Structural Modeling of Measurement Error in Generalized Linear Models with Rasch Measures as Covariates

This paper proposes a structural analysis for generalized linear models when some explanatory variables are measured with error and the measurement error variance is a function of the true variables. The focus is on latent variables investigated on the basis of questionnaires and estimated using item response theory models. Latent variable estimates are then treated as observed measures of the true variables. This leads to a two-stage estimation procedure which constitutes an alternative to a joint model for the outcome variable and the responses given to the questionnaire. Simulation studies explore the effect of ignoring the true error structure and the performance of the proposed method. Two illustrative examples concern achievement data of university students. Particular attention is given to the Rasch model.     

Instrumental variables in factor analysis

The factor analysis model is rewritten as a system of linear structural relations with errors in variables. The method of instrumental variables is applied to this revised form of the model to obtain estimates of the factor loading matrix. The relation between this method and interbattery analysis, proportional profile analysis, and canonical factor analysis is pointed out. In addition, an estimation procedure based on replicated sampling different from proportional profile analysis is given.     

Discriminant Analysis Using Mixed Continuous,Dichotomous, and Ordered Categorical Variables

This article proposes an intuitive approach for predictive discriminant analysis with mixed continuous, dichotomous, and ordered categorical variables that are defined via an underlying multivariate normal distribution with a threshold specification. The classification rule is based on the comparison of the observed data logarithm probability density functions. To reduce the computational burden, the analysis is conducted in the context of a confirmatory factor analysis model with independent error measurements. Identification of the dichotomous and ordered categorical variables is discussed. Results are obtained by implementations of a Monte Carlo expectation maximization (MCEM)algorithm and a path sampling procedure. Probabilities of misclassification are estimated via the idea of the “jackknife” method. A real example is given to illustrate the proposed method.     

SEM with Missing Data and Unknown Population Distributions Using Two-Stage ML: Theory and Its Application

This article provides the theory and application of the 2-stage maximum likelihood (ML) procedure for structural equation modeling (SEM) with missing data. The validity of this procedure does not require the assumption of a normally distributed population. When the population is normally distributed and all missing data are missing at random (MAR), the direct ML procedure is nearly optimal for SEM with missing data. When missing data mechanisms are unknown, including auxiliary variables in the analysis will make the missing data mechanism more likely to be MAR. It is much easier to include auxiliary variables in the 2-stage ML than in the direct ML. Based on most recent developments for missing data with an unknown population distribution, the article first provides the least technical material on why the normal distribution-based ML generates consistent parameter estimates when the missing data mechanism is MAR. The article also provides sufficient conditions for the 2-stage ML to be a valid statistical procedure in the general case. For the application of the 2-stage ML, an SAS IML program is given to perform the first-stage analysis and EQS codes are provided to perform the second-stage analysis. An example with open- and closed-book examination data is used to illustrate the application of the provided programs. One aim is for quantitative graduate students/applied psychometricians to understand the technical details for missing data analysis. Another aim is for applied researchers to use the method properly.     

A feasible method for standard errors of estimate in maximum likelihood factor analysis

A jackknife-like procedure is developed for producing standard errors of estimate in maximum likelihood factor analysis. Unlike earlier methods based on information theory, the procedure developed is computationally feasible on larger problems. Unlike earlier methods based on the jackknife, the present procedure is not plagued by the factor alignment problem, the Heywood case problem, or the necessity to jackknife by groups. Standard errors may be produced for rotated and unrotated loading estimates using either orthogonal or oblique rotation as well as for estimates of unique factor variances and common factor correlations. The total cost for larger problems is a small multiple of the square of the number of variables times the number of observations used in the analysis. Examples are given to demonstrate the feasibility of the method.The research done by R. I. Jennrich was supported in part by NSF Grant MCS 77-02121. The research done by D. B. Clarkson was supported in part by NSERC Grant A3109.     

A modified procedure for parallel analysis of ordered categorical data

Parallel analysis has been well documented to be an effective and accurate method for determining the number of factors to retain in exploratory factor analysis. The O'Connor (2000) procedure for parallel analysis has many benefits and is widely applied, yet it has a few shortcomings in dealing with missing data and ordinal variables. To address these technical issues, we adapted and modified the O'Connor procedure to provide an alternative method that better approximates the ordinal data by factoring in the frequency distributions of the variables (e.g., the number of response categories and the frequency of each response category per variable). The theoretical and practical differences between the modified procedure and the O'Connor procedure are discussed. The SAS syntax for implementing this modified procedure is also provided.     

