Evaluating model fit for growth curve models: Integration of fit indices from SEM and MLM frameworks.

Evaluating overall model fit for growth curve models involves 3 challenging issues. (a) Three types of longitudinal data with different implications for model fit may be distinguished: balanced on time with complete data, balanced on time with data missing at random, and unbalanced on time. (b) Traditional work on fit from the structural equation modeling (SEM) perspective has focused only on the covariance structure, but growth curve models have four potential sources of misspecification: within-individual covariance matrix, between-individuals covariance matrix, marginal mean structure, and conditional mean structure. (c) Growth curve models can be estimated in both the SEM and multilevel modeling (MLM) frameworks; these have different emphases for the evaluation of model fit. In this article, the authors discuss the challenges presented by these 3 issues in the calculation and interpretation of SEM- and MLM-based fit indices for growth curve models and conclude by identifying some lines for future research. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Psychological measurement.

L. L. Thurstone's (see PA, Vol 2:527; see also PA, Vol 81:28135) article developed a representational measurement model of comparative judgment; estimated discrimination probabilities yield scale values that imply values of other probabilities not yet observed, if the model provides a true representation. In practice, the accuracy of such inferences is captured by "goodness-of-fit" statistics. The specific representational measurement model developed can yield magnitude measurement on psychological dimensions for which no corresponding physical dimensions exist (e.g., favorability of "attitude toward"). This revolutionary article led to the development of many other representational measurement models. As opposed to psychophysics, however, the introduction of "true measurement" in social, attitudinal, and personality psychology did not yield the rapid progress Thurstone envisioned, and currently this specific model is seldom used in these areas. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Thurstone and sensory scaling: Then and now.

Following a brief summary of the major ideas and assumptions of L. L. Thurstone's (see PA, Vol 2:527; see also PA, Vol 81:28135) "A law of comparative judgment" and a discussion of its historical context, this article reviews the subsequent major developments of this model in the sensory area. They are grouped as (1) response criteria, which Thurstone incorrectly believed to be of little significance, (2) several interpretations of the underlying continuum, (3) extensions of his formalism to choice and absolute identification, (4) the possibility of non-Gaussian distributions, and (5) extensions to multidimensional representations. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Models Quantify the Total Maximum Daily Load Process

Mathematical models have been used for many years to assist in the management of water quality. The total maximum daily load (TMDL) process is no exception; models represent the means by which the assimilative capacity of a water body can be quantified and a waste load allocation can be determined such that the assimilative capacity is not exceeded. Unfortunately, in many TMDLs, the use of models has not always adhered to the best modeling practices that have been developed over the past half-century. This paper presents what are felt to be the most important principles of good modeling practice relative to all of the steps in developing and applying a model for computing a TMDL. These steps include: Problem definition and setting management objectives; data synthesis for use in modeling; model selection; model calibration and, if possible confirmation; model application; iterative modeling; and model postaudit. Since mathematical modeling of aquatic systems is not an exact science, it is essential that these steps be fully transparent to all TMDL stakeholders through comprehensive documentation of the entire process, including specification of all inputs and assumptions. The overriding consideration is that data richness and quality govern the level of model complexity that can be applied to a given system. The model should never be more complex than the data allow. Also, in applying a model, one should always attempt to quantify the uncertainty in predictions. In general, quantifying uncertainty is easier with simple models, which is another reason to begin with a simple framework.     

Scaling procedures based on the method of paired comparisons.

Thurstone's Case III and Case V, and Guilford's Short-Cut approaches to scaling paired-comparison data were experimentally compared. The stimuli were 10 teacher-approved and 10 teacher-disapproved behavior categories presented in paired-comparison form to 4 groups of school children. Each group contained a sample of 80 Ss and represented a particular sex, experimental condition and an age-grade level in the range grades 6-12. The intercorrelation between the scale values obtained by the 3 methods were approximately unity for both sexes under both experimental conditions. "The results are interpreted as corroborative… . investigations demonstrating the power of less complicated… approaches to scaling… ." 33 references. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling.

