what if we took our models seriously?
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What If We Took Our Models Seriously?. Estimating Latent Scores in Individuals. W . Joel Schneider Department of Psychology. - PowerPoint PPT PresentationTRANSCRIPT
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!