Assessing Dimensionality in Non-Positive Definite Tetrachoric Correlation Matrices: Does Matrix Smoothing Help?

We performed two simulation studies that investigated dimensionality recovery in NPD tetrachoric correlation matrices using parallel analysis. In each study, the NPD matrices were rehabilitated by three smoothing algorithms. In Study 1, we replicated the work by Debelak and Tran on the assessment of dimensionality in one- or two-dimensional common factor models. In Study 2, we extended the Debelak and Tran design in three important ways. Specifically, we investigated: (a) a wider range of factors; (b) models with varying amounts of model error; and (c) models generated from more realistic population item parameters. Our results indicated that matrix smoothing of NPD tetrachoric correlation matrices improves the performance of parallel analysis with binary data. However, these improvements were modest and often of trivial size. To demonstrate the effect of matrix smoothing on an empirical data set, we applied parallel analysis and factor analysis to Adjective Checklist data from the California Twin Registry.

A note on the identification of change detection task models to measure storage capacity and attention in visual working memory.

The change detection task is a common method for assessing the storage capacity of working memory, but estimates of memory capacity from this task can be distorted by lapses of attention. When combined with appropriate mathematical models, some versions of the change detection task make it possible to separately estimate working memory and the probability of attentional lapses. In principle, these models should allow researchers to isolate the effects of experimental manipulations, group differences, and individual differences on working memory capacity and on the rate of attentional lapses. However, the present research found that two variants of a widely accepted model of the change detection task are not mathematically identified.

The PROMIS fatigue item bank has good measurement properties in patients with fibromyalgia and severe fatigue.

PURPOSE: Efficient management of fibromyalgia (FM) requires precise measurement of FM-specific symptoms. Our objective was to assess the measurement properties of the Patient-Reported Outcome Measurement Information System (PROMIS) fatigue item bank (FIB) in people with FM. METHODS: We applied classical psychometric and item response theory methods to cross-sectional PROMIS-FIB data from two samples. Data on the clinical FM sample were obtained at a tertiary medical center. Data for the U.S. general population sample were obtained from the PROMIS network. The full 95-item bank was administered to both samples. We investigated dimensionality of the item bank in both samples by separately fitting a bifactor model with two group factors; experience and impact. We assessed measurement invariance between samples, and we explored an alternate factor structure with the normative sample and subsequently confirmed that structure in the clinical sample. Finally, we assessed whether reporting FM subdomain scores added value over reporting a single total score. RESULTS: The item bank was dominated by a general fatigue factor. The fit of the initial bifactor model and evidence of measurement invariance indicated that the same constructs were measured across the samples. An alternative bifactor model with three group factors demonstrated slightly improved fit. Subdomain scores add value over a total score. CONCLUSIONS: We demonstrated that the PROMIS-FIB is appropriate for measuring fatigue in clinical samples of FM patients. The construct can be presented by a single score; however, subdomain scores for the three group factors identified in the alternative model may also be reported.

Bayesian Modal Estimation of the Four-Parameter Item Response Model in Real, Realistic, and Idealized Data Sets.

In this study, we explored item and person parameter recovery of the four-parameter model (4PM) in over 24,000 real, realistic, and idealized data sets. In the first analyses, we fit the 4PM and three alternative models to data from three Minnesota Multiphasic Personality Inventory-Adolescent form factor scales using Bayesian modal estimation (BME). Our results indicated that the 4PM fits these scales better than simpler item Response Theory (IRT) models. Next, using the parameter estimates from these real data analyses, we estimated 4PM item parameters in 6,000 realistic data sets to establish minimum sample size requirements for accurate item and person parameter recovery. Using a factorial design that crossed discrete levels of item parameters, sample size, and test length, we also fit the 4PM to an additional 18,000 idealized data sets to extend our parameter recovery findings. Our combined results demonstrated that 4PM item parameters and parameter functions (e.g., item response functions) can be accurately estimated using BME in moderate to large samples (N ⩾ 5, 000) and person parameters can be accurately estimated in smaller samples (N ⩾ 1, 000). In the supplemental files, we report annotated [Formula: see text] code that shows how to estimate 4PM item and person parameters in [Formula: see text] (Chalmers, 2012 ).

Fungible Correlation Matrices: A Method for Generating Nonsingular, Singular, and Improper Correlation Matrices for Monte Carlo Research.

