Assessing Goodness of Fit: Is Parsimony Always Desirable?
TL;DR: In this article, a heuristic counterexample is demonstrated in which parsimony as typically operationalized in indices of fit may be undesirable, in simplex models of longitudinal data, and the failure to include correlated uniquenesses relating the same indicators administered on different occasions will typically lead to systematically inflated estimates of stability.
Abstract: Many mechanistic rules of thumb for evaluating the goodness of fit of structural equation models (SEM) emphasize model parsimony; all other things being equal, a simpler, more parsimonious model with fewer estimated parameters is better than a more complex model Although this is usually good advice, in the present article a heuristic counterexample is demonstrated in which parsimony as typically operationalized in indices of fit may be undesirable. Specifically, in simplex models of longitudinal data, the failure to include correlated uniquenesses relating the same indicators administered on different occasions will typically lead to systematically inflated estimates of stability. Although simplex models with correlated uniquenesses are substantially less parsimonious and may be unacceptable according to mechanistic decision rules that penalize model complexity, it can be argued a priori that these additional parameter estimates should be included. Simulated data . are used to support this claim a...
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TL;DR: In this article, the adequacy of the conventional cutoff criteria and several new alternatives for various fit indexes used to evaluate model fit in practice were examined, and the results suggest that, for the ML method, a cutoff value close to.95 for TLI, BL89, CFI, RNI, and G...
Abstract: This article examines the adequacy of the “rules of thumb” conventional cutoff criteria and several new alternatives for various fit indexes used to evaluate model fit in practice. Using a 2‐index presentation strategy, which includes using the maximum likelihood (ML)‐based standardized root mean squared residual (SRMR) and supplementing it with either Tucker‐Lewis Index (TLI), Bollen's (1989) Fit Index (BL89), Relative Noncentrality Index (RNI), Comparative Fit Index (CFI), Gamma Hat, McDonald's Centrality Index (Mc), or root mean squared error of approximation (RMSEA), various combinations of cutoff values from selected ranges of cutoff criteria for the ML‐based SRMR and a given supplemental fit index were used to calculate rejection rates for various types of true‐population and misspecified models; that is, models with misspecified factor covariance(s) and models with misspecified factor loading(s). The results suggest that, for the ML method, a cutoff value close to .95 for TLI, BL89, CFI, RNI, and G...
76,383 citations
TL;DR: A revised two-factor version of the Study Process Questionnaire (R-SPQ-2F) suitable for use by teachers in evaluating the learning approaches of their students is produced, using fewer items.
Abstract: Aim. To produce a revised two-factor version of the Study Process Questionnaire (R-SPQ-2F) suitable for use by teachers in evaluating the learning approaches of their students. The revised instrument assesses deep and surface approaches only, using fewer items. Method. A set of 43 items was drawn up for the initial tests. These were derived from: the original version of the SPQ, modified items from the SPQ, and new items. A process of testing and refinement eventuated in deep and surface motive and strategy scales each with 5 items, 10 items per approach score. The final version was tested using reliability procedures and confirmatory factor analysis. Sample. The sample for the testing and refinement process consisted of 229 students from the health sciences faculty of a university in Hong Kong. A fresh sample of 495 undergraduate students from a variety of departments of the same university was used for the test of the final version. Results. The final version of the questionnaire had acceptable Cronbach alpha values for scale reliability. Confirmatory factor analysis indicated a good fit to the intended two-factor structure. Both deep and surface approach scales had well identified motive and strategy subscales. Conclusion. The revision process has resulted in a simple questionnaire which teachers can use to evaluate their own teaching and the learning approaches of their students.
1,823 citations
Cites background or methods from "Assessing Goodness of Fit: Is Parsi..."
...In the 3P model, student factors, teaching context, on-task approaches to learning, and the learning outcomes, mutually interact, forming a dynamic system (Figure 1)....
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...The goodness of t of the model to the data can be assessed by many t indexes with conventionally accepted cut-off criteria (see, for example, Bentler, 1990; Hoyle & Panter, 1995; Marsh & Hau, 1996)....
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01 Jan 2002
1,315 citations
TL;DR: ESEM, an overarching integration of the best aspects of CFA/SEM and traditional EFA, provides confirmatory tests of a priori factor structures, relations between latent factors and multigroup/multioccasion tests of full (mean structure) measurement invariance.
Abstract: Exploratory factor analysis (EFA) and confirmatory factor analysis (CFA), path analysis, and structural equation modeling (SEM) have long histories in clinical research. Although CFA has largely superseded EFA, CFAs of multidimensional constructs typically fail to meet standards of good measurement: goodness of fit, measurement invariance, lack of differential item functioning, and well-differentiated factors in support of discriminant validity. Part of the problem is undue reliance on overly restrictive CFAs in which each item loads on only one factor. Exploratory SEM (ESEM), an overarching integration of the best aspects of CFA/SEM and traditional EFA, provides confirmatory tests of a priori factor structures, relations between latent factors and multigroup/multioccasion tests of full (mean structure) measurement invariance. It incorporates all combinations of CFA factors, ESEM factors, covariates, grouping/multiple-indicator multiple-cause (MIMIC) variables, latent growth, and complex structures that typically have required CFA/SEM. ESEM has broad applicability to clinical studies that are not appropriately addressed either by traditional EFA or CFA/SEM.
1,052 citations
TL;DR: The positive effects of academic self- Concept on a variety of academic outcomes and integrate self-concept with the developmental motivation literature are demonstrated.
