Quality & Quantity 37: 71–89, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
The Multilevel Approach to Repeated Measures for
Complete and Incomplete Data
CORA J. M. MAAS
and TOM A. B. SNIJDERS
Department of Methodology and Statistics, University of Utrecht;
Department of Statistics and
Measurement Theory, University of Groningen
Abstract. Repeated measurements often are analyzed by multivariate analysis of variance (MAN-
OVA). An alternative approach is provided by multilevel analysis, also called the hierarchical linear
model (HLM), which makes use of random coefﬁcient models. This paper is a tutorial which indicates
that the HLM can be speciﬁed in many different ways, corresponding to different sets of assumptions
about the covariance matrix of the repeated measurements. The possible assumptions range from the
very restrictive compound symmetry model to the unrestricted multivariate model. Thus, the HLM
can be used to steer a useful middle road between the two traditional methods for analyzing repeated
measurements. Another important advantage of the multilevel approach to analyzing repeated meas-
ures is the fact that it can be easily used also if the data are incomplete. Thus it provides a way to
achieve a fully multivariate analysis of repeated measures with incomplete data.
Key words: MANOVA, incomplete data, missing at random, hierarchical linear model, Hotelling’s
test, Wald test, compound symmetry model.
Repeated measures data are common in many disciplines. Procedures for analysing
such data are treated, e.g., in O’Brien and Kaiser (1985), Maxwell and Delaney
(1990), and Stevens (1996). In the period before 1985, mainly the compound sym-
metry model and the closely related sphericity model were used. The compound
symmetry model represents the dependence between the several data obtained from
a single individual by a random main effect of the individual. The paper by O’Brien
and Kaiser (1985) marks the transition to the use of procedures based on mul-
tivariate analysis of variance (MANOVA). In the MANOVA model, no assumptions
are made about the covariance matrix of the repeated measurements. The only
assumptions are independence and identical distributions within treatment groups,
homoscedasticity between groups, multivariate normality, and complete data. The
last assumption means that, if there are p measurement occasions, for each subject
in the data set the measurements on all p occasions are available.
Since the seminal paper by Laird and Ware (1982), random coefﬁcient mod-
els, or linear mixed models, have been increasingly used for analysing repeated
measurements. These models have also been used in multilevel analysis (Bryk and
Raudenbush, 1992; Goldstein, 1995; Snijders and Bosker, 1999), a methodology