Meta-analysis and structural equation modeling (SEM) are two important statistical methods in the behavioral, social, and medical sciences. They are generally treated as two unrelated topics in the literature. The present article proposes a model to integrate fixed-, random-, and mixed-effects meta-analyses into the SEM framework. By applying an appropriate transformation on the data, studies in a meta-analysis can be analyzed as subjects in a structural equation model. This article also highlights some practical benefits of using the SEM approach to conduct a meta-analysis. Specifically, the SEM-based meta-analysis can be used to handle missing covariates, to quantify the heterogeneity of effect sizes, and to address the heterogeneity of effect sizes with mixture models. Examples are used to illustrate the equivalence between the conventional meta-analysis and the SEM-based meta-analysis. Future directions on and issues related to the SEM-based meta-analysis are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

An introduction to using structural equation models in rehabilitation psychology.

Objective: To provide an overview of structural equation modeling (SEM) using an example drawn from the rehabilitation psychology literature. Design: To illustrate the 5 steps in SEM (model specification, identification, estimation methods, interpretation of results, and model modification), an example is presented, with details on determining whether alternative models result in a significant improvement to fit to the observed data. Data are from a sample of 274 people with spinal cord injury. Issues commonly encountered in preparing data for SEM analyses (e.g., missing data, nonnormality) are reviewed, as is the debate surrounding some aspects of SEM (e.g., acceptable sample size). Conclusion: SEM can be a powerful procedure for empirically representing complex and sophisticated theoretical models of interest to rehabilitation psychologists. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Testing Mediational Models With Longitudinal Data: Questions and Tips in the Use of Structural Equation Modeling.

R. M. Baron and D. A. Kenny (1986; see record 1987-13085-001) provided clarion conceptual and methodological guidelines for testing mediational models with cross-sectional data. Graduating from cross-sectional to longitudinal designs enables researchers to make more rigorous inferences about the causal relations implied by such models. In this transition, misconceptions and erroneous assumptions are the norm. First, we describe some of the questions that arise (and misconceptions that sometimes emerge) in longitudinal tests of mediational models. We also provide a collection of tips for structural equation modeling (SEM) of mediational processes. Finally, we suggest a series of 5 steps when using SEM to test mediational processes in longitudinal designs: testing the measurement model, testing for added components, testing for omitted paths, testing the stationarity assumption, and estimating the mediational effects. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Structural Equation Models for Interchangeable Dyads: Being the Same Makes a Difference.

Structural equation modeling (SEM) offers a flexible method for studying the patterns of interdependence in partners' behavior, which lie at the heart of interactions and relationships. Although SEM has been applied to the study of distinguishable dyads, in which partners are distinguishable by type, such as male and female, it has rarely been applied to the study of interchangeable dyads, such as male-male or female-female pairs. The authors integrate a wide range of dyadic interdependence models--including actor-partner interdependence models, mutual-influence models, and common-fate or dyadic personality models--into an SEM framework for use with interchangeable dyads. The authors also address the use of latent variables at both the dyadic and individual levels, whereby substantive relationships in these models can be corrected for errors of measurement. Furthermore, the authors discuss the conceptual underpinnings of dyadic models and give examples of their application. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Hierarchical diffusion models for two-choice response times.

Two-choice response times are a common type of data, and much research has been devoted to the development of process models for such data. However, the practical application of these models is notoriously complicated, and flexible methods are largely nonexistent. We combine a popular model for choice response times—the Wiener diffusion process—with techniques from psychometrics in order to construct a hierarchical diffusion model. Chief among these techniques is the application of random effects, with which we allow for unexplained variability among participants, items, or other experimental units. These techniques lead to a modeling framework that is highly flexible and easy to work with. Among the many novel models this statistical framework provides are a multilevel diffusion model, regression diffusion models, and a large family of explanatory diffusion models. We provide examples and the necessary computer code. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Analyzing longitudinal data with multilevel models: An example with individuals living with lower extremity intra-articular fractures.

