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Comparing Bayesian parametric and semiparametric estimation of nonlinear relationships in structural equation models with ordinal data
Qin, Lu
Qin, Lu
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Abstract
The Bayesian parametric and semiparametric approaches are compared to recover the polynomial and nonpolynomial relationships among latent factors in the structural equation model (SEM). In earlier studies, the semiparametric approach has been demonstrated to be a more advanced approach to estimate the nonnormally distributed densities. However, its performance in recovering nonlinearity among factors has not been widely studied. The objectives of this dissertation are (1) to compare the recovery performances between the parametric and semiparametric approaches in capturing the polynomial and nonpolynomial relationships among latent factors in the structural model and (2) to investigate the recovery performance of the semiparametric approach in capturing the nonpolynomial relationships when the polynomial function is misspecified. The Bayesian semiparametric approach is applied using the truncated Dirichlet process with a stick-breaking prior to track the nonlinearity under different combinations of nonlinear functions (e.g., exponential, logarithmic, and sine) in the simulation study. Several important results were revealed. First, in study 1, both the parametric and semiparametric approaches achieved good convergence rates under the exponential and sine conditions. The polynomial conditions had greater difficulty in convergence due to the quadratic and interaction effects. Second, regarding the nonlinearity recoveries, the parametric approach performed similarly to the semiparametric approach at large truncation levels (200) in recovering the polynomial nonlinearity. The semiparametric approach had better recovery of nonpolynomial nonlinearity than the parametric approach. Third, in study 2, the semiparametric approach had a fairly good convergence rate at truncation level 5 under the exponential and sine conditions. Fourth, the semiparametric approach barely recovered the nonpolynomial nonlinearity with a misspecified polynomial function. A large truncation level did not improve the recovery performance when a nonlinear function is incorrectly presumed. The results implied that when latent factors or data is normally distributed, parametric approach is sufficient to provide an accurate recovery of nonlinear relationships among latent factors. However, when latent factors or data is non-normally distributed, the semiparametric approach provides more accurate estimations and a higher accuracy in capturing nonlinear relationships among latent factors. Considering the capacity of computer memory and running time, a small truncation level is suggested to capture the polynomial and nonpolynomial nonlinearity.
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Date
2018-12-31
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University of Kansas
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Keywords
Educational psychology, nonlinear relationship, parametric approach, semiparametric approach