A course in statistical modelling. session 09: Modelling count variables

Transcription

1 A Course in Statistical Modelling SEED PGR methodology training December 08, 2015: 12 2pm session 09: Modelling count variables blackboard: RSCH80000 SEED PGR Research Modules internet: Manchester Institute of Education, University of Manchester 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 A scatterplot showing the relationship between number of children and disposable income children salary The observed data suggest that those with 0 and 1 child tend to have more disposable income; although this is not an obvious relationship to depict. The green straight line is an OLS regression model of these data and the red curved line is a line of local best-fit. It is noticeable that the OLS regression line of this model does not appear to be a particularly accurate fit (see the effect display below). An OLS regression model of number of children... OLS regression children salary The effect display of the OLS regression model suggests 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 salaries above 125); something that is not possible. OLS regression is, therefore, NOT a good technique to use to model these data...

4 A Poisson regression model of number of children... This model is essentially the same as the OLS regression model computed earlier, but with a log link (from the Poisson distribution). children children Poisson regression: default plot salary Poisson regression: rescale.axis salary The top graphic shows the default effect plot (Y is depicted using a log scale) and shows clearly that the Poisson model is linear. Of particular note is the non-linear scaling of the Y-axis. The lower graphic shows the effect plot drawn with the addition of the command rescale.axis=false which depicts Y as a count. This graphic 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 these data than the OLS model and more closely fits the line of local best-fit shown previously.

5 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...

6 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...

7 Running a poisson regression model... Poisson regression model: standard output (TYPE III tests) 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]

8 Poisson regression model: Analysis of Deviance table (TYPE II tests) 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 Note the big differences between the significance values for some of the parameters provided by the TYPE II and TYPE III tests (for example, the main effect term for the variable colour ). 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

9 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. For example, 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.

10 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, which shows the group someone belongs to and the region in which they live, 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

11 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. The model tells us how to structure the data. 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

12 Poisson model of contingency table 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

13 Poisson model of contingency table: effect display group*region effect plot region : C count region : A region : B 4 2 A B C group 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 likelihood-ratio (G 2 ) test, which is provided as part of the standard chi-square output. This analysis is available as part of the Deducer library that can be installed onto your system using the packages tab in the lower right window of the R-studio. The likelihood-ratio test uses the following command... likelihood.test(group, region) which suggest that the data need to be represented in two columns - one giving information about group and the other information about region.

14 Rearranging the data-frame The data-frame used for the poisson regression model above (the data frame that included cell-count) can be re-arranged to include just information about region and group. The original data-frame (CONTINGENCYtable01) can be rearranged in R using the command... CONTINGENCYtable01LONG <- as.data.frame (lapply(contingencytable01, function(x) rep(x, CONTINGENCYtable01$count))) or can be downloaded from the course web-site in file CONTINGENCYtable01LONG.csv. Computing the likelihood-ratio statistic Load Deducer and run the likelihood test... library(deducer) likelihood.test(contingencytable01long$group, CONTINGENCYtable01LONG$region) Log likelihood ratio (G-test) test of independence without correction Log likelihood ratio statistic (G) = , X-squared df = 4, p-value =

15 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. A multinomial model can be obtained using the following commands from the Rcmdr. run the MNL model

16 get the analysis of deviance table Results from the MNL model Analysis of Deviance Table (Type II tests) Response: group LR Chisq Df Pr(>Chisq) region

17 EXERCISES... 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...

18 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...