Abstract Previous research using the Cattell-Horn-Carroll (CHC) theory of cognitive abilities has shown a relationship between cognitive ability and academic achievement. Most of this research, however, has been done using the Woodcock-Johnson family of instruments with a higher order factor model. For CHC theory to grow, research should be done with other assessment instruments and tested with other factor models. This study examined the relationship between different factor models of CHC theory and the factors' relationships with language-based academic achievement (i.e., reading and writing). Using the co-norming sample for the Wechsler Intelligence Scale for Children--4th Edition and the Wechsler Individual Achievement Test--2nd Edition, we found that bifactor and higher order models of the subtests of the Wechsler Intelligence Scale for Children-4th Edition produced a different set of Stratum II factors, which, in turn, have very different relationships with the language achievement variables of the Wechsler Individual Achievement Test--2nd Edition. We conclude that the factor model used to represent CHC theory makes little difference when general intelligence is of major interest, but it makes a large difference when the Stratum II factors are of primary concern, especially when they are used to predict other variables. PsycINFO Database Record (c) 2014 APA, all rights reserved.

This article recommends an alternative method for testing multifaceted constructs. Researchers often have to choose between two problematic approaches for analyzing multifaceted constructs: the total score approach and the individual score approach. Both approaches can result in conceptual ambiguity. The proposed bifactor model assesses simultaneously the general construct shared by the facets and the specific facets, over and above the general construct. We illustrate the bifactor model by examining the construct of Extraversion as measured by the Revised NEO Personality Inventory (NEO-PI-R; Costa & McCrae, 1992), with two college samples (N65=65383 and 378). The analysis reveals that the facets of the NEO-PI-R Extraversion correlate with criteria in opposite directions after partialling out the general construct. The direction of gender differences also varies by facets. Bifactor models combine the advantages but avoid the drawbacks of the 2 existing methods and can lead to greater conceptual clarity.

AbstractResearchers often debate about whether there is a meaningful differentiation between psychological well-being and subjective well-being. One view argues that psychological and subjective well-being are distinct dimensions, whereas another view proposes that they are different perspectives on the same general construct and thus are more similar than different. The purpose of this investigation was to examine these two competing views by using a statistical approach, the bifactor model, that allows for an examination of the common variance shared by the two types of well-being and the unique variance specific to each. In one college sample and one nationally representative sample, the bifactor model revealed a strong general factor, which captures the common ground shared by the measures of psychological well-being and subjective well-being. The bifactor model also revealed four specific factors of psychological well-being and three specific factors of subjective well-being, after partialling out the general well-being factor. We further examined the relations of the specific factors of psychological and subjective well-being to external measures. The specific factors demonstrated incremental predictive power, independent of the general well-being factor. These results suggest that psychological well-being and subjective well-being are strongly related at the general construct level, but their individual components are distinct once their overlap with the general construct of well-being is partialled out. The findings thus indicate that both perspectives have merit, depending on the level of analysis.

Bifactor and second-order factor models are two alternative approaches for representing general constructs comprised of several highly related domains. Bifactor and second-order models were compared using a quality of life data set (N = 403). The bifactor model identified three, rather than the hypothesized four, domain specific factors beyond the general factor. The bifactor model fit the data significantly better than the second-order model. The bifactor model allowed for easier interpretation of the relationship between the domain specific factors and external variables, over and above the general factor. Contrary to the literature, sufficient power existed to distinguish between bifactor and corresponding second-order models in one actual and one simulated example, given reasonable sample sizes. Advantages of bifactor models over second-order models are discussed.

Four item response theory (IRT) models were compared using data from tests where multiple items were grouped into testlets focused on a common stimulus. In the bifactor model each item was treated as a function of a primary trait plus a nuisance trait due to the testlet; in the testlet-effects model the slopes in the direction of the testlet traits were constrained within each testlet to be proportional to the slope in the direction of the primary trait; in the polytomous model the item scores were summed into a single score for each testlet; and in the independent-items model the testlet structure was ignored. Using the simulated data, reliability was overestimated somewhat by the independent-items model when the items were not independent within testlets. Under these nonindependent conditions, the independent-items model also yielded greater root mean square error (RMSE) for item difficulty and underestimated the item slopes. When the items within testlets were instead generated to be independent, the bi-factor model yielded somewhat higher RMSE in difficulty and slope. Similar differences between the models were illustrated with real data.

