Retrieving the correlation matrix from a truncated PCA solution: The inverse principal component problem

Whenr Principal Components are available fork variables, the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose. The approximation is generally not perfect unlessr =k. In the present paper it is shown that, whenr is at or above the Ledermann bound,r principal components are enough to perfectly reconstruct the correlation matrix, albeit in a way more involved than taking the loading matrix times its transpose. In certain cases just below the Ledermann bound, recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information.     

A simple Gauss-Newton procedure for covariance structure analysis with high-level computer languages

An implementation of the Gauss-Newton algorithm for the analysis of covariance structures that is specifically adapted for high-level computer languages is reviewed. With this procedure one need only describe the structural form of the population covariance matrix, and provide a sample covariance matrix and initial values for the parameters. The gradient and approximate Hessian, which vary from model to model, are computed numerically. Using this approach, the entire method can be operationalized in a comparatively small program. A large class of models can be estimated, including many that utilize functional relationships among the parameters that are not possible in most available computer programs. Some examples are provided to illustrate how the algorithm can be used.We are grateful to M. W. Browne and S. H. C. du Toit for many invaluable discussions about these computing ideas. Thanks also to Scott Chaiken for providing the data in the first example. They were collected as part of the U.S. Air Force's Learning Ability Measurement Project (LAMP), sponsored by the Air Force Office of Scientific Research (AFOSR) and the Human Resource Directorate of the Armstrong Laboratory (AL/HRM).     

Simultaneous classification and multidimensional scaling with external information

For the exploratory analysis of a matrix of proximities or (dis)similarities between objects, one often uses cluster analysis (CA) or multidimensional scaling (MDS). Solutions resulting from such analyses are sometimes interpreted using external information on the objects. Usually the procedures of CA, MDS and using external information are carried out independently and sequentially, although combinations of two of the three procedures (CA and MDS, or multidimensional scaling and using external information) have been proposed in the literature. The present paper offers a procedure that combines all three procedures in one analysis, using a model that describes a partition of objects with cluster centroids represented in a low-dimensional space, which in turn is related to the information in the external variables. A simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure. Also, to illustrate the method, it is applied to two empirical data sets.     

On the estimation of polychoric correlations and their asymptotic covariance matrix

A general theory for parametric inference in contingency tables is outlined. Estimation of polychoric correlations is seen as a special case of this theory. The asymptotic covariance matrix of the estimated polychoric correlations is derived for the case when the thresholds are estimated from the univariate marginals and the polychoric correlations are estimated from the bivariate marginals for given thresholds. Computational aspects are also discussed.The research was supported by the Swedish Council for Research in the Humanities and Social Sciences (HSFR) under the programMultivariate Statistical Analysis. The author thanks a reviewer for pointing out an error in the original version of the paper.     

Analyzing sequential categorical data: Individual variation in markov chains

Markov chains are probabilistic models for sequences of categorical events, with applications throughout scientific psychology. This paper provides a method for anlayzing data consisting of event sequences and covariate observations. It is assumed that each sequence is a Markov process characterized by a distinct transition probability matrix. The objective is to use the covariate data to explain differences between individuals in the transition probability matrices characterizing their sequential data. The elements of the transition probability matrices are written as functions of a vector of latent variables, with variation in the latent variables explained through a multivariate regression on the covariates. The regression is estimated using the EM algorithm, and requires the numerical calculation of a multivariate integral. An example using simulated cognitive developmental data is presented, which shows that the estimation of individual variation in the parameters of a probability model may have substantial theoretical importance, even when individual differences are not the focus of the investigator's concerns.Research contributing to this article was supported by B.R.S. Subgrant 5-35345 from the University of Virginia. I thank the DADA Group, Bill Fabricius, Don Hartmann, William Griffin, Jack McArdle, Ivo Molenaar, Ronald Schoenberg, Simon Tavaré, and several anonymous reviewers for their discussion of these points.     

An alternative two stage least squares (2SLS) estimator for latent variable equations

The Maximum-likelihood estimator dominates the estimation of general structural equation models. Noniterative, equation-by-equation estimators for factor analysis have received some attention, but little has been done on such estimators for latent variable equations. I propose an alternative 2SLS estimator of the parameters in LISREL type models and contrast it with the existing ones. The new 2SLS estimator allows observed and latent variables to originate from nonnormal distributions, is consistent, has a known asymptotic covariance matrix, and is estimable with standard statistical software. Diagnostics for evaluating instrumental variables are described. An empirical example illustrates the estimator. I gratefully acknowledge support for this research from the Sociology Program of the National Science Foundation (SES-9121564) and the Center for Advanced Study in the Behavioral Sciences, Stanford, California. This paper was presented at the Interdisciplinary Consortium for Statistical Applications at Indiana University at Bloomington (March 2, 1994) and at the RMD Conference on Causal Modeling at Purdue University, West Lafayette, Indiana (March 3-5, 1994).     

