Horn's Parallel Research Articles

Considering Horn’s Parallel Analysis from a Random Matrix Theory Point of View

Horn's parallel analysis is a widely used method for assessing the number of principal components and common factors. We discuss the theoretical foundations of parallel analysis for principal components based on a covariance matrix by making use of arguments from random matrix theory. In particular, we show that (i) for the first component, parallel analysis is an inferential method equivalent to the Tracy-Widom test, (ii) its use to test high-order eigenvalues is equivalent to the use of the joint distribution of the eigenvalues, and thus should be discouraged, and (iii) a formal test for higher-order components can be obtained based on a Tracy-Widom approximation. We illustrate the performance of the two testing procedures using simulated data generated under both a principal component model and a common factors model. For the principal component model, the Tracy-Widom test performs consistently in all conditions, while parallel analysis shows unpredictable behavior for higher-order components. For the common factor model, including major and minor factors, both procedures are heuristic approaches, with variable performance. We conclude that the Tracy-Widom procedure is preferred over parallel analysis for statistically testing the number of principal components based on a covariance matrix.

Psychometric properties and refinement of the Reproductive Coercion Scale

ObjectiveIdentification and refinement of psychometric properties of the Reproductive Coercion Scale (RCS) for use in survey research and clinical practice. Study designYoung women aged 16–29 years seeking services in 24 Pennsylvania and 5 California family planning clinics completed questionnaires. Data were pooled for analysis (n=4674), and underlying domains were assessed using Horn's Parallel Analysis and Exploratory Factor Analysis. Multidimensional Item Response Theory was used to refine the scale and assess reliability and validity of a short-form RCS. ResultsThe full, nine-item RCS had two underlying domains: pregnancy coercion and condom manipulation. Five items were retained in the short form: three about pregnancy coercion (e.g., “told you not to use birth control…”) and two for condom manipulation (e.g., “taken off the condom while you were having sex…”; one of these items is the combination of two original items on damaging the condom that were combined because of similar statistical properties and face validity and a third item on removing the condom was retained on its own). Recent reproductive coercion was reported by 6.7% and 6.3% of the sample with the full and short-form RCS, respectively. Characteristics of women reporting reproductive coercion were similar with both forms. ConclusionFindings indicate that reproductive coercion includes pregnancy coercion and deliberate manipulation of condoms to promote pregnancy. Moreover, women experience reproductive coercion across a continuum of severity. We selected items that varied in RC severity and discrimination to generate a five-item short-form RCS for survey research and clinical practice. ImplicationsThis study assesses the psychometric properties of the RCS, identifying pregnancy coercion and condom manipulation as underlying domains of reproductive coercion. Recommendations for using the RCS in research and clinical practice are discussed.

Development of the Older Adult Prescription Drug Assessment Questionnaire for Case Workers.

Instruments assessing geriatric case managers' perceptions of prescription abuse are limited. Based on a review of current literature and pilot testing feedback, 22 items were initially assembled to form an older adult Prescription Drug Assessment Questionnaire (PDAQ). Case managers ( N = 161) at an older adult agency located in a Midwestern state were surveyed using the instrument. Data were subjected to exploratory factor analysis using principal axis factoring (PAF) and Promax rotation, and Horn's parallel analysis determined the number of extracted factors. PAF generated a final 11-item three-factor model accounting for 51% of total variance explained: Standard Assessment (Factor 1; four items; ∞ = .83), Assessment Belief (Factor 2; three items; ∞ = .74), and Problem Scope (Factor 3; four items; ∞ = .62). The older adult PDAQ is a brief, data-driven instrument with acceptable psychometric properties for assessing perceptions of prescription drug abuse and misuse assessment and counseling.

Examining the factor structure of MUIS-C scale among baby boomers with hepatitis C

BackgroundBaby boomers account for two out of every three cases of hepatitis C infection in the U.S. PurposeTo conduct an exploratory factor analysis directed at supporting the use of the MUIS-C as a reliable instrument in measuring illness uncertainty among baby boomers with hepatitis C. MethodsThe steps of conducting a typical principal component analysis (PCA) with an oblique rotation were used on a sample of 146 participants, the sampling adequacy of items was examined via the Kaiser–Meyer–Olkin (KMO) measure, and the Bartlett's sphericity test was used for appropriateness of conducting a factor analysis. ResultsA two-factor structure was obtained by using Horn's parallel analysis method. The two factors explained a cumulative total of 45.8% of the variance. ConclusionsThe results of the analyses indicated that the MUIS-C was a valid and reliable instrument and potentially suitable for use in baby boomer population diagnosed with hepatitis C.

