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Richard ScheinesCarnegie Mellon University
Searching for Statistical Causal Models:
Theory and Practice
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Goals
1) Policy, Law, and Science: How can we use data to answer
a) subjunctive questions (effects of future policy interventions), or
b) counterfactual questions (what would have happened had things been
done differently (law)?
c) scientific questions (what mechanisms run the world)
2) Rumsfeld Problem: Do we know what we don’t know: Can we tell when
there is or is not enough information in the data to answer causal questions?
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Early Progenitors
Charles Spearman (1904)
Statistical Constraints Causal Structure
g
m1 m2 r1 r2
rm1 * rr1 = rm2 * rr2
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Early Progenitors
The Method of Path Coefficients (1934).
Annals of Mathematical Statistics, 5, 161-215.
Sewall Wright (1920s,1930s)
Graphical Model Causal & Statistical
Interpretation
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Social Sciences: 1940s 1970s
Economics:
• Cowles Commission
• Franklin Fisher
• Art Goldberger
• Clive Granger
• Herb Simon
• Haavelmo
• R. Strotz
• H. Wold
Sociology, Psychometrics, etc.
• Hubert Blalock
• Herb Costner
• Otis Dudley Duncan
• David Heise
• David Kenny
• Ken Bollen
• Factor Analysis
• Instrumental VariableEstimators
• Structural Equation Models
• Simultaneous Equation Models
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Population level Counterfactuals
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1970s & 1980s:Graphical Models & Independence Structures
• S. Lauritzen
• J. Darroch
• T. Speed
• H. Kiiveri
• N. Wermuth
• D. Hausman
• D. Papineau
• P. Dawid
• D. Cox
• J. Robins
• J. Whittaker
• Judea Pearl 1988
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1988 1993:Axioms, Intervention, and
Latent Variable Model Search
• P. Spirtes, C. Glymour, and R. Scheines
• Causal Markov Axiom
• Full model of interventions, both surgical and non-surgical
• Equivalence classes for latent variable models, with search
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Modern Non-Parametric Theory of Statistical Causal Models
Intervention & Manipulation
Graphical Models
Counterfactuals
Constraints(Independence)
Causal Bayes Nets
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Semantics of SCMs Choice 1: Take direct causation as primitive, axiomatize
Causal systems over V
Probabilistic Independence Relations in P(V)
Choice 2: Define direct causation in terms of
intervention, i.e., (hypothetical) treatment)
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Choice 1: Causal Markov Axiom
If G is a causal graph, and P a probability distribution
over the variables in G, then in <G,P> satisfy the
Markov Axiom iff:
every variable V is independent of its non-effects,
conditional on its immediate causes.
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Causal Graphs
Causal Graph G = {V,E} Each edge X Y represents a direct causal claim:
X is a direct cause of Y relative to V
Exposure Rash
Exposure Infection Rash
Chicken Pox
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Causal Graphs
Not Cause Complete
Common Cause Complete
Exposure Infection Symptoms
Omitted Causes
Exposure Infection Symptoms
Omitted
Common Causes
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Interventions & Causal GraphsModel an intervention by adding an “intervention” variable outside the
original system as a direct cause of its target.
