Topic Outline

1) Motivation

2) Representing/Modeling Causal Systems

3) Estimation and Updating

4) Model Search

5) Linear Latent Variable Models

6) Case Study: fMRI

1

2

Richard ScheinesCarnegie Mellon University

Discovering Pure Measurement Models

Ricardo Silva*University College London

Clark Glymour and Peter SpirtesCarnegie Mellon University

3

Outline

1. Measurement Models & Causal Inference

2. Strategies for Finding a Pure Measurement Model

3. Purify

4. MIMbuild

5. Build Pure Clusters

6. Examples

a) Religious Coping

b) Test Anxiety

4

Goals:

• What Latents are out there?

• Causal Relationships Among Latent Constructs

DepressionRelationshipSatisfaction

DepressionRelationshipSatisfaction

or

or ?

5

Needed:

Ability to detect

conditional independence

among latent variables

6

Lead and IQ

Parental Resources

IQLeadExposure

PR ~ N(m=10, s = 3)

Lead = 15 -.5*PR + e2

e2 ~ N(m=0, s = 1.635)

IQ = 90 + 1*PR + e3

e3 ~ N(m=0, s = 15)

e2 e3

Lead _||_ IQ | PR

7

Psuedorandom sample: N = 2,000

Parental Resources

IQLead

Exposure

IndependentVariable

Coefficient Estimate p-value Screened-off at .05?

PR 0.98 0.000 No

Lead -0.088 0.378 Yes

Regression of IQ on Lead, PR

8

Measuring the Confounder

Lead Exposure

Parental Resources IQ

X1 X2 X3

e1 e2 e3

X1 = g1* Parental Resources + e1

X2 = g2* Parental Resources + e2

X3 = g3* Parental Resources + e3

PR_Scale = (X1 + X2 + X3) / 3

9

Scales don't preserve conditional independence

Lead Exposure

Parental Resources IQ

X1 X2 X3

PR_Scale = (X1 + X2 + X3) / 3

Independent

Variable

Coefficient

Estimate

p-value Screened-off

at .05?

PR_scale 0.290 0.000 No

Lead -0.423 0.000 No

10

Indicators Don’t Preserve Conditional Independence

Lead Exposure

Parental Resources IQ

X1 X2 X3

IndependentVariable

Coefficient Estimate

p-value Screened-off at .05?

X1 0.22 0.002 No

X2 0.45 0.000 No

X3 0.18 0.013 No

Lead -0.414 0.000 No

Regress IQ on: Lead, X1, X2, X3

11

Structural Equation Models Work

Lead Exposure

Parental Resources

IQ

X1 X2 X3

b

Structural Equation Model

•

• (p-value = .499)

• Lead and IQ “screened off” by PR

0)ˆ( bE07.ˆ b

12

F1

x1 x2

F2 F3

x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

Local Independence / Pure Measurement Models

• For every measured item xi:

xi _||_ xj | latent parent of xi

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F1

x1 x2

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Truth

Local Independence Desirable

1

x1 x2

2 3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Specified Model b31

0)ˆ( 31 bE

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F1

x1 x2

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Truth

F4

Correct Specification Crucial

1

x1 x2

2 3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Specified Model b31

0)ˆ( 31 bE

15

Strategies

• Find a Locally Independent Measurement Model

• Correctly specify the MM, including deviations from Local Independence

16

F1

x1 x2

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Truth

F4

Correctly Specify Deviations from Local Independence

1

x1 x2

2 3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Specified Model b31

ex4 ez4

0)ˆ( 31 bE

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F1

x1 x2

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Truth

F4 F5 F6

Correctly Specifying Deviations from Local Independence is Often Very Hard

18

Finding Pure Measurement Models - Much Easier

F1

x1 x2

F2 F3

x3 y1 y2 y3 y4 z3 z4

Truth

F5 F6

F1

x1 x2

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Truth

F4 F5 F6

Tetrad Constraints

• Fact: given a graph with this structure

• it follows that

L

X Y ZW

W = 1L + e1

X = 2L + e2

Y = 3L + e3

Z = 4L + e4

CovWXCovYZ = (12s2L) (34s2

L) = = (13s2

L) (24s2L) = CovWYCovXZ

sWXsYZ = sWYsXZ = sWZsXY

tetradconstraints

1 2 3

4

Early Progenitors

Charles Spearman (1904)

