Patterns of Strengths and Weaknesses on the WISC-V,DAS-II,and KABC-II and Their Relationship to Students’ Errors in Oral Language,Reading,Writing,Spelling,and Math

Abstract

This study investigated the relationship between specific cognitive patterns of strengths and weaknesses and the errors children make on oral language, reading, writing, spelling, and math subtests from the Kaufman Test of Educational Achievement–Third Edition (KTEA-3). Participants with scores from the KTEA-3 and either the Wechsler Intelligence Scale for Children–Fifth Edition (WISC-V), Differential Ability Scales–Second Edition (DAS-II), or Kaufman Assessment Battery for Children–Second Edition (KABC-II) were selected based on their profile of scores. Error factor scores for the oral and written language tests were compared for three groups: High Gc paired with low processing speed, long-term memory, and/or reasoning abilities; Low Gc paired with high speed, memory, and/or reasoning; and Low orthographic and/or phonological processing. Error factor scores for the math tests were compared for three groups: High Gc profile; High Gf paired with low processing speed and/or long-term memory; and Low Gf paired with high processing speed and/or long-term memory. Results indicated a difference in Oral Expression and Written Expression error factor scores between the group with High Gc paired with low processing speed, long-term memory, and/or reasoning abilities; and the group with Low Gc paired with high speed, memory, and/or reasoning.

Keywords

KTEA-3 assessment of intelligence education assessment academic achievement language and literacy mathematics PSW

Considering the relationships between an individual’s academic performance and cognitive abilities and neuropsychological processes is helpful for understanding why a student is experiencing academic difficulties. Research suggests that cognitive and achievement abilities are highly related but distinct constructs, especially among school-age children, and specific cognitive factors are important for explaining academic achievement (S. B. Kaufman, Reynolds, Liu, Kaufman, & McGrew, 2012). For this reason, evaluating patterns of cognitive–achievement relationships is an important consideration for learning disability diagnostic evaluations, especially when utilizing a cross-battery assessment approach (Flanagan, Ortiz, & Alfonso, 2013). The present study sought to extend the research on cognitive–achievement relations by investigating the reading, writing, and math errors made by groups of students with differing cognitive profiles.

Relationships Between Academic Errors and Cognitive Abilities

Pattern of Strengths and Weaknesses (PSW) and Reading

The Cattell–Horn–Carroll (CHC) theory has been utilized to advance much of the research on cognitive–achievement relationships (e.g., Flanagan, Alfonso, & Mascolo, 2011; Flanagan et al., 2013). According to Flanagan et al. (2013), the narrow abilities most consistently and significantly related to reading achievement are subsumed by Ga (Phonetic Coding), Gc (Language Development, Lexical Knowledge, Listening Ability, and General Information), Glr (Associative Memory, Naming Facility, and Meaningful Memory), Gsm/Gwm (Memory Span, Working Memory), and Gs (Perceptual Speed). Specific CHC abilities, in particular Ga and Gc, have been shown to significantly explain reading achievement above and beyond the powerful effect of g. Developmentally, the relations between reading and the Ga, Gs, and Glr abilities are strongest in the elementary years but decline thereafter, whereas the relations between reading and Gc increase with age. To a lesser extent, the Gv abilities of orthographic processing and visual memory are related to basic reading skills and reading comprehension, respectively. Gf abilities are also related to reading comprehension (Flanagan et al., 2013).

Evans, Floyd, McGrew, and Leforgee (2002) found that verbal ability/comprehension knowledge and working memory demonstrated significant relationships with reading achievement during both childhood and adolescence, whereas processing speed and long-term retrieval showed significant effects on reading only during childhood. There is an especially large literature base for the contribution of foundational oral language skills contributing to reading ability (Carver, 1998; Catts, Hogan, & Adolf, 2005; Catts, Hogan, & Fey, 2003; National Reading Panel, 2000; San Chen & Vellutino, 1997; Tilstra, McMaster, Van den Broek, Kendeou, & Rapp, 2009). The simple view of reading proposes that reading comprehension is comprised of decoding and language comprehension ability, and research has demonstrated these two abilities account for as much as 71% to 85% of the variance in reading comprehension (Hoover & Gough, 1990).

The contributions of working memory to reading ability have also been well-documented in the research literature. The Automatic Information Processing theory (LaBerge & Samuels, 1974) states that there is a limited amount of cognitive capacity available during reading. When readers have developed automaticity and fluency and are no longer decoding each individual sound in a word, they have additional working memory available to allot for comprehending text. Conversely, if students are not fluent readers, they do not have additional working memory capacity available for comprehension. A recent study found that working memory uniquely contributed to word reading, reading fluency, and reading comprehension, even after controlling for reading-related skills such as rapid naming and phonological processing (Jacobson et al., 2016). In addition, research has demonstrated that different kinds of working memory can differentially affect reading skills. Pham and Hasson (2014) found that both verbal working memory and visual-spatial working memory were uniquely predictive of reading ability, verbal working memory was especially predictive of reading fluency, and visual-spatial working memory was especially predictive of reading comprehension.

Compton, Fuchs, and Fuchs (2012) compared different types of learning disabilities on distinct patterns of cognitive abilities in language, working memory, processing speed, and problem solving. Children without learning disabilities were found to have a relatively flat profile of abilities, that is, they performed consistently across measures. However, they found that children with specific word reading difficulties had lower working memory and oral language skills compared with other skills. For those with specific difficulties in reading comprehension, the authors found these children had specific difficulties with language, including vocabulary and oral language comprehension.