A Polychoric Instrumental Variable (PIV) Estimator for Structural Equation Models with Categorical Variables

This paper presents a new polychoric instrumental variable (PIV) estimator to use in structural equation models (SEMs) with categorical observed variables. The PIV estimator is a generalization of Bollen’s (Psychometrika 61:109–121, 1996) 2SLS/IV estimator for continuous variables to categorical endogenous variables. We derive the PIV estimator and its asymptotic standard errors for the regression coefficients in the latent variable and measurement models. We also provide an estimator of the variance and covariance parameters of the model, asymptotic standard errors for these, and test statistics of overall model fit. We examine this estimator via an empirical study and also via a small simulation study. Our results illustrate the greater robustness of the PIV estimator to structural misspecifications than the system-wide estimators that are commonly applied in SEMs. Kenneth Bollen gratefully acknowledges support from NSF SES 0617276, NIDA 1-RO1-DA13148-01, and DA013148-05A2. Albert Maydeu-Olivares was supported by the Department of Universities, Research and Information Society (DURSI) of the Catalan Government, and by grant BSO2003-08507 from the Spanish Ministry of Science and Technology. We thank Sharon Christ, John Hipp, and Shawn Bauldry for research assistance. The comments of the members of the Carolina Structural Equation Modeling (CSEM) group are greatly appreciated. An earlier version of this paper under a different title was presented by K. Bollen at the Psychometric Society Meetings, June, 2002, Chapel Hill, North Carolina.     

The polyserial correlation coefficient

The polyserial and point polyserial correlations are discussed as generalizations of the biserial and point biserial correlations. The relationship between the polyserial and point polyserial correlation is derived. The maximum likelihood estimator of the polyserial correlation is compared with a two-step estimator and with a computationally convenient ad hoc estimator. All three estimators perform reasonably well in a Monte Carlo simulation. Some practical applications of the polyserial correlation are described.By coincidence, the first author and the second and third authors learned that they were working independently on closely related problems and, consequently, decided to write a jointly authored paper.     

Enhancing Interpretability in Factor Analysis by Means of Mathematical Optimization

Abstract

Exploratory Factor Analysis (EFA) is a widely used statistical technique to discover the structure of latent unobserved variables, called factors, from a set of observed variables. EFA exploits the property of rotation invariance of the factor model to enhance factors’ interpretability by building a sparse loading matrix. In this paper, we propose an optimization-based procedure to give meaning to the factors arising in EFA by means of an additional set of variables, called explanatory variables, which may include in particular the set of observed variables. A goodness-of-fit criterion is introduced which quantifies the quality of the interpretation given this way. Our methodology also exploits the rotational invariance of EFA to obtain the best orthogonal rotation of the factors, in terms of the goodness-of-fit, but making them match to some of the explanatory variables, thus going beyond traditional rotation methods. Therefore, our approach allows the analyst to interpret the factors not only in terms of the observed variables, but in terms of a broader set of variables. Our experimental results demonstrate how our approach enhances interpretability in EFA, first in an empirical dataset, concerning volumes of reservoirs in California, and second in a synthetic data example.     

Mediation analysis and categorical variables: The final frontier

Many scholars are interested in understanding the process by which an independent variable affects a dependent variable, perhaps in part directly and perhaps in part indirectly, occurring through the activation of a mediator. Researchers are facile at testing for mediation when all the variables are continuous, but a definitive answer had been lacking heretofore as to how to analyze the data when the mediator or dependent variable is categorical. This paper describes the problems that arise as well as the potential solutions. In the end, a solution is recommended that is both optimal in its statistical qualities as well as practical and easily implemented: compute z_Mediation.     

Mediation with categorical variables compleat

This rejoinder addresses the thoughtful issues raised by the commentators. We hold to the analytical solution of fitting X → M (to obtain a and sa) and X&;M→ (to obtain b and sb) via regression or logistic regression, depending on whether M and Y are continuous or categorical. Then researchers compute za, zb, and ZMediation.     

Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients

The method of finding the maximum likelihood estimates of the parameters in a multivariate normal model with some of the component variables observable only in polytomous form is developed. The main stratagem used is a reparameterization which converts the corresponding log likelihood function to an easily handled one. The maximum likelihood estimates are found by a Fletcher-Powell algorithm, and their standard error estimates are obtained from the information matrix. When the dimension of the random vector observable only in polytomous form is large, obtaining the maximum likelihood estimates is computationally rather labor expensive. Therefore, a more efficient method, the partition maximum likelihood method, is proposed. These estimation methods are demonstrated by real and simulated data, and are compared by means of a simulation study.     

