Psychological Assessment 2015, Vol. 27, No. 4, 1312–1323

© 2015 American Psychological Association 1040-3590/15/$12.00 http://dx.doi.org/10.1037/pas0000105

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Digit Span Is (Mostly) Related Linearly to General Intelligence: Every Extra Bit of Span Counts Gilles E. Gignac

Lawrence G. Weiss

University of Western Australia

Pearson Clinical & Talent Assessment, San Antonio, Texas

Historically, Digit Span has been regarded as a relatively poor indicator of general intellectual functioning (g). In fact, Wechsler (1958) contended that beyond an average level of Digit Span performance, there was little benefit to possessing a greater memory span. Although Wechsler’s position does not appear to have ever been tested empirically, it does appear to have become clinical lore. Consequently, the purpose of this investigation was to test Wechsler’s contention on the Wechsler Adult Intelligence Scale-Fourth Edition normative sample (N ⫽ 1,800; ages: 16 – 69). Based on linear and nonlinear contrast analyses of means, as well as linear and nonlinear bifactor model analyses, all 3 Digit Span indicators (LDSF, LDSB, and LDSS) were found to exhibit primarily linear associations with FSIQ/g. Thus, the commonly held position that Digit Span performance beyond an average level is not indicative of greater intellectual functioning was not supported. The results are discussed in light of the increasing evidence across multiple domains that memory span plays an important role in intellectual functioning. Keywords: memory span, Digit Span, general intelligence, working memory

the association between memory span, as measured by Digit Span from the Wechsler Adult Intelligence Scale-Fourth Edition (WAIS-IV; Wechsler, 2008a), and g. If Digit Span scores were observed to share a meaningful and largely linear association with g, then the relatively poor view of the Digit Span test within the clinical assessment community would need to be reconsidered. By contrast, if Digit Span scores were observed to share a nonlinear (quadratic and negative) association with g, then Wechsler’s position would be substantiated. Either position would be an important one to demonstrate empirically. From an applied clinical perspective, it was considered important to address this question, because Digit Span is one of the 10 core subtests within the Wechsler scales that are used to estimate an individual’s FSIQ. If Digit Span were observed to share a nonlinear association with g, as contended by Wechsler, then perhaps it would be appropriate to consider substituting it for another subtest, one that has a respectable association with g across the whole spectrum of ability.

“Memory span for digits has been underrated as a psychometric test by most clinical psychologists.” —(Jensen, 1970, p. 71)

The assessment of individual differences in memory span has a long history (Blankenship, 1938; Dempster, 1981). Furthermore, the reputation of memory span as an indicator of intellectual functioning has varied substantially over the years, from a poorly regarded cognitive ability considered seriously for exclusion from the Wechsler scales (Matarazzo, 1972) to a construct that may be isomorphic, or nearly so, with general intelligence (g; Colom, Rebollo, Palacios, Juan-Espinosa, & Kyllonen, 2004). Wechsler (1939, 1958; see also Matarazzo, 1972) contended that, beyond an average level of memory span performance, there was little in the way of benefits with respect to intellectual functioning. However, such a postulation does not appear to have ever been tested empirically, despite the fact that it has largely become clinical lore (Fruchter & Fruchter, 1973; Glasser & Zimmerman, 1967; Halstead, 1944; Whimbey & Whimbey, 1975). Consequently, the purpose of this investigation was to evaluate the precise nature of

Digit Span: Description and Historical Reputation Historically, memory span has been measured across a number of modalities with various stimuli (e.g., verbal, spatial, nonsense words, etc.), however, over time, memory for digits became the preferred method of memory span assessment within the conventional cognitive ability assessments (Blankenship, 1938; Dempster, 1981; Wambach, Lamar, Swenson, Penney, Kaplan, & Libon, 2011). The Wechsler scales have included a measure of memory span for digits, known as Digit Span, since the inception of the Wechsler-Bellevue scale (Wechsler, 1939). With respect to the first four editions of the Wechsler scales (Wechsler, 1939; 1955; 1981; 1997), Digit Span consisted of two subtests: Digit Span Forward and Digit Span Backward. Digit Span Forward requires the participant to recall a series of random single digits in the order

This article was published Online First March 16, 2015. Gilles E. Gignac, School of Psychology, University of Western Australia; Lawrence G. Weiss, Pearson Clinical & Talent Assessment, San Antonio, Texas. Lawrence G. Weiss was involved in the research and development of the WAIS-IV as an employee of Pearson, which is the publisher of numerous psychological tests including the Wechsler scales. We thank Mark Hurlstone and Klaus Oberauer for answering some questions during the preparation of this article. Correspondence concerning this article should be addressed to Gilles E. Gignac, School of Psychology, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia, 6009, Australia. E-mail: gilles.gignac@ uwa.edu.au 1312

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DIGIT SPAN AND g

with which they were read. The digits are read to the participant at a rate of one per second and the participant must repeat the digits orally. The sequence of digits vary in length from three to nine.1 By contrast, Digit Span Backward requires the participant to recall the random single digits in the reverse order with which they were read. The sequence of digits varies in length from two to eight. Digit Span Forward has been categorized as a measure of shortterm memory (STM) capacity (STMC), whereas Digit Span Backward, which requires the reverse ordering of the digits before recall (i.e., mental manipulation), has been categorized as a measure of working memory capacity (WMC; Oberauer, Sü␤, Schulze, Wilhelm, & Wittmann, 2000). In the latest version of the adult Wechsler scale (WAIS-IV; Wechsler, 2008a), an additional measure of WMC, Digit Span Sequencing, has been added to the Digit Span subtest. Digit Span Sequencing requires the participant to recall the randomly presented single digits in ascending numeric order (lowest to highest). Although memory span tests have been included in the early intelligence batteries (e.g., Stanford-Binet; Terman, 1917), the inclusion of memory span tests at such a time was considered largely unsubstantiated, as there was no obvious link between scores from such tests and school grades (Estes, 1981). Bronner, Healy, Lowe, and Shimberg (1927) recognized some utility associated with Digit Span test scores, however, they ultimately contended that “the value of the whole test has been greatly overestimated” (p. 197). In his description and evaluation of the Digit Span subtest, Wechsler (1939, 1958; see also Matarazzo, 1972) contended that Digit Span test scores were largely independent of g, a view that continues to be expressed in some relatively contemporaneous clinical assessment texts (e.g., Sbordone & Saul, 2000). Perhaps most critically, Wechsler suggested that having an above average memory span ability was of little benefit: Rote memory more than any other capacity seems to be one of those abilities of which a certain absolute minimum is required, but excesses of which seemingly contribute relatively little to the capacities of the individual as a whole. (Wechsler, 1958, p. 71)