PSW and Writing

In comparison with the number of studies conducted on cognitive–reading relations, relatively few studies have explored the underlying cognitive processes that contribute to writing skills. Researchers have long understood, however, that writing is not the inverse of reading (Frith, 1980; Read, 1981). Rather, writing shares some processes in common with other kinds of language (listening, speaking, and reading) and also utilizes unique processes (Berninger et al., 2006).

Beginning spelling involves making connections between the spoken and written forms of words. As Berninger et al. (2006) explained, typical spelling development generally proceeds from a reliance on phonological information to greater reliance on orthographic and morphological information; however, children begin integrating phonological, orthographic, and morphological information from an early age. Research by Berninger and colleagues (Berninger, 2006; Berninger, 2009) found that orthographic coding, phonological coding, and vocabulary knowledge were the best predictors of spelling ability in elementary through middle school.

Hayes and Flower’s (1980) model identified planning, translating, and reviewing as the key activities involved in skilled writing. Skilled execution of these key writing activities is dependent to a large extent on working memory (e.g., Hayes, 2000; Kellogg et al., 2013; McCutchen, 2000) and language ability, especially verbal comprehension (Bourke & Adams, 2011; Gathercole & Alloway, 2008). Although not within the scope of the present study, noncognitive factors including self-efficacy, anxiety, and motivation are also considerably influential in determining writing performance (Bourke & Adams, 2011; Hayes, 2000).

The CHC literature suggests that seven broad domains contribute to writing achievement—Gc, Gs, Gwm, Glr, Gf, Ga, and Gv (see Flanagan et al., 2013, for a review)—although some research (e.g., Floyd, McGrew, & Evans, 2008) suggests that the first four of these broad abilities are most explanatory of writing ability: Gc, Gs, Gwm, and Glr. As Flanagan et al. (2013) explained, Gc (Comprehension-Knowledge) abilities become increasingly important with age for developing writing skills; Gs: Perceptual Speed contributes to spelling and writing automaticity; Gwm contributes to spelling achievement (Memory Span) as well as written expression (Working Memory Capacity); Glr: Naming Facility relates to writing fluency; Gf relates to basic writing skills in the early grades and written expression at all ages; Ga (Phonetic Coding) is particularly important for young children’s development of basic writing and spelling skills; and Gv is associated with orthographic processing, which contributes to spelling achievement.

PSW and Math

Aside from the undeniable importance of g, research suggests that four broad abilities contribute most consistently to mathematics achievement. These abilities include Fluid Reasoning, Comprehension-Knowledge, Processing Speed, and Short-Term Working Memory. McGrew and Wendling (2010) provided a summary of the extant CHC research on cognitive–achievement relations, relying primarily on studies pertaining to the Woodcock–Johnson III Cognitive and Achievement test batteries. For both math computation and math problem solving, three broad abilities were consistently significant in predicting scores at one or more age groups: Comprehension-Knowledge (Gc), Fluid Reasoning (Gf), and Processing Speed (Gs). Short-Term Working Memory (Gwm) was a consistently significant, albeit low, predictor of math problem-solving ability among high school students. However, Swanson and Beebe-Frankenberger (2004) found a moderate correlation between working memory and math problem solving that was stable across grades, and working memory contributed unique variance to problem solving even when the influence of measures such as processing speed and phonological processing was partialed from the analysis.

Although long-term retrieval (Glr) was not a significant predictor in the McGrew and Wendling (2010) study, several Glr narrow abilities were shown to be important predictors of mathematics achievement: Naming automaticity was consistently predictive of math computation, and associative memory and meaningful memory were predictive of math computation and problem solving at one or more age levels. The relationship between Glr and math achievement has not been consistently supported by CHC-based studies (Flanagan et al., 2013); however, other research has shown Glr to be important for rapidly retrieving math facts (e.g., Geary, Hoard, & Bailey, 2012) and facilitating calculation ability by retrieving mathematical knowledge of algorithms and strategies (Swanson & Beebe-Frankenberger, 2004).

Research on the cognitive processes important for reading, writing, and math abilities suggests the importance of several broad abilities, especially Gc, Gwm, Glr, Gs, and Gf. However, very little is known about whether differences in broad ability cognitive profiles affect the kinds of errors that children make in reading, writing, and math domains.

Error Analysis

Research on the analysis of student errors made on reading, writing, spelling, and math tasks dates back to the early 1900s (e.g., Buswell & Judd, 1925). Numerous studies have promoted the importance of error analysis as a means to provide insight into how individuals learn to read and the strategies that they rely upon to read (see Greenberg, Ehri, & Perin, 2002). McGeown, Medford, and Moxon (2013) used error analysis to identify individual differences in children’s reading and spelling strategies, and to identify the cognitive skills underlying their strategy use. Through analyzing children’s errors, they found that different cognitive skills predicted reliance upon different strategies. For example, if children had more well-developed decoding skills, they were more likely to rely upon a phonological strategy to read new words and less likely to reply upon an orthographic (visual) strategy.

As explained by Greenberg et al. (2002), the analysis of children’s reading errors has been the basis for the development of the field’s foremost theories of literacy by researchers such as Linnea Ehri (1986) and Uta Frith (1985). Error analysis has also been instrumental for developing appropriate intervention strategies to remediate struggling readers (Greenberg et al., 2002; Sawyer, Wade, & Kim, 1999; Shaughnessy, 1979.

Greenberg et al. (2002) used error analysis to identify differences in the way adults and children learn to read. Adults were not as likely as children to rely upon phonological decoding processes to help them read and instead were more likely to rely on visual memory. Greenberg and colleagues took these findings to suggest that adults and children who are learning to read tend to rely upon different cognitive processes and approaches, findings which have important instructional implications.