Objective: The use and quality of longitudinal research designs has increased over the past 2 decades, and new approaches for analyzing longitudinal data, including multilevel modeling (MLM) and latent growth modeling (LGM), have been developed. The purpose of this article is to demonstrate the use of MLM and its advantages in analyzing longitudinal data. Research Method: Data from a sample of individuals with intra-articular fractures of the lower extremity from the University of Alabama at Birmingham's Injury Control Research Center are analyzed using both SAS PROC MIXED and SPSS MIXED. Results: The authors begin their presentation with a discussion of data preparation for MLM analyses. The authors then provide example analyses of different growth models, including a simple linear growth model and a model with a time-invariant covariate, with interpretation for all the parameters in the models. Implications: More complicated growth models with different between- and within-individual covariance structures and nonlinear models are discussed. Finally, information related to MLM analysis, such as online resources, is provided at the end of the article. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Making trade-offs: A probabilistic and context-sensitive model of choice behavior.

The stochastic difference model assumes that decision makers trade normalized attribute value differences when making choices. The model is stochastic, with choice probabilities depending on the normalized difference variable, d, and a decision threshold, delta. The decision threshold indexes a person's sensitivity to attribute value differences and is a free estimated parameter of the model. Depending on the choice context, a person may be more or less sensitive to attribute value differences, and hence delta may be used to measure context effects. With proportional difference used as the normalization, the proportional difference model (PD) was tested with 9 data sets, including published data (e.g., J. L. Myers, M. M. Suydam, & B. Gambino, 1965; A. Tversky, 1969). The model accounted for individual and group data well and described violations of stochastic dominance, independence, and weak and strong stochastic transitivity. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Looking back and ahead in psychometrics.

Reviews developments in 5 areas of psychometrics (computers, testing, scaling, factor analysis, and mathematical learning theory) beginning with L. L. Thurstone's work in the 1930s through the present. Recommendations for further research in measurement in each area are presented, including improvements in (a) the criteria against which aptitude tests are evaluated (e.g., college grades and achievements after college); (b) factor analyses of areas in learning, memory, physiology, or political science; and (c) the relationship between aptitude tests and learning tasks. Characteristics of recently proposed learning models are discussed, and questions in need of experimental investigation are noted (e.g., the use of stochastic or finite step models with learning data, the development of meaningful learning parameters, parameter estimation for individual and group learning curves, and evidence on reliability by replication in learning studies). (2 p ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Causal inference in randomized experiments with mediational processes.

This article links the structural equation modeling (SEM) approach with the principal stratification (PS) approach, both of which have been widely used to study the role of intermediate posttreatment outcomes in randomized experiments. Despite the potential benefit of such integration, the 2 approaches have been developed in parallel with little interaction. This article proposes the cross-model translation (CMT) approach, in which parameter estimates are translated back and forth between the PS and SEM models. First, without involving any particular identifying assumptions, translation between PS and SEM parameters is carried out on the basis of their close conceptual connection. Monte Carlo simulations are used to further clarify the relation between the 2 approaches under particular identifying assumptions. The study concludes that, under the common goal of causal inference, what makes a practical difference is the choice of identifying assumptions, not the modeling framework itself. The CMT approach provides a common ground in which the PS and SEM approaches can be jointly considered, focusing on their common inferential problems. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Examining assumptions about item responding in personality assessment: Should ideal point methods be considered for scale development and scoring?

The present study investigated whether the assumptions of an ideal point response process, similar in spirit to Thurstone's work in the context of attitude measurement, can provide viable alternatives to the traditionally used dominance assumptions for personality item calibration and scoring. Item response theory methods were used to compare the fit of 2 ideal point and 2 dominance models with data from the 5th edition of the Sixteen Personality Factor Questionnaire (S. Conn & M. L. Rieke, 1994). The authors' results indicate that ideal point models can provide as good or better fit to personality items than do dominance models because they can fit monotonically increasing item response functions but do not require this property. Several implications of these findings for personality measurement and personnel selection are described. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Modeling Controlled Water Systems

The particular challenges of modeling controlled water systems are discussed. The high degree of freedom due to the control structures increases the risk of producing the right output for the wrong reasons. On the other hand, many controlled water systems are (partly) manually operated or at least supervised by an operational water manager. The decisions of these managers are not as rigid as a computer simulated control strategy. Therefore, getting a very close fit with a water-system control model is mostly not possible. A modeling framework is proposed that takes advantage of the vast availability of measurement data in controlled water systems. The water level and flow data at control structures allow for intensive validation and subsystem calibration to reduce the degree of modeling freedom and to model separately the natural rainfall-runoff and hydrodynamic processes. The framework is successfully applied to improve a simulation model of the controlled water system of Rijnland, The Netherlands. The yearly volume error was reduced from 11% to less than 1% and as a consequence, the short-term peak events were modeled more accurately as well. The resulting water-system control model is more reliable for both design studies and operational decision support. The framework will contribute to prepare more reliable simulation models of controlled water systems.     