Assessing the Structure of the Ways of Coping Questionnaire in Fibromyalgia Patients Using Common Factor Analytic Approaches.

The Ways of Coping Questionnaire (WCQ) is a widely used measure of coping processes. Despite its use in a variety of populations, there has been concern about the stability and structure of the WCQ across different populations. This study examines the factor structure of the WCQ in a large sample of individuals diagnosed with fibromyalgia. The participants were 501 adults (478 women) who were part of a larger intervention study. Participants completed the WCQ at their 6-month assessment. Foundational factoring approaches were performed on the data (i.e., maximum likelihood factoring [MLF], iterative principal factoring [IPF], principal axis factoring (PAF), and principal components factoring [PCF]) with oblique oblimin rotation. Various criteria were evaluated to determine the number of factors to be extracted, including Kaiser's rule, Scree plot visual analysis, 5 and 10% unique variance explained, 70 and 80% communal variance explained, and Horn's parallel analysis (PA). It was concluded that the 4-factor PAF solution was the preferable solution, based on PA extraction and the fact that this solution minimizes nonvocality and multivocality. The present study highlights the need for more research focused on defining the limits of the WCQ and the degree to which population-specific and context-specific subscale adjustments are needed.

Validación del Modelo Unidimensional de la Escala de Valoración de la Relación en Personas Casadas y en Unión Libre de Monterrey, México

El estudio de las relaciones de pareja cobra especial importancia, ya que esta unión constituye la base de la familia. La satisfacción con la relación se puede definir como la actitud global hacia la relación y la pareja. La escala de Valoración de la Relación (RAS) de Hendrick es un instrumento comúnmente usado para evaluar el constructo. Esta escala ha demostrado tener consistencia interna alta y una estructura unidimensional. Aunque existen estudios de validación de la RAS, estos han empleado técnicas estadísticas inadecuadas para analizar sus ítems tipo Likert y para determinar el número de factores; además, previamente no se ha contrastado su invarianza factorial por sexo. Por tanto, en este estudio se plantean las siguientes preguntas de investigación: ¿la RAS tiene ítems consistentes y discriminativos? Basados los análisis en la matriz de correlaciones policóricas, ¿cuál es su nivel de consistencia interna?, ¿cuál es su número de factores usando métodos empíricos rigurosos?, y ¿el modelo de un factor es invariante por sexo? A fin de responder a estas preguntas de investigación, se usó un muestreo probabilístico de rutas aleatorias en este estudio instrumental de validación de la RAS. La muestra se extrajo de la población de personas casadas o en unión libre de la ciudad de Monterrey en México. Hubo 431 participantes femeninas y 376 participantes masculinos en este estudio. Los siete ítems de la RAS fueron consistentes y discriminativos. La consistencia interna de la escala fue excelente en la muestra conjunta (α ordinal = .93), en la de mujeres (α ordinal = .94) y en la de hombres (α ordinal = .92). El análisis paralelo de Horn y la prueba de la correlación parcial media mínima de Velicer sugirieron una solución de un factor. Además, el modelo de un factor (con una correlación entre los residuos de los dos ítems inversos) tuvo buen ajuste a los datos y sus propiedades de invarianza por sexo fueron aceptables por el método de Mínimos Cuadrados No Ponderados. Se concluye que la escala presenta consistencia interna y validez cruzada por sexo. Se recomienda el uso de la escala en la población estudiada de parejas casadas y que viven juntas.