Within the specific context of intellectual functioning assessment in children, Glasser and Zimmerman (1967) expressed a very similar critical view: [Digit Span] makes the assumption that rote memory is one of those abilities of which a certain absolute minimum is required for all levels of intellectual functioning. However, excesses above this absolute minimum seemingly contribute relatively little to the capacity . . . to function as a whole. It appears then to be one of those abilities which enter into intellectual functioning only as necessary minima.” (p. 96)

Given the perceived relative lack of validity associated with the Digit Span subtest as an indicator of g, it was considered seriously for complete removal from the WAIS (Matarazzo, 1972). Other critical views include Estes (1974) who categorized Digit Span as a basic associative capacity measure of lower levels of intellectual functioning. Sattler (1965, 1982) referred to Digit Span as a measure of nonmeaningful memory. In the context of the Differential Ability Scales (Elliott, 1990), Hughes and McIntosh (2002) also referred to Digit Span as a measure of nonmeaningful memory. Finally, several sources have contended that Digit Span Forward is not a test of memory span at all, but, instead, a test of elementary attention (Hannay, Howieson, Loring, Fischer, &

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Lezak, 2004; Hebben & Milberg, 2009; Rapaport, Gill, & Schafer, 1945; Sbordone & Saul, 2000). In light of the above, it seems clear that Digit Span has been largely considered a relatively poor measure of intellectual functioning within much of the clinical assessment literature. In addition to issues relevant to validity, another psychometric factor that likely helped facilitate Digit Span’s poor reputation was the fact that it was, initially, one of the subtests which yielded the least reliable scores. In the Wechsler-Bellevue (Wechsler, 1939) and the WAIS (Wechsler, 1955), Digit Span was reported to be associated with retest reliabilities of .67 and .68, respectively. Even worse, Digit Span within the WISC (Wechsler, 1949) was reported to be associated with retest reliabilities of .50 to .60 across age groups. McNemar (1942) also noted the relatively low reliability (⬇.70) associated with the memory span test scores associated with the Stanford-Binet. Thus, the relatively low initial reliability estimates associated with memory span test scores were not unique to the Wechsler scales. More important, however, Blackburn and Benton (1957) demonstrated that the reliability of Digit Span test scores could be enhanced by always administering both trials within each digit series, irrespective of whether the participant recalled the first trial correctly. Consequently, within the WAIS-R (Wechsler, 1981) and subsequent Wechsler adult editions, the Digit Span instructions required both trials associated with each item to be administered, until the discontinue rule was reached (failure of both trials of any item). As would be expected based on the results of Blackburn and Benton (1957), Digit Span within the WAIS-R (Wechsler, 1981) was reported to be associated with a respectable retest reliability of .83, which was very comparable with the retest reliabilities associated with the other Wechsler subscales. Digit Span within the WAIS-IV (Wechsler, 2008) was reported to be associated with retest reliability of .83 and an internal consistency reliability of .93. Thus, much more respectable levels of reliability have been achieved with the revised Digit Span subtest in the WAIS-R and later editions. Consequently, the relatively poor view of Digit Span test scores on the grounds of poor reliability is no longer justifiable. Furthermore, as a respectable level of reliability is a necessary (but not sufficient) condition for validity (Bendig, 1952; Mehrens & Lehmann, 1969), it is plausible to consider Digit Span test scores as possibly associated with respectable levels of validity.

Memory Span and Cognitive Science In contrast to the commonly held impression of memory span as a relatively poor indicator of cognitive functioning within much of the intellectual assessment literature, both STM and working memory are highly regarded constructs in the area of cognition (Conway, Jarrold, Kane, Miyake, & Towse, 2007; Hurlstone, Hitch, & Baddeley, 2014). For example, serial recall (of which Digit Span Forward is an operational example) is not considered simply a measure of individual differences in attention or nonmeaningful memory, in contrast to how Digit Span test scores are often considered within the intellectual assessment community (e.g., Hannay, Howieson, Loring, Fischer, & Lezak, 2004; Hebben & 1 Within the WAIS-IV (Wechsler, 2008a), Digit Span Forward includes an additional item of two trials with two digits.

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Milberg, 2009; Rapaport, Gill, & Schafer, 1945; Sbordone & Saul, 2000). Instead, a large body of empirical and theoretical cognitive research has accumulated to substantiate serial recall as one of the fundamental human memory systems (Jonides, Lewis, Nee, Lustig, Berman, & Moore, 2008). Furthermore, detailed computational models have been proposed to represent the cognitive processes involved with serial recall (e.g., Brown, Neath, & Chater, 2007; Farrell, & Lewandowsky, 2002; Lewandowsky, & Farrell, 2008). More important, although an element of attention may form a part of a model of serial recall (e.g., Burgess & Hitch, 1992; Oberauer & Lewandowsky, 2011), at the core of serial recall models is the capacity to maintain and recall objects in STM (Hurlstone et al., 2014). Thus, the commonly expressed view of serial recall (i.e., Digit Span) in the clinical assessment literature as an indicator of elementary attention or nonmeaningful memory is fundamentally at odds with the cognitive science literature. Additionally, as recently reviewed by Conway and Kovacs (2013), individual differences in both serial recall (a.k.a., STMC) and WMC have been found to relate substantially to fluid intelligence and g. For example, based on a meta-analytic review of 14 samples (N ⫽ 3,168), Kane, Hambrick, and Conway (2005) estimated that WMC and fluid intelligence shared 50% of their true score variance. Because the investigations included in Kane et al. were almost all based on range restricted university samples, Gignac (2014a) estimated the WMC and fluid intelligence latent variable association based on the WAIS-IV (Wechsler, 2008b) normative sample (N ⫽ 2, 220). Based on a correlated two-factor model, Gignac (2014a) found that WMC and fluid intelligence shared closer to 60% of their true score variance. As the Digit Span Backward subtest had a standardized loading of .72 on the WMC latent variable, it may be suggested that the correlation between Digit Span Backward and fluid intelligence was .55 (.72 ⴱ 77 ⫽ .55).2 Additionally, based on a bifactor model of the WAIS-IV, Gignac (2014a) reported a Digit Span Backward g loading of .58. Thus, working memory capacity, as measured by Digit Span Backward, may be considered a moderately strong correlate of fluid intelligence and g. In contrast to Digit Span Backward, Digit Span Forward is not typically considered as strong a correlate of fluid intelligence and/or g (e.g., Engle, Tuholski, Laughlin, & Conway, 1999; Jensen & Figueroa, 1975; Prokosch, Yeo, & Miller, 2005). Both theoretically and empirically, the forward recall of objects is considered representative of STMC, as it does not require the manipulation or transformation of information (Oberauer et al., 2000). Consequently, as forward recall does not require the same level of cognitive resources (Li & Lewandowsky, 1995; St ClairThompson & Allen, 2013), it is naturally expected to be a less strong correlate of fluid intelligence and g. In a footnote within the Kane et al. investigation, a moderate meta-analytically derived correlation of .30 between STMC and fluid intelligence was reported, which suggests that only ⬃10% of their variance is shared. However, based on the WAIS-IV normative sample correlation matrix (N ⫽ 2,200), I estimated the correlation between a fluid intelligence latent variable (Matrix Reasoning, Figure Weights, and Block Design) and the Digit Span Forward observed variable at .51, which suggests that ⬃26% of their variance is shared. Furthermore, based on a bifactor model of the WAIS-IV, Gignac (2014a) reported a Digit Span Forward g loading of .46. Thus, although Digit Span Backward is a more

