Estimating and Visualizing Nonlinear Relations Among Latent Variables: A Semiparametric Approach

What Can We Learn from a Semiparametric Factor Analysis of Item Responses and Response Time? An Illustration with the PISA 2015 Data

It is widely believed that a joint factor analysis of item responses and response time (RT) may yield more precise ability scores that are conventionally predicted from responses only. For this purpose, a simple-structure factor model is often preferred as it only requires specifying an additional measurement model for item-level RT while leaving the original item response theory (IRT) model for responses intact. The added speed factor indicated by item-level RT correlates with the ability factor in the IRT model, allowing RT data to carry additional information about respondents' ability. However, parametric simple-structure factor models are often restrictive and fit poorly to empirical data, which prompts under-confidence in the suitablity of a simple factor structure. In the present paper, we analyze the 2015 Programme for International Student Assessment mathematics data using a semiparametric simple-structure model. We conclude that a simple factor structure attains a decent fit after further parametric assumptions in the measurement model are sufficiently relaxed. Furthermore, our semiparametric model implies that the association between latent ability and speed/slowness is strong in the population, but the form of association is nonlinear. It follows that scoring based on the fitted model can substantially improve the precision of ability scores.

Efficient Likelihood Estimation of Generalized Structural Equation Models with a Mix of Normal and Nonnormal Responses

A maximum likelihood estimation routine is presented for a generalized structural equation model that permits a combination of response variables from various distributions (e.g., normal, Poisson, binomial, etc.). The likelihood function does not have a closed-form solution and so must be numerically approximated, which can be computationally demanding for models with several latent variables. However, the dimension of numerical integration can be reduced if one or more of the latent variables do not directly affect any nonnormal endogenous variables. The method is demonstrated using an empirical example, and the full estimation details, including first-order derivatives of the likelihood function, are provided.

The Effect of Local Government Debt on Regional Economic Growth in China: A Nonlinear Relationship Approach

By drawing on the concept of sustainable economic development, this study advances the research on debt sustainability in the economic literature. We explore the correlation between local government debt and regional economic growth in 30 provinces in China. Previous studies have established that the development of economic growth between regions is not independent and we, therefore, investigate the spatial effect of regional economic growth due to the existence of a spatial spillover effect or spatial expansion among regions. Using Moran’s scatter plot, a Local Indicator of Spatial Association (LISA) map, and a semiparametric spatial model (SE-SDM), our results demonstrate the following: (1) the spatial agglomeration effect has a significant influence on regional economic growth; (2) the relationship between local government debt and regional economic growth presents nonlinear characteristics, rather than having an inverted U-shaped relationship; (3) the semiparametric spatial model more accurately characterizes in the nonlinear relationship between local government debt and regional economic growth compared to a basic regression model and the spatial Durbin model; and (4) when the scale of local government debt exceeds a certain level, economic growth will be suppressed by the crowding-out of private investment and the reduction of public expenditure.

The relationship between multilevel models and non-parametric multilevel mixture models: Discrete approximation of intraclass correlation, random coefficient distributions, and residual heteroscedasticity

Multilevel data structures are common in the social sciences. Often, such nested data are analysed with multilevel models (MLMs) in which heterogeneity between clusters is modelled by continuously distributed random intercepts and/or slopes. Alternatively, the non-parametric multilevel regression mixture model (NPMM) can accommodate the same nested data structures through discrete latent class variation. The purpose of this article is to delineate analytic relationships between NPMM and MLM parameters that are useful for understanding the indirect interpretation of the NPMM as a non-parametric approximation of the MLM, with relaxed distributional assumptions. We define how seven standard and non-standard MLM specifications can be indirectly approximated by particular NPMM specifications. We provide formulas showing how the NPMM can serve as an approximation of the MLM in terms of intraclass correlation, random coefficient means and (co)variances, heteroscedasticity of residuals at level 1, and heteroscedasticity of residuals at level 2. Further, we discuss how these relationships can be useful in practice. The specific relationships are illustrated with simulated graphical demonstrations, and direct and indirect interpretations of NPMM classes are contrasted. We provide an R function to aid in implementing and visualizing an indirect interpretation of NPMM classes. An empirical example is presented and future directions are discussed.

The Standardization of Linear and Nonlinear Effects in Direct and Indirect Applications of Structural Equation Mixture Models for Normal and Nonnormal Data

The application of mixture models to flexibly estimate linear and nonlinear effects in the SEM framework has received increasing attention (e.g., Jedidi et al., 1997b; Bauer, 2005; Muthén and Asparouhov, 2009; Wall et al., 2012; Kelava and Brandt, 2014; Muthén and Asparouhov, 2014). The advantage of mixture models is that unobserved subgroups with class-specific relationships can be extracted (direct application), or that the mixtures can be used as a statistical tool to approximate nonnormal (latent) distributions (indirect application). Here, we provide a general standardization procedure for linear and nonlinear interaction and quadratic effects in mixture models. The procedure can also be applied to multiple group models or to single class models with nonlinear effects like LMS (Klein and Moosbrugger, 2000). We show that it is necessary to take nonnormality of the data into account for a correct standardization. We present an empirical example from education science applying the proposed procedure.

A Heterogeneous Growth Curve Model for Nonnormal Data

The heterogeneous growth curve model (HGM; Klein & Muthén, 2006 ) is a method for modeling heterogeneity of growth rates with a heteroscedastic residual structure for the slope factor. It has been developed as an extension of a conventional growth curve model and a complementary tool to growth curve mixture models. In this article, a robust version of the heterogeneous growth curve model (HGM-R) is presented that extends the original HGM with a mixture model to allow for an unbiased parameter estimation under the condition of nonnormal data. In two simulation studies, the performance of the method is examined under the condition of nonnormality and a misspecified heteroscedastic residual structure. The results of the simulation studies suggest an unbiased estimation of the heterogeneity by the HGM-R when sample size was large enough and a good approximation of the heteroscedastic residual structure even when the functional form of the heteroscedasticity was misspecified. The practical application of the approach is demonstrated for a data set from HIV-infected patients.

A model of psychosis and its relationship with impairment

PURPOSE

Some studies suggest that positive symptoms of psychosis-clinical and sub-clinical alike-reflect a single, continuously distributed dimension in the population. It is unknown, however, whether such a spectrum of positive psychotic experiences is non-linearly related to outcomes such as daily functioning. This work aims to characterize the relationship between positive psychosis and impairment.

METHODS

Data from the Office of National Statistics National Psychiatric Morbidity Surveys of Great Britain were used to establish measurement models of psychosis and impairment. Competing linear and nonlinear models of the relationship between the two latent variables were evaluated using mixture structural equation models.

RESULTS

Positive psychosis is best modeled by a continuous, normal distribution. Increases in positive psychosis correlate with roughly linear increases in impairment.

CONCLUSIONS

Positive psychotic symptoms occur throughout the population without a discrete, pathological threshold. Functional deficits are linearly associated with the psychosis at all points along the continuum, and a significant portion of the population experiences subclinical psychosis.