Utilizing error analysis in mathematics has significant value on a number of levels for educators, researchers, and students. Math error analysis can inform instruction, assist in the diagnosis of mathematical difficulties, support the development of effective remediation strategies, and be used by students as a tool to more actively engage in their own learning and develop a deeper understanding of math concepts (Borasi, 1994; Herholdt & Sapire, 2014; Ketterlin-Geller & Yovanoff, 2009; Radatz, 1979; Riccomini, 2005). With respect to instruction, math error analysis is useful for identifying the common systematic errors and misconceptions that effect most students (Radatz, 1979; Riccomini, 2005). According to Ketterlin-Geller and Yovanoff (2009), math error analysis may be used by teachers to provide timely information which can be used to adjust instruction to meet students’ learning needs and develop appropriate remediation strategies for students (Ketterlin-Geller &Yovanoff, 2009; Radatz, 1979). Involving students in the process of analyzing their own errors may also support their comprehension of math concepts (Borasi, 1994).

The growing body of research pertaining to error analysis in the areas of reading, writing, and math supports the utility of error analysis for understanding the cognitive strengths and weaknesses that underlie students’ academic skill acquisition. For this reason, assessment tools such as the Kaufman Test of Educational Achievement–Third Edition (KTEA-3; A. S. Kaufman & Kaufman, 2014) that incorporate error analysis systems are especially valuable for a comprehensive evaluation of academic skills.

Study Purpose

The goal of the present study was to identify groups of students based on their pattern of cognitive strengths and weaknesses, and then compare the groups in terms of the kinds of errors made on KTEA-3 reading, writing, and math subtests. Three cognitive profiles related to reading and writing achievement were of interest for this study:

High Crystallized Ability, Low Memory/Speed/Reasoning (abbreviated as High Gc): Students with high crystallized ability coupled with low long-term memory, short-term working memory, processing speed, and/or fluid reasoning.

Low Crystallized Ability, High Memory/Speed/Reasoning (abbreviated as Low Gc): Students with low crystallized ability coupled with high long-term memory, short-term working memory, processing speed, and/or fluid reasoning.

Low Phonological and/or Orthographic Processing (abbreviated as Low PP/OP): Students with low phonological processing and/or low orthographic processing.

In addition, three cognitive profiles related to mathematics achievement were of interest for this study:

High Fluid Reasoning, Low Memory/Speed (abbreviated as High Gf): Students with high fluid reasoning coupled with low long-term memory, short-term working memory, and/or processing speed.

Low Fluid Reasoning, High Memory/Speed (abbreviated as Low Gf): Students with low fluid reasoning coupled with high long-term memory, short-term working memory, and/or processing speed.

High Crystallized Ability, Low Memory/Speed/Reasoning (abbreviated as High Gc): Children with high crystallized ability coupled with low long-term memory, short-term working memory, processing speed, and/or fluid reasoning.

Two primary research questions guided the research methodology and analyses:

Research Question 1: Do the High Gc, Low Gc, and Low PP/OP groups differ in the kinds of errors they make on reading, writing, and spelling subtests?

Research Question 2: Do the High Gf, Low Gf, and High Gc groups differ in the kinds of errors they make on math computation and math problem-solving subtests?

These PSW groups were expected to show differing patterns of performance on KTEA-3 error factors. The High Gc group was expected to perform more strongly than the Low Gc and Low PP/OP groups on multiple reading, writing, and spelling error factors. The Low PP/OP group was expected to show weak performance on one or more basic reading and writing error factors. The High Gf group was expected to perform more strongly than the Low Gf and High Gc groups on one or more mathematics error factors.

Method

Participants

The total combined sample for these analyses included 222 students in grades prekindergarten (PK) through 12 and between the ages of 4 and 18. This sample included 119 students tested on both the Wechsler Intelligence Scale for Children–Fifth Edition (WISC-V; Wechsler, 2014) and KTEA-3 as part of the WISC-V standardization (Wechsler, 2014), and two additional samples collected as part of the KTEA-3 standardization: a sample of 50 tested on both the Kaufman Assessment Battery for Children–Second Edition (KABC-II; A. S. Kaufman & Kaufman, 2004). KABC-II and KTEA-3 and a sample of 53 tested on both Differential Ability Scales–Second Edition (DAS-II; Elliott, 2007) and KTEA-3 (A. S. Kaufman & Kaufman, 2014). These samples were combined for coding and data analyses. A total of 199 cases were included in one or more math profile groups (referred to as the math sample) and one or more language profile groups (referred to as the language sample). The demographic characteristics of the groups are shown in Table 1 .

Table 1.

Sample Demographics.

	Language sample			Math sample
	High Gc	Low Gc	Low PP/OP	High Gf	Low Gf	High Gc
N	87	63	49	72	40	48
Age
M	10.3	9.6	9.1	9.9	11.8	9.9
SD	3.9	3.7	3.4	3.8	3.7	3.6
Range	4-18	4-17	5-17	4-18	4-18	4-17
Grade
Median	4	5	5	4	5	3
Range	PK-12	PK-12	PK-12	PK-12	PK-12	PK-12
Sex (% male)	63.2	41.3	32.7	52.8	45.0	70.8
Ethnicity
White	63.2	28.6	55.1	43.1	52.5	68.8
Hispanic	18.4	33.3	20.4	23.6	22.5	14.6
African American	6.9	27.0	16.3	20.8	12.5	8.3
Asian	5.7	3.2	2.0	2.8	2.5	6.3
Other	5.7	7.9	6.1	9.7	10.0	2.1
Parent education
<9 years	1.1	7.9	4.1	2.8	2.5	2.1
9-12 years	8.0	22.2	14.3	16.7	15.0	12.5
12 years with diploma	27.6	36.5	34.7	33.3	32.5	29.2
13-15 years	47.1	25.4	40.8	38.9	42.5	45.8
16+ years	16.1	7.9	6.1	8.3	7.5	10.4
Region
Midwest	23.0	22.2	16.3	19.4	35.0	22.9
Northeast	16.1	6.3	14.3	13.9	15.0	16.7
South	47.1	42.9	49.0	54.2	35.0	43.8
West	13.8	28.6	20.4	12.5	15.0	16.7

