The new book by Skrondal and Rabe-Hesketh (2004) is reviewed.GLLAMM, generalized linear latent and mixed models, latent variables

Roger Newson

English

Research Papers in Economics

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trademark of StataCorp LP.The Stata Journal (2005)
5, Number 1, pp. 130–133
Review of Generalized Latent Variable
Modeling by Skrondal and Rabe-Hesketh
Roger Newson
King’s College London, UK
roger.newson@kcl.ac.uk
Abstract. The new book by Skrondal and Rabe-Hesketh (2004) is reviewed.
Keywords: gn0025, GLLAMM, generalized linear latent and mixed models, latent
variables
1 Introduction
This is a very impressive book. The authors introduce a class of statistical models,
known as generalized latent variable models (GLVMs), which contains, as a special
case, the generalized linear models (GLMs) introduced by McCullagh and Nelder (1989).
GLMs have the feature that the conditional mean of the outcome Y -variable, given val-
ues of a list of X-variables, can be transformed by a link function to give a linear
predictor equal to a linear combination of these X-variables. The estimated param-
eters are typically interpreted as proportions, odds, probits, or arithmetic, geometric,
harmonic, or algebraic means and their diﬀerences or ratios. This new book extends
GLMs by allowing the linear predictor to contain not only terms containing X-variables
whose values we actually know, but also terms containing “latent variables”, or hidden
variables, whose values we can only imagine. The estimated parameters are therefore
typically interpreted as conditional proportions, odds, probits, or arithmetic, geometric,
harmonic, or algebraic means (and their diﬀerences or ratios), conditioning on the val-
ues both of the observed X-variables and of the unobserved latent variables. The latent
variables are usually assumed to be sampled from a distribution with a zero mean, and
to have a value, in each observation, speciﬁc to a cluster or subcluster to which that
observation belongs. Typically, therefore, the latent variables are cluster or subcluster
eﬀects, representing a diﬀerence between a particular cluster or subcluster and the “av-
erage” cluster or subcluster. For instance, if the Y -variable is an exam mark produced
by a particular student in a particular class in a particular school on a particular day,
then the observations might correspond to exam scripts, the clusters might be schools,
and the subclusters might be classes within schools and students within classes. Most
(but not all) of the models discussed can be ﬁtted to data using the gllamm package in
Stata (see Rabe-Hesketh, Skrondal, and Pickles [2004]), downloadable from SSC. The au-
thors argue that many statistical methods, such as common factor analysis, latent class
models, frailty models, random-eﬀects models, and multilevel modeling, can be seen as
special cases of GLVMs and implemented using gllamm. However, this book is primarily
about GLVMs (the generic family of methods), and not about gllamm (the means for
implementing these methods speciﬁcally in Stata), which will be covered more fully in
a forthcoming book from Stata Press (Rabe-Hesketh, Pickles, and Skrondal 2005).
c  2005 StataCorp LP gn0025R. Newson 131
GLVMs are subtly diﬀerent from generalized estimating equation (GEE) models, which
are an alternative extension of GLMs to clustered data (see Hardin and Hilbe [2003]).
The parameters of a GLVM corresponding to the X-variables describe the conditional
distribution of the Y -variable, given the values both of the observed X-variables and of
the unobserved latent variables. The corresponding parameters of a GEE model describe
the distribution of the Y -variable, conditional on the X-values but marginal to the latent
variables (assuming that these exist). If the parameters are risk diﬀerences, arithmetic
mean diﬀerences, arithmetic mean ratios, geometric mean ratios, or risk ratios, the GEE
parameters are the same as the corresponding GLVM parameters. However, this identity
does not hold for all parameters. For instance, if the parameters are odds ratios, then
the GEE parameters will tend systematically to be closer to one than the corresponding
GLVM parameters. Therefore, if the observations belong to students, the clusters are
schools, the Y -variable indicates whether an exam is passed, the X-variable of interest
is a student nonsmoking status indicator, and a logit link function is used; then the GEE
smoking-related odds-ratio estimates the factor whereby the odds of passing an exam
would change in the population if the whole world changed from smokers to nonsmokers.
The GLVM odds ratio on the other hand would estimate the factor by which the exam-
passing rate in each school would change if all the students in that school were to
change from smokers to nonsmokers. The latter odds would be further from one than
the former and might arguably be more useful for the head of each school to know,
particularly if each school head had some idea of his/her own school’s baseline odds,
assuming that all the students smoked. Also the GLVM parameters are more comparable
to the corresponding parameters of the ﬁxed-eﬀects model that we would probably have
used if we had data only on a small number of schools.
Latent variables are controversial because it is not often easy to measure them di-
rectly, and this might lead many people to wonder in what sense they can be said to
exist. A GLVM enthusiast might argue that genotypes were once latent variables (in the
days of Gregor Mendel) and that they were useful for some decades before we developed
the technology to measure them directly, and that today’s latent variables might sim-
ilarly correspond to something that our descendants will be able to measure directly.
More realistically, as the authors argue in section 1.7, latent variable models can be used
to generate a very wide range of within-cluster dependence structures for the outcome.
For instance, if there are subclusters nested within clusters, then outcomes in the same
subcluster will be more correlated than outcomes in diﬀerent subclusters within the
same cluster, and hierarchial structures of that nature do not seem to be available using
Stata’s GEE programs. The Gauss–Markov theorem seems to imply that the better you
