Maximum Likelihood and Bayesian Estimation for Nonlinear Structural Equation Models

A comparison of methods for estimating quadratic effects in nonlinear structural equation models.

“…That is, for the measurement model in Equation 3, the three products of the observed variables x mi that would need to be constructed in the data set are $$x_{1i}^2=x_{1i}\cdot x_{1i}{x}_{2i}^2=x_{2i}\cdot x_{2i}{x}_{3i}^2=x_{3i}\cdot x_{3i}.$$ Note that the observed variables x mi are not mean centered before constructing their products. Although multicollinearity will certainly exist between the first-order terms and their corresponding derived quadratic terms (Aiken & West, 1991), Wall (2009) demonstrated the equivalence of the regression coefficient of the highest order term (in this case the quadratic term) and its standard error when the observed variables were mean centered and when they were not. Because our focus is on reliably estimating the coefficient of the quadratic term, we proceed with not mean centering the indicators of the first-order predictor, f 2 .…”

“…, Z n can be written as in Equation 3. Following the notation in Wall (2009), let θ m represent the measurement model parameters (i.e., parameters in Λ , Θ ) and θ s denote the nonlinear structural parameters (i.e., γ 0 , γ 1 , γ 2 , ${\sigma }_d^2$). Note that $\theta ={({\theta }_m^{\prime },{\theta }_s^{\prime })}^{\prime }$.…”

“…MCMC methods are useful in sampling from the multivariate densities, which are not easy to sample from, usually by breaking these densities down into more tractable univariate or multivariate densities (Lynch, 2007). That is, MCMC methods sidestep sampling directly from the joint posterior distribution by sampling from the full conditional posteriors of each parameter conditioned on the data and the most recent value of all other parameters (see, e.g., Lee, Song, & Tang, 2007; Wall, 2009). …”

“…In the latter case, these prior distributions are known as diffuse or noninformative. Following Wall (2009), we used the following noninformative priors for the simulation: τ k ~ N (0, 0.0001); λ k ~ N (0, 0.0001); ${\theta }_{ek}^{-1}\sim \text{Gamma}(0.1,0.1)$; γ j ~ N (0, 0.0001); μ 0 f ~ N (0, 0.0001); ${\phi }_{0f}^{-1}\sim \text{Gamma}(0.1,0.1)$; d ~ N (0, φ 0 d and ${\phi }_{0d}^{-1}\sim \text{Gamma}(0.1,0.1)$.…”

Ordinal Regression Models for Continuous Scales

“…Use of an non-informative prior results in the posterior distribution being proportional to the likelihood, and for large n the same estimate is achieved with ML and Bayesian estimation. Several authors (for example Huber and Train, 2001, Kuhner and Smith, 2007, Wall, 2009) have pointed out the similarity between ML and Bayesian estimates, in the contexts of choice modelling, genealogy and structural equation modelling, respectively. This suggests that the choice between ML and Bayesian estimation then becomes one of computational feasibility and convenience, rather than a philosophical choice between methodologies.…”