Cognitive Assessment

WISC-V Expanded Composite Scores

I have long complained that making custom composite scores should not be difficult. The ability to combine scores as one wishes should be a feature of every scoring program for every cognitive battery.

No matter which tests I have given, I would like to be able to combine them into theoretically valid composite scores. For example, on the WISC-V, the Verbal Comprehension Index (VCI) consists of two subtest scores, Vocabulary and Similarities. However, the Information and Comprehension subtests measure verbal knowledge just as well as the other two tests. We should be able to combine them with the two VCI subtests to make a more valid estimate of verbal knowledge.

The good news is that the WISC-V now allows us to do just that: It now has two expanded composite scores:

  1. Verbal Expanded Crystallized Index (VECI)
    • Similarities
    • Vocabulary
    • Information
    • Comprehension
  2. Expanded Fluid Index (EFI)
    • Matrix Reasoning
    • Picture Concepts
    • Figure Weights
    • Arithmetic

At the risk of sounding greedy, I would like to have an expanded working memory index (Digit Span, Picture Span, and Letter-Number Sequencing) and an expanded processing speed index (Coding, Symbol Search, and Cancellation). Even so, I am grateful for this improvement in the WISC-V.

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Cognitive Assessment, R

Is the structure of intelligence different for people with learning disorders? Let’s hope so!

I do not relish criticizing published studies. However, if a paper uses flawed reasoning to arrive at counterproductive recommendations for our field, I believe that it is proper to respectfully point out why the paper’s conclusions should be ignored. This study warrants such a response:

Giofrè, D., & Cornoldi, C. (2015). The structure of intelligence in children with specific learning disabilities is different as compared to typically developing children. Intelligence, 52, 36–43.

The authors of this study ask whether children with learning disorders have the same structure of intelligence as children in the general population. This might seem like an important question, but it is not—if the difference in structure is embedded in the very definition of learning disorders.

An Analogously Flawed Study

Imagine that a highly respected medical journal published a study titled Tall People Are Significantly Greater in Height than People in the General Population. Puzzled and intrigued, you decide to investigate. You find that the authors solicited medical records from physicians who labelled their patients as tall. The primary finding is that such patients have, on average, greater height than people in the general population. The authors speculate that the instruments used to measure height may be less accurate for tall people and suggest alternative measures of height for them.

This imaginary study is clearly ridiculous. No researcher would publish such a “finding” because it is not a finding. People who are tall have greater height than average by definition. There is no reason to suppose that the instruments used were inaccurate.

Things That Are True By Definition Are Not Empirical Findings.

It is not so easy to recognize that Giofrè and Cornoldi applied the same flawed logic to children with learning disorders and the structure of intelligence. Their primary finding is that in a sample of Italian children with clinical diagnoses of specific learning disorder, the four index scores of the WISC-IV have lower g-loadings than they do in the general population in Italy. The authors believe that this result implies that alternative measures of intelligence might be more appropriate than the WISC-IV for children with specific learning disorders.

What is the problem with this logic? The problem is that the WISC-IV was one of the tools used to diagnose the children in the first place. Having unusual patterns somewhere in one’s cognitive profile is part of the traditional definition of learning disorders. If the structure of intelligence were the same in this group, we would wonder if the children had been properly diagnosed. This is not a “finding” but an inevitable consequence of the traditional definition of learning disorders. Had the same study been conducted with any other cognitive ability battery, the same results would have been found.

People with Learning Disorders Have Unusual Cognitive Profiles.

A diagnosis of a learning disorder is often given when a child of broadly average intelligence has low academic achievement due to specific cognitive processing deficits. To have specific cognitive processing deficits, there must be a one or more specific cognitive abilities that are low compared to the population and also to the child’s other abilities. For example, in the profile below, the WISC-IV Processing Speed Index of 68 is much lower than the other three WISC-IV index scores, which are broadly average. Furthermore, the low processing speed score is a possible explanation of the low Reading Fluency score.

WISC IV LD Profile

The profile above is unusual. The Processing Speed (PS) score is unexpectedly low compared to the other three index scores. This is just one of many unusual score patterns that clinicians look for when they diagnose specific learning disorders. When we gather together all the unusual WISC-IV profiles in which at least one score is low but others are average or better, it comes as no surprise that the structure of the scores in the sample is unusual. Because the scores are unusually scattered, they are less correlated, which implies lower g-loadings.

A Demonstration That Selecting Unusual Cases Can Alter Structural Coefficients

Suppose that the WISC-IV index scores have the correlations below (taken from the U.S. standardization sample, age 14).

VC PR WM PS
VC 1.00 0.59 0.59 0.37
PR 0.59 1.00 0.48 0.45
WM 0.59 0.48 1.00 0.39
PS 0.37 0.45 0.39 1.00

Now suppose that we select an “LD” sample from the general population all scores in which

  • At least one score is less than 90.
  • The remaining scores are greater than 90.
  • The average of the three highest scores is at least 15 points higher than the lowest score.

