Identification with deficient rank loading matrices in confirmatory factor analysis: Multitrait-multimethod models

This paper presents some results on identification in multitrait-multimethod (MTMM) confirmatory factor analysis (CFA) models. Some MTMM models are not identified when the (factorial-patterned) loadings matrix is of deficient column rank. For at least one other MTMM model, identification does exist despite such deficiency. It is also shown that for some MTMM CFA models, Howe's (1955) conditions sufficient for rotational uniqueness can fail, yet the model may well be identified and rotationally unique. Implications of these results for CFA models in general are discussed.     

A multitrait-multimethod model with minimal assumptions

A new model of confirmatory factor analysis (CFA) for multitrait-multimethod (MTMM) data sets is presented. It is shown that this model can be defined by only three assumptions in the framework of classical psychometric test theory (CTT). All other properties of the model, particularly the uncorrelated-ness of the trait with the method factors are logical consequences of the definition of the model. In the model proposed there are as many trait factors as different traits considered, but the number of method factors is one fewer than the number of methods included in an MTMM study. The covariance structure implied by this model is derived, and it is shown that this model is identified even under conditions under which other CFA-MTMM models are not. The model is illustrated by two empirical applications. Furthermore, its advantages and limitations are discussed with respect to previously developed CFA models for MTMM data.     

Confirmatory Factor Analysis of Multitrait-Multimethod Data: The Construct Validation of Multidimensional Self-Concept Responses

ABSTRACT The multitrait-multimethod (MTMM) design is frequently used to test construct validity and is particularly appropriate for evaluating multidimensional instruments in personality research Despite its popularity there are important problems with both the traditional and confirmatory factor analysis (CFA) approaches to MTMM data In previous research (1989a, 1989b) I described a new CFA approach that I claimed to be relatively immune to many of these problems This approach is applied here to test the construct validity of preadolescent responses to three multidimensional self-concept instruments The substantive contribution of the present investigation is to demonstrate good support for the construct validity of two of the three instruments The methodological contributions are to further substantiate my claims about the new CFA approaches and to demonstrate this approach in a substantively meaningful context that has broad applicability to other personality research     

The use of configural frequency analysis for explorative data analysis

Configural frequency analysis (CFA) is a widely used method of explorative data analysis. It tries to detect patterns in the data that occur significantly more or significantly less often than expected by chance. Patterns which occur more often than expected by chance are called CFA types, while those which occur less often than expected by chance are called CFA antitypes. The patterns detected are used to generate knowledge about the mechanisms underlying the data. We investigate the ability of CFA to detect adequate types and antitypes in a number of simulation studies. The basic idea of these studies is to predefine sets of types and antitypes and a mechanism which uses them to create a simulated data set. This simulated data set is then analysed with CFA and the detected types and antitypes are compared to the predefined ones. The predefined types and antitypes together with the method to generate the data are called a data generation model. The results of the simulation studies show that CFA can be used in quite different research contexts to detect structural dependencies in observed data. In addition, we can learn from these simulation studies how much data is necessary to enable CFA to reconstruct the predefined types and antitypes with sufficient accuracy. For one of the data generation models investigated, implicitly underlying knowledge space theory, it was shown that zero‐order CFA can be used to reconstruct the predefined types (which can be interpreted in this context as knowledge states) with sufficient accuracy. Theoretical considerations show that first‐order CFA cannot be used for this data generation model. Thus, it is wrong to consider first‐order CFA, as is done in many publications, as the standard or even only method of CFA.     

Bayesian analysis of longitudinal multitrait–multimethod data with ordinal response variables

A new multilevel latent state graded response model for longitudinal multitrait–multimethod (MTMM) measurement designs combining structurally different and interchangeable methods is proposed. The model allows researchers to examine construct validity over time and to study the change and stability of constructs and method effects based on ordinal response variables. We show how Bayesian estimation techniques can address a number of important issues that typically arise in longitudinal multilevel MTMM studies and facilitates the estimation of the model presented. Estimation accuracy and the impact of between‐ and within‐level sample sizes as well as different prior specifications on parameter recovery were investigated in a Monte Carlo simulation study. Findings indicate that the parameters of the model presented can be accurately estimated with Bayesian estimation methods in the case of low convergent validity with as few as 250 clusters and more than two observations within each cluster. The model was applied to well‐being data from a longitudinal MTMM study, assessing the change and stability of life satisfaction and subjective happiness in young adults after high‐school graduation. Guidelines for empirical applications are provided and advantages and limitations of a Bayesian approach to estimating longitudinal multilevel MTMM models are discussed.     

