Rotational Indeterminacy Research Articles

Regularized Exploratory Factor Analysis as an Alternative to Factor Rotation

Abstract: Exploratory factor analysis (EFA) is widely used in psychological (assessment) research. Due to its exploratory nature, several researcher degrees of freedom exist on how to conduct the analysis. While simulation studies can provide meaningful insights into which factor retention methods to use to determine the number of latent factors, or which estimation methods recover parameter values most precisely given certain data characteristics, the issue of rotational indeterminacy makes it very difficult to decide which rotation method to apply. An alternative to the two-stage approach of extracting factors and subsequently rotating them to foster interpretability is the so-called regularized EFA. In this paper, we contrast both approaches and demonstrate how regularized EFA can be applied. In doing so, we want to encourage researchers to try out the approach themselves and help them find a way of EFA that appears less arbitrary compared to classical factor rotation.

Clustered Common Factor Exploration in Factor Analysis.

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 part of the factor scores can be uniquely determined. Thus, as a whole, they can be viewed as a sum of determined and undetermined parts. The paper proposes to select the undetermined part, such that the resulting common factor scores have the following feature: the rows (i.e., individuals) of the common factor score matrix are as well classified as possible into few clusters. The clear benefit is that we can easily interpret the factor scores simply by focusing on the clusters. The procedure is called clustered common factor exploration (CCFE). An alternating least squares algorithm is developed for CCFE. It is illustrated with real data examples. The proposed approach can be viewed as a parallel to the rotation techniques in FA. They exploit another FA indeterminacy, the rotation indeterminacy, which is resolved by choosing the rotation that transforms the loading matrix into the 'most' interpretable one according to a pre-specified criterion. In contrast to the rotational indeterminacy, the factor indeterminacy is utilized to achieve well-clustered factor scores by CCFE. To the best of our knowledge, such an approach to the FA interpretation has not been studied yet.

Simple Structure Detection Through Bayesian Exploratory Multidimensional IRT Models

In modern validity theory, a major concern is the construct validity of a test, which is commonly assessed through confirmatory or exploratory factor analysis. In the framework of Bayesian exploratory Multidimensional Item Response Theory (MIRT) models, we discuss two methods aimed at investigating the underlying structure of a test, in order to verify if the latent model adheres to a chosen simple factorial structure. This purpose is achieved without imposing hard constraints on the discrimination parameter matrix to address the rotational indeterminacy. The first approach prescribes a 2-step procedure. The parameter estimates are obtained through an unconstrained MCMC sampler. The simple structure is, then, inspected with a post-processing step based on the Consensus Simple Target Rotation technique. In the second approach, both rotational invariance and simple structure retrieval are addressed within the MCMC sampling scheme, by introducing a sparsity-inducing prior on the discrimination parameters. Through simulation as well as real-world studies, we demonstrate that the proposed methods are able to correctly infer the underlying sparse structure and to retrieve interpretable solutions.

