Hierarchical Structure of the Wechsler Adult Intelligence Scale–Fourth Edition with an Egyptian Sample

Method

Results and Discussion

Table 1 provides an overview of descriptive statistics for the WAIS-IV subscale (M = 26.36; SD = 5.5) and four-factor (M = 65.91; SD = 11.03). The skewness and kurtosis values are less than the criterion of |2.0| to prove normal univariate distribution (acceptable; George & Mallery, 2010).

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Table 1.

Descriptive Statistics and Internal Consistency Coefficients (Cronbach’s α, ω , ω h , and ω s ) for the WAIS-IV.

	M	SD	Skewness	Kurtosis	α	ω	ω h ( ω s )
Subscales
SIM	20.19	3.73	.40	0.82	.7
V	40.33	6.15	−.517	0.38	.86
INF	11.56	2.82	.47	−0.28	.78
BD	31.11	7.98	.76	0.82	.7
MR	16.82	4.24	−.22	−0.122	.82
VP	12.37	3.73	.38	−0.76	.83
DS	27.55	5.14	.29	−0.59	.81
AR	13.50	3.30	.35	−0.63	.82
SS	30.08	6.757	.45	1.289	.89
CD	60.12	11.17	−.09	−0.19	.95
Factors
VC	72.09	9.25	−.06	−0.13	.83	.51	.27
PR	60.30	12.53	.45	0.25	.82	.74	.20
WM	41.06	6.73	.25	−0.63	.81	.5	.27
PS	90.20	15.61	.11	−0.12	.95	.74	.68
General factor					.91	.79	.61

Note. WAIS-IV = Wechsler Adult Intelligence Scale–Fourth Edition; VC = Verbal Comprehension; PR = Perceptual Reasoning; WM = Working Memory; PS = Processing Speed; SIM = Similarities; V= Vocabulary; INF = Information; BD = Block Design; MR = Matrix Reasoning; VP = Visual Puzzles; DS = Digit Span; AR = Arithmetic; SS = Symbol Search; CD = Coding; ω = omega; ω h = omega hierarchical.

The top of Table 2 shows the fit statistics for the eight WAIS-IV models first proposed. The fit statistics suggested that the one-factor model and the two-factor model (verbal and nonverbal) were inadequate (χ² significant, χ²/df < 2; RMSEA and >.05, TLI and CFI < .95). For the four-factor structure, the fit statistics were achieved for the correlated four first-order factor model (M3), the direct hierarchical model (Bifactor, M4), the four-factor loading on a higher order factor (second-order, M5) model. Moreover, no significant differences were found between these three models in terms of χ² value or AIC coefficient, but M3 and M4 gave better values of CFI and RMSEA than M5.

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Table 2.

Confirmatory Factor Analysis Fit Statistics for the WAIS-IV: Egyptian Sample.

