Extending the Debate Between Spearman and Wilson 1929: When do Single Variables Optimally Reproduce the Common Part of the Observed Covariances?

The covariances of observed variables reproduced from conventional factor score predictors are generally not the same as the covariances reproduced from the common factors. We sought to find a factor score predictor that optimally reproduces the common part of the observed covariances. It was found algebraically that—under some conditions—the single observed variable with highest loading on a factor reproduces the non-diagonal elements of the observed covariance matrix more exactly than the conventional factor score predictors. This finding is linked to Spearman's and Wilson's 1929 debate on the use of single variables as factor score predictors. A population-based and a sample-based simulation study confirmed the algebraic result that taking a single variable can outperform conventional factor score predictors in reproducing the non-diagonal covariances when the nonzero loading size and the number of nonzero loadings per factor are small. The results indicated that a weighted aggregation of variables does not necessarily lead to an improvement of the score over the variable with the highest loading.     

CFA Models with a General Factor and Multiple Sets of Secondary Factors

We propose a class of confirmatory factor analysis models that include multiple sets of secondary or specific factors and a general factor. The general factor accounts for the common variance among manifest variables, whereas multiple sets of secondary factors account for the remaining source-specific dependency among subsets of manifest variables. A special case of the model is further proposed which constrains the specific factor loadings to be proportional to the general factor loadings. This proportional model substantially reduces the number of model parameters while preserving the essential structure of the general model. Furthermore, the proportional model allows for the interpretation of latent variables as the expected values of the observed manifest variables, decomposition of the variances, and the inclusion of interactions, similar to generalizability theory. We provide two applications to illustrate the utility of the proposed class of models.     

The Recoverability of P-technique Factor Analysis

It seems that just when we are about to lay P–technique factor analysis finally to rest as obsolete because of newer, more sophisticated multivariate time-series models using latent variables—dynamic factor models—it rears its head to inform us that an obituary may be premature. We present the results of some simulations demonstrating that even though it does not explicitly model lagged information, P–technique's ability to recover the parameters of underlying dynamic processes involving lagged relations among the manifest variables is apparently robust and accurate. An empirical example is presented using 103 days of affective mood self-ratings from a young pregnant woman. Implications of the simulation and empirical findings are briefly discussed.     

结构方程建模中的题目打包策略

结构方程建模中题目打包法的优缺点包括：指标数据质量变好、模型拟合程度提高; 估计偏差不大, 可校正; 估计稳定, 但降低了敏感性与可证伪性。打包法的前提条件是单维、同质, 适合结构模型分析, 不适合测量模型分析。对于单维测验, 给出了一个打包流程。对于通常的多个子量表(多维结构)测验, 推荐在子量表内打包, 每个子量表打包成1个指标或者3个指标, 用于结构方程建模。     

Semi-sparse PCA

It is well known that the classical exploratory factor analysis (EFA) of data with more observations than variables has several types of indeterminacy. We study the factor indeterminacy and show some new aspects of this problem by considering EFA as a specific data matrix decomposition. We adopt a new approach to the EFA estimation and achieve a new characterization of the factor indeterminacy problem. A new alternative model is proposed, which gives determinate factors and can be seen as a semi-sparse principal component analysis (PCA). An alternating algorithm is developed, where in each step a Procrustes problem is solved. It is demonstrated that the new model/algorithm can act as a specific sparse PCA and as a low-rank-plus-sparse matrix decomposition. Numerical examples with several large data sets illustrate the versatility of the new model, and the performance and behaviour of its algorithmic implementation.

     

Clustered Common Factor Exploration in Factor Analysis

Psychometrika - The factor analysis (FA) model does not permit unique estimation of the common and unique factor scores. This weakness is notorious as the factor indeterminacy in FA. Luckily, some...     