For a fixed set of standardized regression coefficients and a fixed coefficient of determination (R-squared), an infinite number of predictor correlation matrices will satisfy the implied quadratic form. I call such matrices fungible correlation matrices. In this article, I describe an algorithm for generating positive definite (PD), positive semidefinite (PSD), or indefinite (ID) fungible correlation matrices that have a random or fixed smallest eigenvalue. The underlying equations of this algorithm are reviewed from both algebraic and geometric perspectives. Two simulation studies illustrate that fungible correlation matrices can be profitably used in Monte Carlo research. The first study uses PD fungible correlation matrices to compare penalized regression algorithms. The second study uses ID fungible correlation matrices to compare matrix-smoothing algorithms. R code for generating fungible correlation matrices is presented in the supplemental materials.

The Recaptured Scale Technique: A Method for Testing the Structural Robustness of Personality Scales.

Tellegen and Waller advocated a complex and time-consuming scale construction method that they called "exploratory test construction." Scales that are constructed by this method-such as the Multidimensional Personality Questionnaire (MPQ)-are presumed to be more "psychologically coherent" and "robust" than scales constructed by other means. Using a novel procedure that we call the "recaptured scale technique," we tested this conjecture by conducting a megafactor analysis on data from the 411 adult participants of the Minnesota Study of Twins Reared Apart who completed the MPQ, the MMPI, and the CPI. We extracted and obliquely rotated 21 factors from a matrix of gender-corrected tetrachoric correlations for the 1,102 nonredundant items of the three omnibus inventories. Robustness of the 11 MPQ scales was assessed by the degree to which these factors recaptured the MPQ item groupings. Our results showed that nine factors were clearly recognizable as MPQ scales and two additional factors represented a bifurcation of an MPQ scale. A higher-order factor analysis of all 21 factor scales yielded five factors that clearly resembled the Big Five. Our results provide strong support for (a) the method of exploratory test construction, (b) the structural robustness of most MPQ scales, and

Local minima and factor rotations in exploratory factor analysis.

In exploratory factor analysis, factor rotation algorithms can converge to local solutions (i.e., local minima) when they are initiated from different starting points. To better understand this problem, we performed three studies that investigated the prevalence and correlates of local solutions with five factor rotation algorithms: varimax, oblimin, entropy, and geomin (orthogonal and oblique). In total, we simulated 16,000 data sets and performed more than 57 million factor rotations to examine the influence of (a) factor loading size, (b) number of factor indicators, (c) factor cross loadings, (d) factor correlation size, (e) factor loading standardization, (f) sample size, and (g) model approximation error on the frequency of local solutions in factor rotation. We also examined local solutions in an exploratory factor analysis of an open source data set that included 54 personality items. Across three studies, all five algorithms converged to local solutions under some conditions with geomin (orthogonal and oblique) producing the highest number of local solutions. Follow-up analyses showed that, when factor rotations produced multiple solutions, the factor pattern with the maximum hyperplane count (rather than the lowest complexity value) was typically closest in mean squared error to the population factor pattern. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Heywood you go away! Examining causes, effects, and treatments for Heywood cases in exploratory factor analysis.

Exploratory factor analysis (EFA) is a popular method for elucidating the latent structure of data. Unfortunately, EFA models can sometimes produce improper solutions with nonsensical results. For example, improper EFA solutions can include one or more Heywood cases, where common factors account for 100% or more of an observed variable's variance. To better understand these senseless estimates, we conducted four Monte Carlo studies that illuminate the (a) causes, (b) consequences, and (c) effective treatments for Heywood cases in EFA models. Studies 1 and 2 showed that numerous model and data characteristics are associated with Heywood cases, such as small sample sizes, poorly defined factors with low factor score determinacy values, and factor overextraction. In Study 3, we examined the consequences of Heywood cases for EFA model interpretation and found that Heywood cases increase factor loading variances and upwardly bias factor score determinacy values. Study 4 compared the model recovery of several EFA algorithms that were designed to avoid Heywood cases. Our results indicated that, among the algorithms compared, regularized common factor analysis (Jung & Takane, 2008) was the most reliable method for avoiding Heywood cases and producing EFA parameter estimates with small mean squared errors. We discuss best practices for conducting EFA with data sets that might yield Heywood cases. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

The Geometry of Enhancement in Multiple Regression.

In linear multiple regression, "enhancement" is said to occur when R (2)=b'r>r'r, where b is a p×1 vector of standardized regression coefficients and r is a p×1 vector of correlations between a criterion y and a set of standardized regressors, x. When p=1 then b≡r and enhancement cannot occur. When p=2, for all full-rank R xx≠I, R xx=E[xx']=V Λ V' (where V Λ V' denotes the eigen decomposition of R xx; λ 1>λ 2), the set [Formula: see text] contains four vectors; the set [Formula: see text]; [Formula: see text] contains an infinite number of vectors. When p≥3 (and λ 1>λ 2>⋯>λ p ), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B 1 occurs at the intersection of two hyper-ellipsoids in ℝ (p) . Equations are provided for populating the sets B 1 and B 2 and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with λ p (the smallest eigenvalue of the predictor correlation matrix). These equations are used to illustrate the logic and the underlying geometry of enhancement in population, multiple-regression models. R code for simulating population regression models that exhibit enhancement of any degree and any number of predictors is included in Appendices A and B.

