A reliability coefficient for maximum likelihood factor analysis

Maximum likelihood factor analysis provides an effective method for estimation of factor matrices and a useful test statistic in the likelihood ratio for rejection of overly simple factor models. A reliability coefficient is proposed to indicate quality of representation of interrelations among attributes in a battery by a maximum likelihood factor analysis. Usually, for a large sample of individuals or objects, the likelihood ratio statistic could indicate that an otherwise acceptable factor model does not exactly represent the interrelations among the attributes for a population. The reliability coefficient could indicate a very close representation in this case and be a better indication as to whether to accept or reject the factor solution. This research was supported by the Personnel and Training Research Programs Office of the Office of Naval Research under contract US NAVY/00014-67-A-0305-0003. Critical review of the development and suggestions by Richard Montanelli were most helpful.     

The Self-Consciousness Scale in Chinese College Students

Fenigstein, Scheier, and Buss's (1975) 3-factor model, Burnkrant and Page's (1984) 4-factor model, and Mittal and Balasubramanian's (1987) 5-factor model of the Self-Consciousness Scale (SCS) were tested using the revised SCS (Scheier & Carver, 1985b) on 2 convenience samples of 201 and 179 Chinese college students. Burnkrant and Page's (1984) 4-factor model showed superiority over the other 2 models in confirmatory factor analysis. It is concluded that the division of the private self-consciousness subscale is adequate, whereas the division of the public self-consciousness subscale is not justified.     

Examining the Factor Structure of the Self-Compassion Scale in Four Distinct Populations: Is the Use of a Total Scale Score Justified?

This study examined the factor structure of the Self-Compassion Scale (SCS) using a bifactor model, a higher order model, a 6-factor correlated model, a 2-factor correlated model, and a 1-factor model in 4 distinct populations: college undergraduates (N = 222), community adults (N = 1,394), individuals practicing Buddhist meditation (N = 215), and a clinical sample of individuals with a history of recurrent depression (N = 390). The 6-factor correlated model demonstrated the best fit across samples, whereas the 1- and 2-factor models had poor fit. The higher order model also showed relatively poor fit across samples, suggesting it is not representative of the relationship between subscale factors and a general self-compassion factor. The bifactor model, however, had acceptable fit in the student, community, and meditator samples. Although fit was suboptimal in the clinical sample, results suggested an overall self-compassion factor could still be interpreted with some confidence. Moreover, estimates suggested a general self-compassion factor accounted for at least 90% of the reliable variance in SCS scores across samples, and item factor loadings and intercepts were equivalent across samples. Results suggest that a total SCS score can be used as an overall mesure of self-compassion.     

Individual differences in embarrassment: Self-consciousness,self-monitoring and embarrassibility

Embarrassment has been conceptualized as a self-presentational difficulty resulting from a concern with our observable behaviour and a desire to conform and to please others. Thus, individuals particularly concerned with these aspects of their behaviour (i.e. publicly self-conscious and other-directed) are likely to be particularly susceptible to embarrassment. This experiment was designed to investigate this possibility. The Self-consciousness Scale (SCS), Self-monitoring Scale (SMS) and the Embarrassibility Scale (ES) were administered to over 100 Ss. The factor structures obtained in the present study were compared with those obtained in previous studies and revealed an almost exact correspondence for the SCS, a close correspondence for the SMS and little correspondence for the ES. There was a predicted significant positive correlation between the ES and the Public Self-consciousness subscale of the SCS (i.e. a concern for observable behaviour) and the Other-directedness subscale of the SMS (i.e. a desire to conform and please others). Two aspects of the results are discussed: the possibility that the ES is unidimensional and implications of individual differences in the process of embarrassment for investigating coping strategies.     

Relationship of maternal report of feminine behaviors and extraversion to clinician's rating of gender disturbance

50 boys, aged 3 to 16 yr., referred for psychotherapy for a gender-identity disorder were assessed by two parent-report instruments completed by their mothers--The Gender Behavior Inventory for Boys for preadolescent subjects and a modified form of the Parents' Evaluation of Child Behavior. Without access to these data, an independent clinical psychologist provided a diagnosis of the severity of a gender disturbance on a 5-point rating scale, based on clinical interviews and a psychological test battery. The gender-disturbed boys scored significantly above the mean of the standardized normal boys on the Gender Behavior Inventory for Boys Feminine Behavior subscale and significantly below the mean on the Extraversion subscale of the Gender Behavior Inventory. The ratings of severity of gender disturbance were significantly correlated .34 with Feminine Behavior subscale scores and -.50 with the Extraversion subscale scores. On the Parents' Evaluation the number of observed feminine behaviors listed was significantly correlated .57 with rated severity of gender disturbance but not with the Feminine Behavior subscale scores on the Gender Behavior Inventory.     