Model	χ2	df	RMSEA (90% CI)	CFI	TLI	SRMR	AIC	M9 vs. models
								Δχ2	Δdf	p	ΔAIC
M1: One-factor	110.9**		.1 [.082, .125]	.73	.65	.076	12,226.73
M2: Two-factor (verbal and nonverbal)	95.58**		.094 [.072, .117]	.78	.71	.070	12,213.33
Four-factor structure
M3: Four first-order factors	38.53	30	.037 [.0, .068]	.97	.95	.039	12,164.28	−4.12	1	<.05	−2.12
M4: Bifactor four-factor	38.0	29	.036 [.0, .073]	.97	.96	.041	12,163.76	—	—	—	−1.6
M5: Second-order four-factor	42.29	32	.040 [.0, .069]	.96	.95	.043	12,164.04	−7.88	3	<.05	−1.88
Five-factor structure
M6: Five first-order factors (GC, GV, GF, GSM, GS)	36.73	27	.042 [.0, .073]	.97	.94	.04	12,168.49	—	—	—	−3.76
M7: Bifactor five-factor	38.95	30	.038 [.00, .069]	.97	.95	.042	12,164.70	—	—	—	−3.88
M8: Second-order five-factor	47.4*	33	.046 [.00, .074]	.95	.93	.056	12,167.15	—	—	—	−6.33
Modification of four-factor models
M9: Modified four first-order factor (arithmetic loading on WM and PR)	34.4	29	.03 [.00, .064]	.98	.97	.037	12,162.16	—	—	—	—
M10: Modified bifactor four-factor (arithmetic loading on WM and PR)	38.16	29	.039 [.00, .07]	.97	.95	.041	12,165.92	—	—	—	−2.54
M11: Modified second-order four-factor (arithmetic loading on WM and PR)	42.29	31	.042 [.00, .07]	.96	.94	.043	12,166.04	−12.99	4	<.05	−4.99
M12: Modified four first-order factor (arithmetic loading on WM, PR, and VC)	34.4	28	.033 [.00, .067]	.97	.96	.037	12,164.13	−0.01	1	<.95	−1.97
M13: Modified bifactor four-factor (arithmetic loading on WM, PR, and VC)	38.16	28	.042 [.00, .07]	.96	.94	.041	12,167.9	−3.75	1	<.1	−5.74
M14: Modified second-order four-factor (arithmetic loading on WM, PR, and VC)	42.179	30	.045 [.00, .074]	.96	.93	.044	12,167.93	−7.769	1	<.01	−5.77
Modification of five-factor models
M15: Modified five first-order factors (arithmetic loading on GC, GF, and GSM)	36.83	27	.042 [.00, .074]	.97	.94	.041	12,168.48	—	—	—	−6.32
M16: Modified bifactor five-factor (arithmetic loading on GC, GF, and GSM)	38.83	29	.041 [.00, .071]	.96	.95	.042	12,166.58	—	—	—	−4.42
M17: Modified second-order five-factor (arithmetic loading on GC, GF, and GSM)	47.46*	32	.049 [.012, .076]	.94	.92	.049	12,169.21	—	—	—	−7.05

Note. WAIS-IV = Wechsler Adult Intelligence Scale–Fourth Edition; RMSEA = root mean square error of approximation; CFI = comparative fit index; TLI = Tucker–Lewis index; SRMR = standardized root mean square residual; AIC = Akaike information criterion.

*

p < .05. **p < .01.

The top of Table 3 summarizes the standardized loadings for the three plausible four-factor models: M3, M4 and M5. The 10 WAIS-IV subscales show a significant positive standardized loading value on the four factors (VC, PR, WM and PS) for M3 and M5 models. Similarly, the four-factor model (VC, PR, WM and PS) gives a significant positive loading on second-order factor, for M5. By contrast, in the bifactor model (M4), Matrix Reasoning yielded nonsignificant loadings on PR, and the Symbol Search subscale showed nonsignificant loading on general factor.

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Table 3.

Standardized Loadings for Four- and Five-Factor Models.

	M3				M4					M5				M6					M7						M8
	VC	PR	WM	PS	G	VC	PR	WM	PS	VC	PR	WM	PS	GC	GV	GF	GSM	GS	G	GC	GV	GF	GSM	GS	GC	GV	GF	GSM	GS
SIM	.48				.37	.27				.49				.46					.36	.27					.46
V	.41				.26	.56				.43				.40					.25	.57					.45
INF	.59				.42	.27				.57				.61					.41	.29					.58
BD		.62			.55		.17				.61				.62				.55		.13					.63
MR		.55			.54		.09				.75					.46			.57			.06					.64
VP		.74			.65		.76				.56				.77				.68		.68					.76
DS			.32		.21			.20				.31					.20		.19				.20					.20
AR			.73		.55			.62				.78				.31	.20		.53			.30	.69				.49	.82
SS				.49	.12				.47				.49					.49	.11					.48					.49
CD				.99	.31				.95				.99					.99	.31					.95					.99
G										.74	.85	.73	.31												.64	.98	.79	.10	.30

Note. Bold values indicate p > .05 (nonsignificant loading). VC = Verbal Comprehension; PR = Perceptual Reasoning; WM = Working Memory; PS = Processing Speed; G = General Factor; GC = Crystallized Intelligence; GV = Visual Processing; GF= Fluid Intelligence/Reasoning; GSM = Short-Term Memory; GS= Processing Speed; SIM = Similarities; V= Vocabulary; INF = Information; BD = Block Design; MR = Matrix Reasoning; VP = Visual Puzzles; DS = Digit Span; AR = Arithmetic; SS = Symbol Search; CD = Coding.