substantial correlate of fluid intelligence and g, Digit Span Forward should be considered a moderate correlate, as well. These results are consistent with several other investigations (e.g., Colom, Abad, Rebello, & Chun Shih, 2005; Colom, Flores-Mendoza, Quiroga, & Privado, 2005). Ultimately, as latent variable modeling research has established a substantial (r ⬇ .80) association between STMC and WMC (Conway & Kovacs, 2013), virtually any effects relevant to WMC would likely be shared, to some degree, by STMC.

Memory Span and Neuroscience From a more neuroscientific perspective, the nonlinear effect postulated by Wechsler may be suggested to be unlikely. For example, individual differences in intelligence have been suggested to be mediated, in part, by individual differences in the efficiency with which information can be processed by the brain (the neural efficiency hypothesis; Neubauer & Fink, 2009). The evidence in favor of the neural efficiency hypothesis does not suggest that advantages of efficient information processing diminish across the spectrum of ability. In the context of Digit Span, specifically, it will be noted that there is some neuroscientific research that suggests that superior numerical memory span performers use a different part of their brain, in comparison to average performers, when executing a numerical memory span task. For example, Tanaka, Michimata, Kaminaga, Honda, and Sadato (2002) examined the activation of the brain during the completion of a numerical memory span task in a group of abacus experts and a group of controls. The abacus experts achieved a mean digit span of 12.2 and the controls 8.5 (education levels were about the same between the groups; 14.7 and 15.8, respectively). Tanaka et al. found that, in superior performers, the cortical areas relevant to visual spatial skills were predominantly activated during the completion of the digit span task. By contrast, in average performers, the cortical areas relevant to verbal skills (e.g., Broca’s area) were predominantly activated during the digit span task. Thus, superior digit span performers appear to use a different part of the brain, in comparison to more average performers. Consequently, the switch from a verbal to a spatial strategy may involve some change in the strength of the association between memory span and general intelligence. However, it is difficult to hypothesize specifically how the nature of the association might change across the spectrum of ability, if at all.

Study Purpose Although the empirical research reviewed above may be argued to be supportive of the notion that memory span is an important indicator of cognitive ability, it does not address the main argument made by Wechsler and others—that memory span is related nonlinearly to g, such that beyond an average level of span there are little or no benefits to intellectual functioning. Consequently, the purpose of this investigation was to examine specifically the nature of the association between Digit Span test scores and general intellectual functioning (FSIQ and g) within the WAIS-IV 2 To estimate the association between observed and latent variables within a structural equation model, one can use the path tracing rule (Mulaik & Quartetti, 1997).

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DIGIT SPAN AND g

(Wechsler, 2008) normative sample. In contrast to past factor analytic investigations of the WAIS-IV (e.g., Canivez & Watkins, 2010; Gignac, 2014a; Gignac & Watkins, 2013), the analyses performed in this investigation will be based on the Longest Digit Span Forward (LDSF), Longest Digit Span Backward (LDSB), and Longest Digit Span Sequencing (LDSS) raw data, which will allow for precise and informative estimates of increases in FSIQ/g for each unit increase in memory span. If a nonlinear (quadratic) association were observed, such that beyond an approximate average level of memory span there was no association with g, then Wechsler’s and others’ relatively critical view of Digit Span would be supported. By contrast, if a largely linear association between Digit Span test scores and g were observed, then the relatively favorable view of memory span test scores observed in the cognitive science literature would be supported.

Method Sample and Measure All of the analyses were based upon a combination of the raw and scaled data associated with the WAIS-IV normative sample (Wechsler, 2008b). The data were made available by the test publisher. The WAIS-IV normative sample was collected based on a stratified sampling strategy to reflect the U.S. census results relevant to gender, age, race/ethnicity, education, and geographic location (Wechsler, 2008b). The Wechsler scales are widely considered valid measures of intelligence (Hunsley & Lee, 2009). Although the total WAIS-IV normative sample is based on 2,200 participants, the participants aged 70 and above did not complete the Letter-Number Sequencing, Figure Weights, and Cancellation subtests (Wechsler, 2008b). Consequently, the total sample size upon which the analyses were performed in this investigation was N ⫽ 1,800. Raw scores were made available for the LDSF, LDSB, and LDSS variables. Scaled scores were made available for the remaining 15 WAIS-IV subtests. As, the LDSF, LDSB, and LDSS variables are, in essence, the Digit Span subtest, the overall Digit Span subtest scores were not included in any of the analyses. An examination of the data revealed that a small number of participants (N ⫽ 3) achieved a very low score of two on LDSF. As these participants also scored very low on the remaining subtests, the data were considered valid. However, for the purposes of analyses, a minimum sample size of 10 was considered required for each memory span group. A memory span group corresponded to the individuals in the normative sample who achieved the same longest digit span score on a particular digit span subtest (Forward, Backward, or Sequence). Consequently, in this case, the three scores equal to two on the LDSF variable were Winsorized3 to a score of three, which yielded N ⫽ 12 for the lowest forward memory span group (i.e., span ⫽ three). Similarly, only six participant scores were equal to one or less on LDSB. Consequently, these six scores were Winsorized to a score of two which yielded N ⫽ 49 for the lowest backward memory span group (i.e., span ⫽ two). Finally, only seven participant scores were equal to one or less on LDSS. Consequently, these seven scores were Winsorized to a score of two which yielded N ⫽ 31 for the lowest sequence memory span group (i.e., span ⫽ two). Thus, the total sample remained 1,800 with some slight adjustments to the very small number of lowest longest digit span scores within the sample.

