Structure of the Wechsler Intelligence Scale for Children—Fourth Edition
Among a National Sample of Referred Students
Marley W. Watkins Arizona State University
The structure of the Wechsler Intelligence Scale for Children—Fourth Edition (WISC–IV; D. Wechsler,
2003a) was analyzed via confirmatory factor analysis among a national sample of 355 students referred
for psychoeducational evaluation by 93 school psychologists from 35 states. The structure of the WISC–
IV core battery was best represented by four first-order factors as per D. Wechsler (2003b), plus a
general intelligence factor in a direct hierarchical model. The general factor was the predominate source
of variation among WISC–IV subtests, accounting for 48% of the total variance and 75% of the common
variance. The largest 1st-order factor, Processing Speed, only accounted for 6.1% total and 9.5%
common variance. Given these explanatory contributions, recommendations favoring interpretation of
the 1st-order factor scores over the general intelligence score appear to be misguided.
Keywords: intelligence, factor analysis, WISC–IV, children Following substantial changes in content and structure (Kaufman, Flanagan, Alfonso, & Mascolo, 2006), the Wechsler Intelligence Scale for Children—Fourth Edition (WISC–IV; Wechsler,
2003a) replaced the Wechsler Intelligence Scale for Children—
Third Edition (WISC–III; Wechsler, 1991) in 2003. For example,
four of the 12 WISC–III subtests were omitted from the WISC–IV
or made optional (Information, Picture Arrangement, Picture Completion, and Object Assembly), whereas three new subtests were
added to the core WISC–IV (Picture Concepts, Matrix Reasoning,
and Letter-Number Sequencing). In addition, artwork was revised
and administration and scoring criteria were modified (Wechsler,
2003b). In total, approximately 60% of the items in the core
WISC–IV subtests are new or revised.
Because test revisions “may assess traits, abilities, and conditions in ways different from earlier versions” (Strauss, Spreen, &
Hunter, 2000, p. 237), psychometric standards (see American
Educational Research Association, American Psychological Association, and National Council on Measurement in Education, 1999)
applicable to new tests should also govern test revisions (Adams,
2000). Of particular importance is evidence about the internal
structure of the test revision, because there should be correspondence between the internal structure of the test and the structure of
the construct assumed to be measured by the test (Messick, 1995).
Wechsler (2003b) reported that the WISC–IV was designed to
measure intellectual functioning in four specific cognitive domains
and to provide an overall composite that represents general ability.
The WISC–IV structure was investigated with exploratory (EFA)
and confirmatory (CFA) factor analyses of the normative sample
with both core (10 subtest) and supplemental (15 subtest) batteries (Wechsler, 2003b). Four first-order oblique factors were consistently identified: Verbal Comprehension (VC), Perceptual Reasoning (PR), Working Memory (WM), and Processing Speed (PS).
These four dimensions were adopted as the scoring structure of the
WISC–IV. The same structure was found in an EFA reanalysis of
the normative data (Sattler, 2008), and measurement invariance
across gender (H. Chen & Zhu, 2008) and age (Keith, Fine, Taub,
Reynolds, & Kranzler, 2006) was subsequently demonstrated.
Inexplicably, Wechsler (2003b) did not conduct a higher order
factor analysis to verify and describe the proposed multilevel
structure of the WISC–IV. This oversight was addressed by three
independent analyses of the WISC–IV standardization sample. In
the first, Keith (2005) applied CFA to all 15 subtests but only
considered two models that included a general intelligence factor.
Both models exhibited good fit to the data. The second analysis
applied CFA to all 15 WISC–IV subtests and concluded that a
five-factor model based on the Cattell-Horn-Carroll (CHC;
McGrew, 2009) theory might be the most appropriate model
(Keith et al., 2006). However, Keith et al. (2006) did not analyze
the WISC–IV core battery; the superiority of the CHC model over
the Wechsler (2003b) four-factor model for the 15 subtest battery
was not substantial in terms of fit (F. F. Chen, 2007); simple
structure was abandoned in the CHC model; and the loading of the
second-order Gf factor on the third-order general factor was 1.00,
which was a replication of Gustafsson’s (1984) results that questions the independence of these two factors. The third analysis
applied the Schmid and Leiman (1957) orthogonal transformation
of EFA results from the WISC–IV core battery to reveal the
importance of the general intelligence factor in comparison to the
four first-order factors (Watkins, 2006). Specifically, the general
factor accounted for 71% of the common variance, whereas the
largest first-order factor only contributed 12% of the common
variance.