Screening for emotional and behavioral risk has become more prevalent with young children; however, not much is known about the latent structure for many of the brief instruments currently being used to assess children. Information concerning the latent structure of an instrument is necessary to support use of the scoring information. The underlying structure of the Behavioral and Emotional Screening System, Teacher Rating Form-Preschool (Kamphaus & Reynolds, 2007) was investigated. A sample of ratings of more than 1,400 preschoolers was used to test 4 different confirmatory factor analysis models: (a) unidimensional model, (b) correlated factor model, (c) higher order factor model, and (d) bifactor model. Results showed support for the bifactor model, suggesting that an overall score is useful for interpreting preschoolers' level of emotional and behavioral risk. Relationships between items and the maladaptive risk general factor varied according to content area.

在心理、行为、管理和教育等社科领域,经常使用多维测验。本文评介并比较了各种多维测验的测量模型;总结了基于双因子模型计算得到的统计指标;根据不同研究目的,提出了两个兼顾简洁性和精确性的多维测验分析流程。作为例子,在马基雅维利主义人格量表的研究中,通过双因子模型分析了如何报告、解释多维测验分数以及如何利用多维建模进行后续分析。

61The bifactor ESEM model provided the best fit to the MPS data.61The Machiavellian Personality Scale was dominated by the global Machiavellianism.61It is suggested to report the total score and thedesire for controlsubscale score when using the MPS in practice.

(2017). Examining and Controlling for Wording Effect in a Self-Report Measure: A Monte Carlo Simulation Study. Structural Equation Modeling: A Multidisciplinary Journal. Ahead of Print. doi: 10.1080/10705511.2017.1286228

Inconsistency exists in the empirical literature with respect to the underlying factor structure of the Rosenberg Self-Esteem Scale (RSES; Rosenberg, 1989). In research contexts the RSES is considered a unidimensional measure of self-esteem. Empirical findings have undermined this conceptualisation with factor analytic findings favouring a variety of one-factor solutions (with correlated measurement errors) or multidimensional representations. The current study applied a bifactor modelling approach to provide a theoretical and methodologically satisfying resolution to the current inconsistency. Three alternative factor models of the RSES were tested among a large sample of the adult population (N=6082). Results indicated that a bifactor model was the best fit of the data. This model was demonstrated to be factorially invariant among males and females. The reliability of the scale was established using composite reliability. Results are discussed in terms of resolving the debate about the appropriate factor structure of the RSES.

Goodness-of-fit (GOF) indexes provide "rules of thumb" ecommended cutoff values for assessing fit in structural equation modeling. Hu and Bentler (1999) proposed a more rigorous approach to evaluating decision rules based on GOF indexes and, on this basis, proposed new and more stringent cutoff values for many indexes. This article discusses potential problems underlying the hypothesis-testing rationale of their research, which is more appropriate to testing statistical significance than evaluating GOF. Many of their misspecified models resulted in a fit that should have been deemed acceptable according to even their new, more demanding criteria. Hence, rejection of these acceptable-misspecified models should have constituted a Type 1 error (incorrect rejection of an "acceptable" model), leading to the seemingly paradoxical results whereby the probability of correctly rejecting misspecified models decreased substantially with increasing N. In contrast to the application of cutoff values to evaluate each solution in isolation, all the GOF indexes were more effective at identifying differences in misspecification based on nested models. Whereas Hu and Bentler (1999) offered cautions about the use of GOF indexes, current practice seems to have incorporated their new guidelines without sufficient attention to the limitations noted by Hu and Bentler (1999).

Although several studies have investigated the factor structure of the Rosenberg Self-Esteem Scale (RSES), there are still disagreements about it. The present study assessed: a) the goodness of fit of nine competing factor models for the RSES using data from a clinical sample of 855 women with eating/weight disorders; and b) its measurement invariance across clinical and non-clinical (n = 943) samples. A bifactor model, with a general self-esteem factor, plus positive and negative method factors, provided a better fit with the data than alternative models. However, the results showed the high reliability of the general self-esteem factor, and a low reliability of the two method factors. Furthermore, the full metric invariance of the RSES, as well as a partial scalar invariance and partial strict invariance across clinical and non-clinical groups, was supported by our findings. The factor variances and means differed significantly across groups. Overall, the findings of this study showed that the factor structure of the RSES is contaminated by method effects due to item wording, also with clinical samples, and that respondents from clinical and non-clinical groups interpret the self-esteem construct of the RSES items in a substantially similar way.