Three-Dimensional Micromechanics-Based Constitutive Framework for Analysis of Pultruded Composite Structures

A new 3D micromechanics-based framework is proposed for the nonlinear analysis of pultruded fiber-reinforced polymeric composites. The proposed 3D modeling framework is a nested multiscale approach that explicitly recognizes the response of the composite systems (layers) within the cross section of the pultruded member. These layers can have reinforcements in the form of roving, continuous filament mat (CFM), and∕or woven fabrics. Different 3D micromechanical models for the layers can be used to recognize the basic response of the fiber and matrix materials. The framework is implemented with both shell and 3D finite elements. The 3D lamination theory is used to generate a homogenized nonlinear effective response for a through-thickness representative stacking sequence. The proposed modeling framework for pultruded composites is used to predict the stiffness and nonlinear stress-strain response of E-glass∕vinylester pultruded materials reinforced with roving and CFM. The roving layer is idealized using a 3D nonlinear micromechanics model for a unidirectional fiber-reinforced material. A simple nonlinear micromechanics model for the CFM layer is also applied. The proposed model shows very good predictive capabilities of the overall effective properties and the nonlinear response of pultruded composites, based on the in situ material properties, and the volume fractions of the constituents. Experimental data from off-axis tests of pultruded plates under uniaxial compression are used to verify the proposed model. The proposed framework can be easily incorporated within displacement-based finite-element models of composite structures.     

A hierarchical process-dissociation model.

In fitting the process-dissociation model (L. L. Jacoby, 1991) to observed data, researchers aggregate outcomes across participant, items, or both. T. Curran and D. L. Hintzman (1995) demonstrated how biases from aggregation may lead to artifactual support for the model. The authors develop a hierarchical process-dissociation model that does not require aggregation for analysis. Most importantly, the Curran and Hintzman critique does not hold for this model. Model analysis provides for support of process dissociation--selective influence holds, and there is a dissociation in correlation patterns among participants and items. Items that are better recollected also elicit higher automatic activation. There is no correlation, however, across participants; that is, participants with higher recollection have no increased tendency toward automatic activation. The critique of aggregation is not limited to process dissociation. Aggregation distorts analysis in many nonlinear models, including signal detection, multinomial processing tree models, and strength models. Hierarchical modeling serves as a general solution for accurately fitting these psychological-processing models to data. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Odor preference and urine-marking scales in male and female rats: Effects of gonadectomy and sexual experience on responses to conspecific odors.

Investigation and urine-marking responses of 112 male and female Long-Evans rats toward conspecific urine odors were recorded in pair-wise comparison tests. Each of 16 S groups was given 15 preference tests, 1 for each of the possible pairs of 5 urine odors and a no-odor control (N). The urine sources were own group (G), intact male (M), castrated male (Mc), ovariectomized female (Fo), and estrous females (F). Results were scaled by using a technique based on L. L. Thurstone's (1927) law of comparative judgment. Intact males with sexual experience ordered their odor preferences N?     

Testing parameters in structural equation modeling: Every "one" matters.

A problem with standard errors estimated by many structural equation modeling programs is described. In such programs, a parameter's standard error is sensitive to how the model is identified (i.e., how scale is set). Alternative but equivalent ways to identify a model may yield different standard errors, and hence different Z tests for a parameter, even though the identifications produce the same overall model fit. This lack of invariance due to model identification creates the possibility that different analysts may reach different conclusions about a parameter's significance level even though they test equivalent models on the same data. The authors suggest that parameters be tested for statistical significance through the likelihood ratio test, which is invariant to the identification choice. (PsycINFO Database Record (c) 2010 APA, all rights reserved)