On the basis of these analyses, the factor structure of the
WISC–IV among a nationally representative sample of children
appears to consist of four to five first-order factors and a single
782 WISC–IV FACTOR STRUCTURE broad or superordinate factor associated with general intelligence.
However, as one of the main uses of intelligence tests is the
assessment of children with suspected disabilities, it must be
demonstrated that the WISC–IV is appropriate for those children.
Only two published studies have been conducted with a referral or
clinical sample. In one study, a four-factor oblique EFA solution
similar to that reported by Wechsler (2003b) was supported for the
WISC–IV core battery in a sample of Pennsylvania students referred for evaluation to determine eligibility for special education
services (Watkins, Wilson, Kotz, Carbone, & Babula, 2006). Similar to the results of Watkins (2006), transformation to an orthogonalized higher order model demonstrated that the general factor
accounted for more than 75% of the common variance, whereas
the largest first-order factor contributed only 10.5% of the common variance. The second study included 344 children with a
variety of disorders who received a comprehensive neuropsychological evaluation, including the core WISC–IV battery, at a pediatric hospital (Bodin, Pardini, Burns, & Stevens, 2009). Similar
to Keith (2005), a higher order four-factor model best fit the data
of this clinical sample. The higher order general factor accounted
for 77% of the common variance and, it is interesting to note,
accounted for 99% of variance in the WM factor.
Although they are informative, these clinical studies were restricted to geographically local samples, and analytic methods
were circumscribed. Additional studies with different clinical samples and varied methods are needed (Bodin et al., 2009; Strauss,
Sherman, & Spreen, 2006), because the validity of a structural
model is enhanced if the same model can be replicated in new
samples (Raykov & Marcoulides, 2000). In addition, the structure
of the core WISC–IV battery should receive further attention,
because most clinicians only administer the 10 required subtests.
For example, Watkins et al. (2006) found that fewer than 6% of
their clinical sample received supplemental subtests. Consequently, the structure of the WISC–IV core battery was analyzed
with CFA methods among a national sample of students referred
for psychoeducational evaluation. Method
Participants
Participants included 355 students (218 male and 137 female)
who ranged in age from 6 to 16 years (M = 9.78 years, SD =
2.54). Grade placement ranged from kindergarten to Grade 11,
with a median of fourth grade. Ethnic background of participants,
if reported, was 62.7% White, 20.6% Black, 12.3% Hispanic, 2.4%
Asian/Pacific Islander, and 2.0% other. All students were assessed
to determine eligibility for special education services, but approximately 30% of the participants were determined not to be disabled. In contrast, 70% of the participants were reported to be
eligible for special education services (41% with learning disabilities, 7% with mental retardation, 6% with emotional disabilities,
2% as gifted, 4% with speech disabilities, 9% with other health
impairments, and 1% with autism spectrum disabilities). To ensure
anonymity, no other demographic data were collected. Instrument
The WISC–IV is a revised version of the WISC–III that was
standardized on a nationally representative sample of 2,200 chil- 783 dren aged 6 –16 years closely approximating the 2000 U.S. Census
on sex, race, parent education level, and geographic region. The
WISC–IV core battery contains 10 core subtests (M = 10, SD =
3) that form the four (VC, PR, WM, and PS) factor indices (M =
100, SD = 15). The Full Scale IQ (M = 100, SD = 15) is based
on the sum of scores from the 10 core subtests. Reliability and
validity evidence was provided by Wechsler (2003b) and by Williams, Weiss, and Rolfhus (2003). Additional information on the
WISC–IV was presented by Flanagan and Kaufman (2009) and by
O’Donnell (2009). Procedure
Following IRB approval, data for this study were solicited and
collected electronically. A commercial marketing firm contacted
2,384 school psychologists via e-mail and invited them to anonymously contribute WISC–IV scores from anonymous students
recently evaluated in their schools. Ninety-three (19 male and 74
female) school psychologists from 34 states responded by entering
WISC–IV data for 355 students onto a Web form. On average,
each school psychologist contributed 2.9 cases (SD = 3.7). Of the
responding school psychologists, 19% held a master’s degree, 65%
a specialist degree, and 16% a doctoral degree. Years of experience of contributing psychologists ranged from 1 to 34 (Mdn =
5.5, SD = 10.1). Analyses
CFA using maximum likelihood estimation was applied to the
covariance matrix using EQS for Windows (Version 6.1). The
obtained solutions were checked for convergence, and the adequacy of the parameter estimates and their associated standard
errors were examined prior to considering the reported fit indices.