Obviously, LD diagnosis is more complex than this. The point is that we are selecting from the general population a group of people with unusual profiles and observing that the correlation matrix is different in the selected group. Using the R code at the end of the post, we see that the correlation matrix is:

VC PR WM PS
VC 1.00 0.15 0.18 −0.30
PR 0.15 1.00 0.10 −0.07
WM 0.18 0.10 1.00 −0.20
PS −0.30 −0.07 −0.20 1.00

A single-factor confirmatory factor analysis of the two correlation matrices reveals dramatically lower g-loadings in the “LD” sample.

Whole Sample “LD” Sample
VC 0.80 0.60
PR 0.73 0.16
WM 0.71 0.32
PS 0.53 −0.51

Because the PS factor has the lowest g-loading in the whole sample, it is most frequently the score that is out of sync with the others and thus is negatively correlated with the other tests in the “LD” sample.

In the paper referenced above, the reduction in g-loadings was not nearly as severe as in this demonstration, most likely because clinicians frequently observe specific processing deficits in tests outside the WISC. Thus many people with learning disorders have perfectly normal-looking WISC profiles; their deficits lie elsewhere. A mixture of ordinary and unusual WISC profiles can easily produce the moderately lowered g-loadings observed in the paper.

Conclusion

In general, one cannot select a sample based on a particular measure and then report as an empirical finding that the sample differs from the population on that same measure. I understand that in this case it was not immediately obvious that the selection procedure would inevitably alter the correlations among the WISC-IV factors. It is clear that the authors of the paper submitted their research in good faith. However, I wish that the reviewers had noticed the problem and informed the authors that the paper was fundamentally flawed. Therefore, this study offers no valid evidence that casts doubt on the appropriateness of the WISC-IV for children with learning disorders. The same results would have occurred with any cognitive battery, including those recommended by the authors as alternatives to the WISC-IV.

R code used for the demonstration

# Correlation matrix from U.S. Standardization sample, age 14
WISC <- matrix(c(
  1,0.59,0.59,0.37, #VC
  0.59,1,0.48,0.45, #PR
  0.59,0.48,1,0.39, #WM
  0.37,0.45,0.39,1), #PS
  nrow= 4, byrow=TRUE)
colnames(WISC) <- rownames(WISC) <- c("VC", "PR", "WM", "PS")

#Set randomization seed to obtain consistent results
set.seed(1)

# Generate data
x <- as.data.frame(mvtnorm::rmvnorm(100000,sigma=WISC)*15+100)
colnames(x) <- colnames(WISC)

# Lowest score in profile
minSS <- apply(x,1,min)

# Mean of remaining scores
meanSS <- (apply(x,1,sum) - minSS) / 3

# LD sample
xLD <- x[(meanSS > 90) & (minSS < 90) & (meanSS - minSS > 15),]

# Correlation matrix of LD sample
rhoLD <- cor(xLD)

# Load package for CFA analyses
 library(lavaan)
# Model for CFA
m <- "g=~VC + PR + WM + PS"

# CFA for whole sample
summary(sem(m,x),standardized=TRUE)

# CFA for LD sample
summary(sem(m,xLD),standardized=TRUE)
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Cognitive Assessment, Research Link

No, the WISC-IV doesn’t underestimate the intelligence of children with autism.

The title of a new study asks “Does WISC-IV underestimate the intelligence of autistic children?” The authors’ answer is that it probably does. I believe that the reasoning behind this conclusion is faulty.

This study gives the unwarranted impression that it is a disservice to children with autism to use the WISC-IV. Let me be clear—I want to be helpful to children with autism. I certainly do not wish to do anything that hurts anyone. A naive reading of this article leads us to believe that there is an easy way to avoid causing harm (i.e., use the Raven’s Progressive Matrices test instead of the WISC-IV). In my opinion, acting on this advice does no favors to children with autism and may even result in harm.

Based on the evidence presented in the study, the average score differences between children with and without autism is smaller on Raven’s Progressive Matrices (RPM) and larger on the WISC-IV. The rhetoric of the introduction leaves the reader with the impression that the RPM is a better test of intelligence than the WISC-IV. Once we accept this, it is easy to discount the results of the WISC-IV and focus primarily on the RPM.

There is a seductive undercurrent to the argument: If you advocate for children with autism, don’t you want to show that they are more intelligent rather than less intelligent? Yes, of course! Doesn’t it seem harmful to give a test that will show that children with autism are less intelligent? It certainly seems so!

Such rhetoric reveals a fundamental misunderstanding of what individual intelligence tests like the WISC-IV are designed to do. In the vast majority of settings, they are not for certifying how intelligent a person is (whatever that means!). Their primary purpose is to help psychologists understand what a person can and cannot do. They are designed to help explain what is easy and what is difficult for a person so that appropriate interventions can be selected.