What Carries a Mediation Process? Configural Analysis of Mediation

Mediation is a process that links a predictor and a criterion via a mediator variable. Mediation can be full or partial. This well-established definition operates at the level of variables even if they are categorical. In this article, two new approaches to the analysis of mediation are proposed. Both of these approaches focus on the analysis of categorical variables. The first involves mediation analysis at the level of configurations instead of variables. Thus, mediation can be incorporated into the arsenal of methods of analysis for person-oriented research. Second, it is proposed that Configural Frequency Analysis (CFA) can be used for both exploration and confirmation of mediation relationships among categorical variables. The implications of using CFA are first that mediation hypotheses can be tested at the level of individual configurations instead of variables. Second, this approach leaves the door open for different types of mediation processes to exist within the same set. Using a data example, it is illustrated that aggregate-level analysis can overlook mediation processes that operate at the level of individual configurations.     

Separating trait effects from trait-specific method effects in multitrait-multimethod models: a multiple-indicator CT-C(M-1) model

An overview of several models of confirmatory factor analysis for analyzing multitrait-multimethod (MTMM) data and a discussion of their advantages and limitations are provided. A new class of multi-indicator MTMM models combines several strengths and avoids a number of serious shortcomings inherent in previously developed MTMM models. The new models enable researchers to specify and to test trait-specific-method effects. The trait and method concepts composing these models are explained in detail and are contrasted with those of previously developed MTMM models for multiple indicators. The definitions of the models are explained step by step, and a practical empirical application of the models to the measurement of 3 traits x 3 methods is used to demonstrate their advantages and limitations.     

Controlling Correlational Bias via Confirmatory Factor Analysis of MTMM Data

Confirmatory factor analysis of multitrait-multimethod (MTMM) data has proven to be a useful tool for assessing convergent and discriminant validity. However, researchers have not made full use of the results of MTMM analyses in examining the relationship between MTMM factors and variables outside the MTMM. Often, researchers simply average the various measures of each trait. Alternatively, they estimate LISREL MTMM models, but estimate only relationships between MTMM traits and the outside variables. In the present article, we show that these two approaches to analyzing data outside the MTMM produce equally highly biased parameter estimates when the actual correlations between MTMM method factors and the outside variables are substantial. An algebraic explanation and a simulated data illustration are given for the bias due to misspecification. Also, the problem is illustrated with a brief empirical example. Implications for applied research are discussed.     

Generalized structured component analysis

We propose an alternative method to partial least squares for path analysis with components, called generalized structured component analysis. The proposed method replaces factors by exact linear combinations of observed variables. It employs a well-defined least squares criterion to estimate model parameters. As a result, the proposed method avoids the principal limitation of partial least squares (i.e., the lack of a global optimization procedure) while fully retaining all the advantages of partial least squares (e.g., less restricted distributional assumptions and no improper solutions). The method is also versatile enough to capture complex relationships among variables, including higher-order components and multi-group comparisons. A straightforward estimation algorithm is developed to minimize the criterion.The work reported in this paper was supported by Grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the second author. We wish to thank Richard Bagozzi for permitting us to use his organizational identification data and Wynne Chin for providing PLS-Graph 3.0.     

Confirmatory Measurement Model Comparisons Using Latent Means

Confirmatory factor analysis (CFA) is often used to verify measurement models derived from classical test theory: the parallel, tau-equivalent, and congeneric test models. In this application, CFA is traditionally applied to the observed covariance or correlation matrix, ignoring the observed mean structure. But CFA is easily extended to allow nonzero observed and latent means. The use of CFA with nonzero latent means in testing six measurement models derived from classical test theory is discussed. Three of these models have not been addressed previously in the context of CFA. The implications of the six models for observed mean and covariance structures are fully described. Three examples of the use of CFA in testing these models are presented. Some advantages and limitations in using CFA with nonzero latent means to verify classical measurement models are discussed.     