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Data Analysis The data were analyzed using two strategies: (a) linear and nonlinear contrast analyses, and (b) linear and nonlinear bifactor model analyses. Although both strategies were expected to yield similar results, the first analytic strategy, which was simpler in nature, was considered useful so as to facilitate a more accessible presentation of the results. The second strategy was considered more sophisticated, as it eliminated the impact of measurement error because of the inclusion of latent variables. With respect to the first strategy, a series of three contrast analyses were performed. In each case, the LDSF, LDSB, and LDSS variables were used as an independent variable. The dependent variable was FSIQ, which is based on the 10 core WAIS-IV subtests. However, so as to avoid the possibility of an autocorrelation between the longest digit span variables and FSIQ, the Digit Span scaled scores, which are typically included within the 10 core WAIS-IV subtests (Wechsler, 2008b), were substituted with the Letter-Number Sequencing subtest scaled scores for the purposes of FSIQ calculation. Within each contrast analysis, the linear, quadratic, cubic, and quartic trends were tested. Effect sizes 2 for each term within the contrast analyses were estimated via reffect 4 2 (Furr, 2004). In the context of this investigation, a reffect equal to or less than .01 was considered practically nonsignificant, as it would imply that less than 1% of the variance in FSIQ scores could be accounted for by the pattern of contrast weights specified to reflect a particular trend. Finally, a multiple comparison procedure was also used to test the FSIQ mean differences between all contiguous longest digit span groups (e.g., LDSF 3 vs. LDSF 4; LDSF 4 vs. LDSF 5; LDSF 5 vs. LDSF 6, etc.). As the groups sizes were unequal, and there was the realistic possibility of statistically significant unequal variances, the Games-Howell multiple comparison procedure was chosen, as it is robust to heterogeneity of variances in the presence of unequal sample sizes (Games, Keselman, & Rogan, 1983). To supplement the relatively simple contrast analyses, a strictly linear bifactor model and a linear plus nonlinear (quadratic) bifactor model was tested. A bifactor model is a completely first-order factor model with direct links between the (typically orthogonal) latent variables and the indicators (Gignac, 2008; Gustafsson & Balke, 1993; Reise, 2012). In this investigation, the bifactor model consisted of one first-order general factor and four nested group 3 The method of Winsorizing data is most typically applied in the context of dealing with outliers (Wilcox, 2010). However, it should be made clear that there were no outliers in the WAIS-IV normative sample data based on the interquartile range rule (Hoaglin & Iglewicz, 1987). The method of Winsorizing the lowest longest digit span scores was simply considered to most attractive option in this case, as the lowest scores appeared to be valid and the next lowest scoring groups were not particularly large; consequently, they stood to benefit somewhat by the addition of a few extra cases. 4 2 As explained by Furr (2004), reffect represents the squared correlation between the contrast weights associated with a particular term (in the case of LDSF, e.g., linear: ⫺7, ⫺5, ⫺3, ⫺1, 1, 3, 5, 7; quadratic: ⫺7, ⫺1, 3, 5, 5, 3, ⫺1, ⫺7) and the dependent variable (i.e., FSIQ). From this perspective, the contrast analyses conducted in this investigation may be regarded as very similar to a polynomial regression analysis, and, thus, a relatively powerful statistical approach. The analysis of means approach used in this investigation was considered most useful, in this case, to help specify precisely the possible increases in FSIQ for a given unit change in memory span. By contrast, the beta weights associated with a polynomial regression are typically difficult to interpret (Pedhazur, 1997).

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factors: Verbal Comprehension, Perceptual Organisation, Working Memory, and Processing Speed (see Figure 1). As the bifactor models included LDSF, LDSB, and LDSS variables, it was considered redundant to include the Digit Span subtest in the bifactor model. Consequently, the bifactor model general factor was defined by 9 core WAIS-IV subtests (i.e., all except Digit Span), the five supplemental WAIS-IV subtests, and, finally, the raw scores associated with LDSF, LDSB, and LDSS. The bifactor model analyses were conducted with Mplus (Muthén & Muthén, 1998 –2010), which includes the option to estimate nonlinear (quadratic) factor loadings via numerical integration estimation and robust SEs (MLR; see example 5.7 of the User’s Guide). In this case, as the model was relatively large, the Monte Carlo numerical integration estimation technique was selected. If the linear ⫹ nonlinear bifactor model were observed to be associated with a statistically significant improvement in model fit over the linear bifactor model, the data would be suggested to be more consistent with a combination of linear and nonlinear effects. Additionally, if the LDSF, LDSB, and LDSS indicators were observed to be associated with negative, nonlinear loadings, then it would suggest that the strength of the association between memory span and g decreases across the spectrum of ability. To determine if the linear ⫹ nonlinear model fit better than the linear only model, the loglikelihood difference test with scaling correction factors was used. Furthermore, the Akaike information criterion (AIC; Akaike, 1987) and Bayesion information criterion (Schwarz, 1978) values were reported. Smaller AIC and BIC values, which include a penalty for model complexity, indicate better fit. All latent variables were constrained to 1.0 for the purposes of model identification. See Reynolds (2013) for more technical details on the performance of a nonlinear confirmatory factor analysis. Finally, latent variable strength was estimated with omega hierarchical (␻h; McDonald, 1999; Zinbarg, Revelle, Yovel, & Li, 2005) and omega specific (␻s; Reise, Bonifay, & Haviland, 2013). Unfortunately, bootstrapping is not possible in conjunction with random numerical integration estimation in Mplus. Consequently, only the omega point estimates were calculated, based on the point estimate standardized factor loadings (see Table 3 in Gignac, 2014b, for an accessible example of the relevant ␻h and ␻s calculations).