The WISC-IV provides a Full Scale IQ, which gives an overall summary of cognitive functions. However, it also gives more detailed information about various aspects of ability. Here is a graph I constructed from Figure 1 in the paper. In my graph, I converted percentiles to index scores and rearranged the order of the scores to facilitate interpretation.

asdf

Average Raven’s Progressive Matrices (RPM) and WISC-IV scores for children with and without autism

It is clear that the difference between the two groups of children is small for the RPM. It is also clear that the difference is also small for the WISC-IV Perceptual Reasoning Index (PRI). Why is this? The RPM and the PRI are both nonverbal measures of logical reasoning (AKA fluid intelligence). Both the WISC-IV and the RPM tell us that, on average, children with autism perform relatively well in this domain. The RPM is a great test, but it has no more to tell us. In contrast, the WISC-IV not only tells us what children with autism, on average, do relatively well, but also what they typically have difficulty with.

It is no surprise that the largest difference is in the Verbal Comprehension Index (VCI), a measure of verbal knowledge and language comprehension. Communication problems are a major component of the definition of autism. If children with autism had performed equally well on the VCI, we would wonder whether the VCI was really measuring what it was supposed to measure. Note that I am not saying that a low score on VCI is a requirement for the diagnosis of autism or that the VCI is the best measure of the kinds of language problems that are characteristic of autism. Rather, I am saying that children with autism, on average, have difficulties with language comprehension and that this difference is manifest to some degree in the WISC-IV scores.

The WISC-IV scores also suggest that, on average, children with autism not only have lower scores in verbal knowledge and comprehension, they are more likely to have other cognitive deficits, including in verbal working memory (as measured by the WMI) and information processing speed (as measured by the PSI).

Thus, as a clinical instrument, the WISC-IV performs its purpose reasonably well. Compared to the RPM, it gives a more complete picture of the kinds of cognitive strengths and weaknesses that are common in children with autism.

If the researchers wish to demonstrate that the WISC-IV truly underestimates the intelligence of children with autism, they would need to show that it underpredicts important life outcomes among this population. For example, suppose we compare children with and without autism who score similarly low on the WISC-IV. If the WISC-IV underestimated the intelligence of children with autism, they would be expected to do better in school than the low-scoring children without autism. Obviously, a sophisticated analysis of this matter would involve a more complex research design, but in principle this is the kind of result that would be needed to show that the WISC-IV is a poor measure of cognitive abilities for children with autism.

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CHC Theory, Cognitive Assessment

Intelligence and the Modern World of Work: A Special Issue of Human Resource Management Review

Charles Scherbaum and Harold Goldstein took an innovative approach to editing a special issue of Human Resource Management Review. They asked prominent I/O psychologists to collaborate with scholars from other disciplines to explore how advances in intelligence research might be incorporated into our understanding of the role of intelligence in the workplace.

It was an honor to be invited to participate, and it was a pleasure to be paired to work with Daniel Newman of the University of Illinois at Urbana/Champaign. Together we wrote an I/O psychology-friendly introduction to current psychometric theories of cognitive abilities, emphasizing Kevin McGrew‘s CHC theory. Before that could be done, we had to articulate compelling reasons I/O psychologists should care about assessing multiple cognitive abilities. This was a harder sell than I had anticipated.

Formal cognitive testing is not a part of most hiring decisions, though I imagine that employers typically have at least a vague sense of how bright job applicants are. When the hiring process does include formal cognitive testing, typically only general ability tests are used. Robust relationships between various aspects of job performance and general ability test scores have been established.

In comparison, the idea that multiple abilities should be measured and used in personnel selection decisions has not fared well in the marketplace of ideas. To explain this, there is no need to appeal to some conspiracy of test developers. I’m sure that they would love to develop and sell large, expensive, and complex test batteries to businesses. There is also no need to suppose that I/O psychology is peculiarly infected with a particularly virulent strain of g zealotry and that proponents of multiple ability theories have been unfairly excluded.

To the contrary, specific ability assessment has been given quite a bit of attention in the I/O psychology literature, mostly from researchers sympathetic to the idea of going beyond the assessment of general ability.  Dozens (if not hundreds) of high-quality studies were conducted to test whether using specific ability measures added useful information beyond general ability measures. In general, specific ability measures provide only modest amounts of additional information beyond what can be had from general ability scores (ΔR2 ≈ 0.02–0.06). In most cases, this incremental validity was not large enough to justify the added time, effort, and expense needed to measure multiple specific abilities. Thus it makes sense that relatively short measures of general ability have been preferred to longer, more complex measures of multiple abilities.

However, there are several reasons that the omission of specific ability tests in hiring decisions should be reexamined:

  • Since the time that those high quality studies were conducted, multidimensional theories of intelligence have advanced, and we have a better sense of which specific abilities might be important for specific tasks (e.g., working memory capacity for air traffic controllers). The tests measuring these specific abilities have also improved considerably.
  • With computerized administration, scoring, and interpretation, the cost of assessment and interpretation of multiple abilities is potentially far lower than it was in the past. Organizations that make use of the admittedly modest incremental validity of specific ability assessments would likely have a small but substantial advantage over organizations that do not. Over the long run, small advantages often accumulate into large advantages.
  • Measurement of specific abilities opens up degrees of freedom in balancing the need to maintain the predictive validity of cognitive ability assessments and the need to reduce the adverse impact on applicants from disadvantaged minority groups that can occur when using such assessments. Thus, organizations can benefit from using cognitive ability assessments in hiring decisions without sacrificing the benefits of diversity.