According to Hu and Bentler (1998), values > .95 for comparative
fit index (CFI), < .08 for standardized root-mean-square residual
(SRMR), and < .06 for root-mean-square error of approximation
(RMSEA) indicate that there is a good fit between the hypothesized model and the sample data. Although Marsh, Hau, and Wen
(2004) cautioned against overgeneralization, these cutoff values
seemed appropriate given the variables analyzed in the current
study (Bodin et al., 2009). In addition, Akaike’s information
criterion (AIC) was consulted. The AIC considers statistical
goodness-of-fit as well as model parsimony, with smaller values
representing a better fit.
Each of the six models selected for this study was designed to
evaluate a specific hypothesis about the structure of the WISC–IV
(Bodin et al., 2009; Keith, 2005; Keith et al., 2006; Wechsler,
2003b). As illustrated in Table 2, the first four models contained
one to four first-order oblique factors, as per Wechsler (2003b).
The final two models conceptualized general intelligence in disparate ways, as per Keith (2005). The traditional higher order
model specified that general intelligence had a direct influence on
the first-order factors but only influenced the subtests indirectly
through the first-order factors (see Figure 1). That is, the association between the general intelligence factor and the subtests was
mediated by the first-order factors (Yung, Thissen, & McLeod,
1999). Gignac (2008) called this an indirect hierarchical model. In
contrast, the direct hierarchical model (see Figure 2) specified a
first-order general intelligence factor that had a direct effect on the 784 WATKINS Figure 1. Indirect hierarchical structure, with standardized coefficients,
of the Wechsler Intelligence Scale for Children—Fourth Edition (Wechsler, 2003a) for 355 referral students. SI = Similarities; VC = Vocabulary;
CO = Comprehension; BD = Block Design; PCn = Picture Concepts;
MR = Matrix Reasoning; DS = Digit Span; LN = Letter-Number Sequencing; CD = Coding; SS = Symbol Search; VC = Verbal Comprehension; PR = Perceptual Reasoning; WM = Working Memory; PS =
Processing Speed; g = General Intelligence. 10 subtests. Each of the four first-order factors also had a direct
effect on specific subtests, but the general factor had no effect on
the first-order factors. Thus, each subtest contributed variance
directly to the general intelligence factor and to a narrower group
factor. This model has variously been called the bi-factor model
(Holzinger & Swineford, 1937), nested factor model (Gustafsson
& Balke, 1993), and direct hierarchical model (Gignac, 2008).
These final two models contrasted general intelligence as a higher
order superordinate factor versus a first-order breadth factor, respectively (Gignac, 2008). Results
Participants’ mean WISC–IV subtest, factor, and IQ scores were
slightly lower and somewhat more variable than the normative
sample (see Table 1). Similar patterns have been found with other
samples of referred students (Canivez & Watkins, 1998). Nevertheless, score distributions appeared to be relatively normal, with
.51 the largest skew and .89 the largest kurtosis (Onwuegbuzie &
Daniel, 2002). Two other conditions for multivariate normality are
that all linear combinations of variables follow a normal distribution, and all subsets of variables in the data set are normally
distributed (Stevens, 2009). This was verified by examining the
scatterplots of all variable pairs. All scatterplots had an elliptical
shape. In addition, multivariate kurtosis was examined with the Figure 2. Direct hierarchical structure, with standardized coefficients, of
the Wechsler Intelligence Scale for Children—Fourth Edition (Wechsler,
2003a) for 355 referral students. SI = Similarities; VC = Vocabulary;
CO = Comprehension; BD = Block Design; PCn = Picture Concepts;
MR = Matrix Reasoning; DS = Digit Span; LN = Letter-Number Sequencing; CD = Coding; SS = Symbol Search; VC = Verbal Comprehension; PR = Perceptual Reasoning; WM = Working Memory; PS =
Processing Speed; g = General Intelligence. normalized value calculated by EQS. In practice, normalized values greater than |5.0| indicate that the data are multivariate nonnormal (Byrne, 2006). The normalized value for this sample was
3.78, indicating a reasonable degree of multivariate normality.