Education Income Taxes Pre-intervention graph
Intervene on Income
“Soft” Intervention Education Income Taxes
I
“Hard” Intervention Education Income Taxes
I
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Structural Equation Models
1. Structural Equations2. Statistical Constraints
Education
LongevityIncome
Statistical Causal Model
Causal Graph
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Structural Equation Models
• Structural Equations: One Assignment Equation for each variable V V := f(parents(V), errorV)
for SEM (linear regression) f is a linear function
• Statistical Constraints: Joint Distribution over the Error terms
Education
LongevityIncome
Causal Graph
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Structural Equation Models
Equations: Education := ed
Income :=Educationincome
Longevity :=EducationLongevity
Statistical Constraints: (ed, Income, Income ) ~N(0, 2) 2diagonal - no variance is zero
Education
LongevityIncome
Causal Graph
Education
Income Longevity
1 2
LongevityIncome
SEM Graph
(path diagram)
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Two Routes to the Causal Markov Condition
• Assumption 1: Weak Causal Markov Assumption
V1,V2 causally disconnected V1 _||_ V2
• Assumption 2a:
Causal Markov Axiom
• Assumption 2b: Determinism, e.g.,
Structural Equations
For each Vi V, Vi := f(parents(Vi))
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X is a direct cause of Y relative to S, iff z,x1 x2 P(Y | X set= x1 , Z set= z)
P(Y | X set= x2 , Z set= z)
where Z = S - {X,Y}
Choice 2: Define Direct Causation from Intervention
X is a cause of Y iff x1 x2 P(Y | X set= x1) P(Y | X set= x2)
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Structural Equations: Education := ed
Longevity := f (Education) Longevity
Income := f (Education) income
Education
LongevityIncome
Modularity of Intervention/Manipulation
Causal Graph
Manipulated Structural Equations: Education := ed
Longevity := f (Education) Longevity
Income := f3(M1)
ManipulatedCausal Graph
Education
Longevity Income
M1
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• Manipulation conception of causation and Modularity
--> weak version of CMA
• Zhang, Jiji and Spirtes, Peter (2007) Detection of
Unfaithfulness and Robust Causal Inference. In [2007] LSE-
Pitt Conference: Confirmation, Induction and Science
(London, 8 - 10 March, 2007)
http://philsci-archive.pitt.edu/archive/00003188/01/Detection_of_Unfaithfulness_and_Robust_Causal_Inference.pdf
Manipulation --> Causal Markov
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SCM SearchStatistical Data Causal Structure
Background Knowledge
- X2 before X3
X3 | X2 X1
Independence Relations
Data
Statistical Inference
X2 X3 X1
Equivalence Class of Causal Graphs
X2 X3 X1
X2 X3 X1
Discovery Algorithm
Causal Markov Axiom (D-separation)
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Faithfulness
Constraints on a probability distribution P generated by a causal structure G hold for all parameterizations of G.
Revenues = aRate + cEconomy + Rev.
Economy = bRate + Econ.
Faithfulness: a ≠ -bcTax Revenues
Economyc
ba
Tax Rate
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Equivalence Classes
• Independence (d-separation equivalence)• DAGs : Patterns• PAGs : Latent variable models• Intervention Equivalence Classes
• Measurement Model Equivalence Classes• Linear Non-Gaussian Model Equivalence Classes
Equivalence:• Independence (M1 ╞ X _||_ Y | Z M2 ╞ X _||_ Y | Z)
• Distribution (q1 q2 M1(q1) = M2(q2))
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Patterns
X2
X4 X3
X1
X2
X4 X3
Represents
Pattern
X1 X2
X4 X3
X1
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Patterns: What the Edges Mean
X2 X1
X2 X1X1 X2 in some members of theequivalence class, and X2 X1 inothers.
X1 X2 (X1 is a cause of X2) inevery member of the equivalenceclass.
X2 X1 X1 and X2 are not adjacent in anymember of the equivalence class
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PAGs: Partial Ancestral Graphs
X2
X3
X1
X2
X3
Represents
PAG
X1 X2
X3
X1
X2
X3
T1
X1
X2
X3
X1
etc.
T1
T1 T2
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PAGs: Partial Ancestral Graphs
X2 X1
X2 X1
X2 X1
X2 There is a latent commoncause of X1 and X2
No set d-separates X2 and X1
X1 is a cause of X2
X2 is not an ancestor of X1
X1
X2 X1 X1 and X2 are not adjacent
What PAG edges mean.
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Overview of Search Methods
• Constraint Based Searches• TETRAD (SGS, PC, FCI)• Very fast – max N ~ 1,000• Pointwise Consistent
• Scoring Searches• Scores: BIC, AIC, etc.• Search: Hill Climb, Genetic Alg., Simulated Annealing• Difficult to extend to latent variable models• Meek and Chickering Greedy Equivalence Class (GES)• Very slow – max N ~ 30-40
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Tetrad Demo
http://www.phil.cmu.edu/projects/tetrad_download/
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Case Study 1: Foreign Investment
Does Foreign Investment in 3rd World Countries cause Repression?
Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49, 141-146.
N = 72PO degree of political exclusivityCV lack of civil libertiesEN energy consumption per capita (economic development)FI level of foreign investment
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Correlations
po fi en fi -.175 en -.480 0.330 cv 0.868 -.391 -.430
Case Study 1: Foreign Investment
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Regression Results
po = .227*fi - .176*en + .880*cv SE (.058) (.059) (.060)t 3.941 -2.99 14.6
Interpretation: increases in foreign investment increases political exclusion
Case Study 1: Foreign Investment
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Alternatives
.217
FI
PO
CV En
Regression
.88 -.176
FI
PO
CV En
Tetrad - FCI
FI
PO
CV En
Fit: df=2, 2=0.12, p-value = .94
.31 -.23
.86 -.48
Case Study 1: Foreign Investment
No model with testable constraint (df > 0) in which FI has a positive effect on PO
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Aurora Jackson, Richard Scheines
Single Mothers’ Self-Efficacy, Parenting in the Home Environment, and
Children’s Development in a Two-Wave Study(Social Work Research, 29, 1, 7-20)
Case Study 2: Welfare Reform
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Sampling Scheme
• Longitudinal Data
o Time 1: 1996-97 (N = 188)
o Time 2: 1998-99 (N = 178)
• Single black mothers in NYC
• Current and former welfare recipients
• With a child who was 3 – 5 at time 1,
and 6 to 8 at time 2
Case Study 2: Welfare Reform
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Constructs/Scales/Measures
• Employment Status
• Perceived Self-efficacy
• Depressive Symptoms
• Quality of Mother/Father Relationship
• Father/Child Contact
• Quality of Home Environment
• Behavior Problems
• Cognitive Development
Case Study 2: Welfare Reform
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Background Knowledge
Tier 1: • Employment Status
Tier 2: • Depression• Self-efficacy• Mother/Father Relationship• Father/Child Contact• Mother’s Parenting/HOME
Tier 3:• Negative Behaviors• Cognitive Development
Over 22 million path models consistent with these constraints
Case Study 2: Welfare Reform
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Employment Status
(Time 1)
Mother’s self-efficacy
(Time 1)
Mother’s depressive symptoms (Time 1)
Mother/Father Relationship
(Time 1) Father/Child Contact (Time 1)
Mother’s Parenting/ Home Environment
(Time 1)
Negative Behaviors (Time 2)
Cognitive Development
(Time 2)
.215** -.456*
-.129* .407**
.162*
-.291** .166*
* = p < .05 ** = p < .01
.184*
2 (19) = 18.87 P = .46 GFI = .97 AGFI = .95
Employment Status
(Time 1)
Mother’s self-efficacy
(Time 1)
Mother’s depressive symptoms (Time 1)
Mother/Father Relationship
(Time 1) Father/Child Contact (Time 1)
Mother’s Parenting/ Home Environment
(Time 1)
Negative Behaviors (Time 2)
Cognitive Development
(Time 2)
.215** -.472*
-.184* .407**
.150*
-.291** .166*
-.166*
2 (20) = 22.3 P = .32 GFI = .97 AGFI = .95
* = p < .05 ** = p < .01
Tetrad Equivalence Class
Conceptual Model
2 = 22.3, df = 20, p = .32
2 = 18.87, df = 19, p = .46
Case Study 2: Welfare Reform
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Employment Status
(Time 1)
Mother’s self-efficacy
(Time 1)
Mother’s depressive symptoms (Time 1)
Mother/Father Relationship
(Time 1) Father/Child Contact (Time 1)
Mother’s Parenting/ Home Environment
(Time 1)
Negative Behaviors (Time 2)
Cognitive Development
(Time 2)
.215** -.456*
-.129* .407**
.162*
-.291** .166*
* = p < .05 ** = p < .01
.184*
2 (19) = 18.87 P = .46 GFI = .97 AGFI = .95
Employment Status
(Time 1)
Mother’s self-efficacy
(Time 1)
Mother’s depressive symptoms (Time 1)
Mother/Father Relationship
(Time 1) Father/Child Contact (Time 1)
Mother’s Parenting/ Home Environment
(Time 1)
Negative Behaviors (Time 2)
Cognitive Development
(Time 2)
.215** -.472*
-.184* .407**
.150*
-.291** .166*
-.166*
2 (20) = 22.3 P = .32 GFI = .97 AGFI = .95
* = p < .05 ** = p < .01
Tetrad
Conceptual Model
Points of Agreement:• Mother’s Self-Efficacy mediates
the effect of Employment on all other variables.