Exploratory Factor Analysis With Small Sample Sizes

Exploratory factor analysis (EFA) is generally regarded as a technique for large sample sizes (N), with N = 50 as a reasonable absolute minimum. This study offers a comprehensive overview of the conditions in which EFA can yield good quality results for N below 50. Simulations were carried out to estimate the minimum required N for different levels of loadings (λ), number of factors (f), and number of variables (p) and to examine the extent to which a small N solution can sustain the presence of small distortions such as interfactor correlations, model error, secondary loadings, unequal loadings, and unequal p/f. Factor recovery was assessed in terms of pattern congruence coefficients, factor score correlations, Heywood cases, and the gap size between eigenvalues. A subsampling study was also conducted on a psychological dataset of individuals who filled in a Big Five Inventory via the Internet. Results showed that when data are well conditioned (i.e., high λ, low f, high p), EFA can yield reliable results for N well below 50, even in the presence of small distortions. Such conditions may be uncommon but should certainly not be ruled out in behavioral research data.     

Indeterminacy Problems in the Lisrel Model

The latent variables and errors of the Lisrel model are indeterminate even when the parameters of the model are perfectly identified. The reason for the indeterminacy is that the Lisrel model gives a solution in terms of estimation of latent variables by means of observed variables. The indeterminacy is relevant also in practice; the minimum correlation between equivalent latent variables, is often negative in empirical examples. The degree of indeterminacy of the latent variables depends on the data. The average minimum correlation is a linear combination of the eigenvalues of the correlation matrix of solutions and it is always included in weak bounds which depend on the same eigenvalues.     

Ordinary Least Squares Estimation of Parameters in Exploratory Factor Analysis With Ordinal Data

Exploratory factor analysis (EFA) is often conducted with ordinal data (e.g., items with 5-point responses) in the social and behavioral sciences. These ordinal variables are often treated as if they were continuous in practice. An alternative strategy is to assume that a normally distributed continuous variable underlies each ordinal variable. The EFA model is specified for these underlying continuous variables rather than the observed ordinal variables. Although these underlying continuous variables are not observed directly, their correlations can be estimated from the ordinal variables. These correlations are referred to as polychoric correlations. This article is concerned with ordinary least squares (OLS) estimation of parameters in EFA with polychoric correlations. Standard errors and confidence intervals for rotated factor loadings and factor correlations are presented. OLS estimates and the associated standard error estimates and confidence intervals are illustrated using personality trait ratings from 228 college students. Statistical properties of the proposed procedure are explored using a Monte Carlo study. The empirical illustration and the Monte Carlo study showed that (a) OLS estimation of EFA is feasible with large models, (b) point estimates of rotated factor loadings are unbiased, (c) point estimates of factor correlations are slightly negatively biased with small samples, and (d) standard error estimates and confidence intervals perform satisfactorily at moderately large samples.     

Some Mathematical Properties of the Matrix Decomposition Solution in Factor Analysis

A new factor analysis (FA) procedure has recently been proposed which can be called matrix decomposition FA (MDFA). All FA model parameters (common and unique factors, loadings, and unique variances) are treated as fixed unknown matrices. Then, the MDFA model simply becomes a specific data matrix decomposition. The MDFA parameters are found by minimizing the discrepancy between the data and the MDFA model. Several algorithms have been developed and some properties have been discussed in the literature (notably by Stegeman in Comput Stat Data Anal 99:189–203, 2016), but, as a whole, MDFA has not been studied fully yet. A number of new properties are discovered in this paper, and some existing ones are derived more explicitly. The properties provided concern the uniqueness of results, covariances among common factors, unique factors, and residuals, and assessment of the degree of indeterminacy of common and unique factor scores. The properties are illustrated using a real data example.     

