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.     

Efficient estimation in image factor analysis

The image factor analytic model (IFA), as related to Guttman's image theory, is considered as an alternative to the traditional factor analytic model (TFA). One advantage with IFA, as compared with TFA, is that more factors can be extracted without yielding a perfect fit to the observed data. Several theorems concerning the structural properties of IFA are proved and an iterative procedure for finding the maximum likelihood estimates of the parameters of the IFA-model is given. Substantial experience with this method verifies that Heywood cases never occur. Results of an artificial experiment suggest that IFA may be more factorially invariant than TFA under selection of tests from a large battery. The first part of this work was done at the University of Uppsala, Sweden and supported by the Swedish Council for Social Science Research. The second part was done at Educational Testing Service and supported by a grant (NSF-GB-1985) from the National Science Foundation to Educational Testing Service. The author is indebted to Mr. G. Gruvaeus, who wrote many of the computer programs, checked the mathematical derivations and gave other invaluable assistance throughout the work.     

Exploratory Factor Analysis With Small Sample Sizes

Exploratory factor analysis (EFA) is generally regarded as a technique for large sample sizes (N), with N = 50 as a reasonable absolute minimum. This study offers a comprehensive overview of the conditions in which EFA can yield good quality results for N below 50. Simulations were carried out to estimate the minimum required N for different levels of loadings (λ), number of factors (f), and number of variables (p) and to examine the extent to which a small N solution can sustain the presence of small distortions such as interfactor correlations, model error, secondary loadings, unequal loadings, and unequal p/f. Factor recovery was assessed in terms of pattern congruence coefficients, factor score correlations, Heywood cases, and the gap size between eigenvalues. A subsampling study was also conducted on a psychological dataset of individuals who filled in a Big Five Inventory via the Internet. Results showed that when data are well conditioned (i.e., high λ, low f, high p), EFA can yield reliable results for N well below 50, even in the presence of small distortions. Such conditions may be uncommon but should certainly not be ruled out in behavioral research data.     

Awareness and Metacognition

Kentridge and Heywood (this issue) extend the concept of metacognition to include unconscious processes. We acknowledge the possible contribution of unconscious processes, but favor a central role of awareness in metacognition. We welcome Shimamura's (this issue) extension of the concept of metacognitive regulation to include aspects of working memory, and its relation to executive attention.     

Temporal properties of human information processing: Tests of discrete versus continuous models

Cognitive psychologists have characterized the temporal properties of human information processing in terms of discrete and continuous models. Discrete models postulate that component mental processes transmit a finite number of intermittent outputs (quanta) of information over time, whereas continuous models postulate that information is transmitted in a gradual fashion. These postulates may be tested by using an adaptive response-priming procedure and analysis of reaction-time mixture distributions. Three experiments based on this procedure and analysis are reported. The experiments involved varying the temporal interval between the onsets of a prime stimulus and a subsequent test stimulus to which a response had to be made. Reaction time was measured as a function of the duration of the priming interval and the type of prime stimulus. Discrete models predict that manipulations of the priming interval should yield a family of reaction-time mixture distributions formed from a finite number of underlying basis distributions, corresponding to distinct preparatory states. Continuous models make a different prediction. Goodness-of-fit tests between these predictions and the data supported either the discrete or the continuous models, depending on the nature of the stimuli and responses being used. When there were only two alternative responses and the stimulus-response mapping was a compatible one, discrete models with two or three states of preparation fit the results best. For larger response sets with an incompatible stimulus-response mapping, a continuous model fit some of the data better. These results are relevant to the interpretation of reaction-time data in a variety of contexts and to the analysis of speed-accuracy trade-offs in mental processes.     

Derived relational responding: a comparison of match-to-sample and the relational completion procedure

Previous research suggests that the Relational Completion Procedure may be an effective alternative procedure for studying derived relational responding. However, the parameters that make it effective, relative to traditional match-to-sample, remain to be determined. The present experiment compared the Relational Completion Procedure and match-to-sample protocols for training and testing Same and Opposite derived stimulus relations. Trials to criterion and overall pass rate (i.e., yield) in both procedures were compared across three variables: presence versus absence of a confirmatory response requirement, three versus five comparison stimuli, and top-to-bottom versus left-to-right presentation format. Findings demonstrated a facilitative effect of the confirmatory response requirement in both procedures. Training trials to criterion were nominally but not significantly lower during the nonarbitrary training phase in the Relational Completion Procedure compared to match-to-sample, and the overall yield on the arbitrary relational test was greater in the former procedure compared to the latter. The present findings support the further development of the Relational Completion Procedure as an efficient alternative procedure for establishing Same and Opposite relations with adult humans, and with potential applicability to other types of derived relations.     