• HOME environment mediates the effect of all other factors on outcomes: Cog. Develop and Prob. Behaviors
Points of Disagreement:• Depression key cause vs. only
an effect
Case Study 2: Welfare Reform
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Online Course in Causal & Statistical Reasoning
Case Study 3: Online Courseware
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Variables
Pre-test (%)
Print-outs (% modules printed)
Quiz Scores (avg. %)
Voluntary Exercises (% completed)
Final Exam (%)
9 other variables
Case Study 3: Online Courseware
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Printing and Voluntary Comprehension Checks: 2002 --> 2003
.302*
-.41**
.75**
.353* .323*
pre
print voluntary questions
quiz
final
2002
-.08
-.16
.41* .25*
pre
print voluntary questions
final
2003
Case Study 3: Online Courseware
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Variables
Tangibility/Concreteness (Exp manipulation)
Imaginability (likert 1-7)
Impact (avg. of 2 likerts)
Sympathy (likert)
Donation ($)
Case Study 4: Charitable Giving
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Theoretical Model
Case Study 4: Charitable Giving
Imaginability Tangibility
Impact
Sympathy
Donation
study 1 (N= 94) df = 5, 2 = 52.0, p= 0.0000
study 2 (N= 115) df = 5, 2 = 62.6, p= 0.0000
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GES Outputs
Case Study 4: Charitable Giving
Imaginability Tangibility
Impact
Sympathy
Donation
study 1: df = 5, 2 = 5.88, p= 0.32
Imaginability Tangibility
Impact
Sympathy
Donation
study 2: df = 5, 2 = 8.23, p= 0.14
study 1: df = 5, 2 = 3.99, p= 0.55
study 2: df = 5, 2 = 7.48, p= 0.18
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The Causal Theory Formation Problem for Latent Variable Models
Given observations on a number of variables, identify the latent variables that underlie these variables and the causal relations among these latent concepts.
Example: Spectral measurements of solar radiation intensities. Variables are intensities at each measured frequency.
Example: Quality of a Child’s Home Environment, Cumulative Exposure to Lead, Cognitive Functioning
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The Most Common Automatic Solution: Exploratory Factor Analysis
• Chooses “factors” to account linearly for as much of the variance/covariance of the measured variables as possible.
• Great for dimensionality reduction• Factor rotations are arbitrary• Gives no information about the statistical and thus the
causal dependencies among any real underlying factors.
• No general theory of the reliability of the procedure
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Other Solutions
• Independent Components, etc• Background Theory• Scales
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Other Solutions: Background Theory
St112
Home
St212
St2112
.
.
T1
Lead
.
.
Cognitive Function
T2
T20
C1 C2 C20 . .
?
Key Causal Question
Thus, key statistical question: Lead _||_ Cog | Home ?
Specified Model
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St112
Home
St212
St2112
.
.
T1
Lead
.
.
Cognitive Function
T2
T20
C1 C2 C20 . .
F
Lead _||_ Cog | Home ?
Yes, but statistical inference will say otherwise.