AXES ROTATION FOR MAXIMIZATION OF INDIVIDUAL DIFFERENCES IN INDIVIDUAL SCALES METHOD

This study examines the methodology of "individual scaling method" in order to value each person's vocabulary and viewpoints. "Individual scales" refers to evaluation items made by subjects' own terms. Thus, these scales are different from person to person. In the previous report Part1, we proposed principal component analysis (PCA) method for evaluation data measured by individual scales. In this PCA method, evaluation object is regarded as observations, and individual scales of all subjects is regarded as variables. And, individual difference in vocabulary and viewpoint is described as difference in distribution of factor loadings vectors in principal component space. However, suitable method for analyzing individual differences that are described as such has not yet been proposed because of so-called "rotational indeterminacy" related problem. In this paper, first, when analyzing individual differences based on proposed PCA method, we discuss the problems caused by so-called "rotational indeterminacy". Next, as a solution to this problem, the new method of orthogonal rotation is proposed. This rotation method has the following features. ·By the criterion in this method, R-squares matrix of [each person × rotated axes] is directed to simple structure. ·By this rotation, individual differences in semantic space is maximized. R-squares is measured by personal-PCR (principal component regression analysis), i.e. PCA is performed by each subject, and then by using their each PCA scores as explanatory variables, multiple regression analysis is performed by each subject. Thus, the number of personal PCA scores is an important option in the analysis. The criteria to be maximized , we propose the following three. These are another important option. a) Sum of variances of columns of R-squares matrix b) Sum of variances of columns of Normalized R-squares matrix divided by row sum. c) Simply, variance of R-squares When replacing R-squares matrix to factor loadings squares matrix, each of the criteria is equivalent to the following well-known criteria. a) Row varimax criterion, b) Normal varimax criterion, c) Quartimax criterion As a case study, the proposed rotation method was applied on evaluation data of townscape in Kanda area. The results obtained and discussion are as follows . 1) In this case of data, the results by criterion c) seems to be most useful. 2) It's recommended to use personal PCA scores of slightly many numbers. In this case, six score is necessary. 3) Rotated axes are interpreted as follows; R1: History and taste, R2: Disorderly-Orderly, R3: Harmony and familiar, R4: Enjoying 4) Among these axes, R3 and R4 are large individual differences of R-squares. About these individual differences, the following tendencies were found. (1) Correlation between R-squares of R3 and R-squares of R4 is negative. Thus, terms to express R3 and terms to express R4, there seem to be not so many people with the both of vocabulary. (2) Subjects majoring in architecture have larger R-squares of R3 than other majors, and have smaller R-squares of R4 than other majors. Furthermore, from the viewpoint of validity of R-square as indicator of explanatory power, diagnostics of the number of personal PCA scores was discussed. Finally, it was noted that there are cases not to require rotation proposed in this paper. And some cases of analysis on such data were presented.

Graphical tool of sparse factor analysis

Factor analysis is a popular statistical model for analyzing correlation structures among observed variables. It is well known that this model has a rotational indeterminacy. Traditionally, the model parameters are estimated by the maximum likelihood method; then, factor rotation methods are applied to obtain an interpretable factor loading matrix. Recently, several sparse estimation procedures via penalization have been developed. Sparse estimation via penalization is an alternative to the factor rotation; it leads to an interpretable and sufficiently sparse solution. In this paper, we give an overview of several sparse factor analysis models, followed by a discussion of a relation between ordinary factor rotation and penalized maximum likelihood approaches. Then, we introduce a novel analyzing tool wherein a user can select a model that is easy to interpret and also possesses desirable values of goodness-of-fit indices based on the graphical representation of solution path.

Parallel psychometric and cognitive modeling analyses of the Penn Face Memory Test in the Army Study to Assess Risk and Resilience in Servicemembers

Objective: The psychometric properties of the Penn Face Memory Test (PFMT) were investigated in a large sample (4,236 participants) of U.S. Army Soldiers undergoing computerized neurocognitive testing. Data were drawn from the initial phase of the Army Study to Assess Risk and Resilience in Servicemembers (Army STARRS), a large-scale study directed towards identifying risk and resilience factors for suicidal behavior and other stress-related disorders in Army Soldiers. In this paper, we report parallel psychometric and cognitive modeling analyses of the PFMT to determine whether ability estimates derived from the measure are precise and valid indicators of memory in the Army STARRS sample. Method: Single-sample cross-validation methodology combined with exploratory factor and multidimensional item response theory techniques were used to explore the latent structure of the PFMT. To help resolve rotational indeterminacy of the exploratory solution, latent constructs were aligned with parameter estimates derived from an unequal-variance signal detection model. Results: Analyses suggest that the PFMT measures two distinct latent constructs, one associated with memory strength and one associated with response bias, and that test scores are generally precise indicators of ability for the majority of Army STARRS participants. Conclusions: These findings support the use of the PFMT as a measure of major constructs related to recognition memory and have implications for further cognitive–psychometric model development.

