Chapter 1 Introduction
5.2 The Generalized Linear Mixed Model
5.2.6 The methods of Schall and Breslow and Clayton
In this derivation u denotes the vector of random effects. If we assume the random effects are u∼ N(0,G), the normal “errors” linear mixed model is given by
y =Xβ +Zu+e.
the conditional mean of the observations given the random model effects is E[y|u] = Xβ+Zu
and the conditional variance
var[y|u] =R =var(e).
The observations can then be described as y =µ+e
whereµdenotes the conditional meanE[y|µ]. The Generalized Linear Mixed Model (GLMM) can also be described in terms of the conditional means. It takes the form:
η = Xβ+Zu
whereηis the linear predictor through the link function g(µ).Thus we could write the GLMM as
g{E[y|u]} = Xβ+Zu
As in the conventional mixed model, the random model effectsuare assumed to have a multivariate normal distribution with mean 0and variance covari- ance matrix G while as in the conventional generalized linear model, the underlying distribution of y is assumed to be a member of the exponential family (for any given u)
To see the difficulty that arises with GLMM’s, suppose that we have a cluster sampled survey data with a random sample of clusters indexed i= 1, . . . , N with elementary unitsj = 1, . . . , niwithin each cluster. The observations sat- isfy generalized linear models with common distributions and a link function.
We now introduce a random term ui corresponding to the random cluster ef- fect, and assume that the random effects ui are normally distributed with a mean 0 and a variance σu2. We put ui =γzi where zi ∼N ID(0,1)
As an example we consider a logistic regression model for binary outcomes.
Let
pij = Prob(yij = 1)
so that
(pij|zi) = Prob(yij = 1|zi)
gives the probability of “success” for the jth unit in clusteri, given the value of the random cluster effect Zi in cluster i. Then the model can be written as
logit(pij|zi) = xijβ+γzi. (5.8) Overall therefore regardless of the cluster effect, pij is found by integrating over the random cluster effects to get
pij = Z ∞
zi=−∞
(pij|zi)φ(zi)dzi (5.9) where φ is the density function the standard normal random variable (N(0,1)). For model Eq. (5.8), the joint likelihood for yij, j = 1, . . . , ni and i= 1, . . . , N involves the integral in Eq. (5.9) since
l(β, γ) = X
ij
{yijlogpij + (nij −yij)log(1−pij)}+ contsant. (5.10) The difficulty is that the integral has to be evaluated numerically. This can be done for example using the Gaussian Hermite formula for numerical quadrature, (discussed in detail in Section 5.7.10 of this chapter) but briefly under this method, an integral such as above is approximated by means of sums as follows:
Z ∞
−∞
f(u)e−u2du≈
m
X
j=1
cjf(sj) (5.11)
where the values ofcj andsj are given in standard tables. When we integrate out (numerically) the random effects, we obtain the marginal likelihood l(β, γ) and thismarginallikelihood can be maximized to find the maximum likelihood estimates (MLEs) of β andγ. This approach was used by Hedeker
and Gibbons (1994) who proposed a random effects ordinal regression model for the analysis of clustered response data. Now a binary response can be considered a special case of an ordinal response with only two(ordered) out- come categories. They developed the model for both the probit and the logistic response functions using the “threshold” concept in which it is as- sumed that the observed ordered category is determined by the value of a latent unobservable continuous response that follows a linear model incorpo- rating random effects. Hedeker and Gibbons (1996) in addition developed a program called MIXOR (and an extended version MIXORE) to implement this method of marginal maximum likelihood estimation (MMLE). MIXORE is a public domain computer program that can be downloaded from the inter- net, together with a manual that describes how the data should be prepared for analysis, together with a specification file MIXORE.DEF which describes the setup of the the data in terms of which columns contain the random ef- fects, which columns contain the fixed effects and which column contains the (ordinal) response. In this program, any given set of factors (i.e. categorical explanatory variables) should be first converted into the required number of indicator variables which are then stored as separate columns in the input data set. This could be seen as a drawback for analyses with a large number of factors each having a large number of levels (In the Kilifi data set, factors such as “visit”-44 levels). In addition, a constant term is required in the linear predictor of the model, thus the data file should have a column of 1’s as one of the explanatory variables. For these reasons We will not use the MIXORE program in the analysis of the current data set.
There are a number of issues associated with this approach, among them:
• It is in fact relatively easy to extend NLMIXED analyses for correlated random effects such as those found in random coefficient regression
models. Such models are quite easy to fit in WinBUGS.
• It may be computationally demanding
• It is not widely implemented in existing commercial software packages (it is only freely available in MIXORE and also available in Stata ver- sions 6 and 9)
One might wonder why we opt to use the likelihood conditional on the ran- dom effects u in the case of the non-normal response. The answer is that in the case of a normal response and the identity link, the random effects do not appear explicitly in the likelihood, but only appear through the vari- ance covariance parameters σ21, σ22, . . . , σ2k in the case of k random effects or equivalently through the “ratios or gammas”γi = σσ2i2 whereσ2 is the resid- ual variance. In the above example E(zj) = 0, thus E[xijβ+γzj] = xijβ.
However E[g(µ)]6=g[E(µ)] in the case of the non-identity link functiong, so taking the expectations will not cause the random terms to vanish. Hence we will consider a number of alternative approaches based on modifications to the mixed model equations.