Other Solutions: Background Theory
True Model
“Impurities”
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F1
x1 x2
F2 F3
x3 x4 y1 y2 y3 y4 z1 z2 z3 z4
Purify
Specified Model
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F1
x1 x2
F
F2 F3
x3 x4 y1 y2 y3 y4 z1 z2 z3 z4
Purify
True Model
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F1
x1 x2
F
F2 F3
x3 x4 y1 y2 y3 y4 z1 z2 z3 z4
Purify
True Model
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F1
x1 x2
F
F2 F3
x3 x4 y1 y2 y3 y4 z1 z2 z3 z4
Purify
True Model
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F1
x1 x2
F
F2 F3
x3 x4 y1 y2 y3 y4 z1 z2 z3 z4
Purify
True Model
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F1
x1 x2
F
F2 F3
x3 y1 y2 y3 y4 z1 z3 z4
Purify
Purified Model
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Scale = sum(measures of a latent)
Other Solutions: Scales
St112
Home
St212
St2112
.
.
Homescale = i=1 to 21 (Sti)
Homescale
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True Model
Other Solutions: Scales
Pseudo-Random Sample: N = 2,000
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Scales vs. Latent variable Models
Regression:Cognition on Home, Lead
Predictor Coef SE Coef T PConstant -0.02291 0.02224 -1.03 0.303Home 1.22565 0.02895 42.33 0.000Lead -0.00575 0.02230 -0.26 0.797 S = 0.9940 R-Sq = 61.1% R-Sq(adj) = 61.0%
Insig.
True Model
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Scales vs. Latent variable Models
Scales homescale = (x1 + x2 + x3)/3leadscale = (x4 + x5 + x6)/3cogscale = (x7 + x8 + x9)/3
True Model
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Scales vs. Latent variable Models
Cognition = - 0.0295 + 0.714 homescale - 0.178 Lead Predictor Coef SE Coef T PConstant -0.02945 0.02516 -1.17 0.242homescal 0.71399 0.02299 31.05 0.000Lead -0.17811 0.02386 -7.46 0.000
Regression:Cognition on
homescale, Lead
Sig.
True Model
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Scales vs. Latent variable Models
Modeling Latents
True Model
Specified Model
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Scales vs. Latent variable Models
(2 = 29.6, df = 24, p = .19)
B5 = .0075, which at t=.23, is correctly insignificant
True Model
Estimated Model
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Scales vs. Latent variable Models
Mixing Latents and Scales
(2 = 14.57, df = 12, p = .26)
B5 = -.137, which at t=5.2, is incorrectly highly significantP < .001
True Model
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Build Pure Clusters
Output - provably reliable (pointwise consistent):
Equivalence class of measurement models over a pure subset of measures
L1 L2 L3
m1 m2 m3 m4 m5 m6 m7 m8 m9
Stress Dep Health
m1 m2 m3 m4 m5 m6 m7 m8 m9 m11 m10
m
BPC
True Model
Output
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Build Pure ClustersQualitative 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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Case Study 4: Stress, Depression, and Religion
MSW 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 4: Stress, Depression, and Religion
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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 4: Stress, Depression, and Religion
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Case Study 5: 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 5: Test Anxiety
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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 5: Test Anxiety
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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
Unininformative
Scales: No Independencies or Conditional Independencies
Case Study 5: Test Anxiety
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Economics
Bessler, Pork Prices
Hoover, multiple
Cryder & Loewenstein,
Charitable Giving
Other Cases
Educational Research
Easterday, Bias & Recall
Laski, Numerical coding
Climate Research
Glymour, Chu, , Teleconnections
Biology
Shipley,
SGS, Spartina Grass
Neuroscience
Glymour & Ramsey, fMRI
Epidemiology
Scheines, Lead & IQ
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References
Biology
Chu, Tianjaio, Glymour C., Scheines, R., & Spirtes, P, (2002). A Statistical Problem for Inference to Regulatory Structure from Associations of Gene Expression Measurement with Microarrays. Bioinformatics, 19: 1147-1152.
Shipley, B. Exploring hypothesis space: examples from organismal biology. Computation, Causation and Discovery. C. Glymour and G. Cooper. Cambridge, MA, MIT Press.
Shipley, B. (1995). Structured interspecific determinants of specific leaf area in 34 species of
herbaceous angeosperms. Functional Ecology 9.
General
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.
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References
Scheines, R. (2000). Estimating Latent Causal Influences: TETRAD III Variable Selection and Bayesian Parameter Estimation: the effect of Lead on IQ, Handbook of Data Mining, Pat Hayes, editor, Oxford University Press.