Development of a microcomputer system for verbal interaction analysis

Verbal interaction analysis has been demonstrated to be a valuable procedure for research on temporal, noncontent variables in dyadic interviews. Temporal variables include durations of utterance, reaction time latency, initiative time latency, and overlap, which are recorded for both parties in the interview. Instruments for the measurement of these variables have been both complex and expensive. An Apple II microcomputer is programmed to record the four temporal variables in verbal interaction analysis, providing a portable, less expensive, and convenient instrument.     

A monotonically convergent algorithm for factals

Takane, Young, and de Leeuw proposed a procedure called FACTALS for the analysis of variables of mixed measurement levels (numerical, ordinal, or nominal). Mooijaart pointed out that their algorithm does not necessarily converge, and Nevels proposed a new algorithm for the case of nominal variables. In the present paper it is shown that Nevels' procedure is incorrect, and a new procedure for handling nominal variables is proposed. In addition, a procedure for handling ordinal variables is proposed. Using these results, a monotonically convergent algorithm is constructed for FACTALS of any mixture of variables.The authors are obliged to Jos ten Berge for stimulating comments on an earlier version of this paper. The research of H. A. L. Kiers has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. The research of Y. Takane has been supported by the Natural Sciences and Engineering Research Council of Canada, grant number A6394, and by the McGill-IBM Cooperative Grant.     

Tandem criteria for analytic rotation in factor analysis

Two related orthogonal analytic rotation criteria for factor analysis are proposed. Criterion I is based upon the principle that variables which appear on the same factor should be correlated. Criterion II is based upon the principle that variables which are uncorrelated should not appear on the same factor. The recommended procedure is to rotate first by criterion I, eliminate the minor factors, and then rerotate the remaining major factors by criterion II. An example is presented in which this procedure produced a rotational solution very close to expectations whereas a varimax solution exhibited certain distortions. A computer program is provided.     

Some Checking Procedures For Extension Analysis

The extension procedure in common factor analysis is a two-stage least squares estimation procedure which may yield very poor fit to the extension variables and may yield Heywood cases. The three residual matrices, and three measures of goodness of fit that arise from the procedure should be examined in most applications, and the possible presence of Heywood variables in the extension set should be considered.     

Maximizing the validity of a unit-weight composite as a function of relative component lengths with a fixed total testing time

An explicit solution is given to the problem of assigning relative lengths to the subtests of a test so as to maximize the correlation of the unit weight composite with a specified criterion when the total testing time is fixed. This solution is valid and unique whenever it specifies nonnegative times for all variables. A step-down procedure is suggested for cases in which some of the testing times are zero. This procedure does not necessarily provide an optimal allocation. However in examples studied it is found to provide near optimum results. Algorithms are also developed for the determination of the least total testing time required to attain specified multiple and composite correlations. A numerical example is given illustrating the use of the unit weight procedure in combination with the regression weight algorithm. Supported in part by the Personnel and Training Branch of the Office of Naval Research under Contrast Number 000-14-69C-0119, Melvin R. Novick, Principal Investigator. Reproduction, translation, publication, use and disposal in whole or in part by or for the United States Government is permitted.     

A non-parametric test of correlation using rank orders within subgroups

Kendall's rank order test for association between two variables is generalized to the case where the total sample is made up of several subgroups and the data on one or both variables consist of the rank order within each subgroup. The test involves no assumptions concerning scales of measurement, shapes of distributions, or relative level of excellence or amount of variability of the different subgroups. Two empirical examples indicate that the normal approximation to the exact test of significance can be considered adequate for most practical situations. Special consideration is given to the case of tied ranks. If ties occur in but one variable within any given subgroup, only a slight modification in procedure is needed. Extensive ties in both variables within subgroups lead to difficulties in determining the appropriate correction for continuity.     

Simulating Ordinal Data

The increasing use of ordinal variables in different fields has led to the introduction of new statistical methods for their analysis. The performance of these methods needs to be investigated under a number of experimental conditions. Procedures to simulate from ordinal variables are then required. In this article, we deal with simulation from multivariate ordinal random variables. We propose a new procedure for generating samples from ordinal random variables with a prespecified correlation matrix and marginal distributions. Its features are examined and compared with those of its main competitors. A software implementation in R is also provided along with examples of its application.     

