Factor Retention Using Machine Learning With Ordinal Data

Determining the number of factors in exploratory factor analysis is probably the most crucial decision when conducting the analysis as it clearly influences the meaningfulness of the results (i.e., factorial validity). A new method called the Factor Forest that combines data simulation and machine learning has been developed recently. This method based on simulated data reached very high accuracy for multivariate normal data, but it has not yet been tested with ordinal data. Hence, in this simulation study, we evaluated the Factor Forest with ordinal data based on different numbers of categories (2–6 categories) and compared it to common factor retention criteria. It showed higher overall accuracy for all types of ordinal data than all common factor retention criteria that were used for comparison (Parallel Analysis, Comparison Data, the Empirical Kaiser Criterion and the Kaiser Guttman Rule). The results indicate that the Factor Forest is applicable to ordinal data with at least five categories (typical scale in questionnaire research) in the majority of conditions and to binary or ordinal data based on items with less categories when the sample size is large.     

题目因素分析：基于SEM和基于IRT的两类方法

心理和教育测量一般只能达到顺序量表的水平,其测量数据与被测因子间并非简单线性关系。题目因素分析是用来描述测量题目与因子间非线性关系的统计模型。题目因素分析主要有基于结构方程模型和基于项目反应理论两类方法,两类方法之间存在紧密的联系,甚至可以看作是同一模型的两种表现形式。本文详细阐述了该关系,同时对两类方法在参数估计、模型拟合指标、测量一致性检验和支撑软件等方面的特点进行了分析和比较,以便研究者选择最为适合其研究的方法。     

Covariance structure analysis of ordinal ipsative data

Data are ipsative if they are subject to a constant-sum constraint for each individual. In the present study, ordinal ipsative data (OID) are defined as the ordinal rankings across a vector of variables. It is assumed that OID are the manifestations of their underlying nonipsative vector y, which are difficult to observe directly. A two-stage estimation procedure is suggested for the analysis of structural equation models with OID. In the first stage, the partition maximum likelihood (PML) method and the generalized least squares (GLS) method are proposed for estimating the means and the covariance matrix of A_cy, where A_c is a known contrast matrix. Based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix, the generalized least squares method is used to estimate the structural parameters in the second stage. A goodness-of-fit statistic is given for testing the hypothesized covariance structure. Simulation results show that the proposed method works properly when a sufficiently large sample is available.This research was supported by National Institute on Drug Abuse Grants DA01070 and DA10017. The authors are indebted to Dr. Lee Cooper, Dr. Eric Holman, Dr. Thomas Wickens for their valuable suggestions on this study, and Dr. Fanny Cheung for allowing us to use her CPAI data set in this article. The authors would also like to acknowledge the helpful comments from the editor and the two anonymous reviewers.     

Analysis of contingency tables by ideal point discriminant analysis

Cross-classified data are frequently encountered in behavioral and social science research. The loglinear model and dual scaling (correspondence analysis) are two representative methods of analyzing such data. An alternative method, based on ideal point discriminant analysis (DA), is proposed for analysis of contingency tables, which in a certain sense encompasses the two existing methods. A variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and/or as direct constraints on model parameters. This, along with maximum likelihood estimation of the model parameters, allows interesting model comparisons. This is illustrated by the analysis of several data sets.Presented as the Presidential Address to the Psychometric Society's Annual and European Meetings, June, 1987. Preparation of this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Chikio Hayashi of University of the Air in Japan for providing the ISM data, and to Jim Ramsay and Ivo Molenaar for their helpful comments on an earlier draft of this paper.     

Factor analysis for clustered observations

Classical factor analysis assumes a random sample of vectors of observations. For clustered vectors of observations, such as data for students from colleges, or individuals within households, it may be necessary to consider different within-group and between-group factor structures. Such a two-level model for factor analysis is defined, and formulas for a scoring algorithm for estimation with this model are derived. A simple noniterative method based on a decomposition of the total sums of squares and crossproducts is discussed. This method provides a suitable starting solution for the iterative algorithm, but it is also a very good approximation to the maximum likelihood solution. Extensions for higher levels of nesting are indicated. With judicious application of quasi-Newton methods, the amount of computation involved in the scoring algorithm is moderate even for complex problems; in particular, no inversion of matrices with large dimensions is involved. The methods are illustrated on two examples.Suggestions and corrections of three anonymous referees and of an Associate Editor are acknowledged. Discussions with Bob Jennrich on computational aspects were very helpful. Most of research leading to this paper was carried out while the first author was a visiting associate professor at the University of California, Los Angeles.     

