Multivariate 2 Level Generalised Linear Models

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Multivariate 2 Level Generalised Linear Models
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Multivariate 2 Level Generalised Linear Models

• We now introduce the superscript r to enable us
  to distinguish the different models, variates,
  random effects etc of a multivariate response.
• 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, e.g. number of consultations with a
  doctor or specialist and the number of
  prescriptions
• An event history example occurs in the modelling
  the sequence of months of job vacancies, which
  last until either they are successfully filled or
  withdrawn from the market. This data leads to a
  correlated competing risk model as the firm
  effects are present in both the filled and lapsed
  durations,

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Multivariate 2 Level Generalised Linear Models

• A trivariate example is the joint (simultaneous
  equation) modelling of wages, training and
  promotion of individuals over time present in
  a panel survey such as the British Household
  Panel Survey (BHPS)
• Joint modelling of simultaneous responses allows
  us to disentangle the direct effects of the
  different responses on each other from any
  correlation that occurs in the random effects.
• Without a multivariate multilevel GLM for complex
  social process like these we risk inferential
  errors.

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Multivariate 2 Level Generalised Linear Models

• The multivariate GLM is obtained from the
  univariate GLM (using superscipts)
• scale parameter
• conditional mean
• Variance
• linear predictor

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Multivariate 2 Level Generalised Linear
ModelsLikelihood

• is a multivariate Normal distribution
  of dimension R with mean zero and variance
  covariance structure

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Example Bivariate Poisson model

• Cameron and Trivedi (1988,1998) example is from
  the Australian Health survey for 1977-1978.
• We use a version of the Cameron and Trivedi
  (1988) data set (called racd.tab) for a bivariate
  model. In this example we have a single response
  as we only have one pair of response ( dvisits,
  prescrib) for each sampled individual.
• Data description
• Number of observations (rows) 5190
• Number of variables (columns) 21

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Poisson Model Example C6

• 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

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Bivariate Poisson Model Example C6
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Demo example c6
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Results Bivariate Model (racd.dat)

• This shows different level of overdispersion in
  the different responses and a large correlation
  between the random intercepts.
• If we had not been interested in obtaining the
  correlation between the responses we could have
  done a separate analysis of each response and
  made adjustments to the SEs.
• This is legitimate here as there are no
  simultaneous direct effects (e.g. visits on
  prescribe) in this model

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Bivariate linear and probit Example L9

• The data we use is a version of the NLSY data as
  used in various Stata Manuals (to illustrate the
  xt commands). The data is for young women who
  were aged 14-26 in 1968.
• The women were surveyed each year from 1970 to
  1988, except for 1974, 1976, 1979, 1981, 1984 and
  1986.
• We have removed records with missing values on
  one or more of the response and explanatory
  variables we want use in our analysis of the
  joint determinants of wages and trade union
  membership.
• There are 4132 women (idcode) with between 1 and
  12 years of observation on wages being in
  employment (i.e. not in full time education) and
  earning more than 1/hour but less than
  700/hour.
• The direct effect of trade union membership on
  wages is dealt with including trade union
  membership as a covariate in the wage equation
  linear predictor.

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Path Diagrams

• This picture shows the dependence between trade
  union membership and wages, there are no
  multilevel random effects affecting either wages
  or trade union membership. This model can be
  estimated by any software that estimates basic
  GLMs.
• This picture also also shows the dependence
  between trade union membership and wages, this
  time there are multilevel random effects
  affecting both wages and trade union membership.
  However the multilevel random effects are
  independent This model can be estimated by any
  software that estimates multilevel GLMs by
  treating the wage and trade union models as
  independent.

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Path Diagrams

• This picture shows the dependence between trade
  union membership and wages, this time there is a
  correlation between the multilevel random effects
  affecting wages and trade union membership, this
  is shown by the curved line linking them
  together. This model can be estimated by Sabre as
  a bivariate multilevel GLMs by allowing for a
  correlation between the wage and trade union
  responses at each wave of the panel

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Bivariate linear and probit Example L9

• Data description
• Number of observations 18995
• Number of cases (columns) 4132
• Variables include
• ln_wageln(wage/GNP deflator) in a particular
  year are
• black1 if woman is black, 0 otherwise
• msp1 if woman is married and spouse is present,
  0 otherwise
• grade years of schooling completed (0-18)
• not_smsa1 if woman was living outside a standard
  metropolitan statistical area (smsa), 0
  otherwise
• south1 if the woman was living in the South, 0
  otherwise
• union1 if a member of a trade union, 0
  otherwise
• tenure job tenure in years (0-26).
• age respondents age
• age2 age age

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Bivariate linear and probit Example L9

• We take ln_wage (linear model) and union (probit
  link) as the response variables and model them
  with a randon intercept and a range of
  explanatory variables.

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Demo Bivariate linear and probit Example L9

• Model for tunion on its own
• Model for ln_wage on its own
• Then estimate a joint model allowing for the
  overdispersion in ln_wage and tunion and a
  correlation between them,
• Also the log wage equation contains union as an
  explanatory variable.

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Bivariate linear and probit Example L9

• This shows different levels of overdispersion in
  the different responses and a positive
  correlation between the random intercepts.
• The value of trade union membership in the wage
  equation of the homogenous model changes
• Sabre 5.0 can model up to 3 different panel
  responses simultaneously.

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Exercise

• There is an exercise to accompany this section,
  this is the bivariate linear and logit exercise
  L10.