Regarding the five-factor CHC-based structure derived from terms of the CHC theory of cognitive abilities through (McGrew, 2009) alongside the results of previous research (van Aken et al., 2017; Benson et al., 2010; Weiss et al., 2013b), the three five-factor models again fitted the data (see Table 2). For these models, the Arithmetic subscale was loaded significantly on both the WM and GF factors. M7 was a better fit than M6 and M8, giving the lowest value of AIC. Moreover, ΔΔχ2 was statistically significant between M7 and M8 (Δχ² = −8.45, df = 3, p < .05) but not between M6 and M7 (Δχ² = −2.22, df = 3, p < .75). For standardized loadings of the five-factor models (M6, M7, M8), a very similar pattern is observed with respect to what is mentioned in the four-factor models, as well as the same problem on bifactor model regarding the nonsignificant loadings of specific scales (see the bottom of Table 3).

To optimize the fit, some modifications are proposed (see the bottom of Table 2). For the four-factor structure, following the addition of a path between the Arithmetic subscale and PR, in addition to WM, the fit of M9 was improved due to multiple loadings of the Arithmetic subscale on PR and WM, but M10 and M11 did not show any improvement of fit statistics relative to M4 and M5. Interestingly, the Arithmetic subscale showed nonsignificant loading on PR and VC in multiple attempts to find the best model fit when a path was added from the Arithmetic subscale to PR, WM, and VC for models M12, M13, and M14. For the five-factor structure, no improvement in fit was observed in the loadings of the Arithmetic subscale on GC, GF and GSM for models M15, M16, or M17. In addition, no significant loadings were found between Arithmetic and GC for the three modified models M15 (.11), M16 (.041), and M17 (.14).

The modified models reproduce the previous observations for the unmodified models. As shown in Figure 1 for M9, the loadings of 10 subscales on the four-factor structure were very similar to M3, which produced a significant positive standardized loading value on the four factors (VC, PR, WM, and PS). Interestingly, comparisons of M9 with nested models using Δχ² suggested that M9 provided better overall fit (see Table 2). In addition, ΔAIC also supported the use of M9 over the other models. In summary, the best fit appeared to be obtained for Model 9 (see Table 2).

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Figure 1.

Path diagram for M9.

In addition, the 10 subscales and four factors were shown to have acceptable internal consistency (see Table 1). The mean Cronbach’s alpha values for the 10 subscales and four factors were .81 and .85, respectively. Cronbach’s alpha for the general factor was very high (.91) and Coding and Symbol Search gave the highest internal consistency values (.95 and .89, respectively), while Similarities and Block Design gave the lowest (.7). The values of ω and ω h were low (0.79 and 0.61, respectively). For the four factors, ω ranged from 0.5 to 0.74 and ω h from 0.20 to 0.68. Zinbarg, Yovel, Revelle, and McDonald (2006) suggested that ω ≥ ω h and that equality occurs in the case of a unidimensional scale only. In addition, this result may be explained by the fact that the omega hierarchical is very limited for unique constructs due to the general factor (Reise, 2012). These results are in agreement with Gignac and Watkins (2013), who reported that the estimated reliability of the WAIS-IV according to ω h was very low for all age groups. However, the internal consistency findings must be interpreted with caution for the Coding and Symbol Search subscales because they are speeded scales.

