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Generalized Linear Latent and Mixed Models

We describe a Stata program called gllamm that can fit a large number of generalised linear latent and mixed models. These models are extensions of the random intercept models that may be estimated in Stata 6 using xtreg, xtlogit, xtpois etc. for cross-sectional time-series or other clustered data.

All these models include a random intercept for clusters in the linear predictor. If there is a single explanatory variable x, the linear predictor is given by

η_ij=β₀+β₁x_ij+u_i

where the index ij refers to the ith “level 1” unit clustered within the jth “level 2” unit, e.g. observation times within subjects or pupils within schools, and the random effect u_i is usually assumed to have a normal distribution with mean zero.

The program gllamm allows any combination of five basic extensions to the random intercept model: (1) discrete random effects distributions, (2) multi-level models, (3) random coefficients (4) factor loadings and (5) mixed responses.

Similarly to the xt programs, gllamm requires the data to be in “long” form with all responses stacked into a single variable and the cluster index (or indices) stored in separate variable(s).

The program simply uses Stata’s ml commands with method deriv0 to maximise the likelihood which is evaluated by numerical integration. The five extensions to the random intercept model are described below using the examples that are also used in the talk.

1. Discrete random effects distributions: We may wish to assume that there are several latent classes or groups of subjects where each group is homogeneous in its random effect.
2. Multi-level models: The level 2 units may be clustered within level 3 units. For example, there may be multiple observations (k) per person (j) clustered in families (i). The linear predictor now includes two random effects, one for families (level 3) and one for subjects (level 2):
   η_ijk=β₀+β₁x_ijk+u_i⁽³⁾+u_j⁽²⁾.
3. Random coefficients: The coefficient of an explanatory variable may differ between level 2 units. For example, the effect of pupil's (j) maths results in year 3 on maths results in year 5 may differ between schools (i). The linear predictor now has a random intercept u_i⁽⁰⁾ and a random slope u_i⁽¹⁾ where the two random effects may be correlated,
   η_ij=β₀+ u_i⁽⁰⁾+(β₁+ u_i⁽¹⁾)math3_ij.
4. Factor loadings: Several variables may load on a latent variable. For example, on a psychometric test, the results on the items (j) for each subject (i) could be modelled as a 2-parameter Rasch model
   η_ij=β_j+ u_iλ_j.
   Here the random effect u_i is a measure of the subject’s ability, -β_j is a measure of the difficulty of item j and λ_j is a “factor loading” representing the effect of the subject’s ability on their performance on item j. A separate loading is estimated for each item.
5. Mixed responses: If the items or variables are of different types, e.g. continuous and dichotomous, then different generalised linear models (families and links) need to be specified for different `observations'. An example of this is logistic regression with a continuous explanatory or `exposure' variable which is subject to measurement error. We need to model the measured exposure and dichotomous outcome simultaneously using
   μ_ij=β₀+ u_i
   and

   logit(π_ij)=β₁+λu_i

   respectively.