61Developing appropriate instruments to measure student engagement in math and science is needed.61Student engagement is a multidimensional construct, including behavioral, emotional, cognitive, and social components.61The use of multiple informants to assess student engagement provides a more comprehensive perspective.61Math and science engagement predicted academic performance and career aspirations.

讨论了Hu和Bentler(1998,1999)推荐的检验结构方程模型的 7个拟合指数准则 ,对这 7个指数的历史、特点和表现做了比较详细的述评。指出了他们基于这 7个指数的单指数准则和 2 -指数准则的不足之处。提出了超低显著性水平下的卡方准则 ,并部分重复他们的模拟例子 ,将卡方准则与这 7个指数准则比较 ,结果说明新的卡方准则优于其中的 6个 ,与另一个相当。最后简要说明了应当如何检视拟合指数进行模型检验和模型比较。

This Monte Carlo simulation study investigated different strategies for forming product indicators for the unconstrained approach in analyzing latent interaction models when the exogenous factors are measured by unequal numbers of indicators under both normal and nonnormal conditions. Product indicators were created by (a) multiplying parcels of the larger scale by items of the smaller scale, and (b) matching items according to reliability to create several product indicators, ignoring those items with lower reliability. Two scaling approaches were compared where parceling was not involved: (a) fixing the factor variances, and (b) fixing 1 loading to 1 for each factor. The unconstrained approach was compared with the latent moderated structural equations (LMS) approach. Results showed that under normal conditions, the LMS approach was preferred because the biases of its interaction estimates and associated standard errors were generally smaller, and its power was higher than that of the unconstrained approach. Under nonnormal conditions, however, the unconstrained approach was generally more robust than the LMS approach. It is recommended to form product indicators by using items with higher reliability (rather than parceling) in the matching and then to specify the model by fixing 1 loading of each factor to unity when adopting the unconstrained approach.

双因子模型和高阶因子模型,作为既有全局因子又有局部因子的两个竞争模型,在研究中得到了广泛应用。本文采用Monte Carlo模拟方法,在模型拟合比较的基础上,比较了效标分别为外显变量和内潜变量时,两个模型在各种负荷水平下预测准确度的差异。结果发现,两种模型在拟合效果方面无显著差异;但在预测效度方面,当效标为显变量时,两个模型的结构系数估计值皆为无偏估计;而效标为潜变量时,高阶因子模型表现优于双因子模型：高阶因子模型的结构系数为无偏估计,双因子模型的结构系数估计值则在50%左右的情况下存在偏差。

Multidimensional tests are frequently applied to the studies of psychology, education, society and management. Before aggregating all item scores to form a composite score of a multidimensional test, we should consider the homogeneity of the test. Homogeneity coefficient which reflects the extent that all test items measure the same trait can be employed to evaluate test homogeneity. If homogeneity coefficient is low, the composite score is meaningless and cannot be used for further analyses. Homogeneity coefficient is the proportion of variability in composite score that is accounted for by the general factor, which is viewed as common to all items. Any multidimensional test can be represented by a bifactor model that contains a general factor and local factors. Hence homogeneity coefficient can be calculated based on a bifactor model. A unidimensional test with positively worded items and negatively worded items can also be represented by a bifactor model, where the assessed construct is the general factor and method factors are local factors. The confidence interval of homogeneity coefficient provides more information than its point estimate. There are three approaches to estimate the confidence interval of composite reliability: Bootstrap method, Delta method and direct use of the standard error generated from an SEM software output (e.g., LISREL). It has been found that the interval estimates that obtained by Delta method and Bootstrap method were almost the same, whereas the results obtained by LISREL software and by Bootstrap method had large differences. Delta method was recommended when estimating the confidence interval of composite reliability. In order to compute the confidence interval of homogeneity coefficient, we deduced a formula by using Delta method for computing the standard error of homogeneity coefficient. Based on the standard error, the confidence interval can be obtained easily. We used an example to illustrate how to calculate homogeneity coefficient and its confidence interval by using the proposed Delta method with LISREL software. We also illustrated how to get the same result with Mplus software that automatically calculates the standard error with Delta method and presents the confidence interval. Before composite scores of a test are aggregated for further statistical analysis, it is recommended to report homogeneity coefficient so that readers could evaluate the extent that the statistical results are reliable.

The relationship between the higher-order factor model and the hierarchical factor model is explored formally. We show that the Schmid-Leiman transformation produces constrained hierarchical factor...