A review of model fit statistics (see Table 3) indicates that the
final three models met a priori guidelines for a close fit (i.e., CFI >
.95, SRMR < .08, and RMSEA < .06). Thus, the one-, two-, and
Table 1
Descriptive Statistics for 355 Referral Students on the Wechsler
Intelligence Scale for Children—Fourth Edition
WISC–IV component M SD Skew Kurtosis Verbal Comprehension Index
Perceptual Reasoning Index
Working Memory Index
Processing Speed Index
Full Scale IQ
Block Design
Similarities
Digit Span
Picture Concepts
Coding
Vocabulary
Letter-Number Sequencing
Matrix Reasoning
Comprehension
Symbol Search 92.4
95.5
89.7
90.8
90.6
8.9
8.8
8.1
9.7
8.3
8.6
8.4
9.0
8.7
8.3 16.5
16.9
15.6
15.3
16.9
3.3
3.4
3.0
3.3
3.2
3.1
3.3
3.3
3.2
3.1 +.30
-.04
+.11
+.06
+.09
+.23
+.35
+.24
-.26
+.35
+.27
-.33
+.28
-.05
-.33 +.86
+.02
+.52
+.08
+.76
-.36
+.03
+.44
-.03
+.09
+.53
+.02
+.04
-.02
+.23 Note. WISC–IV = Wechsler Intelligence Scale for Children—Fourth
Edition (Wechsler, 2003a). 785 WISC–IV FACTOR STRUCTURE Table 2
Six Structural Models of the Wechsler Intelligence Scale for Children—Fourth Edition
Subtest Model SI VC CO BD PCn MR DS LN CD SS One factor
Two factor
Three factor
Four factor
Indirect hierarchical
Direct hierarchical I
I
I
I
I
I I
I
I
I
I
I I
I
I
I
I
I I
II
II
II
II
II I
II
II
II
II
II I
II
II
II
II
II I
I
III
III
III
III I
I
III
III
III
III I
II
III
IV
IV
IV I
II
III
IV
IV
IV Note. SI = Similarities; VC = Vocabulary; CO = Comprehension; BD = Block Design; PCn = Picture Concepts; MR = Matrix Reasoning; DS = Digit
Span; LN = Letter-Number Sequencing; CD = Coding; SS = Symbol Search; I is factor one, II is factor II, III is factor III, and IV is factor four. Wechsler
Intelligence Scale for Children—Fourth Edition (Wechsler, 2003a). three-factor models were inadequate. RMSEA and AIC values for
the first-order four-factor model were inferior to those of the
hierarchical models. Beyond statistical fit, the oblique four-factor
model was unsatisfactory because it did not include general intelligence as specified by Wechsler (2003b) and others (Carroll,
1993, 2003; Gustafsson, 1994; Jensen & Weng, 1994). Of the two
remaining models, the direct hierarchical model was statistically
superior to the indirect hierarchical model (df = 2, �X2 = 6.68,
p
= .048).
On the basis of these statistical and theoretical arguments, the
direct hierarchical model was deemed preferable and is presented
with standardized loadings in Figure 2. Overall, the influence of
general intelligence dwarfed the contributions made by the four
WISC–IV first-order factors. The general factor accounted for
75% of the common variance and 48% of the total variance. The
VC factor accounted for 7.5% common and 4.8% total variance,
the PR factor for 4.9% common and 3.1% total variance, the WM
factor for 2.9% common and 1.9% total variance, and the PS factor
for 9.5% common and 6.1% total variance. Altogether, the general
and broad factors accounted for approximately 64% of the total
variance, leaving 36% unique variance. Discussion
Factor analyses of the WISC–IV scores of a national sample of
students referred to school psychologists for evaluation to determine eligibility for special education services essentially replicated
the results of Wechsler (2003b), Keith (2005), and Watkins (2006) for the general population as well as Watkins et al. (2006) and
Bodin et al. (2009) for clinical samples. The structure of the
WISC–IV core battery among this sample of students was best
represented by the four first-order factors named VC, PR, WM,
and PS by Wechsler (2003b), plus a general intelligence factor.