Combining Methodologies in the Construct Validation Process: An Empirical Illustration

Confirmatory Factor Analysis (CFA) has received considerable support as a methodology for assessing construct validity. As with other methodologies, however, numerous problems can be encountered when using CFA to assess construct validity. Given the limitations of the CFA, Multitrait-Multimethod (MTMM), and analysis of variance (ANOVA) methodologies, a set of guidelines was developed. The guidelines recommend that the characteristics of the data set be used to determine how the three alternative construct validation methodologies can be used in combination. Examples from the research literature are discussed in order to demonstrate the application of these guidelines.     

Temporal patterns of variable relationships in person-oriented research: longitudinal models of configural frequency analysis

This article reviews the premises of configural frequency analysis (CFA), including methods of choosing significance tests and base models, as well as protecting alpha, and discusses why CFA is a useful approach when conducting longitudinal person-oriented research. CFA operates at the manifest variable level. Longitudinal CFA seeks to identify those temporal patterns that stand out as more frequent (CFA types) or less frequent (CFA antitypes) than expected with reference to a base model. A base model that has been used frequently in CFA applications, prediction CFA, and a new base model, auto-association CFA, are discussed for analysis of cross-classifications of longitudinal data. The former base model takes the associations among predictors and among criteria into account. The latter takes the auto-associations among repeatedly observed variables into account. Application examples of each are given using data from a longitudinal study of domestic violence. It is demonstrated that CFA results are not redundant with results from log-linear modeling or multinomial regression and that, of these approaches, CFA shows particular utility when conducting person-oriented research.     

Structural equation modeling of multitrait-multimethod data: different models for different types of methods

The question as to which structural equation model should be selected when multitrait-multimethod (MTMM) data are analyzed is of interest to many researchers. In the past, attempts to find a well-fitting model have often been data-driven and highly arbitrary. In the present article, the authors argue that the measurement design (type of methods used) should guide the choice of the statistical model to analyze the data. In this respect, the authors distinguish between (a) interchangeable methods, (b) structurally different methods, and (c) the combination of both kinds of methods. The authors present an appropriate model for each type of method. All models allow separating measurement error from trait influences and trait-specific method effects. With respect to interchangeable methods, a multilevel confirmatory factor model is presented. For structurally different methods, the correlated trait-correlated (method-1) model is recommended. Finally, the authors demonstrate how to appropriately analyze data from MTMM designs that simultaneously use interchangeable and structurally different methods. All models are applied to empirical data to illustrate their proper use. Some implications and guidelines for modeling MTMM data are discussed.     

On various causes of improper solutions in maximum likelihood factor analysis

In the applications of maximum likelihood factor analysis the occurrence of boundary minima instead of proper minima is no exception at all. In the past the causes of such improper solutions could not be detected. This was impossible because the matrices containing the parameters of the factor analysis model were kept positive definite. By dropping these constraints, it becomes possible to distinguish between the different causes of improper solutions. In this paper some of the most important causes are discussed and illustrated by means of artificial and empirical data.The author is indebted to H. J. Prins for stimulating and encouraging discussions.     

Modeling Common Traits and Method Effects in Multitrait-Multimethod Analysis

Method effects often occur when constructs are measured by different methods. In traditional multitrait-multimethod (MTMM) models method effects are regarded as residuals, which implies a mean method effect of zero and no correlation between trait and method effects. Furthermore, in some recent MTMM models, traits are modeled to be specific to a certain method. However, often we are not interested in a method-specific trait but in a trait that is common to all methods. Here we present the Method Effect model with common trait factors, which allows modeling “common” trait factors and method factors that represent method “effects” rather than residuals. The common trait factors are defined as the mean of the true-score variables of all variables measuring the same trait and the method factors are defined as differences between true-score variables and means of true-score variables. Because the model allows estimating mean method effects, correlations between method factors, and correlations between trait and method factors, new research questions may be investigated. The application of the model is demonstrated by 2 examples studying the effect of negative, as compared with positive, item wording for the measurement of mood states.     