Results The descriptive statistics associated with the longest digit span variables were as follows: LDSF, M ⫽ 6.71, SD ⫽ 1.32, skew ⫽ ⫺.08; LDSB, M ⫽ 4.86, SD ⫽ 1.39, skew ⫽ .34; and LDSS, M ⫽ 5.85, SD ⫽ 1.32, skew ⫽ ⫺.19. The FSIQ scores were associated with a M ⫽ 100.04 and SD ⫽ 14.98. Thus, the FSIQ data were representative of the normal population, as expected, and were also normally distributed (skew ⫽ ⫺.31). As can be seen in Table 1, there were numerical increases in the FSIQ means across the LDSF, LDSB, and LDSS memory span groups. The assumption of homogeneity of variances was not satisfied across all three LDS variables; consequently, a robust analysis of variance was performed. Based on a series of Welch’s one-way between Groups analysis of variances (ANOVAs), the null hypothesis of equal FSIQ means was rejected for all three longest digit span variables: LDSF, F(6, 133.22) ⫽ 85.45, p ⬍ .001, ␻2 ⫽ .22; LDSB, F(6, 344.59) ⫽ 110.33, p ⬍ .001, ␻2 ⫽ .27; LDSS, F(7, 217.58) ⫽ 107.32, p ⬍ .001, ␻2 ⫽ .29. Thus, between 22% and 29% of the variance in FSIQ scores was accounted for by memory span, which is a large effect size based on Cohen’s (1992) guidelines. As can be seen in Table 2, nearly all of the contiguous memory span levels were associated with statistically significant differences in FSIQ means, based on the Games-Howell multiple comparison procedure. The mean Cohen’s d effects across LDSF, LDSB, and LDSS corresponded to ⫺.64, ⫺.51, and ⫺.55, respectively, which corresponds to a medium sized effect based on Cohen’s (1992) guidelines. In raw score units, the differences in LDSF from 6 to 7 digits, 7 to 8 digits, and 8 to 9 digits corresponded to FSIQ increases of 4.68, 4.51, and 3.51 points, respectively. Next, a contrast analysis approach was used to examine the patterns in the means. As can be seen in Table 3, the linear and quadratic effects were statistically significant (p ⬍ .001) and associated with minimally practically significant effect sizes 2 ⬎ .01). However, the linear effects (.13 to .25) were much (reffect larger than the quadratic effects (⬇.02) across all three longest digit span variables. Although some of the cubic and quartic contrast results were statistically significant (p ⬍ .05), none of them were associated with minimally practically significant effect

Figure 1. Bifactor model tested in this investigation. VC ⫽ Verbal Comprehension; PO ⫽ Perceptual Organization; WM ⫽ Working Memory; PS ⫽ Processing Speed; VOC ⫽ Vocabulary; IN ⫽ Information; CO ⫽ Comprehension; SIM ⫽ Similarities; FW ⫽ Figure Weights; MR ⫽ Matrix Reasoning; BD ⫽ Block Design; VP ⫽ Visual Puzzles; PC ⫽ Picture Completion; LDSF ⫽ Longest Digit Span Forward; LDSB ⫽ Longest Digit Span Backward; LDSS ⫽ Longest Digit Span Sequencing; LN ⫽ Letter-Number Sequencing; AR ⫽ Arithmetic; SS ⫽ Symbol Search; CD ⫽ Coding; CA ⫽ Cancellation.

DIGIT SPAN AND g

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Table 1 Longest Digit Span Descriptive Statistics Across Memory Span Groups (2 to 9)

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LDSF

LDSB

LDSS

Span

M

SD

N

M

SD

N

M

SD

N

2 3 4 5 6 7 8 9

— 57.58 81.32 93.20 97.10 101.78 106.29 109.80

— 11.33 15.31 14.20 13.44 13.02 12.51 12.80

— 12 50 303 392 555 298 190

72.10 88.13 96.77 101.77 106.96 111.12 113.47 —

17.91 13.75 12.54 11.86 12.00 11.18 12.53 —

49 214 543 463 276 183 72 —

64.35 74.71 90.00 95.62 101.86 107.06 110.85 114.29

14.00 14.96 12.33 12.27 12.53 11.78 11.73 12.20

31 34 217 282 790 278 110 58

Note. Column on the left represents the longest digit span achieved; LDSF ⫽ longest digit span forward; LDSB ⫽ longest digit span backward; LDSS ⫽ longest digit span sequence.

sizes (all ⬍ .01). As can be seen in Figure 2, although there was a bend in the pattern of the means indicative of a quadratic effect, the FSIQ means continued to increase across the whole spectrum of memory capacity. Next, the bifactor model analyses were performed. Because the numerical integration estimation procedure does not provide conventional model-fit statistics, the bifactor model was first tested via maximum likelihood estimation and was found to be associated with acceptable levels of model close-fit, ␹2(102, N ⫽ 1,800) ⫽ 463.01, p ⬍ .001, RMSEA ⫽ .044, SRMR ⫽ .029, CFI ⫽ .977, TLI ⫽ .969. Furthermore, all of the factor loadings were statistically significant (available upon request). As estimated via numerical integration, the bifactor model that included only linear factor loading terms yielded a log likelihood value of ⫺65,418.822, AIC ⫽ 130,971.64, BIC ⫽ 131,339.85, scaling correction factor ⫽ 1.0532, and 67 freely estimated parameters. Next, again, as estimated via numerical integration, the second bifactor model that included both linear and quadratic factor loading terms yielded a log likelihood value of ⫺65176.59, AIC ⫽ 130555.19, BIC ⫽ 130789.37, scaling correction factor 1.0668, and 101 freely estimated parameters. Based on the loglikelihood difference test, the bifactor model that included both linear and nonlinear factor loadings was found to be better fitting than the bifactor model that included only linear factor loadings, Trd ⫽ 442, df ⫽ 34, p ⬍ .001. Furthermore, the AIC and BIC values were smaller for the linear ⫹ Table 2 Contiguous Mean Difference Effect Sizes (Cohen’s d) Across Levels of Longest Digit Spans Span 2 3 4 5 6 7 8

vs. vs. vs. vs. vs. vs. vs.