Note. Except for sample size (n), age, and grade, data are reported as percentages. High Gc = High Crystallized Ability, Low Memory/Speed/Reasoning; Low Gc = Low Crystallized Ability, High Memory/Speed/Reasoning; Low PP/OP = Low Phonological and/or Orthographic Processing; High Gf = High Fluid Reasoning, Low Memory/Speed; Low Gf = Low Fluid Reasoning, High Memory/Speed; PK = prekindergarten.

Measures

KTEA-3

The KTEA-3 (A. S. Kaufman & Kaufman, 2014) is an individually administered measure of academic achievement for grades PK through 12, or ages 4 through 25. The KTEA-3 has two independent, parallel forms (A and B); covers a wide range of achievement and language domains; and provides error analysis capabilities. The KTEA-3 is unique among diagnostic achievement tests in that it provides norm-referenced error analysis data to assist the examiner in understanding error patterns for individual students. For KTEA-3 subtests with item-level error analysis, including Phonological Processing, Reading Comprehension, Written Expression, Math Computation, and Math Concepts & Applications, each incorrect item counts as an error in one or more categories to which that item belongs. For KTEA-3 subtests with within-item error analysis, including Letter & Word Recognition, Nonsense Word Decoding, Spelling, and Math Computation (Math Computation includes both item-level and within-item-level error analysis), the examiner determines error classifications based on a qualitative analysis of the student’s response.

The KTEA-3 Letter Naming Facility and Object Naming Facility subtests were used as measures of Gs. The Phonological Processing subtest was used as a measure of phonological processing, and the Letter Naming Facility subtest was used as the measure of orthographic processing.

DAS-II

The DAS-II (Elliott, 2007) is an individually administered clinical instrument for assessing the cognitive abilities of children and adolescents ages 2 years 6 months through 17 years. The DAS-II comprises 20 subtests divided into two batteries: the Early Years (2:6-6:11) and School-Age (7:0-17:11) batteries. Both the Early Years and the School-Age batteries are organized into a set of core subtests that yields a composite score focused on reasoning and conceptual abilities called the General Conceptual Ability (GCA) score. The GCA was used as a measure of global cognitive ability (g). The CHC terms for the remaining scales used in this study include the Nonverbal Reasoning Cluster (Gf), the Working Memory Cluster (Gsm), the Processing Speed Cluster (Gs), the Verbal Cluster (Gc), and the Phonological Processing subtest (Ga—phonetic coding).

WISC-V

The WISC-V (Wechsler, 2014) is an individually administered, comprehensive clinical instrument for assessing the intelligence of children ages 6 through 16. The WISC-V includes 10 primary subtests, six secondary subtests, and five complementary subtests. The 10 primary subtests are used in certain combinations to derive the Full Scale Intelligent Quotient (FSIQ), the five primary index scores, and three of the five ancillary index scores. The CHC terms for the index scores used in this study include the FSIQ as a measure of g, the Verbal Comprehension Index (Gc), the Fluid Reasoning Index (Gf), the Working Memory Index (Gsm), the Symbol Translation Index and the Storage & Retrieval Index (Glr), and the Processing Speed Index (Gs).

KABC-II

The KABC-II (A. S. Kaufman & Kaufman, 2004) is an individually administered measure of the processing and cognitive abilities of children and adolescents ages 3 through 18. The KABC-II subtests are grouped into five scales that correspond to processing areas and broad abilities from the Luria and CHC models. The CHC terms for the scales used in this study include the Fluid-Crystallized Index as a measure of g, Knowledge (Gc), Planning (Gf), Sequential (Gsm), and Learning (Glr).

Analysis

A multistep process was used to investigate the relationship between students’ cognitive profiles and their corresponding KTEA-3 error scores in reading, writing, and mathematics. The first analytic step in this process was the derivation of factor scores.

The KTEA-3 utilizes a unique error analysis methodology based on the specific subskills measured by a given subtest. For 10 of the KTEA-3 subtests, curriculum experts identified the different categories of errors students are likely to make on each subtest. For each category of error on a given subtest, students received a grade-level, normative performance label of weakness, average, or strength based on a comparison of a student’s total errors to the average number of errors made by individuals in the KTEA-3 normative sample (A. S. Kaufman, Kaufman, & Breaux, 2014). This performance label is called the skill status. Based on this error analysis system, students received multiple skill status error scores within each subtest. To facilitate the use of these skill status error scores in further analyses, exploratory factor analysis and principal components analysis were used to create a reduced error score variable set.

To create the factor scores, polychoric correlation matrices were generated for each subtest. An exploratory factor analysis using unweighted least square extraction was conducted for each of the subtests, excluding Reading Comprehension, Written Expression, and Phonological Processing. Because the Reading Comprehension, Written Expression, and Phonological Processing subtests include a small number of error scores, principal components analysis was used to extract the factors for these subtests (for details of the analyses, see Choi et al., 2017; Hatcher et al., 2017; O’Brien et al., 2017).