Clustering binary data in the presence of masking variables

A number of important applications require the clustering of binary data sets. Traditional nonhierarchical cluster analysis techniques, such as the popular K-means algorithm, can often be successfully applied to these data sets. However, the presence of masking variables in a data set can impede the ability of the K-means algorithm to recover the true cluster structure. The author presents a heuristic procedure that selects an appropriate subset from among the set of all candidate clustering variables. Specifically, this procedure attempts to select only those variables that contribute to the definition of true cluster structure while eliminating variables that can hide (or mask) that true structure. Experimental testing of the proposed variable-selection procedure reveals that it is extremely successful at accomplishing this goal.     

Design and analysis of semi-controlled studies

Semi-controlled studies provide a hybrid approach in between controlled experiments and naturalistic driving studies. As in controlled experiments, the researcher can assign participants to groups, select the route and define the tasks, but the participants are given more freedom when it comes to if, when, where and how to perform the tasks. Increased flexibility makes it possible to investigate how drivers use tactical behaviour to accommodate task execution. The disadvantage is decreased control and more complicated analyses. The main objective of this paper is to discuss how to analyse data obtained in semi-controlled studies.The analysis of data from a semi-controlled study include three types of variables: (i) variables that describe the experimental design, (ii) variables that describe the tactical choices of the participants and (iii), operational variables such as speed, lateral position or glance behaviour. To analyse the three types of variables a two-step procedure is suggested. First, the tactical indicators are analysed with regard to the experimental design. Second, the operational indicators are analysed and the tactical indicators are used to divide participants into sub-populations. The semi-controlled design does not need any new statistical procedures to be developed. It is more important that the analysis conditions on the initial properties and not on structures that happen to occur during the experiment, like where the participant chose to do a certain task.We recommend to use the semi-controlled study method when investigating questions involving adaptive and compensatory behaviour on the tactical level. It is especially useful if causal relationships are of interest, if the data collection should be accelerated in comparison to naturalistic studies, and if certain geographical locations definitely should be included.     

Orthogonal inter-battery factor analysis

It is the purpose of this paper to present a method of analysis for obtaining (i) inter-battery factors and (ii) battery specific factors for two sets of tests when the complete correlation matrix including communalities is given. In particular, the procedure amounts to constructing an orthogonal transformation such that its application to an orthogonal factor solution of the combined sets of tests results in a factor matrix of a certain desired form. The factors isolated are orthogonal but may be subjected to any suitable final rotation, provided the above classification of factors into (i) and (ii) is preserved. The general coordinate-free solution of the problem is obtained with the help of methods pertaining to the theory of linear spaces. The actual numerical analysis determined by the coordinate-free solution turns out to be a generalization of the formalism of canonical correlation analysis for two sets of variables. A numerical example is provided.This investigation has been supported by the U.S. Office of Naval Research under Contract Nonr-2752(00).     

Two-step weighted least squares factor analysis of dichotomized variables

A two-step weighted least squares estimator for multiple factor analysis of dichotomized variables is discussed. The estimator is based on the first and second order joint probabilities. Asymptotic standard errors and a model test are obtained by applying the Jackknife procedure.     

Testing Specific Hypotheses Concerning Latent Group Differences in Multi-group Covariance Structure Analysis with Structured Means

This article concerns multi-group covariance structure analysis with structured means. The traditional latent selection model is formulated as a special case of phenotypic selection, that is, selection based not on latent variables, but on observed variables. This formulation has the advantage that it enables one to test very specific hypotheses concerning selection on latent variables. Illustrations are given using simulated and real data.     

Factor Analysis of Ordinal Variables: A Comparison of Three Approaches

Theory and methodology for exploratory factor analysis have been well developed for continuous variables. In practice, observed or measured variables are often ordinal. However, ordinality is most often ignored and numbers such as 1, 2, 3, 4, representing ordered categories, are treated as numbers having metric properties, a procedure which is incorrect in several ways. In this article we describe four approaches to factor analysis of ordinal variables which take proper account of ordinality and compare three of them with respect to parameter estimates and fit. The comparison is made both in terms of their relative methodological advantages and in terms of an empirical data example and two generated data examples. In particular, we discuss the issue of how to test the model and to measure model fit.