Equivalence and Differences Between Structural Equation Modeling and State-Space Modeling Techniques

Abstract

State-space modeling techniques have been compared to structural equation modeling (SEM) techniques in various contexts but their unique strengths have often been overshadowed by their similarities to SEM. In this article, we provide a comprehensive discussion of these 2 approaches' similarities and differences through analytic comparisons and numerical simulations, with a focus on their use in representing intraindividual dynamics and interindividual differences. To demonstrate the respective strengths and weaknesses of the 2 approaches in representing these 2 aspects, we simulated data under (a) a cross-sectional common factor model, (b) a latent difference score model with random effects in intercept and slope, and (c) a bivariate dynamic factor analysis model with auto- and cross-regression parameters. Possible ways in which SEM and state-space modeling can be utilized as complementary tools in representing human developmental and other related processes are discussed.

Notes

¹Equation 1 is sometimes further partitioned into two equations to express the measurement model separately for the dependent (i.e., endogenous) and independent (i.e., exogenous) latent variables. We use a simpler expression here to better highlight the parallels between SEM models and state-space models. Equations eq1 and 2 can also be combined into a single equation involving latent and manifest variables (using the reticular action model or RAM notations; (McArdle & McDonald, 1984) but this simplified form is not considered here.

²In this case, a filter matrix needs to be used to reduce Σ and μ to the appropriate dimensions for each person.

³ SsfPack is one of the many statistical packages implemented using Ox, an object-oriented matrix programming language with a comprehensive mathematical and statistical function library (Doornik 1998). Mkfm6 can be downloaded from Conor Dolan's Web site at http://users.fmg.uva.nl/cdolan/, whereas Ox and SSfPack can be downloaded from http://www.doornik.com/download.html.

⁴Note that because there is only one replication, some systematic biases were present in the parameter estimates in both cases.

⁵Bivariate extensions of this univariate model with coupling between two constructs have also been presented by McArdle and Hamagami (2001) Although the bivariate model is not considered here, results from this article are readily generalizable to other equivalent latent difference equation models.

⁶Another option is to fit a reformulated version of the dual change score model as explicated by McArdle and Hamagami (2001). Using this approach, the dual change score model can be expressed as an oblique two-factor model with the two factors representing latent level (i.e., η₀) and slope (i.e., η _s ). The appropriate parameter constraints are then imposed on the factor loadings of η _s on y_t to obtain β. Thus, the resultant model eliminates all the latent change scores, η _t , and the true score components, η _t , from the path diagram in Figure 2. We did not consider this approach for this illustration.

⁷This is because parameter estimates yielded from the PED are known to be maximum likelihood estimates (Shumway 2000) but estimates obtained using the block Toeplitz method in general are not (Hamaker, Dolan, & Molenaar, 2002).