Regardless of the factor extraction technique, a combination of parallel analysis (Horn, 1965), a visual inspection of the scree plot (Cattell, 1966), and content review of the factor structure were used to determine the number of factors to extract. For the subtests related to the current study, two factors were extracted from the Comprehension and Expression subtests, three factors for Letter & Word Recognition, and two factors for Nonsense Word Decoding, Spelling, and Phonological Processing subtest. R Version 3.2.3 was used to generate Bartlett factor scores for each of the extracted factors.

The next analytic step involved the classification of students into the three reading and writing profile groups and the three math groups based on the students’ cognitive profiles of strengths and weakness. Regardless of whether the scores were obtained on the KTEA-3, DAS-II, KABC-II, or WISC-V, each student’s profile of broad ability scores was reviewed independently by four coders to identify as many cases as possible that fit each cognitive profile. The High Gc profiles included Gc scores greater than 90 and at least 15 points higher than most other broad ability scores. The low Gc profile included Gc scores less than 110 (although the vast majority were below 90) and at least 15 points lower than most other broad ability scores. The Low PP/OP group included PP and/or OP scores that were less than 90 and at least 15 points lower than the overall cognitive ability estimate. A case was assigned to a group if at least two of the four raters agreed. No overlapping cases occurred between the first two reading and writing groups (High Gc, Low Gc) or between the first two math groups (High Gf, Low Gf). However, cases in the High Gc group for either the math or language sample were also allowed to belong to another group if the inclusion criteria were met. Cases were evaluated for group membership separately for the language profiles and the math profiles. For this reason, a case that was included in the High Gc group for the language sample may not have been included in this group for the math sample, or vice versa, if the case was a better fit for one of the other groups. From the original sample of 222 students, 23 students presented with scores that were not consistent with any of the cognitive profiles of interest. A total of 199 cases were included in one or more math profile groups and one or more language profile groups.

It was determined that a case would belong to a group if at least two of the four raters coded the case for that group. No overlapping cases occurred between the first two reading and writing groups (High Gc, Low Gc) or between the first two math groups (High Gf, Low Gf); however, cases in the High Gc, Low Glr/Gwm/Gs/Gf group (for either the math or language sample) were also allowed to belong to another group if the inclusion criteria were met. Cases were evaluated for group membership separately for the language and math profiles. For this reason, a case that was included in the High Gc, Low Glr/Gwm/Gs/Gf group for the language sample may not have been included in this group for the math sample, or vice versa, if the case was a better fit for one of the other groups.

The final analytic step was to investigate whether students with different profiles of strengths and weaknesses have different mean error factor scores on the reading, writing, and mathematics subtests. For the three math groups and the three language groups, an ANOVA was conducted with subtest error factor scores as dependent variables and the profile of strength and weakness as the independent variable.

Prior to conducting the analyses, each set of subtest factor scores was examined for univariate normality issues and outliers. Any extreme cases were analyzed to verify their impact on the distributional properties of each subtest. Using a criteria of |2| skewness and |6| kurtosis (Lix, Keselman, & Keselman, 1996), no violations of normality were observed. To examine the assumption of homogeneity of within-group covariance matrixes, a two-step analysis process was utilized (Huberty & Petoskey, 2000). First, for each analysis, the Box F test was calculated. For each subtest, the Box test was statistically significant. However, as noted by Huberty and Petoskey (2000), the Box test is an extremely powerful test. Therefore, as a follow-up analysis, the natural log of the determinant of the covariance matrix for each level of the independent variable was compared with the natural log determinant of the pooled matrix (Huberty & Petoskey, 2000; Olejnik, 2010) for each subtest. In the judgment of the researchers, the differences were relatively close, with the largest difference between a given group and the pooled natural log determinant equal to −2.00.

Results

Means and standard deviations for the language and math samples were computed for each KTEA-3 subtest and composite scores (see Table 2 for the language sample; see Table 3 for the math sample).

Table 2.

Means and Standard Deviations on Subtests and Composites of Interest for the Language Sample.

	High GcMean (SD)	Low GcMean (SD)	Low PP/OPMean (SD)
Subtest scores
Letter & Word Recognition	101.6 (12.3)	97.8 (11.1)	93.6 (9.2)
Nonsense Word Decoding	103.6 (9.5)	99.5 (12.4)	98.3 (13.1)
Phonological Processing	99.1 (12.8)	96.6 (12.0)	92.2 (9.7)
Reading Comprehension	103.3 (12.2)	96.3 (11.9)	95.6 (8.8)
Reading Vocabulary	104.1 (10.4)	97.5 (10.1)	97.5 (11.3)
Written Expression	103.8 (10.6)	98.1 (14.3)	98.6 (10.8)
Spelling	99.6 (11.1)	97.6 (11.9)	93.0 (9.3)
Oral Expression	97.9 (14.3)	93.8 (11.7)	90.5 (9.4)
Listening Comprehension	103.7 (10.0)	94.2 (11.8)	96.6 (10.7)
Associational Fluency	97.4 (13.0)	95.4 (14.0)	91.3 (14.1)
Object Naming Fluency	99.3 (10.4)	102.1 (15.6)	95.8 (12.4)
Letter Naming Fluency	98.7 (10.4)	100.4 (15.1)	90.0 (11.1)
Composite scores
Reading	106.4 (13.3)	97.3 (11.6)	95.7 (11.9)
Orthographic Processing	102.4 (12.4)	100.1 (13.0)	90.7 (12.0)
Written Language	102.3 (12.3)	98.0 (14.4)	96.6 (12.1)
Oral Language	104.5 (13.0)	94.3 (13.3)	94.3 (12.9)
Sound–Symbol	102.4 (13.1)	97.8 (11.9)	93.8 (12.1)
Decoding	103.6 (12.5)	97.3 (11.9)	93.8 (13.0)
Reading Fluency	103.8 (11.1)	100.9 (11.2)	97.6 (13.3)
Academic Fluency	101.1 (12.6)	101.2 (12.0)	94.0 (12.5)
Reading Understanding	107.8 (12.1)	96.4 (10.9)	98.0 (14.1)
Comprehension	106.9 (12.3)	95.2 (11.8)	97.0 (13.1)
Expression	102.6 (12.7)	95.9 (14.5)	96.7 (12.4)