Statistical Constraints Measurement Model Structure

g

m1 m2 r1 r2

rm1 * rr1 = rm2 * rr2

21

F1

x1 x2 x3 x4

Truth

Impurities/Deviations from Local Independencedefeat tetrad constraints selectively

F1

x1 x2 x3 x4

Truth

F5

rx1,x2 * rx3,x4 = rx1,x3 * rx2,x4

rx1,x2 * rx3,x4 = rx1,x4 * rx2,x3

rx1,x3 * rx2,x4 = rx1,x4 * rx2,x3

rx1,x2 * rx3,x4 = rx1,x3 * rx2,x4

rx1,x2 * rx3,x4 = rx1,x4 * rx2,x3

rx1,x3 * rx2,x4 = rx1,x4 * rx2,x3

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F1

x1 x2

F

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Purify

True Model

F1

x1 x2

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Initially Specified Measurement Model

Purify

Iteratively remove item whose removal most improves measurement model fit (tetrads or c2)

– stop when confirmatory fit is acceptable

F1

x1 x2

F

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Remove x4

F1

x1 x2

F

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Remove z2

24

F1

x1 x2

F

F2 F3

x3 y1 y2 y3 y4 z1 z3 z4

Purify

Detectibly Pure Subset of Items

Detectibly Pure Measurement Model

25

Purify

How a pure measurement model is useful

F1

x1 x2

F

F2 F3

x3 y1 y2 y3 y4 z1 z3 z4

1. Consistently estimate covariances/correlations among latents- test conditional independence with estimated latent correlations

2. Test for conditional independence among latents directly

2. Test conditional independence relations among latents directly

b21

b21 = 0 L1 _||_ L2 | {Q1, Q2, ..., Qn}

Question: L1 _||_ L2 | {Q1, Q2, ..., Qn}

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MIMbuild

MIMbuild

PC algorithm with independence tests

performed directly on latent variables

Output: Equivalence class of structural models

over the latent variables

Input:

- Purified Measurement Model

- Covariance matrix over set of pure items

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Purify & MIMbuild

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Goal 2: What Latents are out there?

• How should they be measured?

31

F1

x1 x2 x3 x4

Model 1

x5

F2

x5

Latents and the clustering of items they measure imply tetrad constraints diffentially

F1

x1 x2 x3 x4

Model 2

x5

F2

x5

F1

x1 x2 x3 x4

Model 4

x5

F2

x6

F3

F1

x1 x2 x3 x4

Model 3

x5

F2

x6

F3

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Build Pure Clusters (BPC)

BPC

1) Cluster (complicated boolean combinations of tetrads)

2) Purify

Output: Equivalence class of measurement models over a pure subset of original Items

Input:

- Covariance matrix over set of original items

33

Build Pure Clusters

34

Build Pure Clusters

Qualitative Assumptions1. Two types of nodes: measured (M) and latent (L)

2. M L (measured don’t cause latents)

3. Each m M measures (is a direct effect of) at least one l L

4. No cycles involving M

Quantitative Assumptions:1. Each m M is a linear function of its parents plus noise

2. P(L) has second moments, positive variances, and no deterministic relations

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Build Pure ClustersOutput - provably reliable (pointwise consistent):

Equivalence class of measurement models over a pure subset of M

For example:

L1 L2 L3

m1 m2 m3 m4 m5 m6 m7 m8 m9

L1 L2 L3

m1 m2 m3 m4 m5 m6 m7 m8 m9 m11 m10 True Model

Output

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Build Pure ClustersMeasurement models in the equivalence class are at most refinements, but never coarsenings or permuted clusterings.