Rotation Techniques in Asymmetric Multidimensional Scaling

Multidimensional scaling (MDS) is a set of techniques, used especially in behavioral and social sciences, that enable a researcher to visualize proximity data in a multidimensional space. This article focuses on a particular class of MDS models proposed to deal with proximities which describe asymmetric relationships (i.e., trade indices for a set of countries, brand switching data, occupational mobility tables, and so on). They are based on the decomposition of the relationships into a symmetric and a skew-symmetric part. In this way the objects are represented as points in a multidimensional space and the intensity of their relationships as scalar products (symmetry) or triangle areas (skew-symmetry). These models are seen as special cases of a general model and their rotational indeterminacy is investigated. The aim is to propose a rotation method that makes easier the visual inspection of the graphical representation, highlighting the simple structure of the data. In particular an orthomax-like family of rotation methods and a general algorithm are proposed. Advantages of the proposal are illustrated by analysis of import-export data.

Understanding and controlling rotations in factor analytic models

Positive Matrix Factorization (PMF) is a least-squares approach for solving the factor analysis problem. It has been implemented in several forms. Initially, a program called PMF2 was used. Subsequently, a new, more flexible modeling tool, the Multilinear Engine, was developed. These programs can utilize different approaches to handle the problem of rotational indeterminacy. Although both utilize non-negativity constraints to reduce rotational freedom, such constraints are generally insufficient to wholly eliminate the rotational problem. Additional approaches to control rotations are discussed in this paper: (1) global imposition of additions among “scores” and subtractions among the corresponding “loadings” (or vice versa), (2) constraining individual factor elements, either scores and/or loadings, toward zero values, (3) prescribing values for ratios of certain key factor elements, or (4) specifying certain columns of the loadings matrix as known fixed values. It is emphasized that application of these techniques must be based on some external information about acceptable or desirable shapes of factors. If no such a priori information exists, then the full range of possible rotations can be explored, but there is no basis for choosing one of these rotations as the “best” result. Methods for estimating the rotational ambiguity in any specific result are discussed.

PARAFAC for Spectral Curve Resolution: A Case Study Using Total Luminescence in Human Dental Tartar

Chromophore identification in biological samples often requires the physical separation of the compounds, which can be difficult. Although there are several advantages to hyphenated spectroscopic techniques for identification of substances, complex mixtures of chromophores presenting overlapped spectra cannot be identified directly through this method. This work presents an application of chemometrics to compound identification in biological samples by a spectroscopic hyphenated technique using a curve resolution method. The PARAllel FACtor analysis model (PARAFAC), which has no rotational indeterminacy, was used for curve resolution of excitation-emission spectra of human dental tartars. PARAFAC was applied under constraints (i.e., unimodality and non-negativity) and evaluated with a validation procedure. The resolved profiles are porphyrinic-like spectra presenting excitation band maxima at 407, 416, and 431 nm in the Soret band region (390–440 nm) of these substances.

Confirmatory compositional approaches to the development of product spaces

The set of compositional approaches to product space development is expanded to include confirmatory methods. Specifically, describes and compares product space development (perceptual mapping) via confirmatory factor analysis and partial least squares with the aid of an empirical example. Both of these procedures are widely used in causal or structural equation modelling. Since they tend to be confirmatory extensions to factor analysis and principal components analysis, the approaches are also well suited to the development of product spaces. Confirmatory approaches have several advantages over exploratory approaches including the incorporation of prior knowledge, the elimination of rotational indeterminacy, and the use of a wide variety of measurement tools to assess the reliability and validity of model results.

The attributes, behavior, and performance of U.S. mutual funds

This article examines the risk and return characteristics of U.S. mutual funds. We employ an equilibrium version of the Arbitrage Pricing Theory (APT) and a principal-components-based statistical technique to identify performance benchmarks. We also consider the Capital Asset Pricing Model (CAPM) as an alternative. We implement a procedure for overcoming the rotational indeterminacy of factor models. This procedure is a hybrid of statistical factor estimation and prespecification of factors. We estimate measures of timing ability for the CAPM and extend it to the APT. We find that this timing test is misspecified due to noninformation-based changes in mutual fund betas. We develop a modification of the timing measure that, under certain conditions, distinguishes true timing ability from noninformation-based beta changes.