Introduction 3
repeated-measures ANOVA models assumed a multivariate normal (MVN) distribution
of the repeated measures and required either estimation of all covariance parameters of
the MVN distribution or an assumption of “sphericity” of the covariance matrix (with
corrections such as those proposed by Geisser and Greenhouse (1958) or Huynh and Feldt
(1976) to provide approximate adjustments to the test statistics to correct for violations
of this assumption). In contrast, LMM software, although assuming the MVN
distribution of the repeated measures, allows users to fit models with a broad selection
of parsimonious covariance structures, offering greater efficiency than estimating the full
variance-covariance structure of the MVN model, and more flexibility than models assum-
ing sphericity. Some of these covariance structures may satisfy sphericity (e.g., indepen-
dence or compound symmetry), and other structures may not (e.g., autoregressive or
various types of heterogeneous covariance structures). The LMM software procedures
considered in this book allow varying degrees of flexibility in fitting and testing covariance
structures for repeated-measures or longitudinal data.
Software for LMMs has other advantages over software procedures capable of fitting
traditional repeated-measures ANOVA models. First, LMM software procedures allow
subjects to have missing time points. In contrast, software for traditional repeated-
measures ANOVA drops an entire subject from the analysis if the subject has missing
data for a single time point (known as complete-case analysis; see Little and Rubin, 2002).
Second, LMMs allow for the inclusion of time-varying covariates in the model (in addition
to a covariate representing time), whereas software for traditional repeated-measures
ANOVA does not. Finally, LMMs provide tools for the situation in which the trajectory
of the outcome varies over time from one subject to another. Examples of such models
include growth curve models, which can be used to make inference about the variability
of growth curves in the larger population of subjects. Growth curve models are examples
of random coefficient models (or Laird–Ware models), which will be discussed when
considering the longitudinal data in Chapter 6 (the Autism data).
In Chapter 5, we consider LMMs for a small repeated-measures data set with two within-
subject factors (the Rat Brain data). We consider models for a data set with features of
both clustered and longitudinal data in Chapter 7 (the Dental Veneer data).
1.1.3 The Purpose of this Book
This book is designed to help applied researchers and statisticians use LMMs appropri-
ately for their data analysis problems, employing procedures available in the SAS, SPSS,
Stata, R, and HLM software packages. It has been our experience that examples are the
best teachers when learning about LMMs. By illustrating analyses of real data sets using
the different software procedures, we demonstrate the practice of fitting LMMs and
highlight the similarities and differences in the software procedures.
We present a heuristic treatment of the basic concepts underlying LMMs in Chapter 2.
We believe that a clear understanding of these concepts is fundamental to formulating an
appropriate analysis strategy. We assume that readers have a general familiarity with
ordinary linear regression and ANOVA models, both of which fall under the heading of
general (or standard) linear models. We also assume that readers have a basic working
knowledge of matrix algebra, particularly for the presentation in Chapter 2.
Nonlinear mixed models and generalized LMMs (in which the dependent variable may
be a binary, ordinal, or count variable) are beyond the scope of this book. For a discussion
of nonlinear mixed models, see Davidian and Giltinan (1995), and for references on
generalized LMMs, see Diggle et al. (2002) or Molenberghs and Verbeke (2005). We also
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