WHAT IF WE TOOK OUR MODELS SERIOUSLY?Estimating Latent

Scoresin Individuals

W. Joel SchneiderDepartment of Psychology

[E]very individual is a singularity, a one-time existing being, nowhere else and never before present. To be sure, certain law-like regularities apply to him, certain types are embodied in him, but the individual is not exhausted by

William Stern

1871–1938

these laws and types; there remains ever something more, through which the individual is distinct from others who conform to the same laws and types.

And this last kernel of being, which reveals the individual to be thus and so, distinct from all others, is not expressible in the language of scientific concepts, it is unclassifiable, incommensurable. In this sense, the

William Stern

1871–1938

individual is a limiting concept, toward which theoretical investigation strives but can never reach; it is, one could say, the asymptote of science.

A Revolutionary IdeaFrancis Galton(1822-1911)

Karl Pearson(1857-1936)

circa 1909

Up to 1889 [scientists] had thought only in terms of causation, in [the] future they were to admit another working category, that of correlation, and thus open to quantitative analysis wide fields of medical, psychological and sociological research. Karl Pearson

[…]we realize to-day how little progress in that direction was possible because [earlier scholars] lacked the key—correlation—to the treasure chamber.

Karl Pearson

[…] Galton turning over two different problems in his mind reached the conception of correlation: A is not the sole cause of B, but it contributes to the production of B; there may be other, many or few, causes, some of which we do not know and may never know. Karl Pearson

Are we then to exclude from mathematical analysis all such cases of incomplete causation?

Galton’s answer was: “No, we must endeavor to find a quantitative measure of this degree of partial causation.”

Karl Pearson

This measure of partial causation was the germ of the broad category—that of correlation, which was to replace not only in the minds of many of us the old category of causation, but deeply to influence our outlook on the universe.

Karl Pearson

YX d.Y

𝜆 √1−𝜆2

YX d.Y

0.5 √1−0.52

X d.Y

0.5 Y √1−0.52

X d.Y0.5 Y √1−0.52

X d.Y0.5 Y √1−0.52

If X = ?

If X = 100

X d.Y

0.7 Y √1−0.72

If X = ?

If X = 100

If X = 70

Y

Does Block Design measure visual-spatial ability?