Abstract: Reciprocal effects models of longitudinal data show that academic self-concept is both a cause and an effect of achievement. In this study this model was extended to juxtapose self-concept with academic interest. Based on longitudinal data from 2 nationally representative samples of German 7th-grade students (Study 1: N = 5,649, M age = 13.4; Study 2: N = 2,264, M age = 13.7 years), prior self-concept significantly affected subsequent math interest, school grades, and standardized test scores, whereas prior math interest had only a small effect on subsequent math self-concept. Despite stereotypic gender differences in means, linkages relating these constructs were invariant over gender. These results demonstrate the positive effects of academic self-concept on a variety of academic outcomes and integrate self-concept with the developmental motivation literature.
1,028 citations
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TL;DR: In this paper, two types of error involved in fitting a model are considered, error of approximation and error of fit, where the first involves the fit of the model, and the second involves the model's shape.
Abstract: This article is concerned with measures of fit of a model. Two types of error involved in fitting a model are considered. The first is error of approximation which involves the fit of the model, wi...
25,611 citations
TL;DR: A new coefficient is proposed to summarize the relative reduction in the noncentrality parameters of two nested models and two estimators of the coefficient yield new normed (CFI) and nonnormed (FI) fit indexes.
Abstract: Normed and nonnormed fit indexes are frequently used as adjuncts to chi-square statistics for evaluating the fit of a structural model A drawback of existing indexes is that they estimate no known population parameters A new coefficient is proposed to summarize the relative reduction in the noncentrality parameters of two nested models Two estimators of the coefficient yield new normed (CFI) and nonnormed (FI) fit indexes CFI avoids the underestimation of fit often noted in small samples for Bentler and Bonett's (1980) normed fit index (NFI) FI is a linear function of Bentler and Bonett's non-normed fit index (NNFI) that avoids the extreme underestimation and overestimation often found in NNFI Asymptotically, CFI, FI, NFI, and a new index developed by Bollen are equivalent measures of comparative fit, whereas NNFI measures relative fit by comparing noncentrality per degree of freedom All of the indexes are generalized to permit use of Wald and Lagrange multiplier statistics An example illustrates the behavior of these indexes under conditions of correct specification and misspecification The new fit indexes perform very well at all sample sizes
21,588 citations
Book•
28 Apr 1989
TL;DR: The General Model, Part I: Latent Variable and Measurement Models Combined, Part II: Extensions, Part III: Extensions and Part IV: Confirmatory Factor Analysis as discussed by the authors.
Abstract: Model Notation, Covariances, and Path Analysis. Causality and Causal Models. Structural Equation Models with Observed Variables. The Consequences of Measurement Error. Measurement Models: The Relation Between Latent and Observed Variables. Confirmatory Factor Analysis. The General Model, Part I: Latent Variable and Measurement Models Combined. The General Model, Part II: Extensions. Appendices. Distribution Theory. References. Index.
19,019 citations
TL;DR: In this article, a general null model based on modified independence among variables is proposed to provide an additional reference point for the statistical and scientific evaluation of covariance structure models, and the importance of supplementing statistical evaluation with incremental fit indices associated with the comparison of hierarchical models.
Abstract: Factor analysis, path analysis, structural equation modeling, and related multivariate statistical methods are based on maximum likelihood or generalized least squares estimation developed for covariance structure models. Large-sample theory provides a chi-square goodness-of-fit test for comparing a model against a general alternative model based on correlated variables. This model comparison is insufficient for model evaluation: In large samples virtually any model tends to be rejected as inadequate, and in small samples various competing models, if evaluated, might be equally acceptable. A general null model based on modified independence among variables is proposed to provide an additional reference point for the statistical and scientific evaluation of covariance structure models. Use of the null model in the context of a procedure that sequentially evaluates the statistical necessity of various sets of parameters places statistical methods in covariance structure analysis into a more complete framework. The concepts of ideal models and pseudo chi-square tests are introduced, and their roles in hypothesis testing are developed. The importance of supplementing statistical evaluation with incremental fit indices associated with the comparison of hierarchical models is also emphasized. Normed and nonnormed fit indices are developed and illustrated.
16,420 citations
Book•
01 Feb 1993
TL;DR: In this paper, Bollen et al. proposed a model fitting metric for Structural Equation Models, which is based on the Monte Carlo evaluation of Goodness-of-Fit measures.
Abstract: Introduction - Kenneth A Bollen and J Scott Long Multifaceted Conceptions of Fit in Structural Equation Models - J S Tanaka Monte Carlo Evaluations of Goodness-of-Fit Indices for Structural Equation Models - David W Gerbing and James C Anderson Some Specification Tests for the Linear Regression Model - J Scott Long and Pravin K Trivedi Bootstrapping Goodness-of-Fit Measures in Structural Equation Models - Kenneth A Bollen and Robert A Stine Alternative Ways of Assessing Model Fit - Michael W Browne and Robert Cudeck Bayesian Model Selection in Structural Equation Models - Adrian E Raftery Power Evaluations in Structural Equation Models - Willem E Saris and Albert Satorra Goodness-of-Fit with Categorical and Other Nonnormal Variables - Bengt O Muthen Some New Covariance Structure Model Improvement Statistics - P M Bentler and Chih-Ping Chou Nonpositive Definite Matrices in Structural Modeling - Werner Wothke Testing Structural Equation Models - Karl G Joreskog
9,377 citations