Note. High Gc = High Crystallized Ability, Low Memory/Speed/Reasoning; Low Gc = Low Crystallized Ability, High Memory/Speed/Reasoning; Low PP/OP = Low Phonological and/or Orthographic Processing.

Table 3.

Means and Standard Deviations on Subtests and Composites of Interest for the Math Sample.

	High GfMean (SD)	Low GfMean (SD)	High GcMean (SD)
Subtest scores
Math Concepts & Applications	105.9 (11.6)	102.6 (10.7)	104.6 (8.6)
Math Computation	106.6 (12.3)	100.3 (10.0)	101.4 (11.4)
Math Fluency	101.2 (9.5)	100.7 (10.4)	98.1 (8.4)
Oral Expression	98.5 (14.5)	95.7 (12.3)	95.3 (14.2)
Listening Comprehension	99.4 (12.2)	97.2 (10.5)	103.4 (10.8)
Associational Fluency	98.1 (15.6)	96.0 (12.2)	96.5 (12.7)
Object Naming Fluency	97.0 (11.3)	103.0 (8.4)	98.2 (10.2)
Letter Naming Fluency	98.5 (11.3)	101.7 (10.6)	98.4 (9.8)
Composite Scores
Math	106.0 (13.6)	101.6 (12.6)	100.5 (11.2)
Reading	104.0 (12.7)	100.7 (11.3)	104.8 (14.5)
Academic Skills Battery	104.0 (12.3)	98.5 (9.6)	100.4 (10.8)
Written Language	102.7 (11.8)	101.1 (11.7)	99.6 (13.7)
Oral Language	100.6 (14.3)	98.3 (11.8)	104.5 (14.2)
Academic Fluency	100.1 (12.7)	105.3 (12.1)	100.0 (13.6)
Comprehension	102.8 (13.3)	98.8 (13.7)	106.3 (13.0)
Expression	101.9 (11.8)	99.1 (12.4)	100.7 (14.0)

Note. High Gf = High Fluid Reasoning, Low Memory/Speed; Low Gf = Low Fluid Reasoning, High Memory/Speed; High Gc = High Crystallized Ability, Low Memory/Speed/Reasoning.

Differences Between Strengths and Weaknesses Profiles and KTEA-3 Oral Language, Reading, Writing, and Spelling Error Scores

All error factors were named and interpreted based on a consolidation of expert reviews (D. Kilpatrick, personal communication, April 2, 2016; N. Mather, personal communication, March 26, 2016; J. Willis & R. Dumont, personal communication, March 26, 2016). For the first set of analysis, different cognitive profiles of strengths and weakness were compared with mean error factor scores on the KTEA-3 oral language, reading, writing, and spelling subtests. To examine the first hypothesis, ANOVA was utilized to examine the mean error factor score differences for each cognitive profile on each subtest. Means and standard deviations are presented in Table 4, and ANOVA results and mean differences can be viewed in Table 5.

Table 4.

Mean Error Scores by Group for the Language Sample.

	High GcMean (SD)	Low GcMean (SD)	Low PP/OPMean (SD)
Letter & Word Recognition
Contextual vowel pronunciation	11.1 (2.0)	10.2 (2.6)	10.4 (2.0)
Intermediate letter–sound knowledge	10.1 (3.0)	9.8 (3.4)	10.2 (2.9)
Consonant–pattern knowledge	11.1 (2.2)	10.5 (2.8)	10.6 (2.6)
Spelling
Sound-to-letter mapping	10.0 (2.7)	10.7 (2.2)	11.0 (1.9)
Phonological awareness	10.0 (3.5)	9.8 (3.15)	9.5 (2.5)
Nonsense Word Decoding
Letter-sound knowledge	9.9 (2.4)	10.7 (2.2)	9.9 (3.3)
Basic phonic decoding	10.0 (3.0)	9.9 (3.2)	10.0 (3.06)
Written Expression
General written expression	11.0 (1.6)	9.7 (2.9)	10.8 (1.8)
Writing mechanics	10.7 (2.0)	10.1 (2.8)	10.6 (1.9)
Oral Expression
Oral expression grammar	11.1 (1.6)	9.9 (2.8)	10.2 (2.7)
General oral expression	10.8 (2.4)	9.9 (3.0)	10.9 (2.0)
Listening Comprehension
Expository—Literal	11.0 (2.1)	9.8 (2.4)	10.2 (2.4)
Narrative—Inferential	10.0 (2.6)	10.2 (2.8)	10.9 (2.8)
Reading Comprehension
Expository—Literal	10.7 (1.8)	9.9 (3.0)	10.3 (1.7)
Narrative—Inferential	10.8 (2.0)	10.6 (2.6)	10.6 (2.0)
Phonological Processing
Basic phonological awareness	10.5 (2.2)	10.1 (2.7)	10.0 (2.4)
Advanced phonological processing	10.3 (2.7)	9.8 (3.1)	9.7 (2.9)

Note. Factor z-scores were converted to scaled scores with a mean of 10 and standard deviation of 3. High Gc = High Crystallized Ability, Low Memory/Speed/Reasoning; Low Gc = Low Crystallized Ability, High Memory/Speed/Reasoning; Low PP/OP = Low Phonological and/or Orthographic Processing.