Validación cruzada de la escala de expresión de ira en mujeres con VIH y población general

The aims of this study were to estimate the internal consistency and contrast the invariance (between a clinical sample and a general population sample) of the 4-factor model for the 24 items of the Anger Expression scale of the revised State-Trait-Anger Expression Inventory, and explore alternative models if there appears problems of internal consistency and factor definition. The scale was applied to 200 women living with HIV, and 120 women and 120 men from the general population. Items 3 and 17 showed problems of internal consistency. The correlation between anger-in factor and anger-out factor was unique in the confirmatory factor analysis. Horn's parallel analysis and Velicer's criterion indicated that the number of factors is three. After their extraction, the 2 expected factors of anger control and a factor of anger expression (union of anger-in factor and anger-out factor without items 3 and 17) were configured. The internal consistency of these 3 factors varied from .76 to .89. The fit to the data and the invariance properties of 3-factor model were acceptable from four indexes: χ2/df (<3), RMSEA and SRMR (<.10), and CFI (>.80, except for the model constrained on structural variances-covariances and the model constrained on measurement residuals when the invariance was contrasted). It is concluded that the scale should be reduced to 22 items and has a structure of three correlated factors with acceptable cross-validity.

Exploratory factor analysis of the Oral Health Impact Profile.

Although oral health-related quality of life (OHRQoL) as measured by the Oral Health Impact Profile (OHIP) is thought to be multidimensional, the nature of these dimensions is not known. The aim of this report was to explore the dimensionality of the OHIP using the Dimensions of OHRQoL (DOQ) Project, an international study of general population subjects and prosthodontic patients. Using the project's Learning Sample (n = 5173), we conducted an exploratory factor analysis on the 46 OHIP items not specifically referring to dentures for 5146 subjects with sufficiently complete data. The first eigenvalue (27·0) of the polychoric correlation matrix was more than ten times larger than the second eigenvalue (2·6), suggesting the presence of a dominant, higher-order general factor. Follow-up analyses with Horn's parallel analysis revealed a viable second-order, four-factor solution. An oblique rotation of this solution revealed four highly correlated factors that we named Oral Function, Oro-facial Pain, Oro-facial Appearance and Psychosocial Impact. These four dimensions and the strong general factor are two viable hypotheses for the factor structure of the OHIP.

A new look at Horn’s parallel analysis with ordinal variables.

Previous research evaluating the performance of Horn's parallel analysis (PA) factor retention method with ordinal variables has produced unexpected findings. Specifically, PA with Pearson correlations has performed as well as or better than PA with the more theoretically appropriate polychoric correlations. Seeking to clarify these findings, the current study employed a more comprehensive simulation study that included the systematic manipulation of 7 factors related to the data (sample size, factor loading, number of variables per factor, number of factors, factor correlation, number of response categories, and skewness) as well as 3 factors related to the PA method (type of correlation matrix, extraction method, and eigenvalue percentile). The results from the simulation study show that PA with either Pearson or polychoric correlations is particularly sensitive to the sample size, factor loadings, number of variables per factor, and factor correlations. However, whereas PA with polychorics is relatively robust to the skewness of the ordinal variables, PA with Pearson correlations frequently retains difficulty factors and is generally inaccurate with large levels of skewness. In light of these findings, we recommend the use of PA with polychoric correlations for the dimensionality assessment of ordinal-level data.

THE DEVELOPMENT OF A NEW APATHY MEASUREMENT SCALE: DIMENSIONAL APATHY SCALE

ObjectiveTo develop a new questionnaire, the Dimensional Apathy Scale (DAS) which aims to be sensitive to Levy and Dubois' 3 subtypes of apathy, Auto-Activation, Emotional-affective and Cognitive. To develop a...

Standardization of the MARISTÁN scale to measure needs in people with schizophrenia and related psychoses

Existing measures of needs in severe mental illness have been developed mainly from professionals' viewpoints and are Eurocentric. Our aim was to standardize a measure of the needs of people with schizophrenia across several cultures and based on users' own viewpoints. An instrument to measure needs, based on qualitative data on users', carers' and professionals' views, was tested in 164 people with schizophrenia or related psychoses in six countries. Participants underwent face-to-face interviews, one third of which were repeated 30 days later. Principal axis factoring and Promax rotation evaluated scale structure; Horn's parallel combined with bootstrapping determined the number of factors; and intra-class correlation assessed test-retest reliability. The instrument contained four factors: (1) health needs; (2) work and leisure needs; (3) existential needs; and (4) needs for support in daily living. Cronbach's α for internal consistency was 0.81, 0.81, 0.77 and 0.76 for factors 1-4 and 0.81 for the scale as a whole. Correlation between factors was of moderate range for the first three factors (0.41-0.50) and low for the fourth factor (0.14-0.29). Intra-class correlation coefficient for test-retest reliability was 0.74 (0.64-0.82) for the whole scale. Mean item score on needs for support in daily living was lower than for the other factors. The MARISTÁN Scale of Needs evaluates needs from the patient perspective and it is a valid instrument to measure the needs of people with severe mental illness across cultures.