Dynamic analysis of multivariate panel data with non-linear transformations

Many models for multivariate data analysis can be seen as special cases of the linear dynamic or state space model. Contrary to the classical approach to linear dynamic systems analysis, in which high-dimensional exact solutions are sought, the model presented here is developed from a social science framework where low-dimensional approximate solutions are preferred. Borrowing concepts from the theory on mixture distributions, the linear dynamic model can be viewed as a multi-layered regression model, in which the output variables are imprecise manifestations of an unobserved continuous process. An additional layer of mixing makes it possible to incorporate non-normal as well as ordinal variables.Using the EM-algorithm, we find estimates of the unknown model parameters, simultaneously providing stability estimates. The model is very general and cannot be well estimated by other estimation methods. We illustrate the applicability of the obtained procedure through an example with generated data.     

A unified approach to multiple‐set canonical correlation analysis and principal components analysis

Multiple‐set canonical correlation analysis and principal components analysis are popular data reduction techniques in various fields, including psychology. Both techniques aim to extract a series of weighted composites or components of observed variables for the purpose of data reduction. However, their objectives of performing data reduction are different. Multiple‐set canonical correlation analysis focuses on describing the association among several sets of variables through data reduction, whereas principal components analysis concentrates on explaining the maximum variance of a single set of variables. In this paper, we provide a unified framework that combines these seemingly incompatible techniques. The proposed approach embraces the two techniques as special cases. More importantly, it permits a compromise between the techniques in yielding solutions. For instance, we may obtain components in such a way that they maximize the association among multiple data sets, while also accounting for the variance of each data set. We develop a single optimization function for parameter estimation, which is a weighted sum of two criteria for multiple‐set canonical correlation analysis and principal components analysis. We minimize this function analytically. We conduct simulation studies to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of functional neuroimaging data to illustrate its empirical usefulness.     

即时通讯软件使用动机的探讨

鉴于当前流行的即时通讯软件(instant messaging,IM)既广泛集成了互联网的功能和服务,又包含了其独特的功能和服务,笔者借鉴互联网使用动机的研究,提出了两个基于IM环境的用户使用动机的假设模型:其一是包含人际沟通、娱乐休闲、自我展示和信息获取动机的四因素模型;其二是包含人际沟通、娱乐休闲和信息获取动机的三因素模型。本研究以主流的大学生IM用户为对象,通过验证性因素分析的检验结果发现,大学生IM用户使用IM主要在于满足其人际沟通、娱乐休闲和信息获取三种动机。进一步的二阶验证性因素分析表明,首要动机是人际沟通,次要动机是娱乐休闲与信息获取。基于上述结果,笔者提出了关于IM产品设计的建议。     

Normative versus ipsative configural frequency analysis in personality research—their use discussed in a reanalysis of data on situation-bound anxiety

Configural frequency analysis (CFA) tests whether certain individual patterns in different variables are observed more frequently in a sample than expected by chance. In normative CFA, these patterns are derived from the subject's specific position in relation to sample characteristics such as the median or the mean. In ipsative CFA, patterns are defined within an individual reference system, e.g. relative to the subject's median of different variable scores. Normative CFA examines dimensionality of scales and is comparable to factor analysis in this respect. Ipsative CFA rather yields information about location of scores in different variables, in a similar way to ANOVA or Friedman testing. However, both normative and ipsative CFA may supply information not obtainable by means of the aforementioned methods. This is illustrated in a reanalysis of data in four scales of an anxiety inventory. © 1997 John Wiley & Sons, Ltd.     

Assessing local influence for specific restricted likelihood: Application to factor analysis

In restricted statistical models, since the first derivatives of the likelihood displacement are often nonzero, the commonly adopted formulation for local influence analysis is not appropriate. However, there are two kinds of model restrictions in which the first derivatives of the likelihood displacement are still zero. General formulas for assessing local influence under these restrictions are derived and applied to factor analysis as the usually used restriction in factor analysis satisfies the conditions. Various influence schemes are introduced and a comparison to the influence function approach is discussed. It is also shown that local influence for factor analysis is invariant to the scale of the data and is independent of the rotation of the factor loadings. The authors are most grateful to the referees, the Associate Editor, and the Editor for helpful suggestions for improving the clarity of the paper.