George A. Marcoulides

Assessing Structural Equation Models by Equivalence Testing With Adjusted Fit Indexes

Conventional null hypothesis testing (NHT) is a very important tool if the ultimate goal is to find a difference or to reject a model. However, the purpose of structural equation modeling (SEM) is to identify a model and use it to account for the relationship among substantive variables. With the setup of NHT, a nonsignificant test statistic does not necessarily imply that the model is correctly specified or the size of misspecification is properly controlled. To overcome this problem, this article proposes to replace NHT by equivalence testing, the goal of which is to endorse a model under a null hypothesis rather than to reject it. Differences and similarities between equivalence testing and NHT are discussed, and new “T-size” terminology is introduced to convey the goodness of the current model under equivalence testing. Adjusted cutoff values of root mean square error of approximation (RMSEA) and comparative fit index (CFI) corresponding to those conventionally used in the literature are obtained to facilitate the understanding of T-size RMSEA and CFI. The single most notable property of equivalence testing is that it allows a researcher to confidently claim that the size of misspecification in the current model is below the T-size RMSEA or CFI, which gives SEM a desirable property to be a scientific methodology. R code for conducting equivalence testing is provided in an appendix.

On the Relationship Between Classical Test Theory and Item Response Theory From One to the Other and Back

The frequently neglected and often misunderstood relationship between classical test theory and item response theory is discussed for the unidimensional case with binary measures and no guessing. It is pointed out that popular item response models can be directly obtained from classical test theory-based models by accounting for the discrete nature of the observed items. Two distinct observational equivalence approaches are outlined that render the item response models from corresponding classical test theory-based models, and can each be used to obtain the former from the latter models. Similarly, classical test theory models can be furnished using the reverse application of either of those approaches from corresponding item response models.

A Direct Latent Variable Modeling Based Method for Point and Interval Estimation of Coefficient Alpha

A direct approach to point and interval estimation of Cronbach’s coefficient alpha for multiple component measuring instruments is outlined. The procedure is based on a latent variable modeling application with widely circulated software. As a by-product, using sample data the method permits ascertaining whether the population discrepancy between alpha and the composite reliability coefficient may be practically negligible for a given empirical setting. The outlined approach is illustrated with numerical data.

Longitudinal models of socio-economic status: Impact on positive parenting behaviors

Parenting research is frequently conducted without a thorough examination of socio-economic characteristics. In this study, longitudinal observations of positive parenting were conducted across six time points. Participants were 219 mothers of children with and without developmental delays. Mothers’ positive parenting increased during early and middle childhood in children with and without developmental delays. Mothers who reported more education had significantly higher levels of positive parenting when their children were 3 years old. Mothers who reported more family income grew at a significantly faster rate in positive parenting. There was preliminary support that mothers with more income were more likely to be members of a class that started off and remained at a higher level of positive parenting over time. Implications are discussed.

Examining Population Heterogeneity in Finite Mixture Settings Using Latent Variable Modeling

A latent variable modeling procedure for examining whether a studied population could be a mixture of 2 or more latent classes is discussed. The approach can be used to evaluate a single-class model vis-à-vis competing models of increasing complexity for a given set of observed variables without making any assumptions about their within-class interrelationships. The method is helpful in the initial stages of finite mixture analyses to assess whether models with 2 or more classes should be subsequently considered as opposed to a single-class model. The discussed procedure is illustrated with a numerical example.

Scale Reliability Evaluation Under Multiple Assumption Violations

A latent variable modeling approach to evaluate scale reliability under realistic conditions in empirical behavioral and social research is discussed. The method provides point and interval estimation of reliability of multicomponent measuring instruments when several assumptions are violated. These assumptions include missing data, correlated errors, nonnormality, lack of unidimensionality, and data not missing at random. The procedure can be readily used to aid scale construction and development efforts in applied settings, and is illustrated using data from an educational study.

Evaluation of Measurement Instrument Criterion Validity in Finite Mixture Settings

A method for evaluating the validity of multicomponent measurement instruments in heterogeneous populations is discussed. The procedure can be used for point and interval estimation of criterion validity of linear composites in populations representing mixtures of an unknown number of latent classes. The approach permits also the evaluation of between-class validity differences as well as within-class validity coefficients. The method can similarly be used with known class membership when distinct populations are investigated, their number is known beforehand and membership in them is observed for the studied subjects, as well as in settings where only the number of latent classes is known. The discussed procedure is illustrated with numerical data.

Do Two or More Multicomponent Instruments Measure the Same Construct? Testing Construct Congruence Using Latent Variable Modeling:

A latent variable modeling procedure is discussed that can be used to test if two or more homogeneous multicomponent instruments with distinct components are measuring the same underlying construct. The method is widely applicable in scale construction and development research and can also be of special interest in construct validation studies. The approach can be readily utilized in empirical settings with observed measure nonnormality and/or incomplete data sets. The procedure is based on testing model nesting restrictions, and it can be similarly employed to examine the collapsibility of latent variables evaluated by multidimensional measuring instruments. The outlined method is illustrated with two data examples.

Scale Reliability Evaluation With Heterogeneous Populations

A latent variable modeling approach for scale reliability evaluation in heterogeneous populations is discussed. The method can be used for point and interval estimation of reliability of multicomponent measuring instruments in populations representing mixtures of an unknown number of latent classes or subpopulations. The procedure is helpful also for evaluation of possible between-class reliability differences as well as of within-class reliability coefficients. The estimation approach can similarly be used in empirical settings with known class membership when distinct populations are investigated, their number is known beforehand and membership in them is observed for the studied subjects, or alternatively in settings where only the number of latent classes is known. A modification and extension of the method for evaluation of maximal reliability or coefficient alpha in heterogeneous populations are also outlined. The discussed procedure is illustrated with numerical data.

The Importance of the Assumption of Uncorrelated Errors in Psychometric Theory

A critical discussion of the assumption of uncorrelated errors in classical psychometric theory and its applications is provided. It is pointed out that this assumption is essential for a number of fundamental results and underlies the concept of parallel tests, the Spearman–Brown’s prophecy and the correction for attenuation formulas as well as the discrepancy between observed and true correlations, and the upper bound property of the reliability index with respect to validity. These relationships are shown not to hold if the errors of considered pairs of tests are correlated. The assumption of lack of error correlation is demonstrated not to be testable using standard covariance structure analysis for pairs of indivisible measures evaluating the same true score with identical error variances.