Standardization of the Maristán Scale of Informal Care in people with schizophrenia and other psychoses

No instrument has been developed and validated across cultures to measure the degree of support provided by informal carers to people with schizophrenia. We aimed to develop such a measure. The Maristán Scale of Informal Care was developed directly from the views of patients with schizophrenia in six countries. Face-to-face interviews were carried out with participants and 103 were repeated after 30 days. Principal Axis Factoring followed by Promax rotation evaluated the structure of the scale. Horn's parallel combined with bootstrapping determined the number of factors. Cronbach's alpha estimated the scale's internal consistency and intra-class correlation its test-retest reliability. A total of 164 interviews were undertaken, 103 with re-test. The Horn's Parallel Analysis and the analysis of the Promax rotation revealed one factor. Cronbach's alpha was 0.89. Intra-class correlation coefficient was 0.56 (95% CI 0.42-0.68) and this increased to 0.64 (95% CI 0.51-0.75) after removing two outlying values. Patients from Argentina recorded the lowest scores (poor informal support/care). The Maristán Scale of Informal Care is a reliable instrument to assess the degree of support provided by informal carers to people with schizophrenia across cultures. A confirmatory factor analysis is needed to evaluate the stability of its factor structure.

Safety climate in trucking industry and its effects on safety outcomes

We developed a safety climate scale for the trucking industry and confirmed its criterion validity by correlations with safety outcome measures. A questionnaire survey was conducted to elicit safety climate in the trucking industry. A total of 1,028 responses (54 % response rate) were collected from truck drivers working for 49 trucking companies in Japan. Incident records between 2004 and 2008 were obtained from 21 out of the 49 companies that we surveyed safety climate. Following a factor extraction criteria (Horn's parallel analysis), six factors on safety climate were elicited by applying principal component analysis with the Promax rotation, yielding 39 % of cumulative variance. Significant correlations with safety outcome measures were identified for the following safety climate factors: strong awareness of safety management/systems, teamwork, and realistic competence-based view. Safety climate contributes positively to the safety performance in a trucking organisation. Safety climate assessment using the factors developed in this paper is applicable as a proactive approach to risk reduction in the trucking industry.

The link between belongingness and depressive symptoms: An exploration in the workplace interpersonal context

Interpersonal factors are crucial to a deepened understanding of depression. Belongingness, also referred to as connectedness, has been established as a strong risk/protective factor for depressive symptoms. To elucidate this link it may be beneficial to investigate the relative importance of specific psychosocial contexts as belongingness foci. Here we investigate the construct of workplace belongingness. Employees at a disability services organisation (N = 125) completed measures of depressive symptoms, anxiety symptoms, workplace belongingness and organisational commitment. Psychometric analyses, including Horn's parallel analyses, indicate that workplace belongingness is a unitary, robust and measurable construct. Correlational data indicate a substantial relationship with depressive symptoms (r = −.54) and anxiety symptoms (r = −.39). The difference between these correlations was statistically significant, supporting the particular importance of belongingness cognitions to the etiology of depression. Multiple regression analyses support the hypothesis that workplace belongingness mediates the relationship between affective organisational commitment and depressive symptoms. It is likely that workplaces have the potential to foster environments that are intrinsically less depressogenic by facilitating workplace belongingness. From a clinical perspective, cognitions regarding the workplace psychosocial context appear to be highly salient to individual psychological health, and hence warrant substantial attention.

Implementing Horn's Parallel Analysis for Principal Component Analysis and Factor Analysis

I present paran, an implementation of Horn's parallel analysis criteria for factor or component retention in common factor analysis or principal component analysis in Stata. The command permits classical parallel analysis and more recent extensions to it for the pca and factor commands. paran provides a needed extension to Stata's built-in factor- and component-retention criteria.

