate (for the same reason, t-tests and ANOVAs computed on count data are also inappropriate). This is particularly obvious when predictions of counts assume negative values (interpreting effect displays are useful for demonstrating this). During this course we will model count data using Poisson regression, which is simply a GLM model with a log link. A full description of the model and examples are provided below... Poisson regression: an example analysis The following example of Poisson regression uses an actual dataset (Arrests) that is available as part of the effects library. These data give information about the number of police data bases a person appears on (checks), their colour, age and sex, the year in which they were arrested and whether they are currently in employment. Load these data using the Rcmdr menu options...

5 Recoding year as categorical... NOTE: this dataset codes the variable year as continuous; a variable it is probably best to consider as categorical. Recode this variable into a categorical variable (yearcat) using the Rcmdr menus... Defining the model... Previous research indicates that a model of interest is... checks sex*yearcat + colour*yearcat + citizen*yearcat + age We are particularly interested in changes in sex, colour and citizen over the years... The following analyses show a Poisson regression model run using the Rcmdr. The task here is to interpret the effect displays... these should give you a clear picture of the relationships in the data and should agree with the results from the standard output...

6 Running a poisson regression model... age effect plot sex*yearcat effect plot checks checks sex : Female sex : Male age yearcat yearcat*colour effect plot yearcat*citizen effect plot colour : Black colour : White 2.0 citizen : No citizen : Yes checks checks yearcat yearcat

7 Estimate Std. Error z value Pr(> z ) (Intercept) ** sex[t.male] e-06 *** yearcat[t.1998] yearcat[t.1999] ** yearcat[t.2000] ** yearcat[t.2001] * yearcat[t.2002] colour[t.white] ** citizen[t.yes] age < 2e-16 *** sex[t.male]:yearcat[t.1998] * sex[t.male]:yearcat[t.1999] *** sex[t.male]:yearcat[t.2000] *** sex[t.male]:yearcat[t.2001] ** sex[t.male]:yearcat[t.2002] yearcat[t.1998]:colour[t.white] yearcat[t.1999]:colour[t.white] ** yearcat[t.2000]:colour[t.white] yearcat[t.2001]:colour[t.white] yearcat[t.2002]:colour[t.white] yearcat[t.1998]:citizen[t.yes] yearcat[t.1999]:citizen[t.yes] yearcat[t.2000]:citizen[t.yes] yearcat[t.2001]:citizen[t.yes] yearcat[t.2002]:citizen[t.yes] LR Chisq Df Pr(>Chisq) sex < 2.2e-16 *** yearcat *** colour < 2.2e-16 *** citizen *** age < 2.2e-16 *** sex:yearcat ** yearcat:colour yearcat:citizen

8 The effect displays and the standard output provide similar impressions of the results. The effect displays are, however, much easier to interpret, particularly in the presence of interaction terms. The effect displays give the same information as the regression parameters. The regression estimate shows that for each unit increase in age, the log of the number of checks increases by This can be verified from the effect plot as the number of checks increases from 1.55 to 2.60 when age increases by 40. A unit increase is therefore (log(2.6) - log(1.55))/40 = The conclusions about significance are similar for both reporting methods, although the effect displays do indicate that the year 1997 may be responsible for most, if not all, of the significance (a result easy to miss from the standard output). The effect displays give more detailed information than the standard output and do not require any mathematical manipulation of parameters. They are easier to understand and more informative. The standard output is mostly useful to verify and quantify certain aspects of the model. It is not required for UNDERSTANDING the model.

9 The analysis of contingency tables... The Poisson regression models are particularly useful as they allow the analysis of contingency table data... Consider the following contingency table, taken from... Hutcheson, G. D. and Schaefer, L. (2012). Test selection in the 21st century. Journal of Modelling in Management, 7,3: Group A B C North Region South West 1 2 1

10 The information we have here is cell count - a count variable. In order to investigate the relationship between region and group, we need to look at the interaction model (to see if region influences group). The model for this is... cell count region*group Which suggests we need three columns of data; one for cell count, one for region and one for group... frequency group region 2 A north 2 A south 1 A west 1 B north 2 B south 2 B west 3 C north 1 C south 1 C west

11 Poisson model of contingency table: effect display group*region effect plot region : C count region : A region : B 4 2 A B C group Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) group[t.b] group[t.c] region[t.south] region[t.west] group[t.b]:region[t.south] group[t.c]:region[t.south] group[t.b]:region[t.west] group[t.c]:region[t.west] Analysis of Deviance Table (Type II tests) Response: count LR Chisq Df Pr(>Chisq) group region group:region

12 This model shows no significant interactions, which is hardly surprising given the small sample size involved. What is particularly interesting, however, are the statistics for the group:region interaction, which are exactly the same as the standard contingency table chi-square test. It is also interesting to note that this analysis gives the same results as a multinomial regression predicting one of the variables (region or group). The model for this is... region group Which suggests a different data structure consisting of just two groups. The structure of the data can easily be changed using the R command... CONTINGENCYtable01LONG <- as.data.frame(lapply(contingencytable01, function(x) rep(x, CONTINGENCYtable01$count))) EXERCISES...

13 HairEyeColor... Load the HairEyeColor contingency table from the datasets library... Plot the effect displays for the three way interaction hair*eye*sex What does this suggest to you? Does it agree with the tabular output of parameter values and significance estimates? rescale the axis for the effect display... What does this suggest to you? Does it agree with the tabular output of parameter values and significance estimates? You may construct an animation - going through all hair colours or genders using the given.values = c(sexmale = 1) command... Compare Poisson analysis to multinomial A useful exercise, if you have tim,e, is to compare the Poisson regression analyses with a multinomial logit model. First, you will need to transform the HairEyeColor contingency table into a long-format data frame... HairEyeColorLong <- as.data.frame(lapply(haireyecolor, function(x) rep(x, HairEyeColor$Freq))) You should get the same significance values for the models...