Exploratory Factor Analysis With Small Samples and Missing Data

Exploratory factor analysis (EFA) is an extremely popular method for determining the underlying factor structure for a set of variables. Due to its exploratory nature, EFA is notorious for being conducted with small sample sizes, and recent reviews of psychological research have reported that between 40% and 60% of applied studies have 200 or fewer observations. Recent methodological studies have addressed small size requirements for EFA models; however, these models have only considered complete data, which are the exception rather than the rule in psychology. Furthermore, the extant literature on missing data techniques with small samples is scant, and nearly all existing studies focus on topics that are not of primary interest to EFA models. Therefore, this article presents a simulation to assess the performance of various missing data techniques for EFA models with both small samples and missing data. Results show that deletion methods do not extract the proper number of factors and estimate the factor loadings with severe bias, even when data are missing completely at random. Predictive mean matching is the best method overall when considering extracting the correct number of factors and estimating factor loadings without bias, although 2-stage estimation was a close second.     

Matched Determiners Vs. Factor Invariance: A Reply To Korth

Korth (1978) does well to describe factor matching as vital to personality research but seriously underestimates the extent of successful matching both within and between cultures. His evaluation of matching of a set of factors is an advance, but the achievement of "diagonalization" of r[SUBc] coefficients in a matching matrix probably has a higher significance than his method would indicate.

Regarding Monte Carlo determinations of r[SUBc] distributions, the writer maintains that treating loadings as random normal deviates is incorrect and that a special distribution (here presented) is required. Further, it is argued that "factor invariance," as commonly defined, is not the required proof of identify of determiners. Instead, the principles of real base factor analysis need to be applied to demonstrate degree of matching of determiners. A numerical illustration shows that when congruence is actually perfect for the real base factor patterns, it is not so for ordinary factor analysis patterns. Even in this framework the congruence, coefficient has weaknesses, and it is suggested that decisions be based on the joint outcome of r[SUBc] and 8, the salient variable similarity index.     

The Social Phobia and Social Interaction Anxiety Scales: Evaluation of Psychometric Properties

The present investigations examined the factor structure and psychometric properties of two new self-report measures of social phobia, the Social Interaction Anxiety Scale (SIAS) and the Social Phobia Scale (SPS). A confirmatory factor analysis in Study I provided support for the fit of a two-factor model of the SIAS and SPS. Internal consistency estimates were high for the original two scales with a sample of 200 undergraduates. Also, using an item parceling procedure, the obtained internal consistency reliability indices for each parcel were acceptable. Results of the CFA in Study II provided support for the factorial stability of the model identified in Study I. Furthermore, multisample analyses showed invariant patterns for factor loadings and factor correlations across 138 men and 272 women. Gender differences were not observed in the mean SIAS and SPS scale and item scores. Both scales correlated negatively and significantly with measures of social desirability. Concurrent validity was established for the scales. The SPS was less specific than the SIAS to symptoms of social phobia.     

The Conservatism Scale: In Search of a Replicable Factor Structure

Abstract

Responses of three independent groups of subjects to the Wilson-Patterson Conservatism Scale were submitted to principal components analyses followed by varimax rotation. Inspection of the unrotated factor loadings provided evidence for the presence of a general factor in all three groups, supporting the contention of the scale's authors (Wilson & Patterson, 1968). Detailed analyses of the present and previously published results revealed a consistent underlying theme of fundamental religious conservatism. Examination of loadings obtained from two-, three-, and four-factor rotations, using the FACTOREP procedure for factor comparison, failed to identify any other consistently replicable factor structure, a result consistent with conclusions based on an examination of previously published research.     

Multilevel Mixture Factor Models

Factor analysis is a statistical method for describing the associations among sets of observed variables in terms of a small number of underlying continuous latent variables. Various authors have proposed multilevel extensions of the factor model for the analysis of data sets with a hierarchical structure. These Multilevel Factor Models (MFMs) have in common that—as in multilevel regression analysis—variation at the higher level is modeled using continuous random effects. In this article, we present an alternative multilevel extension of factor analysis which we call the Multilevel Mixture Factor Model (MMFM). It is based on the assumption that higher level units belong to latent classes that differ in terms of the parameters of the factor model specified for the lower level units. We demonstrate the added value of MMFM compared with MFM, both from a theoretical and applied perspective, and we illustrate the complementarity of the two approaches with an empirical application on students' satisfaction with the University of Florence. The multilevel aspect of this application is that students are nested within study programs, which makes it possible to cluster these programs based on their differences in students' satisfaction.     