can model the within-cluster dependence, the less wide the conﬁdence limits will be for
the same coverage probability. Latent variable models can therefore still be useful, even
if we do not seriously believe in the latent variables themselves. The gllamm package
oﬀers the option of Huber variances for users who are skeptical of the possibility of
guessing the correlation structure right the ﬁrst time.132 Review of Skrondal and Rabe-Hesketh
2 Summary
The book is divided into two parts, of which the ﬁrst presents the methods and the
second gives us a tour of a wide range of possible applications.
In the Methodology part, chapter 1 gives a survey of the uses to which latent vari-
able models have been put (rightly or wrongly) in diverse areas of science. Chapter
2 gives a brief survey of modeling methods used for various outcomes, with particu-
lar reference to GLMs. Chapter 3 surveys a range of existing latent variable methods.
Chapter 4 presents the GLVM family, which includes most of the latent variable models
in the previous chapters as special cases. Chapter 5 presents analytical methods for
establishing, for a particular GLVM, the feasibility of identifying those parameters which
govern the distribution of the latent variables. As the authors state, these parameters
are identiﬁable only through their eﬀects on the conditional covariance of the Y -variable
in observations belonging to the same cluster, which can be measured whether or not
we believe in the latent variables themselves. Chapter 6 surveys estimation methods
for the parameters, including maximum likelihood, quasilikelihood, and Bayesian meth-
ods, with particular reference to numerical integration or summation over the ranges of
the hypothesized latent variables, which is important because it accounts for the high
level of computing resources required by GLVMs. Chapter 7 presents methods whereby,
in some circumstances, the controversial latent variables might actually be measured.
These include maximum likelihood methods (which are most useful if the clusters are
large and which estimate the latent variables as we might estimate ﬁxed eﬀects) and
empirical Bayesian methods (which are most useful if we have a large number of small
clusters and which explicitly pool information from all the clusters to calculate credible
intervals for the cluster-speciﬁc latent variables for each individual cluster). Finally,
chapter 8 gives a survey of inference, diagnostics, and goodness-of-ﬁt tests, including a
discussion of Huber variances.
In the Applications part, the authors give a range of examples, based on datasets
that users can download. These examples are grouped by the type of the outcome
variable, which in these examples may be dichotomous, ordinal, counts, survival data,
polytomous discrete-choice responses, or a mixture of diﬀerent response types.
3 Limitations
This is an excellent book and has fewer limitations than anybody is entitled to expect,
given that the authors and their coworkers have developed a very comprehensive grand
uniﬁed method mostly in the last few years. However, from the Applications part, it
is clear that the main problem with GLVMs is going to be explaining the parameters,
and sometimes the controversial latent variables, to nontechnical people. Some latent
variables are more controversial than others. Most people in the medical sector have no
problem believing in the counterfactual outcomes that a patient would have experienced,
had the patient been allocated to treatments A or B, and we can estimate at least the
mean diﬀerence between these two counterfactuals using a randomized controlled trial
(see Rosenbaum [2003]). At the other end of the spectrum, the polytomous discrete-R. Newson 133
choice models with latent utilities demonstrated in chapter 13 do not seem to correspond
to any mechanism whereby I myself choose a coﬀee maker or an elected representative,
and it does not surprise me that these models lead to the counterintuitive “paradoxes”
that the authors mention. However, some of the estimated model parameters may
still be useful to know. Although this book is intended mainly for readers who can
understand natural logarithms and standard errors, such people usually have to explain
their output to less-technical people, who (in my experience) usually understand odds
ratios with conﬁdence limits better than they understand log odds ratios with standard
errors, as presented in this book. On a minor point of accuracy, it is not strictly true
(as the authors suggest in section 9.5.2) that a ﬁxed-eﬀect meta-analysis requires the
assumption of an equal treatment eﬀect in all studies in order to be valid. If a common
treatment eﬀect is estimated using unclustered full Huber variances, then the conﬁdence
interval is a valid interval estimate of the “weighted average” treatment that would have
been observed, if only we could scale up the size of each study by the same large factor,
and this may be useful to know, even if the eﬀect varies from study to study. However,
I have no hesitation in recommending readers to buy this book, and look forward to
seeing the forthcoming book on gllamm (Rabe-Hesketh, Pickles, and Skrondal 2005).
4 References
Hardin, J. W. and J. M. Hilbe. 2003. Generalized Estimating Equations. Boca Raton,
FL: Chapman & Hall/CRC.
McCullagh, P. and J. A. Nelder. 1989. Generalized Linear Models. 2nd ed. London:
Chapman & Hall.
Rabe-Hesketh, S., A. Pickles, and A. Skrondal. 2005 [forthcoming]. Multilevel and
Structural Equation Modeling for Continuous, Categorical, and Event Data. College
Station, TX: Stata Press.
Rabe-Hesketh, S., A. Skrondal, and A. Pickles. 2004. GLLAMM Manual. Working
paper 160, University of California Berkeley, Division of Biostatistics.
http://www.bepress.com/ucbbiostat/paper160/
Rosenbaum, P. R. 2003. Observational Studies. 2nd ed. New York: Springer.
Skrondal, A. and S. Rabe-Hesketh. 2004. Generalized Latent Variable Modeling: Mul-
tilevel, Longitudinal, and Structural Equation Models. Boca Raton, FL: Chapman &
Hall/CRC.
About the Author
Roger Newson is a Lecturer in Medical Statistics at King’s College London, UK, working
principally in asthma research. He wrote the Stata 5 package rglm for calculating semi-Huber
and full Huber variances for generalized linear models.

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