L1 L2 L3

m1 m2 m3 m4 m5 m6 m7 m8 m9

Output

L1 L2 L3

m1 m2 m3 m4 m5 m6 m7 m8 m9

L4

L1 L2 L3

m1 m2 m3 m4 m5 m6 m7 m8 m9

L1 L3

m1 m2 m3 m4 m5 m6 m7 m8 m9

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Build Pure Clusters

Algorithm Sketch:

1. Use particular rank (tetrad) constraints on the measured correlations to find pairs of items mj, mk that do NOT share a single latent parent

2. Add a latent for each subset S of M such that no pair in S was found NOT to share a latent parent in step 1.

3. Purify

4. Remove latents with no children

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Build Pure Clusters + MIMbuild

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Case Studies

Stress, Depression, and Religion (Lee, 2004)

Test Anxiety (Bartholomew, 2002)

40

Case Study: Stress, Depression, and Religion

Masters Students (N = 127) 61 - item survey (Likert Scale)

• Stress: St1 - St21

• Depression: D1 - D20

• Religious Coping: C1 - C20

p = 0.00

St112

Stress

St212

St2112

.

.

Dep112

Coping

.

. Depression

Dep212

Dep2012

C1 C2 C20 . .

+

- +

Specified Model

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Build Pure Clusters St3

12

Stress

St412 St1612

Dep912

Coping

Depression Dep1312

Dep1912

C9 C12 C15

St1812

St2012

C14

Case Study: Stress, Depression, and Religion

42

Assume Stress temporally prior:

MIMbuild to find Latent Structure: St312

Stress

St412 St1612

Dep912

Coping

Depression Dep1312

Dep1912

C9 C12 C15

St1812

St2012

C14

+

+

p = 0.28

Case Study: Stress, Depression, and Religion

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Case Study : Test Anxiety

Bartholomew and Knott (1999), Latent variable models and factor analysis

12th Grade Males in British Columbia (N = 335)

20 - item survey (Likert Scale items): X1 - X20:

X2

Emotionality Worry

X8

X9

X10

X15

X16

X18

X3

X4

X5

X6

X7

X14

X17

X20

Exploratory Factor Analysis:

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Build Pure Clusters:

X2

Emotionalty

X8

X9

X10

X11

X16

X18

X3

X5

X7

X14

X6

Cares About Achieving

Self-Defeating

Case Study : Test Anxiety

45

Build Pure Clusters:

X2

Emotionalty

X8

X9

X10

X11

X16

X18

X3

X5

X7

X14

X6

Worries About Achieving

Self-Defeating

X2

Emotionality Worry

X8

X9

X10

X15

X16

X18

X3

X4

X5

X6

X7

X14

X17

X20

p-value = 0.00 p-value = 0.47

Exploratory Factor Analysis:

Case Study : Test Anxiety

46

X2

Emotionalty

X8

X9

X10

X11

X16

X18

X3

X5

X7

X14

X6

Worries About Achieving

Self-Defeating

MIMbuild

p = .43

Emotionalty-Scale

Worries About Achieving-Scale

Self-Defeating

Uninformative

Scales: No Independencies or Conditional Independencies

Case Study : Test Anxiety

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Limitations

• In simulation studies, requires large sample sizes to be really reliable (~ 400-500).

• 2 pure indicators must exist for a latent to be discovered and included

• Moderately computationally intensive (O(n6)). • No error probabilities.

48

Open Questions/Projects

• IRT models?• Bi-factor model extensions?• Appropriate incorporation of background knowledge

49

References

• Tetrad: www.phil.cmu.edu/projects/tetrad_download• Spirtes, P., Glymour, C., Scheines, R. (2000). Causation, Prediction,

and Search, 2nd Edition, MIT Press.• Pearl, J. (2000). Causation: Models of Reasoning and Inference,

Cambridge University Press.• Silva, R., Glymour, C., Scheines, R. and Spirtes, P. (2006) “Learning the

Structure of Latent Linear Structure Models,” Journal of Machine Learning Research, 7, 191-246.

• Learning Measurement Models for Unobserved Variables, (2003). Silva, R., Scheines, R., Glymour, C., and Spirtes. P., in Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence , U. Kjaerulff and C. Meek, eds., Morgan Kauffman