The publishers of Human Resource Management Review have made our paper available to download for free until January 25th, 2015.

Broad Abilities in CHC Theory

Broad Abilities in CHC Theory

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Cognitive Assessment

John Willis’ Comments on Reports newsletter makes me happy.

Whenever I find that John Willis has posted a new edition of his Comments on Reports newsletter, I read it greedily and gleefully. Each newsletter is filled with sharp-witted observations, apt quotations, and practical wisdom about writing better psychological evaluation reports.

Recent gems:

From #251

The first caveat of writing reports is that readers will strive mightily to attach significant meaning to anything we write in the report. The second caveat is that readers will focus particularly on statements and numbers that are unimportant, potentially misleading, or — whenever possible — both. This is the voice of bitter experience.

Also from #251

Planning is so important that people are beginning to indulge in “preplanning,” which I suppose is better than “postplanning” after the fact. One activity we often do not plan is evaluations.

From #207:

I still recall one principal telling the entire team that, if he could not trust the spelling in my report, he could not trust any of the information in it. This happened recently (about 1975), so it is fresh in my mind. Names of tests are important to spell correctly. Alan and Nadeen Kaufman spell their last name with a single f and only one n. David Wechsler spelled his name as shown, never as Weschler. The American version of the Binet-Simon scale was developed at Stanford University, not Standford. I have to keep looking it up, but it is Differential Ability Scales even though it is a scale for several abilities. Richard Woodcock may, for all I know, have attended the concert, but his name is not Woodstock.

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Cognitive Assessment

Three inspired presentations at the Richard Woodcock Institute Event

On 10/24/2014, I attended the Richard Woodcock Institute Event hosted at University of Texas at Austin. The three speakers had strikingly different presentation styles but were equally excellent.

Dick Woodcock gave the opening remarks. I loved hearing about the twists and turns of his career and how he made the most of unplanned opportunities. It was rather remarkable how diverse his contributions are (including an electronic Braille typewriter). Then he stressed the importance of communicating test results in ways that non-specialists can understand. He speculated on what psychological testing will look like in the future, focusing on integrative software that will guide test selection and interpretation in more sophisticated ways than has hitherto been possible. Given that he has been creating the future of assessment for decades now, I am betting that he is likely to be right. Later he graciously answered my questions about the WJ77 and how he came up with what I consider to be among the most ingenious test paradigms we have.

After a short break, Kevin McGrew gave us a wild ride of a talk about advances in CHC Theory. Actually it was more like a romp than a ride. I tried to keep track of all the interesting ideas for future research he presented but there were so many I quickly lost count. The visuals were stunning and his energy was infectious. He offered a quick overview of new research from diverse fields about the overlooked importance of auditory processing (beyond the usual focus on phonemic awareness). Later he talked about his evolving conceptualization of the memory factors in CHC theory and role of complexity in psychometric tests. My favorite part of the talk was a masterful presentation of information processing theory, judiciously supplemented with very clever animations.

After lunch, Cathy Fiorello gave one of the most thoughtful presentations I have ever heard. Instead of contrasting nomothetic and idiographic approaches to psychological assessment, Cathy stressed their integration. Most of the time, nomothetic interpretations are good first approximations and often are sufficient. However, there are certain test behaviors and other indicators that a more nuanced interpretation of the underlying processes of performance is warranted. Cathy asserted (and I agree) that well trained and highly experienced practitioners can get very good at spotting unusual patterns of test performance that completely alter our interpretations of test scores. She called on her fellow scholars to develop and refine methods of assessing these patterns so that practitioners do not require many years of experience to develop their expertise. She was not merely balanced in her remarks—lip service to a sort of bland pluralism is an easy and rather lazy trick to seem wise. Instead, she offered fresh insight and nuance in her balanced and integrative approach to cognitive and neuropsychological assessment. That is, she did the hard work of offering clear guidelines of how to integrate nomothetic and idiographic methods, all the while frankly acknowledging the limits of what can be known.

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CHC Theory, Cognitive Assessment

Exploratory model of cognitive predictors of academic skills that I presented at APA 2014

I have many reservations about this model of cognitive predictors of academic abilities that I presented at APA today (along with co-presenters Lee Affrunti, Renée Tobin, and Kimberley Collins) but I think that it illustrates an important point: prediction and explanation of cognitive and academic abilities is so complex that it is impossible to do in one’s head. Eyeballing scores and making pronouncements is not likely to be accurate and will result in misinterpretations. We need good software that can manage the complex calculations for us. We can still think creatively in the diagnostic process but the creativity must be grounded in realistic probabilities.