Asymptotic efficiency of the pseudo‐maximum likelihood estimator in multi‐group factor models with pooled data

A multi‐group factor model is suitable for data originating from different strata. However, it often requires a relatively large sample size to avoid numerical issues such as non‐convergence and non‐positive definite covariance matrices. An alternative is to pool data from different groups in which a single‐group factor model is fitted to the pooled data using maximum likelihood. In this paper, properties of pseudo‐maximum likelihood (PML) estimators for pooled data are studied. The pooled data are assumed to be normally distributed from a single group. The resulting asymptotic efficiency of the PML estimators of factor loadings is compared with that of the multi‐group maximum likelihood estimators. The effect of pooling is investigated through a two‐group factor model. The variances of factor loadings for the pooled data are underestimated under the normal theory when error variances in the smaller group are larger. Underestimation is due to dependence between the pooled factors and pooled error terms. Small‐sample properties of the PML estimators are also investigated using a Monte Carlo study.     

Item-Level and Scale-Level Factor Structures of the MMPI-A

The Minnesota Multiphasic Personality Inventory-Adolescent (MMPI-A) is a 478-item test that represents a substantial revision of the original form of the MMPI. This investigation sought to identify the item-level factor structure of the MMPI-A and also examined the scale-level factor structure of the 69 scales and subscales of this instrument. The study utilized the 1,620 normal adolescents (805 boys and 815 girls) of the normative sample for the MMPI-A. These adolescents ranged in age from 14 to 18 inclusive, with a mean age of 15.54 for boys and 15.60 for girls. A principal factor analysis of item-level responses resulted in extraction of 14 factors that were subjected to promax (oblique) rotation procedures. These 14 factors incorporated 81% of the total MMPI-A item pool and accounted for 44% of the total item-level response variance. For the scale-level analysis, 8 factors were selected for extraction and submitted to promax rotation procedures. These eight factors accounted for a total of 93.5% of the total variance in MMPI-A scale and subscale raw scores, Item-level results were discussed in terms of areas of congruence and dissimilarities from previously reported MMPI factor analyses in adolescent and adult samples, and scale-level factor results were presented in terms of clinical implications for the interpretation of MMPI-A scales and subscales.     

Athletic self-concept and mathematics achievement in girls

Several researchers have suggested that girls' mathematics performance may be mediated by an assertive sex role or "masculine interest." The present study made the assumption that girls' athletic self-confidence reflects "masculine interest" so girls' test scores for perceived athletic competence would be related to their mathematics achievement scores. A total of 207 boys and girls in Grades 4, 5, and 6 were tested for their perceived athletic ability using the Athletic Competence subscale of Harter's 1985 Self-perception Profile, and these scores were correlated with their mathematics achievement as measured on the Metropolitan Achievement Test and term grades. A low but significant correlation with Athletic Competence scores was found for girls on both measures of mathematics achievement. Although boys scored higher on the Athletic Competence subscale, there were no sex differences on either measure of mathematics achievement. Results are discussed in terms of both sex-role theory and cognitive development.     

Heteroscedastic one‐factor models and marginal maximum likelihood estimation

In the present paper, a general class of heteroscedastic one‐factor models is considered. In these models, the residual variances of the observed scores are explicitly modelled as parametric functions of the one‐dimensional factor score. A marginal maximum likelihood procedure for parameter estimation is proposed under both the assumption of multivariate normality of the observed scores conditional on the single common factor score and the assumption of normality of the common factor score. A likelihood ratio test is derived, which can be used to test the usual homoscedastic one‐factor model against one of the proposed heteroscedastic models. Simulation studies are carried out to investigate the robustness and the power of this likelihood ratio test. Results show that the asymptotic properties of the test statistic hold under both small test length conditions and small sample size conditions. Results also show under what conditions the power to detect different heteroscedasticity parameter values is either small, medium, or large. Finally, for illustrative purposes, the marginal maximum likelihood estimation procedure and the likelihood ratio test are applied to real data.     

Social-ecological assessment of environments: Toward a two-factor model

A study was conducted to test the usefulness of the two-factor model in assessing social environments. Subjects were residents and staff at a residential treatment center for emotionally disturbed and/or delinquent boys and girls. Rating instruments consisted of (a) two parallel versions of Moos' Community-Oriented Programs Enivronment Scale (COPES), each worded so as to be appropriate to the setting being measured, and (b) global ratings of satisfaction with the environment. Principal component factor analyses, carried out separately on the youths' COPES-School and the youths' COPES-Cottage, yielded two orthogonal but similar factors in each environment. In a comparison with Moos' three-dimensional formulation of the social milieu, the two factors strongly resembled two of Moos' dimensions but were more independent, had greater validity, and discriminated between the two environments at a higher level of significance. The instruments derived from the factor analyses provide a simple procedure for assessing a variety of treatment programs and populations.     

