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/.     

General theory and methods for matric factoring

Methods are developed for factoring an arbitrary rectangular matrixS of rankr into the formFP, whereF hasr columns andP hasr rows. For the statistical problem of factor analysis,S may be the score matrix of a population of individuals on a battery of tests. ThenF is a matrix of factor loadings,P is a matrix of factor scores, andr is the number of factor variates. (As in current procedures, there remains a subsequent problem of rotation of axes and interpretation of factors, which is not discussed here.) Methods are also developed for factoring an arbitrary Gramian matrixG of rankr into the formFF, whereF hasr columns andF denotesF transposed. For the statistical problem of factor analysis,G may be the matrix of intercorrelations,R, of a battery of tests, with unity, communalities, or other parameters in the principal diagonal.R is proportional toSS, and it is shown thatS can be factored by factoringR. This may usually be the most economical procedure in practice; it should not be overlooked, however, thatS can be factored directly. The general methods build up anF (andP) in as many stages as desired; as many factors as may be deemed computationally practical can be extracted at a time. Perhaps it will usually be found convenient to extract not more than three factors at a time. Current procedures, like the centroid and principal axes, are special cases of a general method presented here for extracting one factor at a time.     

Predictors of university students’ willingness in the USA to use clergy as sources of skilled help

This exploratory study investigated U.S. university students’ perceptions (N = 186) and the predictor variables associated with their willingness to use clergy as a source of help. In the final regression analysis using the predicted variable of willingness to use clergy as a source of help (R = 0.816, R ²= 0.665, Adjusted R ²= 0.650), there were seven significant predictor variables: (a) trust of clergy, (b) empathic ability of clergy, (c) having previously sought help from clergy, (d) respondents’ dominant/minority cultural identification, (e) attendance at religious services at least once a year, (f) believing that spiritual counselling is as effective as psychotherapy, and (g) receiving religious/spiritual education as a child. An eighth variable was retained in the final regression model because of its proximity to significance (Friendships between clergy and people, p = 0.051). Accusations against clergy and the belief that clergy are held to a higher standard did not predict willingness to use clergy as a source of help. A general linear model (F = 125.696, df = 10, p < 0.001) revealed that those who self-identified with Protestant Christianity, Catholic/Orthodox Christianity, and Judaism were more likely to consider seeking help from clergy than those who self-identified with another religious tradition (Hinduism, Islam, or Buddhism), or adherence to spiritual not religious belief. Further, those who self-identified as Jewish or Christian were also more like to perceive clergy as trustworthy and empathic. Finally, African American/Caribbean Black respondents were more likely than either Latino/Latino American respondents or European American respondents to seek help from clergy, to perceive clergy as empathic, and to believe that spiritual counselling is as effective as psychotherapy (F = 1495, df = 12, p < 0.001).     

Can Counterfactuals Really Be about Possible Worlds?

The standard view about counterfactuals is that a counterfactual (A > C) is true if and only if the A‐worlds most similar to the actual world @ are C‐worlds. I argue that the worlds conception of counterfactuals is wrong. I assume that counterfactuals have non‐trivial truth‐values under physical determinism. I show that the possible‐worlds approach cannot explain many embeddings of the form (P > (Q > R)), which intuitively are perfectly assertable, and which must be true if the contingent falsity of (Q > R) is to be explained. If (P > (Q > R)) has a backtracking reading then the contingent facts that (Q > R) needs to be true in the closest P‐worlds are absent. If (P > (Q > R)) has a forwardtracking reading, then the laws required by (Q > R) to be true in the closest P‐worlds will be absent, because they are violated in those worlds. Solutions like lossy laws or denial of embedding won't work. The only approach to counterfactuals that explains the embedding is a pragmatic metalinguistic approach in which the whole idea that counterfactuals are about a modal reality, be it abstract or concrete, is given up.     

Adjustment for the Total Number of Responses in Calculating the Rorschach Apperception Type

In order to adjust for total responses (R) on the Rorschach, apperception (location) scores arc expressed as percentages of R. An alternative procedure is recommended in which adjustments are the differences between obtained scores and those predicted from the regression of the scores on R. Standardization Ss were 200 male and female normals, ages 20 to 40. Percentages of these Ss falling within certain ranges of adjusted scores are given. Caution is recommended in making inferences about apperception types obtained by extrapolating beyond the range of R in the standardization group. Preliminary results obtained by applying these adjustments to disturbed adult and children's groups suggest that the distributions of adjusted scores tend to approximate the score distribution of the standardization group.     

A criterion for selecting variables in a regression analysis

Methods are given for deciding whether to use some or no predictor variables in a regression analysis. Previously obtained results on the more general problem, whether to usek ork — r predictor variables are reviewed with emphasis on applications.     

