Poisson regression models for count data

Poisson regression models for count data

• 1.
  Poisson Regression Models forCount Data
• 2.
  2 Outline • Review • Introductionto Poisson regression • A simple model: equiprobable model • Pearson and likelihood-ratio test statistics • Residual analysis • Poisson regression with a covariate (Poisson time trend model)
• 3.
  3 Review of Regression Youmay have come across: Dependent Variable Regression Model Continuous Linear Binary Logistic Multicategory (unordered) (nominal variable) Multinomial Logit Multicategory (ordered) (ordinal variable) Cumulative Logit
• 4.
  4 Regression In this session: DependentVariable Regression Model Continuous Linear Binary Logistic Multicategory (unordered) (nominal variable) Multinomial Logit Multicategory (ordered) (ordinal variable) Cumulative Logit Count variable Poisson Regression (Log-linear model)
• 5.
  Data Data for thissession are assumed to be: • A count variable Y (e.g. number of accidents, number of suicides) • One categorical variable (X) with C possible categories (e.g. days of week, months) • Hence Y has C possible outcomes y1, y2, …, yC 5
• 6.
  6 Introduction: Poisson regression •Poisson regression is a form of regression analysis model count data (if all explanatory variables are categorical then we model contingency tables (cell counts)). • The model models expected frequencies • The model specifies how the count variable depends on the explanatory variables (e.g. level of the categorical variable)
• 7.
  7 Introduction: Poisson regression •Poisson regression models are generalized linear models with the logarithm as the (canonical) link function. • Assumes response variable Y has a Poisson distribution, and the logarithm of its expected value can be modelled by a linear combination of unknown parameters. • Sometimes known as a log-linear model, in particular when used to model contingency tables (i.e. only categorical variables).
• 8.
  8 Example: Suicides (countvariable) by Weekday (categorical variable) in France Mon 1001 15.2% Tues 1035 15.7% Wed 982 14.9% Thur 1033 15.7% Fri 905 13.7% Sat 737 11.2% Sun 894 13.6% Total 6587 100.0%
• 9.
  9 Introduction: Poisson regression •Let us first look at a simple case: the equiprobable model (here for a 1-way contingency table)
• 10.
  10 Equiprobable Model • Anequiprobable model means that: – All outcomes are equally probable (equally likely). – That is, for our example, we assume a uniform distribution for the outcomes across days of week (Y does not vary with days of week X).
• 11.
  • The equiprobablemodel is given by: P(Y=y1) = P(Y=y2) = … = P(Y=yC) = 1/C i.e. we expect an equal distribution across days of week. • Given the data we can test if the assumption of the equiprobable model (H0) holds 11 Equiprobable Model
• 12.
  12 Example 1: Suicidesby Weekday in France Mon 1001 15.2% Tues 1035 15.7% Wed 982 14.9% Thur 1033 15.7% Fri 905 13.7% Sat 737 11.2% Sun 894 13.6% Total 6587 100.0% H0: Each day is equally likely for suicides (i.e. the expected proportion of suicides is 100/7 = 14.3% each day)
• 13.
  13 Example 2: TrafficAccidents by Weekday H0: Each day is equally likely for an accident (i.e. the expected proportion of accidents is 100/7 = 14.3% each day) Mon 11 11.8% Tues 9 9.7% Wed 7 7.5% Thur 10 10.8% Fri 15 16.1% Sat 18 19.4% Sun 23 24.7% Total 93 100.0%
• 14.
  14 • H0: Eachday is equally likely for an accident. • Alternative null hypotheses are: – H0: Each working day equally likely for an accident. – H0: Saturday and Sunday are equally likely for an accident. • Omitted variables? For example, distance driven each day of the week. Hypothesis Testing
• 15.
  15 • We canexpress this equiprobable model more formally as a Poisson regression model (without a covariate), which models the expected frequency Poisson regression – without a covariate
• 16.
  16 • We assumea Poisson distribution with parameter μ for the random component, i.e. yi ~ Poisson(µ), i.e. • Y is a random variable that takes only positive integer values • Poisson distribution has a single parameter (μ) which is both its mean and its variance. y i i i e P(Y y ) where y 1,2, 3 y ! i i i i m m- = = = Poisson regression
• 17.
  • We aimto model the expected value of Y. It can be shown that this is the parameter μ, hence we aim to model μ. • We can write the equiprobable model defined earlier as a simple Poisson model (no explanatory variables), i.e. mean of Y does not change with month: where is a constant. 17 Poisson regression: Simple Model (No Covariate) i i i E(y ) 1/ log( ) i 1, ,C Cm m a = = = = L log(1/ )Ca =