Jackson, A., and Scheines, R., (2005). Single Mothers' Self-Efficacy, Parenting in the Home Environment, and Children's Development in a Two-Wave Study, Social Work Research , 29, 1, pp. 7-20.
Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49, 141-146.
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ReferencesEconomics
Akleman, Derya G., David A. Bessler, and Diana M. Burton. (1999). ‘Modeling corn exports and exchange rates with directed graphs and statistical loss functions’, in Clark Glymour and Gregory F. Cooper (eds) Computation, Causation, and Discovery, American Association for Artificial Intelligence, Menlo Park, CA and MIT Press, Cambridge, MA, pp. 497-520.
Awokuse, T. O. (2005) “Export-led Growth and the Japanese Economy: Evidence from VAR and Directed Acyclical Graphs,” Applied Economics Letters 12(14), 849-858.
Bessler, David A. and N. Loper. (2001) “Economic Development: Evidence from Directed Acyclical Graphs” Manchester School 69(4), 457-476.
Bessler, David A. and Seongpyo Lee. (2002). ‘Money and prices: U.S. data 1869-1914 (a study with directed graphs)’, Empirical Economics, Vol. 27, pp. 427-46.
Demiralp, Selva and Kevin D. Hoover. (2003) !Searching for the Causal Structure of a Vector Autoregression," Oxford Bulletin of Economics and Statistics 65(supplement), pp. 745-767.
Haigh, M.S., N.K. Nomikos, and D.A. Bessler (2004) “Integration and Causality in International Freight Markets: Modeling with Error Correction and Directed Acyclical Graphs,” Southern Economic Journal 71(1), 145-162.
Sheffrin, Steven M. and Robert K. Triest. (1998). ‘A new approach to causality and economic growth’, unpublished typescript, University of California, Davis.
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ReferencesEconomics
Swanson, Norman R. and Clive W.J. Granger. (1997). ‘Impulse response functions based on a causal approach to residual orthogonalization in vector autoregressions’, Journal of the American Statistical Association, Vol. 92, pp. 357-67.
Demiralp, S., Hoover, K., & Perez, S. A Bootstrap Method for Identifying and Evaluating a Structural Vector Autoregression Oxford Bulletin of Economics and Statistics, 2008, 70, (4), 509-533
- Searching for the Causal Structure of a Vector Autoregression Oxford Bulletin of Economics and Statistics, 2003, 65, (s1), 745-767
Kevin D. Hoover, Selva Demiralp, Stephen J. Perez, Empirical Identification of the Vector Autoregression: The Causes and Effects of U.S. M2*, This paper was written to present at the Conference in Honour of David F. Hendry at Oxford University, 2325 August 2007.
Selva Demiralp and Kevin D. Hoover , Searching for the Causal Structure of a Vector Autoregression, OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 65, SUPPLEMENT (2003) 0305-9049
A. Moneta, and P. Spirtes “Graphical Models for the Identification of Causal Structures in Multivariate Time Series Model”, Proceedings of the 2006 Joint Conference on Information Sciences, JCIS 2006, Kaohsiung, Taiwan, ROC, October 8-11,2006, Atlantis Press, 2006.
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References
• Causation, Prediction, and Search, 2nd Edition, (2000), by P. Spirtes, C. Glymour, and R. Scheines ( MIT Press)
• Causality: Models, Reasoning, and Inference (2000). By Judea Pearl, Cambridge Univ. Press
• Computation, Causation, & Discovery (1999), edited by C. Glymour and G. Cooper, MIT Press
• Eberhardt, F., and Scheines R., (2007).“Interventions and Causal Inference”, in PSA-2006, Proceedings of the 20th biennial meeting of the Philosophy of Science Association 2006 http://philsci.org/news/PSA06
• 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.
• TETRAD IV: www.phil.cmu.edu/projects/tetrad • Web Course on Causal and Statistical Reasoning :
www.phil.cmu.edu/projects/csr/• Causality Lab: www.phil.cmu.edu/projects/causality-lab