The images from the poster are from a single exploratory model based on a clinical sample of 865 college students. The model was so big and complex I had to split the path diagram into two images:

Exploratory Model of WAIS and WJ III cognitive subtests

Exploratory Model of WAIS and WJ III cognitive subtests. Gc = Comprehension/Knowledge, Ga = Auditory processing, Gv = Visual processing, Gl = Long-term memory: Learning, Gr = Long-term memory: Retrieval speed, Gs = Processing speed, MS = Memory span, Gwm = Working memory capacity, g = Anyone’s guess

Exploratory model of cognitive predictors of WJ III academic subtests

Exploratory model of cognitive predictors of WJ III academic subtests. Percentages in error terms represent unexplained variance.

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Cognitive Assessment

Cognitive profiles are rarely flat.

Because cognitive abilities are positively correlated, there is an assumption that cognitive abilities should be evenly developed. When psychologists examine cognitive profiles, they often describe any features that deviate from the expected flat profile.

It is true, mathematically, that the expected profile IS flat. However, this does not mean that flat profiles are common. There is a very large set of possible profiles and only a tiny fraction are perfectly flat. Profiles that are nearly flat are not particularly common, either. Variability is the norm.

Sometimes it helps to get a sense of just how uneven cognitive profiles typically are. That is, it is good to fine-tune our intuitions about the typical profile with many exemplars. Otherwise it is easy to convince ourselves that the reason that we see so many interesting profiles is that we only assess people with interesting problems.

If we use the correlation matrix from the WAIS-IV to randomly simulate multivariate normal profiles, we can see that even in the general population, flat, “plain-vanilla” profiles are relatively rare. There are features that draw the eye in most profiles.

WAISIVProfilesIf cognitive abilities were uncorrelated, profiles would be much more uneven than they are. But even with moderately strong positive correlations, there is still room for quite a bit of within-person variability.

Let’s see what happens when we look at profiles that have the exact same Full Scale IQ (80, in this case). The conditional distributions of the remaining scores are seen in the “violin” plots. There is still considerable diversity of profile shape even though the Full Scale IQ is held constant.

WAISIVProfiles80Note that the supplemental subtests have wider conditional distributions because they are not included in the Full Scale IQ, not necessarily because they are less g-loaded.

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Cognitive Assessment, Psychometrics, R

How common is it to have no academic weaknesses?

I’m afraid that the question posed by the title does not have a single answer. It depends on how we define and measure academic performance.

Let’s sidestep some difficult questions about what exactly an “academic deficit” is and for the sake of convenience pretend that it is a score at least 1 standard deviation below the mean on a well normed test administered by a competent psychologist with good clinical skills.

Suppose that we start with the 9 core WJ III achievement tests (the answers will not be all that different with the new WJ IV):

ReadingWritingMathematics
SkillsLetter-Word IdentificationSpellingCalculation
ApplicationsPassage ComprehensionWriting SamplesApplied Problems
FluencyReading FluencyWriting FluencyMath Fluency

What is the percentage of the population that does not have any score below 85? If we can assume that the scores are multivariate normal, the answer can be found using data simulation or via the cumulative density function of the multivariate normal distribution. I gave examples of both methods in the previous post. If we use the correlation matrix for the 6 to 9 age group of the WJ III NU, about 47% of the population has no academic scores below 85.

Using the same methods we can estimate what percent of the population has no academic scores below various thresholds. Subtracting these numbers from 100%, we can see that fairly large proportions have at least one low score.

Threshold% with no scores below the threshold% with at least one score below the threshold
8547%53%
8063%37%
7577%23%
7087%13%

What proportion of people with average cognitive scores have no academic weaknesses?

The numbers in the table above include people with very low cognitive ability. It would be more informative if we could control for a person’s measured cognitive abilities.

Suppose that an individual has index scores of exactly 100 for all 14 subtests that are used to calculate the WJ III GIA Extended. We can calculate the means and the covariance matrix of the achievement tests for all people with this particular cognitive profile. We will make use of the conditional multivariate normal distribution. As explained here (or here), we partition the academic tests (\mathbf{X}_1) and the cognitive predictor tests (\mathbf{X}_2) like so:

\begin{pmatrix}\mathbf{X}_1 \\ \mathbf{X}_2 \end{pmatrix}\sim\mathcal{N}\left(\begin{pmatrix}\boldsymbol{\mu}_1 \\ \boldsymbol{\mu}_2\end{pmatrix},\begin{pmatrix}\mathbf{\Sigma}_{11} & \mathbf{\Sigma}_{12} \\ \mathbf{\Sigma}_{21} & \mathbf{\Sigma}_{22}\end{pmatrix}\right)

  • \boldsymbol{\mu}_1 and \boldsymbol{\mu}_2 are the mean vectors for the academic and cognitive variables, respectively.
  • \mathbf{\Sigma}_{11} and \mathbf{\Sigma}_{22} are the covariances matrices of academic and cognitive variables, respectively.
  • \mathbf{\Sigma}_{12} is the matrix of covariances between the academic and cognitive variables.