Individual Differences in Different Measures of Opioid Self-Administration in Rats Are Accounted for by a Single Latent Variable.

Individual differences in vulnerability to addiction have been widely studied through factor analysis (FA) in humans, a statistical method that identifies "latent" variables (variables that are not measured directly) that reflect the common variance among a larger number of observed measures. Despite its widespread application in behavioral genetics, FA has not been used in preclinical opioid addiction research. The current study used FA to examine the latent factor structure of four measures of i.v. morphine self-administration (MSA) in rats (i.e., acquisition, demand elasticity, morphine/cue- and stress/cue-induced reinstatement). All four MSA measures are generally assumed in the preclinical literature to reflect "addiction vulnerability," and individual differences in multiple measures of abuse liability are best accounted for by a single latent factor in some human studies. A one-factor model was therefore fitted to the data. Two different regularized FAs indicated that a one-factor model fit our data well. Acquisition, elasticity of demand and morphine/cue-induced reinstatement loaded significantly onto a single latent factor while stress/cue-induced reinstatement did not. Consistent with findings from some human studies, our results indicated a common drug "addiction" factor underlying several measures of opioid SA. However, stress/cue-induced reinstatement loaded poorly onto this factor, suggesting that unique mechanisms mediate individual differences in this vs. other MSA measures. Further establishing FA approaches in drug SA and in preclinical neuropsychopathology more broadly will provide more reliable, clinically relevant core factors underlying disease vulnerability in animal models for further genetic analyses.

Recovering bifactor models: A comparison of seven methods.

The last decade has witnessed a resurgence of interest in exploratory bifactor analysis models and the concomitant development of new methods to estimate these models. Understandably, due to the rapid pace of developments in this area, existing Monte Carlo comparisons of bifactor analysis have not included the newest methods. To address this issue, we compared the model recovery capabilities of 5 existing methods and 2 newer methods (Waller, 2018a) for exploratory bifactor analysis. Our study expands upon previous work in this area by comparing (a) a greater number of estimation algorithms and (b) by including both nonhierarchical and hierarchical bifactor models in our study design. In aggregate, we conducted almost 3 million exploratory bifactor analyses to identify the most accurate methods. Our results showed that, when compared with the alternatives, the rank-deficient Schmid-Leiman and Direct Schmid-Leiman methods were better able to recover both nonhierarchical and hierarchical bifactor structures. (PsycINFO Database Record (c) 2020 APA, all rights reserved).

Item response theory and clinical measurement.

In this review, we examine studies that use item response theory (IRT) to explore the psychometric properties of clinical measures. Next, we consider how IRT has been used in clinical research for: scale linking, computerized adaptive testing, and differential item functioning analysis. Finally, we consider the scale properties of IRT trait scores. We conclude that there are notable differences between cognitive and clinical measures that have relevance for IRT modeling. Future research should be directed toward a better understanding of the metric of the latent trait and the psychological processes that lead to individual differences in item response behaviors.

Direct Schmid-Leiman Transformations and Rank-Deficient Loadings Matrices.

The Schmid-Leiman (S-L; Psychometrika 22: 53-61, 1957) transformation is a popular method for conducting exploratory bifactor analysis that has been used in hundreds of studies of individual differences variables. To perform a two-level S-L transformation, it is generally believed that two separate factor analyses are required: a first-level analysis in which k obliquely rotated factors are extracted from an observed-variable correlation matrix, and a second-level analysis in which a general factor is extracted from the correlations of the first-level factors. In this article, I demonstrate that the S-L loadings matrix is necessarily rank deficient. I then show how this feature of the S-L transformation can be used to obtain a direct S-L solution from an unrotated first-level factor structure. Next, I reanalyze two examples from Mansolf and Reise (Multivar Behav Res 51: 698-717, 2016) to illustrate the utility of 'best-fitting' S-L rotations when gauging the ability of hierarchical factor models to recover known bifactor structures. Finally, I show how to compute direct bifactor solutions for non-hierarchical bifactor structures. An online supplement includes R code to reproduce all of the analyses that are reported in the article.

Carving nature at its joints: Paul Meehl's development of taxometrics.

This article presents a brief summary of Paul Meehl's development of taxometrics. Of special concern is the question, Why did Paul Meehl develop taxometrics? It is shown that taxometrics played an integral role in Meehl's broader research program on the genetics of schizotaxia.