An irt-based model for dichotomous longitudinal data

The LLRA (linear logistic model with relaxed assumptions; Fischer, 1974, 1977a, 1977b, 1983a) was developed, within the framework of generalized Rasch models, for assessing change in dichotomous item score matrices between two points in time; it allows to quantify change on latent trait dimensions and to explain change in terms of treatment effects, treatment interactions, and a trend effect. A remarkable feature of the model is that unidimensionality of the item set is not required. The present paper extends this model to designs with any number of time points and even with different sets of items presented on different occasions, provided that one unidimensional subscale is available per latent trait. Thus unidimensionality assumptions within subscales are combined with multidimensionality of the item set. Conditional maximum likelihood methods for parameter estimation and hypothesis testing are developed, and a necessary and sufficient condition for unique identification of the model, given the data, is derived. Finally, a sample application is presented.To my friend Josef Roppert who has taught me how to apply statistical reasoning to substantive problems.This research was supported in part by Österreichische Forschungsgemeinschaft under grant No. 01/0054. The author wishes to thank B. Wild for the numerical computation of the sample application in section 5.     

Sex differences in grade three students' attitudes toward physical activity.

The purpose of this study was to investigate the attitudes children have towards physical activity and whether boys and girls differ significantly. Sex differences were hypothesized on one or more of the subdomains of Grade 3 Children's Attitudes Toward Physical Activity inventory. The 46 boys and 51 girls ranged from 8 to 10 years of age. A multivariate analysis of variance indicated a significant effect for sex, and subsequent univariate analysis indicated that the boys had significantly lower scores than the girls on the Aesthetic subscale. Results were discussed in terms of attitides about activity in physical education classes and continued research with younger children.     

Psychometric properties of the Chinese version of the End-of-Life Decision-Making and Staff Stress Questionnaire

Background/objectiveThe goal of this study is to establish a Chinese version of the End-of-Life Decision Making and Associated Staff Stress Questionnaire to assess its reliability and validity.MethodA sample of 119 Intensive Care Unit physicians and 485 nurses in China completed the questionnaire, along with questionnaires assessing motional exhaustion subscale, Stress Overload Scale, and other variables associated with end-of-life decision.ResultsSeven factors obtained via exploratory factor analysis could explain 70.61% of the total variance. Confirmatory factor analysis demonstrated an acceptable model fit with Root Mean Square Error of Approximation (RMSEA) being .078 and Standardized Root Mean Square Residual (SRMR) being .066. Validity evidence based on relationships with other variables was provided by positive or negative correlations between the questionnaire subscales and emotional exhaustion, stress overload, and other variables associated with end-of-life decision. The average content validity index was .96. The Cronbach’s α and test–retest reliability was outstanding.ConclusionsThe Chinese version of the End-of-Life Decision Making and Associated Staff Stress Questionnaire is a reliable and valid instrument for measuring the facilitators and hinders to facilitate the end-of-life decision-making, communication and the associated pressure perceived by relevant Intensive Care Unit medical staff among the Chinese population.     

A mixture model for responses and response times with a higher-order ability structure to detect rapid guessing behaviour

Many educational and psychological assessments focus on multidimensional latent traits that often have a hierarchical structure to provide both overall-level information and fine-grained diagnostic information. A test will usually have either separate time limits for each subtest or an overall time limit for administrative convenience and test fairness. In order to complete the items within the allocated time, examinees frequently adopt different test-taking behaviours during the test, such as solution behaviour and rapid guessing behaviour. In this paper we propose a new mixture model for responses and response times with a hierarchical ability structure, which incorporates auxiliary information from other subtests and the correlation structure of the abilities to detect rapid guessing behaviour. A Markov chain Monte Carlo method is proposed for model estimation. Simulation studies reveal that all model parameters could be recovered well, and the parameter estimates had smaller absolute bias and mean squared error than the mixture unidimensional item response theory (UIRT) model. Moreover, the true positive rate of detecting rapid guessing behaviour is also higher than when using the mixture UIRT model separately for each subscale, whereas the false detection rate is much lower than the mixture UIRT model. The deviance information criterion and the logarithm of the pseudo-marginal likelihood are employed to evaluate the model fit. Finally, a real data analysis is presented to demonstrate the practical value of the proposed model.     

