Poster
Reference
Bayesian Structural Equation Modeling of the WISC-IV with a Large Referred US Sample
GOLAY, Philippe, et al.
Abstract
When investigating the structure of cognitive ability measures like the WISC-IV, subtest scores theoretically associated with one latent variable could also be related to other factors.
The objective of this study was to determine whether secondary interpretation of the 10 WISC-IV core subtests in a large referred US sample could be justified or whether a simple and unambiguous interpretation was more appropriate. To achieve this goal, the influence of each latent factor on subtest scores was estimated using Bayesian structural equation modeling (BSEM). A major drawback of classical confirmatory factor analysis (CFA) is that the majority of factor loadings needs to be fixed to zero to estimate the model parameters. This unnecessary strict parameterization can lead to model rejection and cause researchers to perform many exploratory modifications to achieve acceptable model fit. BSEM overcomes this limitation by replacing fixed-to-zero-loadings with “approximate” zeros that translates into small, but not necessary zero, cross-loadings. Because all relationships between factors and subtest scores are estimated, both the [...]
GOLAY, Philippe, et al. Bayesian Structural Equation Modeling of the WISC-IV with a Large Referred US Sample. In: 122nd Annual Convention of the American Psychological Association, Washington, DC, August 7-10, 2014, 2014
Available at:
http://archive-ouverte.unige.ch/unige:39203
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INTRODUCTION
CONCLUSIONS
RESULTS – MODELS COMPARISONS
Contact: Philippe.Golay@unige.ch 122nd AMERICAN PSYCHOLOGICAL ASSOCIATION ANNUAL CONVENTION – WASHINGTON D.C. – AUGUST 7-10, 2014
Bayesian Structural Equation Modeling of the WISC-IV with a Large Referred US Sample
Philippe Golay
1,2, Thierry Lecerf
1, Marley W. Watkins
3& Gary L. Canivez
41University of Geneva, 2University of Lausanne, 3Baylor University, 4Eastern Illinois University
•
The Wechsler Intelligence Scale for children, 4th edition (WISC-IV) remains the most widely used test in the field of intelligence assessment.• General intelligence has been conceptualized as a superordinate factor (higher order models) or as a breadth factor (bifactor models). Numerous studies have supported exploratory and confirmatory bifactor structures of the WISC-IV in US, French, and Irish samples.
• Many controversies remain on the nature of the constructs measured by each subtest score : subtest scores theoretically associated with one latent variable
could also be related to other factors. It is crucial to determine whether secondary interpretation of the 10 WISC-IV core subtests could be justified or whether a
simple and unambiguous interpretation is more appropriate.
1. Results on a sample of 1130 referred US children showed that the bifactor model fit was better than the higher order solution. Models including small cross-loadings were more adequate.
2. BSEM suggested a simple and parsimonious interpretation of the subtest scores
3. Loadings of the subtests scores on the g-factor were systematically higher than their respective loadings on the four index scores. Index scores
represented rather small deviations from unidimensionality and did not
necessarily provide additional and separate information from the Full Scale IQ score (FISQ).
4. BSEM allowed us to estimate models that were closer to theoretical assumptions.
5. BSEM also permits to test more complex models that are not possible to estimate through maximum likelihood estimation.
SAMPLE DESCRIPTION
LIMITATIONS OF CONFIRMATORY FACTOR ANALYSIS (CFA)
With classical confirmatory factor analysis (CFA) the majority of factor loadings need to be fixed to zero to estimate the model parameters.
Although needed for model identification, these restrictions do not always faithfully reflect the researchers’ hypotheses. Small but not necessarily zero loading could be equally or even more compatible with theory.
This unnecessary strict parameterization can contribute to poor model fit, distorted factors and biased factor correlations (Marsh, et al., 2010). It also may cause researchers to perform many exploratory modifications to achieve acceptable model fit. Extensive specification searches can lead to unjustified overfitting of data, with loss of meaning for indices of statistical significance (Carroll, 1995)
• WISC-IV data were obtained from 1130 US children who were referred for evaluation of learning difficulties.
• As it appears to be common in clinical assessments, only the 10 core subtests were
administered. Only children with complete data for all core subtests were included in the analyses.
Sample N % Male Minimum Age Maximum Age Mean Age (SD)
US 1130 62% (696) 6-0 16-11 10.24 (2.51)
GOALS OF THE PRESENT STUDY
o The first goal of this study was to compare indirect (higher-order) versus direct (bifactor) hierarchical models of the 10 WISC-IV core subtests from a large referred US sample.
o The second objective was to determine more precisely which constructs are measured by each core subtest score of the WISC-IV : can secondary interpretation of some subtest scores be supported by the data ?
Model
Number of free parameters
Posterior Predictive
P-Value
Difference between observed &
replicated Χ2 95% C.I.
