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psychometrikavol. 70, no. 1, 147167
march 2005DOI: 10.1007/s11336-003-1070-8ON MUTH ENS MAXIMUM LIKELIHOOD FOR TWO-LEVEL COVARIANCESTRUCTURE MODELSKe-Hai Yuanuniversity of notre dameKentaro Hayashiuniversity of hawaii at manoaData in social and behavioral sciences are often hierarchically organized. Special statistical procedures that take into account the dependence of such observations have been developed. Among procedures
for 2-level covariance structure analysis, Muthens maximum likelihood (MUML) has the advantage of
easier computation and faster convergence. When data are balanced, MUML is equivalent to the maximum likelihood procedure. Simulation results in the literature endorse the MUML procedure also for
unbalanced data. This paper studies the analytical properties of the MUML procedure in general. The
results indicate that the MUML procedure leads to correct model inference asymptotically when level-2
sample size goes to infinity and the coefficient of variation of the level-1 sample sizes goes to zero. The
study clearly identifies the impact of level-1 and level-2 sample sizes on the standard errors and test statistic
of the MUML procedure. Analytical results explain previous simulation results and will guide the design
or data collection for the future applications of MUML.Key words: asymptotics, likelihood ratio statistic, multilevel covariance structure, standard error estimates1. IntroductionData in social and behavioral sciences often exhibit hierarchical structure. For example,
households are nested within neighborhoods, neighborhoods are nested within cities, and cities
are further nested within countries; students are nested within classes, classes are nested within
schools, and schools are further nested within school districts. Cases within a cluster are generally
correlated. Analysis of such data has to explicitly account for these correlations. The development of statistical methods for hierarchical data is documented by monographs and edited books
(Goldstein, 1995; Heck & Thomas, 2000; Hox, 2002; Kreft & de Leeuw, 1998; Raudenbush &
Bryk, 2002; Reise & Duan, 2003; Snijders & Bosker, 1999). Among these are the hierarchical
linear model (HLM) and the multilevel structural equation model (SEM) (Bentler & Liang, 2003;
du Toit & du Toit, 2004 Goldstein & McDonald, 1988; Lee, 1990; Lee & Poon, 1998; Liang &
Bentler, in 2004; Little, Schnabel & Baumert, 2000; Longford, 1993; McArdle & Hamagami,
1996; McDonald & Goldstein, 1989; Muthen, 1994, 1997; Muthen & Satorra, 1995; Poon & Lee,
1994; Yuan & Bentler, 2002, 2003a).In a multilevel covariance structure model, parameters...