A Confirmatory Approach to Examining the Factor Structure of the Strengths and Difficulties Questionnaire (SDQ): A Large Scale Cohort Study

The aim of this study was to examine the factor structure of the Strengths and Difficulties Questionnaire (SDQ) using a Structural Confirmatory Factor Analytic approach. The Danish translation of the SDQ was distributed to 71,840 parents and teachers of 5–7 and 10–12-year-old boys and girls from four large scale cohorts. Three theoretical models were examined: 1. a model with five first order factors (i.e., hyperactivity/inattention, conduct, emotional, peer problems and prosocial), 2. a model adding two internalising and externalising second order factors to model 1, and 3. a model adding a total difficulties second order factor to model 1. Model fits were evaluated, multi-group analyses were carried out and average variance extracted (AVE) and composite reliability (CR) estimates were examined. In this general population sample, low risk sample models 1 and 2 showed similar good overall fits. Best model fits were found when two positively worded items were allowed to cross load with the prosocial scale, and cross loadings were allowed for among three sets of indicators. The analyses also revealed that model fits were slightly better for teachers than for parents and better for older children than for younger children. No convincing differences were found between boys and girls. Factor loadings were acceptable for all groups, especially for older children rated by teachers. Some emotional, peer, conduct and prosocial subscale problems were revealed for younger children rated by parents. The analyses revealed more internal consistency for older children rated by teachers than for younger children rated by parents. It is recommended that model 1 comprising five first order factors, or alternatively model 2 with additionally two internalising/externalising second order factors, should be used when employing the SDQ in low risk epidemiological samples.     

Gender as a moderator variable in the relationship between an intrinsic motivation scale and short-term novelty in children

Harter's (1980) Intrinsic-Extrinsic Orientation scale was examined for evidence of empirical and construct validity. We hypothesized that subscales defining the motivational component of intrinsic motivation would be correlated with novelty, a collative motivational variable. Partial support for the hypothesis was obtained for boys; correlations between novelty and Harter's Curiosity subscale were .57, .64, and .58 for boys in the third, fifth, and combined grades, respectively, and correlations approached significance for Harter's Challenge subscale. Not predicted were the correlations of .40, .68, and .46 obtained for girls in the third, fifth, and combined grades, respectively, between the Independent Judgment subscale (a cognitive-informational scale) and novelty. Results indicated that gender operated as a moderator variable, with boys expressing collative motivation directly in an action-oriented form, and girls demonstrating it somewhat indirectly in a thought-oriented form.     

Children's cognitive ability from 4 to 9 years old as a function of prenatal cocaine exposure, environmental risk, and maternal verbal intelligence

This study examined the effects of prenatal cocaine exposure, environmental risk, and maternal verbal intelligence on children's cognitive ability. Gender and age were examined as moderators of potential cocaine exposure effects. The Stanford-Binet IV intelligence test was administered to 231 children (91 cocaine exposed, 140 unexposed) at ages 4, 6, and 9 years. Neonatal medical risk and other prenatal exposures (alcohol, cigarettes, and marijuana) were also examined for their unique effects on child IQ. Mixed models analysis indicated that prenatal cocaine exposure interacted with gender, as cocaine-exposed boys had lower composite IQ scores. Age at assessment did not moderate this relation, indicating that cocaine-exposed boys had lower IQs across this age period. A stimulating home environment and high maternal verbal IQ also predicted higher composite IQ scores. Cocaine-exposed boys had lower scores on the Abstract/Visual Reasoning subscale, with trends for lower scores on the Short-Term Memory and Verbal Reasoning subscales, as exposure effects were observed across domains. The findings indicate that cocaine exposure continues to place children at risk for mild cognitive deficits into preadolescence. Possible mechanisms for the Exposure x Gender interaction are discussed.     

On normal theory based inference for multilevel models with distributional violations

Data in social and behavioral sciences are often hierarchically organized though seldom normal, yet normal theory based inference procedures are routinely used for analyzing multilevel models. Based on this observation, simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions. For characterizing the distribution of parameter estimates, sandwich-type covariance matrices are derived. Standard errors based on these covariance matrices remain consistent under distributional violations. Implications of various covariance estimators are also discussed. For evaluating the quality of a multilevel model, a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model. The rescaled statistic, improving the likelihood ratio statistic by estimating one extra parameter, approaches the same mean as its reference distribution. A simulation study with a 2-level factor model implies that the rescaled statistic is preferable.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

Reliability and validity of the kiddie mach scale in Chinese children

The present study examined the psychometric properties of a translated Chinese version of the Kiddie Mach Scale in a sample of Chinese students (N=787) ages 9 to 13 years (416 boys, 371 girls; M age= 11.3 yr., SD = 1.3). Results of confirmatory factor analysis showed a three-factor structure to be the most adequate model: Distrust, Lack of faith in human nature, and Dishonesty. These results are similar to the English version, but not the Greek version. The Chinese version exhibited weak internal consistency, but acceptable 4-week test-retest reliability. The Chinese version of the Kiddie Mach Scale was significantly associated with the Basic Empathy Scale. There was a sex effect on Kiddie Mach Scale scores with boys scoring higher on the Total scale and the Distrust subscale. Some items did not seem to function well in Chinese culture. More research will be needed to find items that represent, especially, distrust in Chinese settings.