Count data, contingency tables and log-linear models

1
Count data, contingency tables and log-linear
models

• Expected frequency
• Log-linear models are linear models of the log
  expected frequency
• (log is used as link function)

2
A log-linear model for independence

• The last parameter of each kind can be set to
  zero

3
The saturated log-linear model

• Independence can be tested by relating the
  difference in deviance D2 D1 to a ?2
  distribution with df2 df1 degrees of freedom.
• What is D1 and df1 for the saturated model?

4
The multinomial distribution

• Consider a nominal random variable that takes k
  distinct values with probabilities p1, p2, , pk
• Assume that have made n independent observations
  of that variable
• Then
• wher nj is the number of times the jth value is
  observed
• Note that n is fixed in a multinomial
  distribution.
• If the observations arrive randomly, a Poisson
  distribution is usually preferable.

5
Analysis of example data

• proc genmod datalinear.snoring
• class snore heart
• model count snore heart/linklog distPoisson
• run
• Can a Poisson distribution be justified?

6
Higher order tables

• Consider the following data on drug use
• Model

7
Terminology

• A alcohol C cigarette M marijuana
• Model A C M mutual independence model
• Model A C M AC AM CM homogenous association
  model
• Model A C M AC AM Model in which C and M are
  mutually independent when controlling for A

8
Contingency table with one response variable

• Consider the example data written in the
  following form
• proc genmod datalinear.snoring2
• class snore
• model heart/total snore/linklogit
  distbinomial
• run

9
Poisson regression I

• Poisson distribution
• Log link
• where x is a covariate

10
Poisson regression II

• Poisson distribution
• Log link
• where the parameters are row,
• column and treatment effects