If the cognitive variables have the vector of particular values \mathbf{x}_2, then the conditional mean vector of the academic variables (\boldsymbol{\mu}_{1|2}) is:

\boldsymbol{\mu}_{1|2}=\boldsymbol{\mu}_1+\mathbf{\Sigma}_{12}\mathbf{\Sigma}^{-1}_{22}(\mathbf{x}_2-\boldsymbol{\mu}_2)

The conditional covariance matrix:
\mathbf{\Sigma}_{1|2}=\mathbf{\Sigma}_{11}-\mathbf{\Sigma}_{12}\mathbf{\Sigma}^{-1}_{22}\mathbf{\Sigma}_{21}

If we can assume multivariate normality, we can use these equations, to estimate the proportion of people with no scores below any threshold on any set of scores conditioned on any set of predictor scores. In this example, about 51% of people with scores of exactly 100 on all 14 cognitive predictors have no scores below 85 on the 9 academic tests. About 96% of people with this cognitive profile have no scores below 70.

Because there is an extremely large number of possible cognitive profiles, I cannot show what would happen with all of them. Instead, I will show what happens with all of the perfectly flat profiles from all 14 cognitive scores equal to 70 to all 14 cognitive scores equal to 130.

What proportion of people with flat WJ III cognitive profiles equal to 70 to 130 have no WJ III academic scores below 85
What proportion of people with flat WJ III cognitive profiles equal to 70 to 130 have no WJ III academic scores below 85

Here is what happens with the same procedure when the threshold is 70 for the academic scores:

What proportion of people with flat WJ III cognitive profiles equal to 70 to 130 have no WJ III academic scores below 70
What proportion of people with flat WJ III cognitive profiles equal to 70 to 130 have no WJ III academic scores below 70

Here is the R code I used to perform the calculations. You can adapt it to other situations fairly easily (different tests, thresholds, and profiles).

library(mvtnorm)
WJ <- matrix(c(
  1,0.49,0.31,0.46,0.57,0.28,0.37,0.77,0.36,0.15,0.24,0.49,0.25,0.39,0.61,0.6,0.53,0.53,0.5,0.41,0.43,0.57,0.28, #Verbal Comprehension
  0.49,1,0.27,0.32,0.47,0.26,0.32,0.42,0.25,0.21,0.2,0.41,0.21,0.28,0.38,0.43,0.31,0.36,0.33,0.25,0.29,0.4,0.18, #Visual-Auditory Learning
  0.31,0.27,1,0.25,0.33,0.18,0.21,0.28,0.13,0.16,0.1,0.33,0.13,0.17,0.25,0.22,0.18,0.21,0.19,0.13,0.25,0.31,0.11, #Spatial Relations
  0.46,0.32,0.25,1,0.36,0.17,0.26,0.44,0.19,0.13,0.26,0.31,0.18,0.36,0.4,0.36,0.32,0.29,0.31,0.27,0.22,0.33,0.2, #Sound Blending
  0.57,0.47,0.33,0.36,1,0.29,0.37,0.49,0.28,0.16,0.23,0.57,0.24,0.35,0.4,0.44,0.36,0.38,0.4,0.34,0.39,0.53,0.27, #Concept Formation
  0.28,0.26,0.18,0.17,0.29,1,0.35,0.25,0.36,0.17,0.27,0.29,0.53,0.22,0.37,0.32,0.52,0.42,0.32,0.49,0.42,0.37,0.61, #Visual Matching
  0.37,0.32,0.21,0.26,0.37,0.35,1,0.3,0.24,0.13,0.22,0.33,0.21,0.35,0.39,0.34,0.38,0.38,0.36,0.33,0.38,0.43,0.36, #Numbers Reversed
  0.77,0.42,0.28,0.44,0.49,0.25,0.3,1,0.37,0.15,0.23,0.43,0.23,0.37,0.56,0.55,0.51,0.47,0.47,0.39,0.36,0.51,0.26, #General Information
  0.36,0.25,0.13,0.19,0.28,0.36,0.24,0.37,1,0.1,0.22,0.21,0.38,0.26,0.26,0.33,0.4,0.28,0.27,0.39,0.21,0.25,0.32, #Retrieval Fluency
  0.15,0.21,0.16,0.13,0.16,0.17,0.13,0.15,0.1,1,0.06,0.16,0.17,0.09,0.11,0.09,0.13,0.1,0.12,0.13,0.07,0.12,0.07, #Picture Recognition
  0.24,0.2,0.1,0.26,0.23,0.27,0.22,0.23,0.22,0.06,1,0.22,0.35,0.2,0.16,0.22,0.25,0.21,0.19,0.26,0.17,0.19,0.21, #Auditory Attention
  0.49,0.41,0.33,0.31,0.57,0.29,0.33,0.43,0.21,0.16,0.22,1,0.2,0.3,0.33,0.38,0.29,0.31,0.3,0.25,0.42,0.47,0.25, #Analysis-Synthesis
  0.25,0.21,0.13,0.18,0.24,0.53,0.21,0.23,0.38,0.17,0.35,0.2,1,0.15,0.19,0.22,0.37,0.21,0.2,0.4,0.23,0.19,0.37, #Decision Speed
  0.39,0.28,0.17,0.36,0.35,0.22,0.35,0.37,0.26,0.09,0.2,0.3,0.15,1,0.39,0.36,0.32,0.3,0.3,0.3,0.25,0.33,0.23, #Memory for Words
  0.61,0.38,0.25,0.4,0.4,0.37,0.39,0.56,0.26,0.11,0.16,0.33,0.19,0.39,1,0.58,0.59,0.64,0.5,0.48,0.46,0.52,0.42, #Letter-Word Identification
  0.6,0.43,0.22,0.36,0.44,0.32,0.34,0.55,0.33,0.09,0.22,0.38,0.22,0.36,0.58,1,0.52,0.52,0.47,0.42,0.43,0.49,0.36, #Passage Comprehension
  0.53,0.31,0.18,0.32,0.36,0.52,0.38,0.51,0.4,0.13,0.25,0.29,0.37,0.32,0.59,0.52,1,0.58,0.48,0.65,0.42,0.43,0.59, #Reading Fluency
  0.53,0.36,0.21,0.29,0.38,0.42,0.38,0.47,0.28,0.1,0.21,0.31,0.21,0.3,0.64,0.52,0.58,1,0.5,0.49,0.46,0.47,0.49, #Spelling
  0.5,0.33,0.19,0.31,0.4,0.32,0.36,0.47,0.27,0.12,0.19,0.3,0.2,0.3,0.5,0.47,0.48,0.5,1,0.44,0.41,0.46,0.36, #Writing Samples
  0.41,0.25,0.13,0.27,0.34,0.49,0.33,0.39,0.39,0.13,0.26,0.25,0.4,0.3,0.48,0.42,0.65,0.49,0.44,1,0.38,0.37,0.55, #Writing Fluency
  0.43,0.29,0.25,0.22,0.39,0.42,0.38,0.36,0.21,0.07,0.17,0.42,0.23,0.25,0.46,0.43,0.42,0.46,0.41,0.38,1,0.57,0.51, #Calculation
  0.57,0.4,0.31,0.33,0.53,0.37,0.43,0.51,0.25,0.12,0.19,0.47,0.19,0.33,0.52,0.49,0.43,0.47,0.46,0.37,0.57,1,0.46, #Applied Problems
  0.28,0.18,0.11,0.2,0.27,0.61,0.36,0.26,0.32,0.07,0.21,0.25,0.37,0.23,0.42,0.36,0.59,0.49,0.36,0.55,0.51,0.46,1), nrow= 23, byrow=TRUE) #Math Fluency
WJNames <- c("Verbal Comprehension", "Visual-Auditory Learning", "Spatial Relations", "Sound Blending", "Concept Formation", "Visual Matching", "Numbers Reversed", "General Information", "Retrieval Fluency", "Picture Recognition", "Auditory Attention", "Analysis-Synthesis", "Decision Speed", "Memory for Words", "Letter-Word Identification", "Passage Comprehension", "Reading Fluency", "Spelling", "Writing Samples", "Writing Fluency", "Calculation", "Applied Problems", "Math Fluency")
rownames(WJ) <- colnames(WJ) <- WJNames

