1 Richard Scheines Carnegie Mellon University Causal Graphical Models II: Applications with Search.
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1
Richard ScheinesCarnegie Mellon University
Causal Graphical Models II: Applications with Search
2
1. Foreign Investment
2. Welfare Reform
3. Online Learning
4. Charitable Giving
5. Stress & Prayer
6. Test Anxiety
7. Causal Connectivity among Brain Regions
Case Studies
3
1. Exceedingly simple
2. Background theory weak
3. Claim:
– Not: search output is true
– Is: search adds value
Case Studies
4
Case Study 1: Foreign Investment
Does Foreign Investment in 3rd World Countries cause Political Repression?
Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49, 141-146.
N = 72
PO degree of political exclusivity
CV lack of civil liberties
EN energy consumption per capita (economic development)
FI level of foreign investment
5
Correlations
po fi en fi -.175 en -.480 0.330 cv 0.868 -.391 -.430
Case Study 1: Foreign Investment
6
Regression Results
po = .227*fi - .176*en + .880*cv
SE (.058) (.059) (.060)
t 3.941 -2.99 14.6
Interpretation: foreign investment increases political repression
Case Study 1: Foreign Investment
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
There is no model with testable constraints (df > 0) in which FI has a positive effect on PO that is not rejected by the data.
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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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Two-Wave Longitudinal Study
• 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
11
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
12
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
c2 = 22.3, df = 20, p = .32
c2 = 18.87, df = 19, p = .46
Case Study 2: Welfare Reform
13
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
14
Online Course in Causal & Statistical Reasoning
Case Study 3: Online Courseware
15
Variables
Pre-test (%)
Print-outs (% modules printed)
Quiz Scores (avg. %)
Voluntary Exercises (% completed)
Final Exam (%)
9 other variables
Case Study 3: Online Courseware
Tier 1
Tier 2
Tier 3
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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
17
Variables
Tangibility/Concreteness (Exp manipulation)
Imaginability (likert 1-7)
Impact (avg. of 2 likerts)
Sympathy (likert)
Donation ($)
Case Study 4: Charitable Giving
Cryder & Loewenstein (in prep)
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Theoretical Model
Case Study 4: Charitable Giving
Imaginability Tangibility
Impact
Sympathy
Donation
study 1 (N= 94) df = 5, c2 = 52.0, p= 0.0000
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GES Outputs
Case Study 4: Charitable Giving
Imaginability Tangibility
Impact
Sympathy
Donation
study 1: df = 5, c2 = 5.88, p= 0.32
Imaginability Tangibility
Impact
Sympathy
Donation
study 1: df = 5, c2 = 3.99, p= 0.55
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Theoretical Model
Case Study 4: Charitable Giving
Imaginability Tangibility
Impact
Sympathy
Donation
study 2 (N= 115) df = 5, c2 = 62.6, p= 0.0000
Imaginability Tangibility
Impact
Sympathy
Donation
Imaginability Tangibility
Impact
Sympathy
Donation
study 2: df = 5, c2 = 8.23, p= 0.14
study 2: df = 5, c2 = 7.48, p= 0.18
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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 Assumptions
1. 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 5: 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
St1
12
Stress
St2
12
St21
12
.
.
Dep1
12
Coping
.
.
Depression
Dep2
12
Dep20
12
C1 C2 C20 . .
+
- +
Specified Model
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Build Pure Clusters St3
12
Stress
St4
12 St16
12
Dep9
12
Coping
Depression Dep13
12 Dep19
12
C9 C12 C15
St18
12
St20
12
C14
Case Study 5: Stress, Depression, and Religion
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Assume Stress temporally prior:
MIMbuild to find Latent Structure: St3
12
Stress
St4
12 St16
12
Dep9
12
Coping
Depression Dep13
12 Dep19
12
C9 C12 C15
St18
12
St20
12
C14
+
+
p = 0.28
Case Study 5: Stress, Depression, and Religion
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Case Study 6: 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 6: 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 6: Test Anxiety
29
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 6: Test Anxiety
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• Goals:– Identify relatively BIG brain regions (ROIs).– Figure out how they influence one another, with what
timing sequences, in producing behaviors of interest.– Figure out individual differences.
Case Study 7: fMRI Brain Connectivity
31
• Experiment: (Xue and Poldrack, unpublished)– 13 right handed subjects– On each trial, subject judged whether visual stimuli
rhymed or not– 8 pairs of words/nonwords presented for 2.5 seconds
each in eight 20 second blocks, separated by 20 seconds of visual fixation
– TR = 2000 milliseconds– 160 time points.
Case Study 7: fMRI
32
• Problems:– Criteria for identifying ROIs– Individuals differ
• Brain ROIs• Parameter values
– Brain processing is cyclic– Time:
• Varying time delays of neuron ROI BOLD response• Time series sampling rate vs. processing rate
– Search Space • 11 ROIs – 323 DAGs
Case Study 7: fMRI Brain Connectivity
ROI Construction
• Mean of signal intensity among voxels in a cluster at a time• 1st or ....4th principal component• Average of top X% variance• Maximum variance voxel.• Eyeballs• Etc., etc
Case Study 7: fMRI
Example ROIs
Case Study 7: fMRI
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– Individuals differ• Brain ROIs• Parameter values
Case Study 7: fMRI Brain Connectivity
– Assume • same qualitiative causal structure• different quantitative causal structure (mixed effects)
– iMAGES search • Apply GES to each subject, 1 step• Take step = max(avg. BIC score) to each search• Repeat
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Time Problem 1
• fMRI recordings at time intervals can be analyzed as a collection of independent cases.