Self-Identity and the Theory of Planned Behavior: A Useful Addition or an Unhelpful Artifice?1

Among the variables proposed as useful additions to the structure of the theory of planned behavior (Ajzen, 1991), there is evidence for the predictive usefulness of a measure of self-identity. In the study reported here, members of the general population in the United Kingdom (n= 235) completed a questionnaire addressing the consumption of a diet low in animal fats. The questionnaire was structured in terms of the TPB and contained additional variables that previous research has indicated to be useful predictors of intentions and behavior. A multiple regression of intentions (R²= .80) showed self-identification as a health-conscious consumer to have a predictive effect independent of the effects of these other variables. This effect was also found in replication studies.     

Predicting Occupational Status for Non-College-Bound Males

In a follow-up of selected 1957 Wisconsin male high school graduates (N = 2,230), a prediction of occupational status was attempted. Of the 17 independent variables used, 3 emerged as highly significant and valid: post-high school education (p < .001), community of residence (p < .001), and high school grade-point average (p < .01). Two multiple coefficients were obtained (R = .32, F = 42.43, p < .001, and R = .26, F = 24.97, p < .001). Results suggest that family background factors and aspiration levels are not as important in predicting occupational status as are further education and training, high school achievement, and community of orientation in a highly industrialized urban technocracy.     

A simplified method for approximating multiple regression coefficients

A method of exhaustion has been described for calculating regression coefficients. This method dispenses with the solution of simultaneous equations but utilizes a process of successive extraction in obtaining's, where each successive is maximized. This procedure permits the worker to discard as he goes along those weights which are deemed unsatisfactory for purposes of prediction. The coefficients and theR in a problem involving a criterion and six independent variables were calculated in sixty minutes. TheR's obtained by this method are smaller than those yielded by the Doolittle technique, but in problems which have been considered this discrepancy has not exceeded .05.     

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.     

A model and algorithm for multidimensional scaling with external constraints on the distances

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configurationX relative to the proximity dataP and the loss ofX relative to a pseudo-data matrixR. The matrixR is set up such that the side constraints to be imposed onX's distances are expressed by the relations amongR's numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solutionX. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matricesP andR. Further generalizations are introduced by substitutingR by a set ofR matrices,R _i ,i=1, ...r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.     

The adequacy of the theories of deductive systems with respect to sentential calculi

Summary The sentential calculiR, under discussion, are axiomatizable and implication is among their primitive terms. The modus ponens and the rule of substitution are their primitive rules. ByS _r is denoted the set of sentences obtained from the formulae of the calculusR by substituting sentences of a given language for all variables. The variablesx, y, z ... represent the elements of the setS _r , the variablesX, Y, Z ... represent the subsets ofS _R . The formulacxy designates an implication withx as its antecedent andy as its consequent,cxy is always an element ofS _R δ(X) means, thatX is closed with respect to the modus ponens rule.A _R designates the class of allS _R -substitutions of the axioms of the sentential calculusR.     

A dynamic factor model for the analysis of multivariate time series

As a method to ascertain the structure of intra-individual variation,P-technique has met difficulties in the handling of a lagged covariance structure. A new statistical technique, coined dynamic factor analysis, is proposed, which accounts for the entire lagged covariance function of an arbitrary second order stationary time series. Moreover, dynamic factor analysis is shown to be applicable to a relatively short stretch of observations and therefore is considered worthwhile for psychological research. At several places the argumentation is clarified through the use of examples.I would like to thank WM. van der Molen, G. J. Mellenbergh and L. H. M. Oppenheimer, who provided valuable ideas that led to this formulation.     

Church Attendance,Religious Activities,and the Life Satisfaction of Older Adults in Middletown,U.S.A.

ABSTRACT

The central focus of this study is to examine the patterns and relationships which may or may not exist between religious attendance and activity, sex, health, income, and age with regard to the sense of subjective well-being as expressed by older Middletowners. Based on a random sample of 400 persons, 60 years of age and older, a multiple regression analysis reveals that three variables account for the vast majority of variance in life satisfaction. Those variables are: SUBJECTIVE HEALTH STATUS, SATISFACTION WITH INCOME, and CHURCH ATTENDANCE. For those three variables R = .4780; R² = .2284. When all seven independent variables were included R = .4851; R² = .2351. In addition, further analysis was undertaken to examine the interrelationships between the seven independent variables.     