ANOVA and ANCOVA of pre- and post-test,ordinal data

With random assignment to treatments and standard assumptions, either a one-way ANOVA of post-test scores or a two-way, repeated measures ANOVA of pre- and post-test scores provides a legitimate test of the equal treatment effect null hypothesis for latent variable . In an ANCOVA for pre- and post-test variablesX andY which are ordinal measures of and , respectively, random assignment and standard assumptions ensure the legitimacy of inferences about the equality of treatment effects on latent variable . Sample estimates of adjustedY treatment means are ordinal estimators of adjusted post-test means on latent variable .     

Factor analysis for non-normal variables

Factor analysis for nonnormally distributed variables is discussed in this paper. The main difference between our approach and more traditional approaches is that not only second order cross-products (like covariances) are utilized, but also higher order cross-products. It turns out that under some conditions the parameters (factor loadings) can be uniquely determined. Two estimation procedures will be discussed. One method gives Best Generalized Least Squares (BGLS) estimates, but is computationally very heavy, in particular for large data sets. The other method is a least squares method which is computationally less heavy. In one example the two methods will be compared by using the bootstrap method. In another example real life data are analyzed.This paper has partly been written while the author was a visiting scholar at the Department of Psychology, University of California, Los Angeles. He wants to thank Peter Bentler who made this stay at UCLA possible and for his valuable contributions to this paper. This research was supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O) under number R56-150 and by USPHS Grant DA01070.     

Estimation for the multiple factor model when data are missing

A maximum likelihood method of estimating the parameters of the multiple factor model when data are missing from the sample is presented. A Monte Carlo study compares the method with 5 heuristic methods of dealing with the problem. The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.This paper is based on the author's doctoral dissertation at the Department of Psychology, University of Illinois at Urbana-Champaign. The author gratefully acknowledges the aid of Drs. Robert Bohrer, Charles Lewis, Robert Linn, Maurice Tatsuoka, and Ledyard Tucker.     

Factor analysis and AIC

The information criterion AIC was introduced to extend the method of maximum likelihood to the multimodel situation. It was obtained by relating the successful experience of the order determination of an autoregressive model to the determination of the number of factors in the maximum likelihood factor analysis. The use of the AIC criterion in the factor analysis is particularly interesting when it is viewed as the choice of a Bayesian model. This observation shows that the area of application of AIC can be much wider than the conventional i.i.d. type models on which the original derivation of the criterion was based. The observation of the Bayesian structure of the factor analysis model leads us to the handling of the problem of improper solution by introducing a natural prior distribution of factor loadings.The author would like to express his thanks to Jim Ramsay, Yoshio Takane, Donald Ramirez and Hamparsum Bozdogan for helpful comments on the original version of the paper. Thanks are also due to Emiko Arahata for her help in computing.     

Bayesian data analysis as a tool for behavior analysts

Bayesian approaches to data analysis are considered within the context of behavior analysis. The paper distinguishes between Bayesian inference, the use of Bayes Factors, and Bayesian data analysis using specialized tools. Given the importance of prior beliefs to these approaches, the review addresses those situations in which priors have a big effect on the outcome (Bayes Factors) versus a smaller effect (parameter estimation). Although there are many advantages to Bayesian data analysis from a philosophical perspective, in many cases a behavior analyst can be reasonably well‐served by the adoption of traditional statistical tools as long as the focus is on parameter estimation and model comparison, not null hypothesis significance testing. A strong case for Bayesian analysis exists under specific conditions: When prior beliefs can help narrow parameter estimates (an especially important issue given the small sample sizes common in behavior analysis) and when an analysis cannot easily be conducted using traditional approaches (e.g., repeated measures censored regression).     

Progressing toward data intimacy: a review of within-session data analysis

Visual inspection of data is a common method for understanding, responding to, and communicating important behavior-environment relations in single-subject research. In a field that was once dominated by cumulative, moment-to-moment records of behavior, a number of graphic forms currently exist that aggregate data into larger units. In this paper, we describe the continuum of aggregation that ranges from distant to intimate displays of behavioral data. To aid in an understanding of the conditions under which a more intimate analysis is warranted (i.e., one that provides a richer analysis than that provided by condition or session aggregates), we review a sample of research articles for which within-session data depiction has enhanced the visual analysis of applied behavioral research.     