#Number of tests
k<-length(WJNames)

#Means and standard deviations of tests
mu<-rep(100,k)
sd<-rep(15,k)

#Covariance matrix
sigma<-diag(sd)%*%WJ%*%diag(sd)
colnames(sigma)<-rownames(sigma)<-WJNames

#Vector identifying predictors (WJ Cog)
p<-seq(1,14)

#Threshold for low scores
Threshold<-85

#Proportion of population who have no scores below the threshold
pmvnorm(lower=rep(Threshold,length(WJNames[-p])),upper=rep(Inf,length(WJNames[-p])),sigma=sigma[-p,-p],mean=mu[-p])[1]

#Predictor test scores for an individual
x<-rep(100,length(p))
names(x)<-WJNames[p]

#Condition means and covariance matrix
condMu<-c(mu[-p] + sigma[-p,p] %*% solve(sigma[p,p]) %*% (x-mu[p]))
condSigma<-sigma[-p,-p] - sigma[-p,p] %*% solve(sigma[p,p]) %*% sigma[p,-p]

#Proportion of people with the same predictor scores as this individual who have no scores below the threshold
pmvnorm(lower=rep(Threshold,length(WJNames[-p])),upper=rep(Inf,length(WJNames[-p])),sigma=condSigma,mean=condMu)[1]


Standard
Cognitive Assessment, Psychometrics, Statistics

How unusual is it to have multiple scores below a threshold?

In psychological assessment, it is common to specify a threshold at which a score is considered unusual (e.g., 2 standard deviations above or below the mean). If we can assume that the scores are roughly normal, it is easy to estimate the proportion of people with scores below the threshold we have set. If the threshold is 2 standard deviations below the mean, then the Excel function NORMSDIST will tell us the answer:

=NORMSDIST(-2)

=0.023

In R, the pnorm function gives the same answer:

pnorm(-2)

How unusual is it to have multiple scores below the threshold? The answer depends on how correlated the scores are. If we can assume that the scores are multivariate normal, Crawford and colleagues (2007) show us how to obtain reasonable estimates using simulated data. Here is a script in R that depends on the mvtnorm package. Suppose that the 10 subtests of the WAIS-IV have correlations as depicted below. Because the subtests have a mean of 10 and a standard deviation of 3, the scores are unusually low if 4 or lower.

