Testing Inequality Constrained Hypotheses in SEM Models

Abstract

Researchers often have expectations that can be expressed in the form of inequality constraints among the parameters of a structural equation model. It is currently not possible to test these so-called informative hypotheses in structural equation modeling software. We offer a solution to this problem using Mplus. The hypotheses are evaluated using plug-in p values with a calibrated alpha level. The method is introduced and its utility is illustrated by means of an example.Order-restricted inference has been studied in the frequentist framework (see, e.g., Barlow, Bartholomew, Bremner, & Brunk, 1972;Robertson, Wright, & Dykstra, 1988;Silvapulle & Sen, 2004) as well as in the Bayesian framework (e.g., Hoijtink, Klugkist, & Boelen, 2008;Klugkist, Laudy, & Hoijtink, 2005). However, testing order constraints has received relatively little attention in the structural equation modeling (SEM) literature (Gonzalez & Griffin, 2001; Stoel, Galindo-Garre, Dolan, & Van den Wittenboer, 2006). SEM is often used and its attractiveness is largely due to its flexibility in specifying and testing hypotheses among both observed and latent variables in multiple groups.SEM software can be used to impose inequality constraints among the parameters of interest. More specifically, to evaluate a research question, model parameters such as regression coefficients can be constrained to being greater or smaller than either a fixed value or other regression coefficients. We call hypotheses that contain inequality constraints informative hypotheses. Mplus (Muthén & Muthén, 2007) allows for such user-specified constraints and order constraint parameter estimation is available. The problem is that a null hypothesis test for the evaluation of an informative hypothesis is lacking in SEM software.

MPTinR: Analysis of multinomial processing tree models in R

“…Furthermore, although multiTree comes with a wealth of bootstrap functions, it misses the flexibility of MPTinR. For example, it seems difficult to implement model mimicry analysis or double-bootstrap procedures to assess pvalues in inequality-restricted inference tests (van de Schoot et al, 2010), while such procedures can be implemented in MPTinR in a relatively straightforward manner through the use of the gen.data and sample.data functions.…”

“…These functions can be used separately or jointly in order to obtain parametric and nonparametric bootstrap samples. These are general purpose functions that can be used for a wide variety of goals, such as (1) obtaining confidence intervals for the estimated parameters, (2) sampling distributions of the G 2 statistic and p-values under several types of null-hypotheses (van de Schoot et al, 2010), and (3) model-mimicry analysis (Wagenmakers, Ratcliff, Gomez, & Iverson, 2004). Also, bootstrap simulations assuming individual differences, as implemented by Hu and Phillips (1999) and Moshagen (2010), can be obtained using these functions.…”

“…In this case, the model is assumed to correspond to the true data-generating process, and the bootstrap is designated as parametric. The parametric bootstraps can be used to evaluate the sampling distribution of several statistics such as the G 2 and the p-values under distinct hypotheses or models (Efron & Tibshirani, 1994; see also van de Schoot, Hoijtink, & Dekovic, 2010). The use of the parametric and nonparametric bootstraps provides a way not only to overcome the limitations of asymptotic statistics, but also to evaluate parameter estimates and statistics under distinct assumptions.…”

Bayesian Evaluation of Informative Hypotheses

“…for disclosure the score 4 means the adolescents tell their parents everything about their activities. The relations among these variables can be represented by the path model presented in Figure 2.1 (van de Schoot, Hoijtink, & Deković, 2010). where Z(.)…”

“…These expectations can be represented by inequality constrained hypotheses among the model parameters. Inequality constrained hypotheses can be evaluated using either the frequentist approach by means of p values (see, e.g., Silvapulle & Sen, 2004;van de Schoot et al, 2010) or the Bayesian approach by means of Bayes factors (see, e.g., van de Schoot, Hoijtink, Hallquist, & Boelen, 2012;Klugkist et al, 2005;Hoijtink, 2012). In this paper, the Bayes factor (Kass & Raftery, 1995) is used as a criterion for assessing the hypotheses because p values can only reject a null hypothesis.…”