3 4 5 6 7 8 9

LDSF

LDSB

LDSS

— ⫺1.78ⴱ ⫺.81ⴱ ⫺.28ⴱ ⫺.35ⴱ ⫺.35ⴱ ⫺.28ⴱ

⫺1.01ⴱ ⫺.66ⴱ ⫺.41ⴱ ⫺.44ⴱ ⫺.36ⴱ ⫺.20 —

⫺.72ⴱ ⫺1.12ⴱ ⫺.46ⴱ ⫺.50ⴱ ⫺.43ⴱ ⫺.32 ⫺.29

Note. The statistical significance associated with the Cohen’s d estimates is implied by the estimated p values associated with the Games-Howell procedure. LDSF ⫽ longest digit span forward; LDSB ⫽ longest digit span backward; LDSS ⫽ longest digit span sequence. ⴱ p ⬍ .05.

nonlinear bifactor model (⌬AIC ⫽ 416.45; ⌬BIC ⫽ 550.48). Thus, the addition of the nonlinear terms to the bifactor model improved model fit. As can be seen in Table 4, all of the subtests were associated with statistically significant (p ⬍ .05) positive linear loadings. Furthermore, the linear general factor (␻h ⫽ .84) was found to be particularly strong, whereas, by contrast, the linear nested VC (␻s ⫽ .30), PO (␻s ⫽ .10), WM (␻s ⫽ .26), and PS (␻s ⫽ .41) latent variables were found to be relatively weak. Finally, the LDSB, LDSF, and LDSS variables were associated with statistically significant nonlinear standardized loadings of ⫺.09, ⫺.06, and ⫺.12, respectively. The negative nonlinear g loadings imply an association between a subtest and g that is decreasing in strength across the spectrum of cognitive ability. Arguably, loadings of such a small magnitude imply a very weak nonlinear effect, which is consistent with the relatively weak effects associated with the contrast analyses. More important, it will be noted that the vast majority of the WAIS-IV subtests (14 out of 17) evidenced negative, nonlinear associations with g (see Table 4).

Discussion Based on the results of the contrast analyses, the LDSF, LDSB, and LDSS memory span groups evidenced mean FSIQ differences largely consistent with a linear effect, although there were also small, nonlinear (quadratic) effects, as well. Based on the nonlinear bifactor analysis, LDSF, LDSB, and LDSS were, again, observed to be largely associated with linear, positive effects on g, with additional, small, negative, nonlinear effects with g. However, virtually all of the WAIS-IV subtests were observed to be associated with similar nonlinear, negative, effects with g. Overall, the results of this investigation do not support Wechsler’s (1939, 1958) view that above average levels of memory span are not beneficial with respect to intellectual functioning, as the association between g and LDSF (.40), LDSB (.48), and LDSS (.51) were principally linear in nature (Figure 2). The magnitude of the memory span bifactor g loadings were noticeably smaller than those reported in Gignac (2014a; Digit Span Forward ⫽ .46; Digit Span Backward ⫽ .58; Digit Span Sequencing ⫽ .63); however, it is important to note that Gignac analyzed the scaled WAIS-IV data, which are based on scoring each subtest item from 0 to 2, rather than simply the longest digit span achieved (i.e., less variability in test scores). Thus, the memory span effects

GIGNAC AND WEISS

1318 Table 3 Between-Groups ANOVA Contrast Results LDSF

LDSB

LDSS

Power

F

p

2 reffect

F

p

2 reffect

F

p

2 reffect

Linear Quadratic Cubic Quartic

293.56 45.91 16.92 3.13

⬍.001 ⬍.001 ⬍.001 .077

.128 .020 .007 .001

588.18 55.74 8.63 4.04

⬍.001 ⬍.001 .003 .045

.227 .022 .003 .002

651.13 49.93 1.61 .42

⬍.001 ⬍.001 .205 .519

.247 .019 .000 .000

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Note. N ⫽ 1,800; degrees of freedom associated with the LDSF and LDSB F tests were 1 and 1,793; degrees of freedom associated with the LDSS F tests were 1 and 1,792. LDSF ⫽ longest digit span forward; LDSB ⫽ longest digit span backward; LDSS ⫽ longest digit span sequence.

reported in Gignac may be regarded as more representative of the validity of Digit Span test scores as they are actually calculated and interpreted in practice. The longest digit span data were used in this investigation simply to facilitate a more intuitive interpretation of the effects. From a Cohen’s d perspective, the differences in mean FSIQ across the contiguous LDSF memory span groups was between small and large in magnitude (Cohen, 1992). Although the magnitude of the difference decreased across the spectrum of general ability (i.e., FSIQ), consistent with a nonlinear effect, small to medium FSIQ statistically significant increases were nonetheless observed for LDSF and mostly observed for LDSB and LDSS. Unfortunately, Wechsler (1939, 1958; see also Matarazzo, 1972) did not specify precisely what an absolute minimum memory span was, however, it is plausible to presume that an absolute minimum was not beyond average performance. From the mid- to high-end of the performance spectrum, each unit increase in memory span performance corresponded to an increase of ⬃4 IQ points, in this investigation. Across the three memory span indicators, the difference between an approximate average level of memory span performance (five to six) and the highest level of memory span performance (eight to nine) was equal to ⬃12.2 IQ points. Arguably, a difference between groups approaching a full SD is substantial, one that would be expected to afford meaningful advantages in life (Gottfredson, 2004; Jensen, 1998; Judge, Ilies, & Dimotakis, 2010). Thus, the clinical lore surrounding the Digit Span subtest should probably be reconsidered, based on the results of this investigation. Although there was a small reduction in the magnitude of the mean FISQ differences from low to high ability, which may be regarded as partial support of Wechsler’s view, such

a small effect may be because of a statistical artifact, as described below. From an applied perspective, practitioners should have confidence in the use of the core Digit Span subtest in the estimation of an individual’s FSIQ across the whole spectrum of ability. It will also be noted that although clinicians are recommended to use the Digit Span subtest and the Arithmetic subtest for the purposes of calculating WM index scores (Wechsler, 2008b), Arithmetic was observed to be associated with a rather weak linear loading of only .13 on the nested WM latent variable. By contrast, Letter-Number Sequencing and Digit Span Backward were found to be associated with respective linear loadings of .56 and .42 on the nested WM latent variable. Consequently, it is suggested that a better representation of working memory capacity could be obtained by a combination of Letter-Number Sequencing and Digit Span Backward, rather than the current recommendation of Digit Span and Arithmetic. The observation that all three indicators of memory span examined in this investigation evidenced substantial and largely linear associations with g is consistent with the commonly expressed view within the cognitive science literature that memory span is an important attribute of cognitive functioning (Conway & Kovacs, 2013). Precisely how greater memory span ability facilitates greater cognitive ability across a range of diverse tasks, ranging from fluid intelligence tasks to the accumulation and expression of knowledge, remains an active area of research (Wiley, Jarosz, Cushen, & Colflesh, 2011). In the context of explaining why Digit Span correlates with crystallized knowledge subtests, Jensen (1970) speculated that “seemingly small individual differences in immediate memory span, when multiplied over a lifetime of experiences, make for highly significant differences in acquired

Figure 2. Scatter plots depicting the association between memory span and FSIQ. See the online article for the color version of this figure.