• Or, they can be analyzed as an auto-regressive time series.
• Which is better? – No general answer.– But if you think the neural activities measured at time
t influence the measurements at time t+1 then the data should be treated as a lag 1 auto-regressive time series.
– But then Granger causality isn’t a consistent estimator of causal relations.
Case Study 7: fMRI
Granger Causality Corrected
Causal processes faster than the sampling rate:
Xt Xt+1 X
Yt Yt+1 Y
Zt Zt+1 Z
Regress on t variables
Apply GES to the RESIDUALS of the regression (Demiralp, Hoover)
NO False path
Case Study 7: fMRI
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Time Problem 2
• Varying time delays : neurons BOLD responses
Case Study 7: fMRI
• Try all time shifts of one or two units over all subsets of 3 vars, choose shift that leads to best likelihoods
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Lag 0 result Lag 1 result.
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Simulation Studies:
• 11 ROIs, each consisting of 50 simulated neurons:• Neuron output spikes simulated by thresholding a
tanh function of the sum of neuron inputs.• Excitatory feedback• Random subset of neurons in one ROI input to
random subset of neurons in an “effectively connected ROI”
• Measured variables = BOLD function of sum of ROI neurons + Gaussian error with variance = error variances of empirical measured variables in the X/P experiment.
Case Study 7: fMRI
41
• Repeat 10 times:– Randomly generate a graphical structure with
11 nodes and 11 (feedforward) directed edges– Randomly select a subset of simulated ROIs.– Generate data – Randomly shift 0 to 3 variables one or 2 time
steps forward.– Apply the iMAGES method with 0 lag and 1
lag, with backshifting.• Tabulate the errors.
Simulate the Xue/Poldrack Experiment Time Series:
Case Study 7: fMRI
42
Simulation Results
0 Lag:
Average number of false positive edges: 0.7
Average number of mis-directed edges: 1.6
1 Lag Residuals:
Average number of false positive edges: 1.2
Average number of mis-directed edges: 1.8
Case Study 6: fMRI
43
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
44
Straw Men!
• Model Search ignores theory
• Model Search hides assumptions
• Model Search needs more assumptions than standard statistical models
45
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.
47
References
Economics
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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References
Economics
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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Extra
Lead and IQ: Variable Selection
BackwardsStepwise Regression
Measured Lead +5 Covariates
Measured Lead +39 Covariates
Final Variables (Needleman)
-lead baby teeth
-fab father’s age
-mab mother’s age
-nlb number of live births
-med mother’s education
-piq parent’s IQ
-ciq child’s IQ
Needleman Regression
- standardized coefficient
- (t-ratios in parentheses)
- p-value for significance
ciq = - .143 lead - .204 fab - .159 nlb + .219 med + .237 mab + .247 piq
(2.32) (1.79) (2.30) (3.08) (1.97) (3.87)
0.02 0.09 0.02 <0.01 0.05 <0.01
All variables significant at .1 R2 = .271
TETRAD Variable Selection
Tetradmab _||_ ciq
fab _||_ ciq
nlb _||_ ciq | med
ciq
mab fab nlb
lead piq med
Regressionmab _||_ ciq | { lead, med, piq, nlb fab}
fab _||_ ciq | { lead, med, piq, nlb mab}
nlb _||_ ciq | { lead, med, piq, mab, fab}
Regressions
- standardized coefficient
- (t-ratios in parentheses)
- p-value for significance
Needleman (R2 = .271)
ciq = - .143 lead - .204 fab - .159 nlb + .219 med + .237 mab + .247 piq
(2.32) (1.79) (2.30) (3.08) (1.97) (3.87)
0.02 0.09 0.02 <0.01 0.05 <0.01
TETRAD (R2 = .243)
ciq = - .177 lead + .251 med + .253 piq
(2.89) (3.50) (3.59)
<0.01 <0.01 <0.01
Measurement Error
• Measured regressor variables are proxies that involve measurement error
• Errors-in-all-variables model for Lead’s influence on IQ - underidentified
Actual LeadExposure
EnvironmentalStimulation
ciq
lead 3
2
111
1
ciq
lead
med
med
piq
piq
Geneticfactors
Strategies:
• Sensitivity Analysis
• Bayesian Analysis
Prior over Measurement Error
Proportion of Variance from Measurement Error
• Measured Lead Mean = .2, SD = .1• Parent’s IQ Mean = .3, SD = .15• Mother’s Education Mean = .3, SD = .15
Prior Otherwise uninformative
Actual LeadExposure
EnvironmentalStimulation
ciq
lead 3
2
111
1
ciq
lead
med
med
piq
piq
Geneticfactors
Posterior
Expected if Normal
0
50
100
150
200
250
0
50
100
150
200
250
Expected if Normal
Frequency
LEAD->ciq
Distribution of LEAD->ciq
Zero
Robust over similar priors
Using Needleman’s Covariates
With similar prior, the marginal posterior:
Expected if Normal
0
20
40
60
80
100
120
140
0
2040
6080
100120
140160
Expected if Normal
Frequency
LEAD->ciq
Distribution of LEAD->ciq
Very Sensitive to Prior Over Regressors
TETRAD eliminated
Zero
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