The effect of additional variables on factor indeterminacy in models with a single common factor

Determinate solutions for the indeterminate common factor ofp variables satisfying the single common factor model are not unique. Therefore an infinite sequence of additional variables that conform jointly with the originalp variables to the original single common factor model does not determine a unique solution for the indeterminate factor of thep variables (although the solution is unique for the factor of the infinite sequence). Other infinite sequences may be found to determine different solutions for the factor of the originalp variables. The paper discusses a number of theorems about the effects of additional variables on factor indeterminacy in a model with a single common factor and draws conclusions from them for factor theory in general.     

The Factor Structure of Effortful Control and Measurement Invariance Across Ethnicity and Sex in a High-Risk Sample

Measurement invariance of a one-factor model of effortful control (EC) was tested for 853 low-income preschoolers (M age = 4.48 years). Using a teacher-report questionnaire and seven behavioral measures, configural invariance (same factor structure across groups), metric invariance (same pattern of factor loadings across groups), and partial scalar invariance (mostly the same intercepts across groups) were established across ethnicity (European Americans, African Americans and Hispanics) and across sex. These results suggest that the latent construct of EC behaved in a similar way across ethnic groups and sex, and that comparisons of mean levels of EC are valid across sex and probably valid across ethnicity, especially when larger numbers of tasks are used. The findings also support the use of diverse behavioral measures as indicators of a single latent EC construct.     

Synonymy and the epistemology of linguistics

In Word and Object, Quine argues from the observation that ‘there is no justification for collating linguistic meanings, unless in terms of men's dispositions to respond overtly to socially observable stimulations’ to the conclusion that ‘the enterprise of translation is found to be involved in a certain systematic indeterminacy’. In this paper, I propose to show (1) that Quine's thesis, when properly understood, reveals in the situation of translation no peculiar indeterminacy but merely the ordinary indeterminacy present in any case of empirical investigation; (2) that it is plausible that, because the subject of inquiry is language, we are in a better position with respect to such empirical indeterminacies than we are in other areas of investigation; (3) that, in any case, Quine's arguments are impotent, for they are either contradictory or incoherent; and (4) that Quine is led to his radical conclusions because he confuses a trivial and unexciting indeterminacy, which does obtain, with the striking indeterminacy for which he argues, which does not obtain.     

Coming Full Circle in the History of Factor Indeterminancy

Nearly 70 years ago, eminent mathematician Edwin Bidwell Wilson attended a dinner at Harvard where visitor Charles Spearman discussed the "two-factor theory" of intelligence and his just-released book The Abilities of Man. Wilson, having just discovered factor indeterminacy, attempted to explain to Spearman and the assembled guests that Spearman's two-factor theory might have a non-uniqueness problem. Neither Spearman nor the guests could follow Wilson's argument, but Wilson persisted, first through correspondence, later through a series of publications that spanned more than a decade, involving Spearman and several other influential statisticians in an extended debate. Many years have passed since the Spearman-Wilson debates, yet the fascinating statistical, logical, and philosophical issues surrounding factor indeterminacy are very much alive. Equally fascinating are the sociological issues and historical questions surrounding the way indeterminacy has periodically vanished from basic textbooks on factor analysis. In this article, I delineate some of these historical-sociological issues, and respond to a critique from some recent commentators on the history of factor indeterminacy.     

Factor analysis by instrumental variables methods

Three alternative estimation procedures for factor analysis based on the instrumental variables method are presented. These procedures are justified by the method of least squares. Formulas for asymptotic standard errors of factor loadings are derived. The procedures are empirically compared to the method of maximum likelihood. The conclusion, based on the data used in this study, is that two of the procedures seem to work well.