Exploring the Sensitivity of Horn's Parallel Analysis to the Distributional Form of Random Data

Horn's parallel analysis (PA) is the method of consensus in the literature on empirical methods for deciding how many components/factors to retain. Different authors have proposed various implementations of PA. Horn's seminal 1965 article, a 1996 article by Thompson and Daniel, and a 2004 article by Hayton, Allen, and Scarpello all make assertions about the requisite distributional forms of the random data generated for use in PA. Readily available software is used to test whether the results of PA are sensitive to several distributional prescriptions in the literature regarding the rank, normality, mean, variance, and range of simulated data on a portion of the National Comorbidity Survey Replication (Pennell et al., 2004) by varying the distributions in each PA. The results of PA were found not to vary by distributional assumption. The conclusion is that PA may be reliably performed with the computationally simplest distributional assumptions about the simulated data.

Commentary on “Exploring the Sensitivity of Horn's Parallel Analysis to the Distributional Form of Random Data”

Commentary on “Exploring the Sensitivity of Horn's Parallel Analysis to the Distributional Form of Random Data”

Higher-order exploratory factor analysis of the Reynolds Intellectual Assessment Scales with a referred sample

The factor structure of the Reynolds Intellectual Assessment Scales (RIAS; [Reynolds, C.R., & Kamphaus, R.W. (2003). Reynolds Intellectual Assessment Scales. Lutz, FL: Psychological Assessment Resources, Inc.]) was investigated with a large ( N = 1163) independent sample of referred students (ages 6–18). More rigorous factor extraction criteria (viz., Horn's parallel analysis (HPA); [Horn, J.L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179–185.], and Minimum Average Partial (MAP) analysis; [Velicer, W.F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321–327.]), in addition to those used in RIAS development, were investigated. Exploratory factor analyses using both orthogonal and oblique rotations and higher-order exploratory factor analyses using the Schmid and Leiman [Schmid, J., and Leiman, J.M. (1957). The development of hierarchical factor solutions. Psychometrika, 22, 53–61.] procedure were conducted. All factor extraction criteria indicated extraction of only one factor. Oblique rotations resulted in different results than orthogonal rotations, and higher-order factor analysis indicated the largest amount of variance was accounted for by the general intelligence factor. The proposed three-factor solution was not supported. Implications for the use of the RIAS with similarly referred students are discussed.

Historical increase in the number of factors measured by commercial tests of cognitive ability: Are we overfactoring?

A historical increase in the number of factors purportedly measured by commercial tests of cognitive ability may result from four distinct pressures including: increasingly complex models of intelligence, test publishers' desires to provide clinically useful assessment instruments with greater interpretive value, test publishers' desires to include minor factors that may be of interest to researchers (but are not clinically useful), and liberal statistical criteria for determining the factor structure of tests. The present study examined the number of factors measured by several historically relevant and currently employed commercial tests of cognitive abilities using statistical criteria derived from principal components analyses, and exploratory and confirmatory factor analyses. Two infrequently used statistical criteria, that have been shown to accurately recover the number of factors in a data set, Horn's parallel analysis (HPA) and Minimum Average Partial (MAP) analysis, served as gold-standard criteria. As expected, there were significant increases over time in the number of factors purportedly measured by cognitive ability tests ( r = .56, p = .030). Results also indicated significant recent increases in the overfactoring of cognitive ability tests. Developers of future cognitive assessment batteries may wish to increase the lengths of the batteries in order to more adequately measure additional factors. Alternatively, clinicians interested in briefer assessment strategies may benefit from short batteries that reliably assess general intellectual ability.

Eigenvalue‐based model selection during latent semantic indexing

AbstractIn this study amended parallel analysis (APA), a novel method for model selection in unsupervised learning problems such as information retrieval (IR), is described. At issue is the selection of k, the number of dimensions retained under latent semantic indexing (LSI). Amended parallel analysis is an elaboration of Horn's parallel analysis, which advocates retaining eigenvalues larger than those that we would expect under term independence. Amended parallel analysis operates by deriving confidence intervals on these “null” eigenvalues. The technique amounts to a series of nonparametric hypothesis tests on the correlation matrix eigenvalues. In the study, APA is tested along with four established dimensionality estimators on six standard IR test collections. These estimates are evaluated with regard to two IR performance metrics. Additionally, results from simulated data are reported. In both rounds of experimentation APA performs well, predicting the best values of k on 3 of 12 observations, with good predictions on several others, and never offering the worst estimate of optimal dimensionality.