More than g: Evidence for the Incremental Validity of Performance-Based Assessments for Predicting Training Performance

Past research has consistently shown that tests measuring specific cognitive abilities provide little if any incremental validity over tests of general mental ability when predicting performance on the job. In this study, we suggest that the seeming lack of incremental validity may have been due to the type of content that has traditionally been assessed. Therefore, we hypothesised that incremental validity can be obtained using specific cognitive abilities that are less highly correlated with g and are matched to the tasks performed on the job. To test this, we examined a recently developed performance-based measure that assesses a number of cognitive abilities related to training performance. In a sample of 310 US Navy student pilots, results indicated that performance-based scores added sizeable incremental validity to a measure of g. The significant increases in R² ranged from .08 to .10 across criteria. Similar results were obtained after correcting correlations for range restriction, though the magnitude of incremental validity was slightly smaller (ΔR² ranged from .05 to .07).     

A Logic for Multiple-source Approximation Systems with Distributed Knowledge Base

The theory of rough sets starts with the notion of an approximation space, which is a pair (U,R), U being the domain of discourse, and R an equivalence relation on U. R is taken to represent the knowledge base of an agent, and the induced partition reflects a granularity of U that is the result of a lack of complete information about the objects in U. The focus then is on approximations of concepts on the domain, in the context of the granularity. The present article studies the theory in the situation where information is obtained from different sources. The notion of approximation space is extended to define a multiple-source approximation system with distributed knowledge base, which is a tuple (U,R_P)_{Pßf N}(U,R_P)_{P\ss_f N}, where N is a set of sources and P ranges over all finite subsets of N. Each R _P is an equivalence relation on U satisfying some additional conditions, representing the knowledge base of the group P of sources. Thus each finite group of sources and hence individual source perceives the same domain differently (depending on what information the group/individual source has about the domain), and the same concept may then have approximations that differ with the groups. In order to express the notions and properties related with rough set theory in this multiple-source situation, a quantified modal logic LMSAS ^D is proposed. In LMSAS ^D , quantification ranges over modalities, making it different from modal predicate logic and modal logic with propositional quantifiers. Some fragments of LMSAS ^D are discussed and it is shown that the modal system KTB is embedded in LMSAS ^D . The epistemic logic S5^D_nS5^D_n is also embedded in LMSAS ^D , and cannot replace the latter to serve our purpose. The relationship of LMSAS ^D with first and second-order logics is presented. Issues of expressibility, axiomatization and decidability are addressed.     

Still no demonstration that g is not unitary: Further comment on Kranzler and Jensen

Because they employed a statistical procedure that was inappropriate for the purpose, Kranzler and Jensen (1991) have not demonstrated that a factor of general intelligence, g, depends on a number of independent processes. It is more parsimonious to interpret their results in terms of a g that is unitary. A factorial reanalysis of their data, including some that they did not present (cross-correlations between psychometric variables and variables from elementary cognitive tasks, or ECTs), shows that many ECT variables, especially those reflecting decision time rather than movement time, have substantial loadings on a second-order general factor. This factor is orthogonal to a second-order factor representing general speed of movement from a “home button” to the response button in ECTs. Possible implications for interpreting g in terms of information-processing efficiency are discussed.     

Inspection time and intelligence in 7-yr-old children: A follow-up

Remeasures of inspection time (IT) were obtained from 30 of the 47 children reported in Nettelbeck and Young (Personality and Individual Differences, 10, 605–614, 1989). Ages ranged from 7 yr–0 months to 7–11 and Full Scale IQ (WISC-R) ranged from 93 to 142 (mean = 116.5 SD = 12.2). Correlations between IT and the WISC-R subtests, scales and factors used in the earlier study were generally statistically significant: and similar to those found earlier. The correlation between IT and Full Scale IQ was — 0.49, with those children whose IQs were below the median score of 116 again contributing more to this outcome. Similarly, the IT-IQ correlation was higher among children showing less well-directed attention in the IT task. Once again IT correlated better with Verbal IQ, supporting the hypothesis that IT is better associated with more general intellectual functioning than with specific cognitive abilities. Cross-lagged panel correlations between IT and IQ were —0.40, whether predicting IQ from IT or the reverse. Thus, there was no evidence for a causal relationship between these variables, although both may reflect a common cognitive ability factor.     

A geometrical analysis of the unfolding model: General solutions

Given the complete setR of rank orders obtained from any configuration ofn stimulus points inr dimensions in accordance with the unfolding model, a configuration from which just these orders may be derived will be described as a solution forR. The space is assumed to be Euclidean. Necessary and sufficient conditions are derived for a configuration to be a solution forR. The geometrical constraints which are necessary and sufficient to determine the subset of pairs of orders and opposites contained inR are also identified and constitute the constraint system for the ordinal vector model. The relationship between the two models is discussed.