心理与教育测验中异常反应侦查新技术：变点分析法

变点分析法(change point analysis, CPA)近些年才引入心理与教育测量学, 相较于传统方法, CPA不仅可以侦查异常作答被试, 还能自动精确地定位变点位置, 高效清洗作答数据。其原理在于：判断作答序列中是否存在可将该序列划分为具有不同统计学属性两部分的点(即变点), 并且需使用被试拟合统计量(person-fit statistic, PFS)来量化两个子序列之间的差异。未来可将单变点分析拓展至多变点, 结合反应时等信息, 构建非参数化指标以及将现有指标拓展至多级计分或多维测验, 以提高CPA的适用广度及效力。     

单维项目因素分析：CCFA与IRT估计方法的比较

当观测指标变量为二分分类数据时,传统的因素分析方法不再适用。作者简要回顾了SEM框架下的分类数据因素分析模型和IRT框架下的测验题目和潜在能力的关系模型,并对两种框架下主要采用的参数估计方法进行了总结。通过两个模拟研究,比较了SEM框架下GLSc和MGLSc估计方法与IRT框架下MML/EM估计方法的差异。研究结果表明：（1）三种方法中,GLSc得到参数估计的偏差最大,MGLSc和MML/EM估计方法相差不大;（2）随着样本量增大,各种项目参数估计的精度均提高;（3）项目因素载荷和难度估计的精度受测验长度的影响;（4）项目因素载荷和区分度估计的精度受总体因素载荷（区分度）高低的影响;（5）测验项目中阈值的分布会影响参数估计的精度,其中受影响最大的是项目区分度。（6）总体来看,SEM框架下的项目参数估计精度较IRT框架下项目参数估计的精度高。此外,文章还将两种方法在实际应用中应该注意的问题提供了一些建议。     

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.     

The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features

A method is discussed which extends principal components analysis to the situation where the variables may be measured at a variety of scale levels (nominal, ordinal or interval), and where they may be either continuous or discrete. There are no restrictions on the mix of measurement characteristics and there may be any pattern of missing observations. The method scales the observations on each variable within the restrictions imposed by the variable's measurement characteristics, so that the deviation from the principal components model for a specified number of components is minimized in the least squares sense. An alternating least squares algorithm is discussed. An illustrative example is given.Copies of this paper and of the associated PRINCIPALS program may be obtained by writing to Forrest W. Young, Psychometric Laboratory, Davie Hall 013-A, Chapel Hill, NC 27514.     

因子混合模型:潜在类别分析与因子分析的整合

因子混合模型(FMM)是考虑了群体潜在异质性后的因子分析模型,它将潜在类别分析(LCA)与传统的因子分析(FA)整合在同一框架内,既保留了两种分析技术的优点,同时又展现出独特优势。FMM的应用主要包括描述变量的潜在结构、对被试进行分组以及探测社会称许偏差等。我们建议分别采用FA、LCA与FMM三种模型拟合数据,参考拟合指数和模型可解释性选择最优模型。总结了FMM的分析步骤以及软件使用,并用于探讨大学生社会面子意识的测量模型。未来研究应关注FMM分析过程的简化,继续深化对拟合指数等方面的探讨。     

The analysis of group contingency data

The data analyses utilized in group contingency projects are reviewed. Previous studies are cited to emphasize advantages of nonconsolidated ("individual") over consolidated analyses. Several procedures are described that enable applied researchers to incorporate nonconsolidated data analyses in group contingency studies.     

Factor analysis and Psychometrika: Major developments

Papers on factor analysis appearing inPsychometrika reflect the initial efforts of the Thurstonians to reformulate psychology as a quantitative science. The Thurstonians' emphasis on the development of factor analysis as an exploratory methodology was not new with them but was taken from British statisticians and psychologists who preceded them, whose literature the Thurstonians otherwise tended to ignore. The Thurstonians' rejection of general factors and focus on rotation to simple structure reflected an attempt to avoid statistical artifact and to identify factors with psychological substance. Much of the literature on factor analysis inPsychometrika concerned solving technical problems in the exploratory factor analysis method. Factor analysis took a major shift in direction in the 1970's with the development of confirmatory methodologies, many of which now receive greater attention than the method of exploratory factor analysis, most of the problems of which are now resolved.