DIGIT SPAN AND g

Table 4 Standardized Loadings Associated With the Bifactor Model: Linear Effects and Nonlinear Effects Linear

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g VC IN CO SIM FW MR BD VP PC LDSF LDSB LDSS LN AR SS CD CA

.70 .66 .66 .65 .74 .67 .64 .63 .50 .40 .48 .51 .50 .76 .50 .53 .39

VC PO

Nonlinear (quadratic) WM

.57 .43 .47 .45 .14 .17 .05 .49 .32 .41 .42 .32 .56 .13

PS

g

ⴚ.02 .00 ⫺.07 ⫺.07 .00 ⫺.05 ⴚ.01 .01 ⫺.09 ⫺.09 ⫺.06 ⫺.12 ⫺.12 .00 .67 ⫺.09 .51 ⫺.11 .37 ⫺.07

VC

PO

WM

PS

.06 .03 ⴚ.01 .00 .06 .02 .06 .08 ⫺.08

⫺.07 .03 ⴚ.03 .26 .02 .06 ⴚ.02 .07

Note. Factor loadings in bold are not statistically significant (p ⬎ .05); to obtain the completely standardized parameter estimates for the nonlinear solution, the latent variable variances were constrained to 1 and the variable scores were transformed into z-scores. VC ⫽ Verbal Comprehension; PO ⫽ Perceptual Organization; WM ⫽ Working Memory; PS ⫽ Processing Speed; IN ⫽ Information; CO ⫽ Comprehension; SIM ⫽ Similarities; FW ⫽ Figure Weights; MR ⫽ Matrix Reasoning; BD ⫽ Block Design; VP ⫽ Visual Puzzles; PC ⫽ Picture Completion; LDSF ⫽ Longest Digit Span Forward; LDSB ⫽ Longest Digit Span Backward; LDSS ⫽ Longest Digit Span Sequence; LN ⫽ Letter-Number Sequencing; AR ⫽ Arithmetic; SS ⫽ Symbol Search; CD ⫽ Coding; CA ⫽ Cancellation.

[knowledge]” (p. 73). Additionally, the ability to make conceptual connections between seemingly disparate pieces of information (e.g., Similarities) may be regarded as an important intellectual capacity (Flynn, 2007). Many breakthroughs in science, for example, may be suggested to be achieved by making connections between several pieces of information, all of which would have to be maintained in memory, simultaneously, during the process of analysis and synthesis. From this perspective, an individual who can maintain eight to nine pieces of information in memory, simultaneously, is arguably at a distinct advantage over an individual who can maintain only three or four. Although some evidence suggests that Digit Span test scores are relatively insensitive to diffuse head injury (Axelrod, Fichtenberg, Millis, & Wertheimer, 2006; Donders, Tulsky, & Zhu, 2001), there are some case studies that have found selective memory span deficits in those who sustained a focal brain injury such as an ischemia (e.g., Vallat, Azouvi, Hardisson, Meffert, Tessier, & Pradat-Diehl, 2005). Additionally, the contention that Digit Span is simply an indicator of attention may be suggested to be inconsistent with the observation that some brain injured individuals can have intact short-term spatial memory ability but serious verbal serial recall deficits (De Renzi & Nichelli, 1975; Vallar & Baddeley, 1984). Conversely, a deficit in spatial STM can be observed in the presence of intact verbal STM (Hanley, Young, & Pearson, 1991). The reconciliation of such findings with the notion that serial recall is an indicator of attention would require an unlikely

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model of cognition that included modality specific attentional systems. Ultimately, the notion that Digit Span measures simply attention, rather than a form of memory, is inconsistent with the manner in which serial recall is viewed within the cognitive science literature (Hurlstone et al., 2014). Thus, based on the empirical literature, as well as the results of this investigation, memory span for digits should arguably be viewed as important indicator of short-term verbal memory functioning. Although the small, negative, nonlinear effects observed in this investigation may be considered, to some degree, consistent with Wechsler’ (1939, 1958) view of Digit Span, it may be suggested that the ostensible partial support may possibly be explained by the well-established psychometric principles of regression toward the mean and standard error of measurement, asymmetry (Daniel, 1999). That is, test scores at the lower- and higher-end of a distribution of scores are known to be associated with less reliability than those test scores at or near the mean of the distribution (Preacher, Rucker, MacCallum, & Nicewander, 2005). As a result, relatively extreme observed test scores are known to regress toward the mean upon retesting (Campbell & Kenny, 1999). Consequently, the phenomenon of regression toward the mean may explain the observation that more substantial group differences in FSIQ were observed in this investigation across the lower memory span groups, whereas progressively smaller group differences in FSIQ were observed across the higher memory span groups. To avoid confusion, it should be made clear that the participants included in this investigation were not retested on any of the WAIS-IV subtests. Instead, the regression toward the mean effect is hypothesized to operate, principally, at the construct level of g. That is, because all of the WAIS-IV subtests are associated with substantial common variance (i.e., g factor; Canivez & Watkins, 2010; Gignac & Watkins, 2013), the administration of the WAIS-IV battery may be regarded, to some degree, as consistent with a retest administration. Thus, individuals who may have performed particularly poorly on the Digit Span subtest, an indicator of g, would be expected to perform somewhat better on the remaining WAIS-IV subtests, also indicators of g, because of the regression toward the mean effect. Conversely, those who performed particularly well on the Digit Span subtest would be expected to perform somewhat worse on the remaining WAIS-IV subtests. Taken from a different psychometric perspective, the standard error of measurement associated with scores at the lowerand higher-end of the distribution is not symmetric (Daniel, 1999). Instead, the standard error of measurement is asymmetric with greater confidence interval coverage toward the mean, rather than away from the mean (Nunnally & Bernstein, 1994). Consequently, given the effects of regression toward the mean and the asymmetry of the standard error of measurement, the small nonlinear bifactor g loadings reported for LDSF, LDSB, and LDSS may be considered a statistical artifact. Indirect support for such a contention is that small, negative, nonlinear loadings were observed in this investigation for nearly all of the WAIS-IV subtests. Furthermore, based on an additional analysis, the Spearman rank correlation between the reliability of subtest scores and the magnitude of the nonlinear g loading was ⫺.61, (bootstrapped 95% confidence interval [CI] [⫺.20, ⫺.81]). Thus, greater levels of subtest score reliability were associated with smaller nonlinear g loadings, an observation which is consistent with the fact that the regression