DIC
Estimated number of parameters Lower 2.5% Upper 2.5%
1. WISC-IV - higher order model 34 0.000 34.732 90.485 25983.228 33.785 2. WISC-IV - higher order model with
cross-loadings (priors variance = 0.01) 64 0.151 -15.168 45.585 25941.978 40.471
3. WISC-IV - bifactor model 40 0.000 21.281 76.965 25963.109 27.042
4. WISC-IV - bifactor model with cross-
loadings (priors variance = 0.01) 70 0.388 -27.095 35.638 25913.312 23.009
RESULTS – WISC-IV BIFACTOR MODEL WITH CROSS-LOADINGS
o Higher Posterior Predictive P-Value and Lower DIC indicates better fit to the data.
o The WISC-IV bifactor model with small cross-loadings showed the better fit overall.
Latent variable
Subtest Subtest Subtest Subtest Subtest Subtest
0 0 0
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Diffuse non informative priors zero mean and
infinite variance
Informative priors zero mean and
small variance
Bayesian estimation combines prior distributions for all parameters with the experimental data and forms posterior distributions via Bayes' theorem.
The posterior distribution of Bayesian estimation was achieved through Markov Chain Monte Carlo (MCMC) algorithm with the Gibbs sampler Method.
Three MCMC chains with 50’000 iterations were used with different starting values and different random seeds.
The prior variance was 0.01 which results in 95% credibility interval of ± 0.20 (small cross-loadings).
g
Verbal Comprehension
VCI
Perceptual Reasoning
PRI
Working Memory
WMI
Processing Speed
PSI
Similarities
Comprehension
Block Design
Matrix Reasoning Picture Concept
Digit Span
Letter Number
Coding
Symbol Search Vocabulary
WISC-IV Higher order model WISC-IV Bifactor model
g
Verbal Comprehension
VCI
Perceptual Reasoning
PRI
Working Memory
WMI
Processing Speed
PSI
Similarities
Comprehension
Block Design
Matrix Reasoning Picture Concept
Digit Span
Letter Number
Coding
Symbol Search Vocabulary
Loadings estimates (median) G VCI PRI WMI PSI
95% CI 95% CI 95% CI 95% CI 95% CI
Similarities 0.756* 0.380* 0.052 -0.008 -0.051
0.680 0.823 0.202 0.504 -0.074 0.184 -0.134 0.132 -0.156 0.054
Vocabulary 0.796* 0.488* 0.007 0.043 -0.014
0.713 0.887 0.267 0.606 -0.142 0.145 -0.127 0.187 -0.138 0.102
Comprehension 0.691* 0.394* -0.049 0.007 0.020
0.603 0.779 0.160 0.521 -0.186 0.153 -0.141 0.157 -0.095 0.133
Block Design 0.699* -0.025 0.408 -0.050 0.034
0.592 0.800 -0.176 0.130 -0.198 0.608 -0.209 0.153 -0.109 0.162
Picture Concepts 0.691* 0.012 0.148 0.001 -0.024
0.601 0.767 -0.150 0.163 -0.159 0.394 -0.165 0.188 -0.146 0.087
Matrix Reasoning 0.752* 0.005 0.285 0.029 0.002
0.671 0.822 -0.127 0.125 -0.097 0.479 -0.116 0.162 -0.119 0.112
Digit Span 0.637* 0.022 -0.010 0.256 0.026
0.566 0.710 -0.116 0.149 -0.151 0.134 -0.178 0.502 -0.084 0.131
Letter-Number Sequencing 0.738* 0.039 -0.006 0.311 0.044
0.663 0.816 -0.113 0.175 -0.156 0.141 -0.200 0.600 -0.088 0.162
Coding 0.691* -0.030 -0.027 0.036 0.554*
0.603 0.779 -0.171 0.108 -0.172 0.126 -0.127 0.179 0.341 0.811
Symbol Search 0.621* -0.002 0.071 0.019 0.472*
0.539 0.699 -0.135 0.124 -0.085 0.204 -0.116 0.151 0.302 0.714
Note. Loadings in bold were freely estimated. Other loadings were estimated with small (0.01) variance priors. G = General Factor; VCI = Verbal
Comprehension Index; PRI = Perceptual Reasoning Index; WMI = Working Memory Index; PSI = Processing Speed Index; CI= credibility interval; * = statistically significant because 95% CI does not contain zero.
BAYESIAN STRUCTURAL EQUATION MODELING (BSEM)
•
BSEM overcomes CFA’s limitations by replacing fixed-to-zero-loadings with“approximate” zeros that translates into small, but not necessary zero, cross- loadings.
• Approximate zeros often reflect more accurately theoretical assumptions and facilitate unbiased estimations of the model parameters.
• Because all relationships between factors and subtest scores are estimated this
approach eliminates the need for comparisons of many competing models. It is also possible to determine the precise nature of the constructs measured by the core
subtest scores of the WISC-IV
o The majority of the expected loadings were supported. However the loadings of Block
Design, Picture Concepts, Matrix Reasoning on PRI and the loadings of Digit Span and Letter- Number Sequencing on WMI were not statistically significant.
o Loadings of the subtests scores on the g-factor were systematically higher than their respective loadings on the four index scores.
o No cross-loadings were supported; thus, no secondary interpretation of subtests scores was supported by the data.