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

g

.84

.78

.89

.86

.55

.79

.37

.33

.44

.37

.45

.40

.37

.33

.42

.36

.42

.39

.46

.46

.82

.88

.74

.84

.74

.32

.85

.32

.40

.48

.64

.63

.72

.29

.80

.54

.50

.54

.63.46

.51

.84

.51

e.SI

e.CA

e.SS

e.CD

e.LNS

e.VO

e.CO

e.IN

e.WR

e.PCm

e.BD

e.MR

e.PCn

e.DS

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

e.ARs.AR

.43

.34

.51

.39

.50

.64

.37

.59

.64

.69

.62

.46

.53

.73

.35

d.Gsm

d.Gs

d.Ind

d.Gv

d.Gc

Gs

Gc

Gv

Induction

Gsm

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

g

.84

.78

.89

.86

.55

.79

.37

.33

.44

.37

.45

.40

.37

.33

.42

.36

.42

.39

.46

.46

.82

.88

.74

.84

.74

.32

.85

.32

.40

.48

.64

.63

.72

.29

.80

.54

.50

.54

.63.46

.51

.84

.51

e.SI

e.CA

e.SS

e.CD

e.LNS

e.VO

e.CO

e.IN

e.WR

e.PCm

e.BD

e.MR

e.PCn

e.DS

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

e.ARs.AR

.43

.34

.51

.39

.50

.64

.37

.59

.64

.69

.62

.46

.53

.73

.35

d.Gsm

d.Gs

d.Ind

d.Gv

d.Gc

Gc

Gv

Induction

Gsm

Gs

Gc

Gv

Induction

Gsm

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture Concepts

Digit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

.84

.78

.89

.86

.55

.79

.82

.88

.74

.84

.74

.32

.48

.64

.85

.32

.40

.29

.63

.72

.80

.54

.50

.35

.37

.33

.44

.37

.45

.40

.37

.33

.42

.36

.42

.39

.46

.46

.51

.43

.34

.51

.39

.50

.64

.37

.59

.64

.69

.62

.46

.53

.73

.54

.63.46

.51

.84 Gs

Gc

GvGf

Induction

Gsm

e.SI

e.CA

e.SS

e.CD

e.LNS

e.VO

e.CO

e.IN

e.WR

e.PCm

e.BD

e.MR

e.PCn

e.DS

e.AR

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

s.AR

d.Gsm

d.Gs

d.Ind

d.Gv

d.Gc

g

𝜼= (𝐈−𝛃 )−1𝛄 𝝃𝜼= (𝐈−𝛃 )−1𝛄 𝝃𝜼= (𝐈−𝛃 )−1𝛄 𝝃𝜼= (𝐈−𝛃 )−1𝛄 𝝃𝜼= (𝐈−𝛃 )−1𝛄 𝝃

𝜼= (𝐈−𝛃 )−1𝛄 𝝃

𝚪=(𝛄𝐈 )𝚩=(𝛃 𝟎

𝟎 𝟎)𝝂=(𝜼𝝃 )

𝝂=(𝐈−𝚩 )−1𝚪𝝃

𝐑 ν=(𝐈−𝚩 )−1𝚪𝚽 ((𝐈−𝚩 )−1𝚪 )′

𝚩Subtest × Latent=𝐑Subtest × Subtest−1 𝐑 Subtest× Latent

𝐝=diag (𝚩Subtest× Latent′ 𝐑 Subtest × Latent )

𝛔e=√𝟏−𝐝

�̂�=𝒔𝚩𝑆𝑢𝑏𝑡𝑒𝑠𝑡×𝐿𝑎𝑡𝑒𝑛𝑡

𝝂=�̂� ±𝑧 95%𝛔e

𝑝 (𝝂 ≤𝐳∨�̂� )=𝑁 (𝐳−�̂�𝛔e )

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

g

.84

.78

.89

.86

.55

.79

.37

.33

.44

.37

.45

.40

.37

.33

.42

.36

.42

.39

.46

.46

.82

.88

.74

.84

.74

.32

.85

.32

.40

.48

.64

.63

.72

.29

.80

.54

.50

.54

.63

.46

.51

.84

.51

e.SI

e.CA

e.SS

e.CD

e.LNS

e.VO

e.CO

e.IN

e.WR

e.PCm

e.BD

e.MR

e.PCn

e.DS

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

e.ARs.AR

.43

.34

.51

.39

.50

.64

.37

.59

.64

.69

.62

.46

.53

.73

.35

d.Gsm

d.Gs

d.Ind

d.Gv

d.Gc

Gc

Gv

Induction

Gsm

Gs

Gc

Gv

Induction

Gsm

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

d.Gc

d.Gv

d.Ind

d.Gsm

d.Gs

g

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

s.AR

Gs

Gc

Gv

Induction

Gsm

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

d.Gc

d.Gv

d.Ind

d.Gsm

d.Gs

Estimated Gc

g

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

s.AR

e.Gc

Gs

Gv

Induction

Gsm

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

d.Gc

d.Gv

d.Ind

d.Gsm

d.Gs

g

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

s.AR

e.d.Gc

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

Est. d.Gc

d.Gv

d.Ind

d.Gsm

d.Gs

g

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

s.AR

e.d.Gc

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

Est. d.Gc

d.Gv

d.Ind

d.Gsm

d.Gs

g

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

s.AR

e.d.Gc

Similarities

Vocabulary

Comprehension

Information

Word Reasoning

Picture Completion

Block DesignMatrix

ReasoningPicture

ConceptsDigit Span

Letter-Number Seq.