Warning Against Faking on a Personality Test: Effects on Applicant Reactions and Personality Test Scores

To reduce faking on personality tests, applicants may be warned that a social desirability scale is embedded in the test. Although this procedure has been shown to substantially reduce faking, there is no data that addresses how such a warning may influence applicant reactions toward the selection procedure or the relationships among personality constructs. Using an organizational justice framework, this study examines the effect of warning on procedural justice perceptions. Additionally, the extent to which warning changes the relationships among personality variables, socially desirable responding, and organizational justice variables, was explored. The results suggest that warning did not negatively affect test‐taker reactions. However, the relationships among the justice measures and the personality variables and socially desirable responding differed across the warned and unwarned groups. The organizational justice model fit best and there was less multicollinearity among the personality variables in the warned condition, compared to the unwarned condition. Thus, providing a warning appears to have positive consequences when using personality measures.     

Regression estimation and post-stratification in factor analysis

Regression estimation and poststratification are methods used in survey sampling to estimate a population mean, when additional information is available for some auxiliary variables. The extension of these methods to factor analysis is considered. These methods may be used either to improve the efficiency of estimation or to adjust for the effects of nonrandom selection. The estimation procedure may be formulated in a LISREL framework.The author is grateful to the referees for their comments.     

Achievement of specificational information usage with true and false feedback in learning a visual relative-mass discrimination task

Participants' usage of informational variables in learning visual relative-mass discrimination in collisions was tracked by means of PROBIT correlations. Four groups received feedback that was true or accorded with either of three nonspecificational cue variables. A majority in each group adopted the feedback, but several participants defied the false feedback. Unlike in previous research, the fit to data of the relative-mass invariant could not be bettered by post hoc linear combinations of the cues. Discriminability was lower in the use of the invariant. Analytic complexity was rejected as an explanation for discriminability differences. A "smart mechanism" for pickup of the relative-mass invariant was developed as an extension of G. Johansson's (1950/1994) vector model.     

Positive Definiteness via Off-diagonal Scaling of a Symmetric Indefinite Matrix

Indefinite symmetric matrices that are estimates of positive-definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based methods for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk.     

Convergence in mean square of factor predictors

Sufficient conditions for mean square convergence of factor predictors in common factor analysis are given by Guttman, by Williams, and by Schneeweiss and Mathes. These conditions do not hold for confirmatory factor analysis or when an error variance equals zero (Heywood cases). Two sufficient conditions are given for the three basic factor predictors and a predictor from rotated principal components analysis to converge to the factors of the model for confirmatory factor analysis, including Heywood cases. For certain model specifications the conditions are necessary. The conditions are sufficient for the existence of a unique true factor. A geometric interpretation is given for factor indeterminacy and mean square convergence of best linear factor prediction.     

Parameterizing inequality constraints on unique variances in linear structural models

Current computer programs for analyzing linear structural models will apparently handle only two types of constraints: fixed parameters, and equality of parameters. An important constraint not handled is inequality; this is particularly crucial for preventing negative variance estimates. In this paper, a method is described for imposing several kinds of inequality constraints in models, without the necessity for having computer programs which explicitly allow such constraints. The examples discussed include the prevention of Heywood cases, extension to inequalities of parameters to be greater than a specified value, and imposing ordered inequalities. Work on this project was aided by the City University of New York—Professional Staff Congress Research Award Program Grant Number 13631.     

Reconsidering the Factor Structure of the Beavers Interactional Scales: Do External Ratings Mirror Self-Reports?

Factor analytic studies indicate that family relationships assessed by self-reports may be most clearly and parsimoniously differentiated along three dimensions, namely, Affective Expression, Behavioral Control, and Shared Activity. The present study explored the question of whether this were true of family ratings by outside parties. Confirmatory factor analysis was used with three data sets (n = 97, 91, and 95; N = 283) employing the Beavers Interactional Scales of family functioning. Multicollinearity of the observed variables was a pervasive problem. When the redundancy of the observed variables was pared down so that there were two observed variables for each latent factor, these three factors in oblique relationship provided a good fit to the observed data.     

Constructing Natural Extensions of Propositional Logics

The proofs of some results of abstract algebraic logic, in particular of the transfer principle of Czelakowski, assume the existence of so-called natural extensions of a logic by a set of new variables. Various constructions of natural extensions, claimed to be equivalent, may be found in the literature. In particular, these include a syntactic construction due to Shoesmith and Smiley and a related construction due to ?o? and Suszko. However, it was recently observed by Cintula and Noguera that both of these constructions fail in the sense that they do not necessarily yield a logic. Here we show that whenever the ?o?–Suszko construction yields a logic, so does the Shoesmith–Smiley construction, but not vice versa. We also describe the smallest and the largest conservative extension of a logic by a set of new variables and show that contrary to some previous claims in the literature, a logic of cardinality \({\kappa}\) may have more than one conservative extension of cardinality \({\kappa}\) by a set of new variables. In this connection we then correct a mistake in the formulation of a theorem of Dellunde and Jansana.     