#WAIS-IV subtest names
WAISSubtests <- c("BD", "SI", "DS", "MR", "VO", "AR", "SS", "VP", "IN", "CD")

# WAIS-IV correlations
WAISCor <- rbind(
  c(1.00,0.49,0.45,0.54,0.45,0.50,0.41,0.64,0.44,0.40), #BD
  c(0.49,1.00,0.48,0.51,0.74,0.54,0.35,0.44,0.64,0.41), #SI
  c(0.45,0.48,1.00,0.47,0.50,0.60,0.40,0.40,0.43,0.45), #DS
  c(0.54,0.51,0.47,1.00,0.51,0.52,0.39,0.53,0.49,0.45), #MR
  c(0.45,0.74,0.50,0.51,1.00,0.57,0.34,0.42,0.73,0.41), #VO
  c(0.50,0.54,0.60,0.52,0.57,1.00,0.37,0.48,0.57,0.43), #AR
  c(0.41,0.35,0.40,0.39,0.34,0.37,1.00,0.38,0.34,0.65), #SS
  c(0.64,0.44,0.40,0.53,0.42,0.48,0.38,1.00,0.43,0.37), #VP
  c(0.44,0.64,0.43,0.49,0.73,0.57,0.34,0.43,1.00,0.34), #IN
  c(0.40,0.41,0.45,0.45,0.41,0.43,0.65,0.37,0.34,1.00)) #CD
rownames(WAISCor) <- colnames(WAISCor) <- WAISSubtests

#Means
WAISMeans<-rep(10,length(WAISSubtests))

#Standard deviations
WAISSD<-rep(3,length(WAISSubtests))

#Covariance Matrix
WAISCov<-WAISCor*WAISSD%*%t(WAISSD)

#Sample size
SampleSize<-1000000

#Load mvtnorm package
library(mvtnorm)

#Make simulated data
d<-rmvnorm(n=SampleSize,mean=WAISMeans,sigma=WAISCov)
#To make this more realistic, you can round all scores to the nearest integer (d<-round(d))

#Threshold for abnormality
Threshold<-4

#Which scores are less than or equal to threshold
Abnormal<- d<=Threshold

#Number of scores less than or equal to threshold
nAbnormal<-rowSums(Abnormal)

#Frequency distribution table
p<-c(table(nAbnormal)/SampleSize)

#Plot
barplot(p,axes=F,las=1,
    xlim=c(0,length(p)*1.2),ylim=c(0,1),
    bty="n",pch=16,col="royalblue2",
    xlab="Number of WAIS-IV subtest scores less than or equal to 4",
    ylab="Proportion")
axis(2,at=seq(0,1,0.1),las=1)
text(x=0.7+0:10*1.2,y=p,labels=formatC(p,digits=2),cex=0.7,pos=3,adj=0.5)

The code produces this graph:
Abnormal Scores Simulation

Using the multivariate normal distribution

The simulation method works very well, especially if the sample size is very large. An alternate method that gives more precise numbers is to estimate how much of the multivariate normal distribution is within certain bounds. That is, we find all of the regions of the multivariate normal distribution in which one and only one test is below a threshold and then add up all the probabilities. The process is repeated to find all regions in which two and only two tests are below a threshold. Repeat the process, with 3 tests, 4 tests, and so on. This is tedious to do by hand but only takes a few lines of code do automatically.

AbnormalPrevalance<-function(Cor,Mean=0,SD=1,Threshold){
  require(mvtnorm)
  k<-nrow(Cor)
  p<-rep(0,k)
  zThreshold<-(Threshold-Mean)/SD
  for (n in 1:k){
    combos<-combn(1:k,n)
    ncombos<-ncol(combos)
    for (i in 1:ncombos){
      u<-rep(Inf,k)
      u[combos[,i]]<-zThreshold
      l<-rep(-Inf,k)
      l[seq(1,k)[-combos[,i]]]<-zThreshold 
      p[n]<-p[n]+pmvnorm(lower=l,upper=u,mean=rep(0,k),sigma=Cor)[1]
    }
  }
  p<-c(1-sum(p),p)
  names(p)<-0:k

  barplot(p,axes=F,las=1,xlim=c(0,length(p)*1.2),ylim=c(0,1),
     bty="n",pch=16,col="royalblue2",
     xlab=bquote("Number of scores less than or equal to " * .(Threshold)),
     ylab="Proportion")
  axis(2,at=seq(0,1,0.1),las=1)
  text(x=0.7+0:10*1.2,y=p,labels=formatC(p,digits=2),cex=0.7,pos=3,adj=0.5)
  return(p)
}
Proportions<-AbnormalPrevalance(Cor=WAISCor,Mean=10,SD=3,Threshold=4)

Using this method, the results are nearly the same but slightly more accurate. If the number of tests is large, the code can take a long time to run.

Abnormal Scores Direct Method

Standard