Analyses of the Wechsler Adult Intelligence Scale—Revised
(Wechsler, 1981) and the Wechsler Adult Intelligence Scale—
Third Edition (Wechsler, 1997) favored a direct hierarchical model
(Gignac, 2005, 2006). The direct hierarchical model was also a
statistically better fit than an indirect hierarchical model with the
WISC–IV normative sample, but Keith (2005) preferred the indirect hierarchical model on theoretical grounds. However, it is
difficult to interpret the relative contributions of first-order and
higher order factors within the indirect hierarchical model without
additional computations or transformations (e.g., a Schmid and
Leiman, 1957, orthogonalization as recommended by Carroll,
1993). In addition, Gignac (2006) questioned the appropriateness
of the full mediation implied by the indirect hierarchical model and
suggested that “it is arguably more congruent and reasonable to
specifically model the most significant factor of a battery of tests
(i.e., ‘g’) directly, rather than indirectly, through first-order factors” (p. 85). Gignac (2008) also reasoned that a first-order breadth
factor is more appropriate than a higher order superordinate factor,
because a principal aspect of general intelligence is its breadth
rather than its superordination. Following this logic, Gustafsson
(2001) observed that the direct hierarchical model often “results in
more parsimonious models with fewer latent variables” (p. 30). Table 3
Fit Statistics for Six Structural Models of the Wechsler Intelligence Scale for Children—Fourth Edition Among 355 Students Referred
for Psychoeducational Evaluation
Model df X2 CFI SRMR RMSEA 90% CI RMSEA AIC One factor
Two factor
Three factor
Four factor
Indirect hierarchical
Direct hierarchical 35
34
32
29
31
27 240.87
182.80
120.02
65.59
68.48
58.91 .893
.922
.954
.981
.980
.983 .063
.059
.046
.028
.030
.028 .129
.111
.088
.060
.058
.058 .114–.144
.095–.127
.071–.105
.040–.079
.040–.077
.038–.078 170.87
114.80
56.02
7.59
6.48
4.91 Note. CFI = comparative fit index; SRMR = standardized root-mean-square residual; RMSEA = root-mean-square error of approximation; CI =
confidence interval; AIC = Akaike’s information criterion. All X 2 values p < .001. Loadings of two-subtest factors constrained to be equal to ensure local
independence in the direct hierarchical model. Wechsler Intelligence Scale for Children—Fourth Edition (Wechsler, 2003a). 786 WATKINS F. F. Chen, West, and Sousa (2006) also noted that the direct
hierarchical model offers greater utility in predicting external
criteria. This benefit was theoretically and empirically explicated
by Schmiedek and Li (2004) and Brunner (2008). A linear dependency of parameters imposed by the structure of the indirect
hierarchical model “does not afford simultaneous estimations of
general and specific effects” (Schmiedek & Li, 2004), and results
in “a severe limitation to the understanding of how general and
specific abilities relate to other theoretical constructs” (p. 162).
Thus, the WISC–IV general intelligence factor is best interpreted
as a first-order breadth factor as specified in the direct hierarchical
model (see Figure 2).
The general factor was the predominate source of variation
among WISC–IV subtests, accounting for 75% of the common
variance and 48% of the total variance. In fact, the general factor
explained three times the variance of the four narrow factors
combined. As a comparison, the general factor accounted for
75.7% of the common variance among a sample of referred students in Pennsylvania (Watkins et al., 2006), 77.2% of the common variance among a pediatric neuropsychological sample in the
southeastern United States, and 71.3% of the common variance
among the WISC–IV normative sample (Watkins, 2006). Given
these explanatory contributions, recommendations favoring interpretation of the narrow factor scores over the general intelligence
score (Prifitera, Saklofske, & Weiss, 2008; Wechsler, 2003b;
Weiss, Saklofske, Prifitera, & Holdnack, 2006; Williams et al.,
2003) appear to be misguided.
As with all studies, these conclusions must be tempered by
limitations of sampling and data collection. Although 93 school
psychologists contributed WISC–IV data, this was only 4% of
those solicited for participation. Other studies have also found low
response rates to such requests (Canivez & Watkins, 1998), and
there is no way to know if this particular sample was representative
of the population. Likewise, the use of electronic solicitation and
data collection may have affected the data in unknown ways.
Fortunately, these concerns are mitigated by results that were
consistent with previous studies of the nationally representative
standardization sample and two clinical samples. Therefore, the
best available evidence indicates that clinicians should favor interpretation of the general intelligence score over the narrow factor
scores when using the WISC–IV core battery. :

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