Interestingly, both the four-factor structure and five-factor CHC theory models fit the WAIS-IV with the Egyptian sample, confirming the findings of (van Aken et al., 2017; Weiss et al., 2013b). For the four-factor structure, the clearest finding was that the second-order model and the four-factor structure were the best fit for the data compared with the bifactor model. These findings are comparable with those of (Canivez & Watkins, 2010; D. I. Miller et al., 2013; Reynolds, Ingram, Seeley, & Newby, 2013). One unanticipated finding was that the bifactor model presented nonsignificant loadings of the MR subscale on PR, and nonsignificant loadings of the SS on general factor and these results are consistent with those of Gignac and Watkins (2013). Moreover, for the CHC-based structure, the five first-order factors (GC, GV, GF, GSM, GS), bifactor five-factor, and second-order five-factor models were also suitable, but the bifactor five-factor model presented nonsignificant loadings of the MR on the general factor, similar to the bifactor four-factor model. A possible explanation for this is that the bifactor model does not include mediation between a higher order factor and the observed variables by the first-order factor; the bifactor model focuses on the direct influences of a general factor on those subscales that also are loaded with nested factors directly, making the general factor closer to subscales (Gignac, 2008).

Another important finding was that adding a path from the Arithmetic subscale to the factors PR and WM for three four-factor structure models showed that the modified four first-order factor model was preferable to the other four- and five-factor models. In addition, the most obvious finding to emerge from the analysis is that Arithmetic is a complex subscale requiring multiple abilities (van Aken et al., 2017; Benson et al., 2010; Bowden et al., 2011; Ward, Bergman, & Hebert, 2012; Weiss et al., 2013b). The current study indicated that Arithmetic loaded significantly on WM (.34) and PR (.40) in the modified four first-order factor model and on GSM (.303) and GF (.689) in the bifactor five-factor model, but it showed nonsignificant when another path was added to VC, confirming with those obtained by Ward et al. (2012) who reported Arithmetic loadings of .32 and .35 on GSM and GF, in addition to slight loading on VC (.17) for the five-factor model. These results are also consistent with the findings of Weiss et al. (2013b), who showed that Arithmetic produced loadings of .36 and .27 on WM and PR, respectively, in the four-factor model. Our findings also corroborate the ideas of Nelson et al. (2013), who suggested that the bifactor model was suitable for the WAIS-IV structure using the normative data, although the correlated four first-order factor model gave a better fit. However, the findings presented here do not support those of Nelson et al., according to which the subscale loadings in the direct hierarchical model were significant. For example, the loading of Symbol Search on the general factor in the bifactor model was.362 according to Nelson et al. and 0.120 in this study, but the loading was nonsignificant on the general factor in the bifactor model. On the opposite, these results differ from the study of Benson et al. (2010) who found that the second-order model fit the data but selected the CHC model as the fittest for the data.

This study helps to disseminate knowledge of one of the most up-to-date and widely used measures in the field of intelligence testing, facilitating the development of scales that are suitable for the Egyptian community and its standards, especially Arab speakers for whom English is not a native language. The implications are important for the development of measures like the WAIS-IV that can be used by school psychologists and counselors to diagnose cognitive abilities and thus help to provide essential services for community institutions, particularly for Arabic speakers who do not speak English as a native language.

This study was limited by the absence of supplemental subscales of the WAIS-IV in our analysis. One of these subscales is Figure Weights, which showed the best loading on the GF factor in the five-factor model and also on the PR factor in the four-factor model in many studies comparing other core subscales, such as Ward et al. (2012). One additional limitation is that the WAIS-IV structure was explored with a randomly selected sample that satisfies the normal distribution criteria but was collected from a particular population and restricted by age (18-24). Caution should be exercised when generalizing the results to other age groups and regions.

In summary, the main goal of the current study was to adapt the WAIS-IV for Arabic-speaking populations. In addition, this study examined dimensionality and reliability using ω h and ω for the WAIS-IV with an Egyptian sample. According to our data, we can infer that the four- and five-factor models are suitable but that the modified four first-order factor model should be preferred. One of the more significant findings to emerge from this study is that the indirect hierarchical models of both the five-factor and the four-factor structures provide a better fit than the direct hierarchical models (bifactor models). This finding may support the hypothesis presented in the WAIS-IV Technical and Interpretive Manual (Wechsler, 2008) that selected the second-order model and four first-order factor model as the best models. Despite these promising results, further work is required to establish the viability of dimensionality invariance for WAIS-IV across ages (25-90).