Multilevel Poisson Models

1
Multilevel Poisson Models
Session 6
Damon Berridge
2
Multilevel Poisson Models

• Another important type of discrete data is count
  data.
• For example, for a population of road crossings
  one might count the number of accidents in one
  year or for a population of doctors, one could
  count how often in one year they are confronted
  with a certain medical problem.
• The set of possible outcomes of count data is the
  set of natural numbers
• The standard distribution for counts is the
  Poisson distribution.

3
Multilevel Poisson Models

• The Poisson distribution has some properties that
  we can make use of when modelling our data. For
  example, the mean is equal to the variance
• When we have Poisson distributed data, it is
  usual to use a logarithmic transformation to
  model the mean
• There is no theoretical restriction, however, on
  using other transformations of , so long as
  the mean is positive, as discussed in Dobson
  (1991).

4
Poisson Regression Models

• In Poisson regression it is assumed that the
  response variable has a Poisson distribution
  given the explanatory variables
• where the log of the mean is assumed to be a
  linear function of the explanatory variables
• which implies that the mean is the exponential
  function of independent variables

5
Poisson Regression Models

• In models for counts it is quite usual that there
  is a variable mij
• that is known to be proportional to the expected
  counts.
• For example, if the count is the number of events
  in some time interval of non-constant length, it
  is often natural to assume that the expected
  count is proportional to this length of the time
  period.
• In order to let the expected count be
  proportional to mij there should be a term
  log(mij ) in the linear model with a regression
  coefficient fixed to 1.
• Such a term is called an offset in the linear
  model (see e.g., McCullagh and Nelder, 1989
  Goldstein, 2003). Therefore, the Poisson
  regression model can be written in the following
  form

6
The Two-Level Poisson Model

• is the count for level-1 unit i in level 2 unit
  j,
• Level-1 Model
• Level-2 Model The Random Intercept Model
• Regression model plus a random intercept for the
  logarithm of the expected count,
• So that

7
The Two Level Poisson Model Likelihood

• where
• and

8
Poisson Model Example C5

• Cameron and Trivedi (1988) use various forms of
  overdispersed Poisson model to study the
  relationship between type of health insurance and
  various responses which measure the demand for
  health care, such as the total number of
  prescribed medications used in past 2 days.
• The data set they use in this analysis is from
  the Australian Health survey for 1977-1978.
• Data description
• Number of observations (rows) 5190
• Number of variables (columns) 21

9
Poisson Model Example C5

• Variables
• sex 1 if respondent is female, 0 if male
• age respondent's age in years divided by 100,
• agesq age squared
• income respondent's annual income in Australian
  dollars divided by 1000
• levyplus 1 if respondent is covered by private
  health insurance fund for private patient in
  public hospital (with doctor of choice), 0
  otherwise
• freepoor 1 if respondent is covered by
  government because low income, recent immigrant,
  unemployed, 0 otherwise
• freerepa1 if respondent is covered free by
  government because of old-age or disability
  pension, or because invalid veteran or family of
  deceased veteran, 0 otherwise
• illness number of illnesses in past 2 weeks
  with 5 or more coded as 5
• actdays number of days of reduced activity in
  past two weeks due to illness or injury
• hscore respondent's general health
  questionnaire score using Goldberg's method, high
  score indicates bad health.
• chcond1 1 if respondent has chronic
  condition(s) but not limited in activity, 0
  otherwise
• chcond2 1 if respondent has chronic
  condition(s) and limited in activity, 0 otherwise
• dvisits number of consultations with a doctor
  or specialist in the past 2 weeks
• nondocco number of consultations with
  non-doctor health professionals, (chemist,
  optician, physiotherapist, social worker,
  district community nurse, chiropodist or
  chiropractor in the past 2 weeks
• hospadmi number of admissions to a hospital,
  psychiatric hospital, nursing or convalescent
  home in the past 12 months (up to 5 or more
  admissions which is coded as 5)
• hospdays number of nights in a hospital, etc.
  during most recent admission, in past 12 months
• medicine total number of prescribed and
  nonprescribed medications used in past 2 days
• prescribe total number of prescribed
  medications used in past 2 days

10
Poisson Model Example C5
11
Poisson Model Example C5
Even with a range of explanatory variables, there
is still a highly significant amount of between
respondent variation in, total number of
prescribed medications.