Table 5.

ANOVA Results and Pairwise Comparisons by Error Factors for the Language Sample.

	Mean z-score difference
	F	p	High Gc–Low Gc	High Gc–Low PP/OP	Low Gc–Low PP/OP
Letter & Word Recognition
Contextual vowel pronunciation	2.64	.07	0.30	0.23	−0.07
Intermediate letter–sound knowledge	0.27	.77	0.11	−0.05	−0.16
Consonant-pattern knowledge	1.22	.30	0.20	0.18	−0.02
Spelling
Sound-to-letter mapping	1.75	.18	−0.22	−0.33	−0.10
Phonological awareness	0.20	.82	0.07	0.17	0.10
Nonsense Word Decoding
Letter–sound knowledge	1.22	.30	−0.25	0.01	0.26
Basic phonic decoding	0.06	.94	0.06	0.001	−0.06
Written Expression
General written expression	5.69	.004	0.43**	0.06	−0.37
Writing mechanics	1.21	.30	0.21	0.03	−0.18
Oral Expression
Oral expression grammar	3.58	.03	0.41**	0.29	−0.12
General oral expression	1.79	.17	0.28	−0.06	−0.34
Listening Comprehension
Expository—Literal	2.43	.09	0.40	0.27	−0.13
Narrative—Inferential	0.76	.47	0.26	0.06	−0.21
Reading Comprehension
Expository—Literal	1.29	.28	0.26	0.11	−0.15
Narrative—Inferential	0.15	.86	0.07	0.07	−0.004
Phonological Processing
Basic phonological awareness	0.57	.57	0.12	0.14	0.01
Advanced phonological processing	0.80	.45	0.16	0.22	0.06

Note. High Gc = High Crystallized Ability, Low Memory/Speed/Reasoning; Low Gc = Low Crystallized Ability, High Memory/Speed/Reasoning; Low PP/OP = Low Phonological and/or Orthographic Processing.

p < .05.

Reading subtests

Error factor scores from four reading subtests were analyzed: Letter & Word Recognition, Nonsense Word Decoding, Phonological Processing, and Reading Comprehension. Three error factors were generated for the Letter & Word Recognition subtest: Contextual Vowel Pronunciation, Intermediate Letter–Sound Knowledge, and Consonant-Pattern Knowledge. Two error factors were generated for the Nonsense Word Decoding subtest: Letter–Sound Knowledge and Basic Phonic Decoding. Two error factors were generated for the Phonological Processing subtest: Basic Phonological Awareness (phonological awareness and sound awareness) and Advanced Phonological Processing. Two error factor scores were generated for the Reading Comprehension subtest: Expository—Literal and Narrative—Inferential.

For each of the reading subtests, the three cognitive profile groups did not differ on the error factor scores. The largest difference was found on Contextual Vowel Pronunciation, F(2, 152) = 2.64, p > .05. For this factor, the differences in cognitive profiles explained 3.5% (R² = .035) of the variation in error factor scores. This slight difference can be attributed to the High Gc group making fewer errors on this factor when compared with Low Gc and Low PP/OP groups.

Writing subtests

Error factor scores from two writing subtests were analyzed: Spelling and Written Expression. Spelling had two error factors: Sound-to-Letter Mapping and Phonological Awareness. The differences in the cognitive profile groups for the Spelling error factors were not statistically significant. Written Expression had two error factors: General Written Expression and Writing Mechanics. For Written Expression, Writing Mechanics was nonsignificant at F(2, 152) = 1.21, p > .05, R² = .016; however, the General Written Expression factor was significant at F(2, 152) = 5.69, p < .01, R² = .07. Because the overall F was statistically significant for the General Written Expression factor, post hoc comparisons with a Bonferroni correction were conducted. These results indicated that the difference between the High Gc, Low Memory, Speed, and/or Reasoning group and the Low Gc, High Memory, Speed, and/or Reasoning group was both statistically significant.

Language subtests

Errors from two language subtests were analyzed: Listening Comprehension and Oral Expression. Listening Comprehension error factors included Expository—Literal and Narrative—Inferential, and Oral Expression error factors included Oral Expression Grammar and General Oral Expression. For Listening Comprehension, the Expo Expository—Literal sitory-Literal and the Narrative- Narrative—Inferential Inferential error factor scores were not significantly different across the cognitive profile groups. For the Oral Expression subtest, statistically significant differences were found between cognitive profiles groups on the General Oral Expression error factor, F(2, 121) = 3.58, p < .05, R² = .056. A review of the post hoc comparisons with a Bonferroni correction indicated that the difference in the number of errors for High Gc, Low Memory, Speed, and/or Reasoning group and the Low Gc, High Memory, Speed, and/or Reasoning group was statistically significant.

Differences Between Strengths and Weaknesses Profiles and KTEA-3 Math Error Scores

The same method that was used for the language sample analyses was used for the math sample using the mathematics subtests: Math Concepts & Applications and Math Computation. To examine the second research question, five ANOVAs were conducted with the cognitive profiles serving as independent variables and error factor scores as dependent variables. The analyses showed nonsignificant results between cognitive profiles for each of the mathematics error factor scores. Means and standard deviations are presented in Table 6, and ANOVA results and mean differences can be viewed in Table 7.

Table 6.

Mean Error Scores by Group for the Math Sample.