Influence of school-level and family-level variables on Chinese college students’ aggression

With the frequent occurrence of campus violence, scholars have devoted increasing attention to college students’ aggression. This study aims to estimate the prevalence of aggression in Chinese university students and identify factors that could influence their aggression. We can thus find methods to reduce the incidence of college students’ aggression in the future. A multi-stage stratified sampling procedure was used to select university students (N = 4565) aged 16–25 years in Harbin. The Aggression Questionnaire, the Adolescent Self-Rating Life Events Checklist and the Social Support Revalued Scale were used to collect data. Females reported lower levels of aggression than males (p < .001). A multiple linear regression analysis was conducted to determine the influence of factors of aggression, and the model was highly significant (R² = .233, Ad R² = .230, p < .01). The results show that the aggression is affected by gender, family-level and school-level variables. Aggression scores are significantly correlated with not only family-level or school-level variables independently, but their combination as well. We find that the risk factors for aggression include a dissatisfying profession, higher levels of study pressure, poor parental relationships, poor interpersonal relationships, the presence of siblings, punishment, health maladjustment, less subjective support, and lower levels of utilization of social support.     

Linguistic characteristics of neurotic, borderline and psychotic personality organization

Thirty patients were subjected to a Structural Interview by means of which 10 of them received the diagnosis NPO (Neurotic Personality Organization), 10 BPO (Borderline Personality Organization) and 10 PPO (Psychotic Personality Organization). About 2500 words and groups of words were extracted from the patients' utterances for analysis. The word-frequencies thus found were analyzed by a PLS discriminant analysis which yielded two significant principal components (main dimensions) explaining 57% of the variance. This analysis showed that the three groups of patients are well separated from one another and that there is a definite correlation between personality organization and linguistic variables. The main features of the BPO-patients in this study seem to be that they refer to positions outside themselves and their language is impersonal; we see this as an exemple of a vacillating identity. The NPO-group is characterized by an intense and rich language, signs of a more advanced symbolizing ability, deixis and high level defenses. The language of the PPO-patients is poor and its predominant feature is a lack of words; in our interpretation this indicates foreclosure and a lack of identity.     

无意识目标启动的定义、机制及应用

无意识目标与社会态度、刻板印象和图式等一样, 是行为的满意状态或结果在个体头脑中的一种表征, 包含有目标设定、追求情境和追求手段等内容。这些内容能够被环境中的刺激自动激活, 在意识不到的情况下影响人的心理和行为。无意识目标启动的概念和其他心理表征的启动不同, 无意识目标启动能够促使带有动机性的行为出现, 启动线索的影响不会随着时间的推移而消失。无意识目标启动的自变量与自身心理状态和环境中他人的影响有关, 无意识目标启动的因变量与任务表现和社会关系的处理有关。“知觉-行为模型”、“无意识行为选择模型”、“评价准备模型”和“情感因素模型”能够分别从“无意识”和“目标”两个角度对无意识目标启动的心理机制进行解释。无意识目标启动原理在商业、教育、医疗等领域当中有着广泛的应用前景, 为了更好地完善理论研究和进行实际应用, 今后应该更加重视无意识目标启动的脑机制研究。     

目击证人辨认研究

目击证人辨认对司法审判具有重要的意义,这使其成为心理学研究领域的热点之一。本文重点介绍了影响目击证人辨认准确性的因素,即估计者变量和系统变量,并分析了目击证人辨认研究的现存问题。     

A random forest approach to explore how situational variables affect perceived exertion of elite youth soccer players

The aim of this study was to explore how situational variables affect youth soccer players’ perceived exertion (RPE) after official matches. Thirty-five elite youth male players (14.33 ± 0.86 years; 173.49 ± 6.16 cm; 63.44 ± 5.98 kg) who belonged to two different teams of a professional club participated in this study. Data collection was conducted during two seasons (2016–2017, 2017–2018) and included 60 official matches (30 official matches per team). Ten minutes after each match players rated their RPE and using a modified Borg CR-10 scale. A Random Forest Regression was used to quantify the importance of match-related situational variables in RPE. Afterwards, a linear mixed model analysis was applied to identify the variability in RPE among the situational variables. The game-playing time, the player status (starter or substitute) and the player identity were the strongest predictors of RPE. Moreover, the match outcome and the final scoreline showed significant effects on both starter and substitute players but the main effect of the quality of the opponent was only identified in starter players (p < 0.05). These results allow practitioners to know how situational variables interact and modulate RPE after official matches and help them to prescribe and adapt the players’ training content and load before and after matches.     

国外心理契约破坏及结果变量与调节变量：述评与展望

心理契约破坏是个体对组织未能履行其在心理契约中应承担责任的认知评价。文章首先结合以往研究探讨了心理契约破坏与相应结果变量的关系,然后探讨了调节变量的作用,最后从被试群体、研究方法及理论模型的构建上对未来研究进行了展望。