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GIGNAC AND WEISS

toward the mean effect does not occur when test scores are perfectly reliable (Preacher et al., 2005). Finally, it will be noted briefly that this investigation may be considered relevant to the topic of Spearman’s Law of Diminishing Returns (SLODR), which states that the positive manifold (or g) is greater at lower levels of ability than at higher levels of ability (Deary, & Pagliari, 1991; Jensen, 2003; Spearman, 1927). In this investigation, the nature of the association between memory span scores and FSIQ depicted in the scatter plots within Figure 2 may be suggested to be consistent with SLODR, as the increases in FSIQ means were larger across the lower memory span groups, in comparison to the higher memory span groups. However, again, such an effect may be plausibly explained based on regression toward the mean and the asymmetry of standard error of measurement, (Daniel, 1999). Thus, much of the empirical research ostensibly supportive of SLODR (e.g., Coyle & Rindermann, 2013; Reynolds, 2013; te Nijenhuis & Hartmann, 2006) may be attributable to a statistical artifact. Although, latent variable modeling techniques are emerging to address the regression toward the mean effect (e.g., Marsh & Hau, 2002), none appear to have been proposed in the context of SLODR or the hypothesis relevant to this investigation. Future research in this area is encouraged. It is hypothesized that once the asymmetry of measurement error is controlled and the nonlinear effects between subtests and g may be largely eliminated. It should be emphasized that not all nonlinear effects are suggested here to be because of statistical artifacts. Inverted U-shaped effects would not be explainable from this perspective, nor would nonlinear effects based on scores associated with very high levels of reliability. Finally, nonlinear effects based on scores that do not share a common construct would not apply either.

Limitations Although the memory span subtests examined in this investigation were found to be associated with respectable levels of validity, there are some minor changes that could be implemented for the purposes of improvement. For example, within the WAIS-IV, an individual’s longest digit span is equal to the number of digits that can be recalled correctly on 50% of trials (Wechsler, 2008b). However, there are only two trials associated with each digit series length (Wechsler, 2008a). Consequently, a 50% success rate implies that the participant needs to recall correctly only one trial. Arguably, enhanced levels of reliability could be achieved with the administration of, say, four or five trials. Enhanced reliability would be particularly advantageous at the lower- and higher-end of the memory distribution of test scores, which are known to be associated with less reliability (Preacher et al., 2005). Consequently, enhanced reliability, particularly at the ends of the distribution of test scores, would be expected to help militate against the regression toward the mean effect hypothesized to have been observed in this investigation. Furthermore, in clinical settings, it would be expected that many cases would perform at the lower (less reliable) end of the spectrum of ability. In neuropsychological settings, as many as 10 trials are recommended to be administered for each digit series length to determine an individual’s longest digit span (Lezak, 2004). In the area of cognition, researchers typically administer between there and five trials per item, with thresholds of 67% to 80% correct to demarcate an individual’s

memory span (Conway, Kane, Bunting, Hambrick, Whilhelm, & Engle, 2005). Of course, a distinct disadvantage associated with increasing the number of trials substantially within each digit span series would be a concomitant increase in administration time. Perhaps the ideal would be to move onto an item response theory (IRT) and computer adaptive testing (CAT) framework (van der Linden & Glas, 2000), which would facilitate the administration of a greater number of items/trials closer to an individual’s latent ability relatively quickly. In addition to the possible advantages associated with administering more trials within a digit series item, it may be beneficial to include at least one more additional digit series of greater length within the Digit Span subtest. That is, with respect to Digit Span Forward, there does appear to be a moderate ceiling effect, as ⬃10% of the normal population can recall nine digits (Wechsler, 2008), that is, the largest digit series within the test. Although not reported in the technical manual (Wechsler, 2008), it is highly unlikely that 10% of the population answer correctly the last items within the Vocabulary, Similarities, and Matrix Reasoning subtests, by comparison. Consequently, as range restriction is wellknown to attenuate effect sizes (Ghiselli, 1964; Huck, 1992; Sackett & Yang, 2000), it may be suggested that the factor loadings associated with Digit Span reported in this investigation (and others) are, to some degree, smaller than they would otherwise be with the inclusion of an additional digit series at the higher end of the spectrum of ability.

Conclusion In conclusion, the popularity of memory span individual differences research in the cognitive sciences continues unabated (Conway & Kovacs, 2013). Greater attention on memory span in the clinical assessment community has occurred, as well. For example, the Wechsler scales provide normative sample information for Digit Span Forward and Digit Span Backward, separately. Furthermore, an additional subtest of working memory functioning, Digit Span Sequencing, was added to the WAIS-IV. However, it may be suggested that, overall, the view of memory span within the clinical assessment community is not especially favorable as an indicator of general intellectual functioning, or even memory capacity, in some cases. The results of this investigation suggest that memory span, as measured via Digit Span Forward, Digit Span Backward, and Digit Span Sequencing, are at least moderately good indicators of g. Furthermore, the association is likely best interpreted as largely linear, with every extra bit of memory span counting toward additional intellectual functioning. Consequently, it is perhaps long overdue that Digit Span be regarded as a high quality subtest, as Jensen (1970) suggested nearly 45 years ago.

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Appendix Mplus Syntax Used to Estimate the Bifactor Model Solution Variable: Names are ldsf ldsb ldss bdss siss mrss vcss arss ssss vpss coss cdss lnss fwss inss cass pcss; ANALYSIS: ESTIMATOR ⫽ MLR; TYPE ⫽ RANDOM; ALGORITHM ⫽ INTEGRATION; INTEGRATION ⫽ MONTECARLO(5000); MODEL: g BY vcssⴱ inss coss siss fwss mrss bdss vpss pcss ldsf ldsb ldss lnss arss ssss cdss cass; vc BY vcssⴱ inss coss siss; po BY fwssⴱ mrss bdss vpss pcss; wm BY ldsfⴱ ldsb ldss lnss arss; ps BY ssssⴱ cdss cass; g@1; vc@1; po@1; wm@1;

ps@1; g WITH vc@0 po@0 wm@0 ps@0; vc WITH po@0 wm@0 ps@0; po WITH wm@0 ps@0; wm WITH ps@0; gxg | g XWITH g; ldsf-pcss ON gxg; vcxvc | vc XWITH vc; vcss inss coss siss ON vcxvc; poxpo | po XWITH po; fwss mrss bdss vpss pcss ON poxpo; wmxwm | wm XWITH wm; ldsf ldsb ldss lnss arss ON wmxwm; psxps | ps XWITH ps; ssss cdss cass ON psxps; Output: TECH1 TECH8; Received September 23, 2014 Revision received January 22, 2015 Accepted January 23, 2015 䡲