Some contributions to efficient statistics in structural models: Specification and estimation of moment structures

Current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments (e.g., means and covariances), and to distribution theory based on multivariate normality. In psychometrics the multinormality assumption is often incorrect, so that statistical tests on parameters, or model goodness of fit, will frequently be incorrect as well. It is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary. Structural representations are developed for generalizations of the Bentler-Weeks, Jöreskog-Keesling-Wiley, and factor analytic models. Some asymptotically distribution-free efficient estimators for such arbitrary structural models are developed. Limited information estimators are obtained as well. The special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail. The argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory, which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds. Many open research areas are described.     

Core self-evaluation and life satisfaction: The person-environment fit perspective

This study is the first to test the mediating roles of three person-environment (P-E) fit variables (i.e., person-major fit, person-university fit, and person-society fit), in the relationship between core self-evaluation and life satisfaction. Data from 794 Chinese university students were analyzed using structural equation modeling. Results supported a partial mediating effect for all three P-E fit variables on the relationship between core self-evaluation and life satisfaction. Person-major fit and person-university fit partially mediated the relationship between core self-evaluation and person-society fit. Additionally, person-society fit partially mediated the relationships of life satisfaction with person-major fit and person-university fit. The findings reveal the complicated mechanisms underlying the influence that core self-evaluation has on life satisfaction and suggest important implications for future research.     

The Effectiveness of Different Recruitment Sources: A Mediating Variable Analysis

Previous research suggests that one or more of three mediating variables account for the relationships between recruiting sources and applicant effectiveness. This paper offers a critical test of the three mediating variables: demographic characteristics, realistic expectations, and perceived person-job fit. Using a sample of 242 newly hired marketing representatives, the study found that employees recruited through college placement offices had better initial levels of performance than did employees recruited through newspaper advertisements. While all of the proposed mediating variables were associated with some recruitment sources, none of them mediated the effects of recruitment sources on performance or turnover. This paper suggests that there may be complex contingency relationships between recruitment sources and employee performance and turnover.     

Workplace support, discrimination, and person-organization fit: tests of the theory of work adjustment with LGB individuals

The present study explored the links of 2 workplace contextual variables--perceptions of workplace heterosexist discrimination and lesbian, gay, and bisexual (LGB)-supportive climates--with job satisfaction and turnover intentions in a sample of LGB employees. An extension of the theory of work adjustment (TWA) was used as the conceptual framework for the study; as such, perceived person-organization (P-O) fit was tested as a mediator of the relations between the workplace contextual variables and job outcomes. Data were analyzed from 326 LGB employees. Zero-order correlations indicated that perceptions of workplace heterosexist discrimination and LGB-supportive climates were correlated in expected directions with P-O fit, job satisfaction, and turnover intentions. Structural equation modeling (SEM) was used to compare multiple alternative measurement models evaluating the discriminant validity of the 2 workplace contextual variables relative to one another, and the 3 TWA job variables relative to one another; SEM was also used to test the hypothesized mediation model. Comparisons of multiple alternative measurement models supported the construct distinctiveness of the variables of interest. The test of the hypothesized structural model revealed that only LGB-supportive climates (and not workplace heterosexist discrimination) had a unique direct positive link with P-O fit and, through the mediating role of P-O fit, had significant indirect positive and negative relations with job satisfaction and turnover intentions, respectively. Moreover, P-O fit had a significant indirect negative link with turnover intentions through job satisfaction.     

Necessary Conditions for Mean Square Convergence of the Best Linear Factor Predictor

Several sufficient conditions are available for mean square convergence of factor predictors. A necessary and sufficient condition is given in the Heywood case with respect to (confirmatory) factor analysis. This condition generalizes that of Krijnen (2006) and performs better than a signal-to-noise type of condition (Schneeweiss & Mathes, 1995). The author is obliged to the reviewers for their stimulating remarks. Requests for reprints should be sent to Department of Psychology, Psychological Methods, University of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands.     

Stepwise variable selection in factor analysis

It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in a questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection.In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain numberp of observed variables (thep-variable model is labeled the current model), simple formulas for predicted fit measures such as chi-square, GFI, CFI, IFI and RMSEA, developed in the field of the structural equation modeling, are provided for all models obtained by adding an external variable (so that the number of variables isp + 1) and for those by deleting an internal variable (so that the number isp – 1), provided that the number of factors is held constant.A programSEFA (Stepwise variable selection in Exploratory Factor Analysis) is developed to actually obtain a list of the fit measures for all such models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general.The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.The URL of the programSEFA is http://koko15.hus.osaka-u.ac.jp/~harada/factor/stepwise/.