Coding

Symbol Search

Cancellation

Arithmetic

Est. d.Gc

d.Gv

d.Ind

d.Gsm

d.Gs

g

s.SI

s.VO

s.CO

s.IN

s.WR

s.PCm

s.BD

s.MR

s.PCn

s.DS

s.LNS

s.CD

s.SS

s.CA

s.AR

e.d.Gc

WISC-IV FSIQ

95% CI (g)ω = 0.81ρxx = 0.97

95% CI (IQ)

gSpecific

d.Gc

Error

d.Gsd.G

vd.G

sm

d.Ind

80.7%

3.8%

3.2%

2.7%

1.8% 1.0%

0.6%

6.1%

WISC

-IV FSIQ

g Specificd.G

cE

rror

d.Gs

d.Gv

d.Gsm

d.Ind

86.6%

2.9%

2.5%

0.4%0.6%

1.3%

0.4%

5.3%

Estim

ated g

FSIQ gVCI Gc

d.Gc

PRI Gvd.G

vInd d.I

ndWMI

Gsmd.G

sm PSI Gsd.G

s0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

81%87%

60%

72%

4%

63%67%

2%

83%

2%

46%

79%

2%

30%37%

4%

3%

25%

21%

57%

3%

2%

3%

1%

2%

2%1%

2%

9%

18%

47%

1%

1%

1%

5%

2%

17%

1%

1%

16%

5%

27%

3%

41%

41%

62%

6% 5%

8%

4%

20%

15%8%

25%

9%

55%

29%

11%

52%

16%13%

22%

3% 3% 6% 3%13%

7% 6%

21%

3%

22%

8%3%

15% 12% 9%14%

Composition of Composites and Estimated Latent Variables

g d.Gc d.Gv d.Inductiond.Gsm d.Gs Specific Error

Composites and Estimated Latent Variables

Var

ianc

e

FSIQ

VCI

Similarities

Vocabulary

Com

prehension

Information

Word R

easoning

PRI

Block Design

Picture Com

pletion

Picture Concepts

Matrix R

easoning

WM

I

Letter-Num

ber Sequencing

Digit Span

PSI

Coding

Cancellation

Symbol Search

Arithmetic

1

4

7

10

13

16

19

9.6 9.8 10.4

12.6

5

11

910

12 12

65

1312

1312

5

8

6

10

g s.AR Gc d.Gc s.SI s.VC s.CO s.IN s.WR Gv d.Gv s.PCm

s.BD Ind d.Ind

s.MR s.PCn

Gsm d.Gsm

s.DS s.LN Gs d.Gs s.CD s.SS s.CA

0.04 -0.05 0.14 0.19 0.29

-0.69

-0.12

0.73 0.63

-0.90

-1.49 -1.52

-0.76

0.380.75

1.00 0.820.42

0.750.46

0.90

-1.28-1.56

-0.82-0.41

-0.03

404550556065707580859095100105110115120125130135140145150155160

Estimated Latent Variable Scores

Range Observed Scores

Estimated Latent Scores

Very High

HighHigh Average Gf & Gsm

Average g, Gc g, Gc, Gf, & Gsm

Low Average Gv & Gs

Low Gv & Gs

Very Low

I have nothing to offer but doubt, dread, fuss, and strain.

You ask, what is our aim? I can answer in one word: Validity. Validity at all costs— Validity in spite of all illusions— Validity, however daunting or dreary the task may be, for without validitythere is no purpose.

You changedmy speech!

The Future?

Similarities

Vocabulary

Comprehension

Information

Word ReasoningPicture

CompletionBlock Design

Matrix Reasoning

Picture Concepts

Digit SpanLetter-Number

Seq.Coding

Symbol Search

Cancellation

Arithmetic

.71.43

.50

.42

.39

.42

.31

.43

.26

.48

.27*

.27*

.36*

.36*

.64

.74

.63

.74

.62

.51

.65

.67

.53

.56

.64

.84

.43

.54

.19

.11

.18

.16

.11

.22.41.47

g

Gc

Gv

Gf

Gsm

Gs

Predictions!

Spreadsheets

WISC-IV

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