ANALYSIS OF TWO-LEVEL STRUCTURAL EQUATION MODELS VIA EM TYPE ALGORITHMS

In this paper, the maximum likelihood estimation of a general two-level structural equation model with an unbalanced design is formulated as a missing data problem by treating the latent random vectors at the group level as hypo- thetical missing data. The commonly used EM algorithm is utilized to obtain the solution. Expressions for the E-step are derived and it is shown that the complex optimization of the M -step can be completed conveniently with existing software. Some accelerated procedures such as the EM gradient algorithm and the Quasi- Newton EM algorithm are modified to improve the convergence rate of the basic EM algorithm. Results from simulation studies and analysis of examples illustrate the features and potential of the EM approach.

[1]  Sik-Yum Lee,et al.  Multilevel analysis of structural equation models , 1990 .

[2]  Nicholas T. Longford,et al.  Factor analysis for clustered observations , 1992 .

[3]  K. Lange.  A gradient algorithm locally equivalent to the EM algorithm , 1995 .

[4]  Peter M. Bentler,et al.  EQS : structural equations program manual , 1989 .

[5]  Harvey Goldstein,et al.  Balanced versus unbalanced designs for linear structural relations in two‐level data , 1989 .

[6]  Donald B. Rubin,et al.  Max-imum Likelihood from Incomplete Data , 1972 .

[7]  Donald B. Rubin,et al.  EM and beyond , 1991 .

[8]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[9]  D. Rubin,et al.  The ECME algorithm: A simple extension of EM and ECM with faster monotone convergence , 1994 .

[10]  D. G. Weeks,et al.  Linear structural equations with latent variables , 1980 .

[11]  Robert I. Jennrich,et al.  A study of algorithms for covariance structure analysis with specific comparisons using factor analysis , 1979 .

[12]  W. Poon,et al.  Two-level analysis of covariance structures for unbalanced designs with small level-one samples. , 1992, The British journal of mathematical and statistical psychology.

[13]  Dorothy T. Thayer,et al.  EM algorithms for ML factor analysis , 1982 .

[14]  D. Titterington.  Recursive Parameter Estimation Using Incomplete Data , 1984 .

[15]  Karl G. Jöreskog,et al.  Lisrel 8: User's Reference Guide , 1997 .

[16]  R. Darrell Bock,et al.  Multilevel analysis of educational data , 1989 .

[17]  Stephen W. Raudenbush,et al.  Maximum likelihood estimation for unbalanced multilevel covariance structure models via the EM algorithm , 1995 .

[18]  R. Jennrich,et al.  Conjugate Gradient Acceleration of the EM Algorithm , 1993 .