	High GfMean (SD)	Low GfMean (SD)	High GcMean (SD)
Math Concepts & Applications
Math calculation	10.4 (2.6)	11.5 (1.2)	10.1 (2.4)
Geometric concepts	10.1 (3.8)	10.7 (2.7)	9.4 (4.1)
Complex math problems	9.9 (3.2)	9.8 (3.1)	10.4 (2.9)
Math Computation
Basic math concepts	10.9 (2.6)	12.4 (1.3)	10.5 (3.4)
Addition	9.3 (3.5)	10.3 (2.9)	10.2 (3.0)

Note. Factors z-scores were converted to scaled scores with a mean of 10 and standard deviation of 3. High Gf = High Fluid Reasoning, Low Memory/Speed; Low Gf = Low Fluid Reasoning, High Memory/Speed; High Gc = High Crystallized Ability, Low Memory/Speed/Reasoning.

Table 7.

ANOVA Results and Pairwise Comparisons by Error Factors for the Math Sample.

	Mean z-score difference
KTEA-3 error factors	F	p	High Gf–Low Gf	High Gf–High Gc	Low Gf–High Gc
Math Concepts & Applications
Math calculation	1.86	.16	−0.38	0.10	0.48
Geometric concepts	0.79	.46	−0.23	0.20	0.43
Complex math problems	0.44	.64	0.06	−0.16	−0.22
Math Computation
Basic math concepts	1.75	.19	−0.49	0.17	0.66
Addition	0.48	.62	−0.35	−0.32	0.03

Note. High Gf = High Fluid Reasoning, Low Memory/Speed; Low Gf = Low Fluid Reasoning, High Memory/Speed; High Gc = High Crystallized Ability, Low Memory/Speed/Reasoning.

Discussion

Based on the extant CHC literature, the PSW profiles selected for the present study were expected to differentially affect academic skill acquisition among school-age children. However, this hypothesis was only partially supported. Results indicated a significant difference in error scores between the High Gc group (students with high crystallized ability coupled with low long-term memory, short-term working memory, processing speed, and/or fluid reasoning) and the Low Gc group (students with low crystallized ability coupled with high long-term memory, short-term working memory, processing speed, and/or fluid reasoning) on the General Written Expression factor and the General Oral Expression factor. This finding suggests that a strength in Gc may be especially influential in facilitating expressive language skill acquisition, even more so than relative strengths in long-term storage and retrieval, short-term working memory, processing speed, and/or fluid reasoning. These results are consistent with previous research (e.g., Bourke & Adams, 2011; Evans et al., 2002; Flanagan et al., 2013; Floyd et al., 2008; Gathercole & Alloway, 2008) indicating that crystallized ability is an essential broad ability for developing oral and written expression skills.

The demographic characteristics of the High Gc groups indicated higher parent education level on average relative to the other groups, which could potentially indicate higher average socioeconomic status and cognitive ability levels. However, the High Gc group did not differ from the other groups on the Math Concepts & Applications error factors, and performance on Math Concepts & Applications tends to correlate highly with general intelligence (A. S. Kaufman et al., 2014). Hence, the results of the present study are not believed to be attributable to demographic or cognitive ability differences between the PSW groups.

The lack of significant differences between cognitive profile groups on mathematics error factor scores was unexpected. A similar study conducted by Koriakin et al. (2017) found significant differences between a group with High Gc (paired with low processing speed and/or long-term storage and retrieval) and a group with Low Gc (paired with high processing speed and/or long-term storage and retrieval) on three error factor scores: two error factors on Math Concepts & Applications (Math Calculation and Complex Math Problems) and one error factor on Math Computation (Basic Math Concepts). The Koriakin et al. study used different measures of broad abilities to identify cognitive profiles (all measures were from the KTEA-3), which may have accounted for the lack of significant differences in this study.

Limitations and Future Directions

These findings must be interpreted in light of the limitations of the present study, which may direct future research in this area. One of the limitations of the study was that the authors determined groups based on broad ability scores from a variety of different cognitive ability measures. Broad ability scores do not typically reflect the same underlying narrow abilities from one test to another. As a result, students with similar broad ability profiles may have differed in their profile of narrow abilities, which could have increased heterogeneity within groups. In contrast to the findings of the present study, Liu et al. (2017) found that particular profiles of strengths and weaknesses differentially predict error performance across several tests of reading and writing, and Koriakin et al. (2017) found that particular profiles of strengths and weaknesses differentially predict error performance across tests of mathematics. The results of the present study along with the Liu et al. and Koriakin et al. studies suggest that the measures used to determine profiles may be an important sampling consideration for identifying group differences.

In addition, this study included a group with Low PP/OP. Due to sample size limitations, distinct phonological and orthographic groups were not feasible. However, the error factor scores of students with phonological processing weaknesses may differ from those of students with orthographic processing weaknesses, as shown in Liu et al. (2017). Combining these profiles into one group may have obscured group differences in error factor scores.

Previous research (Flanagan et al., 2013) suggests that Gc becomes increasingly important for reading skill acquisition as age increases; therefore, including samples with narrower age bands may affect the results. Subsequent research could investigate whether error patterns differ between these cognitive profiles at different age/grade levels. Finally, the PSW groups differed on some demographic characteristics, and future studies could test the stability of error factor performance across demographic variables.

Footnotes

Acknowledgements

The authors wish to thank NCS Pearson for providing the standardization and validation data for the Kaufman Test of Educational Achievement–Third Edition (KTEA-3). Copyrights by NCS Pearson, Inc., used with permission. They also wish to thank Alan and Nadeen Kaufman for their supervision of the comprehensive error analysis research program.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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