1. Workshop Files

2. Center for Causal Discovery: Summer Short Course/Datathon - 2018
June 11-15, 2018
Carnegie Mellon University

3. Goals Basic working knowledge of graphical causal models
Basic working knowledge of Tetrad 6
Basic understanding of search algorithms
“Fully started” on using CCD algorithms/tools on real data, preferably your own.
Grow the community of researchers, users, and students interested in causal discovery in biomedical research

4. Monday: Basics of Graphical Causal Models, Tetrad
Morning: 9 AM – Noon, Baker Hall A51 : Giant Eagle Auditorium
Introduction – Greg Cooper
Representing/Modeling Causal Systems
Causal Graphs/Interventions
Parametric Models
Instantiated Models
Afternoon: 1:30 PM – 4 PM, Baker Hall A51 : Giant Eagle Auditorium
Estimation, Inference, and Model fit
Real data examples:
Charitable Giving
College Plans
Dinner: On your own

5. Tuesday: Basics of Search, Break-out Sessions
Morning: 9 AM – Noon, Baker Hall A51 : Giant Eagle Auditorium
Searching for Causal Systems
Model Equivalence
Basic Search Algorithms
Hands-on Work
Afternoon: 1:30 PM – 4 PM, Baker Hall A51  breakout rooms
CCD Software and Bridges Supercomputer (Jeremy Espino)
Break-out Sessions 1:
Brain/fMRI
Cancer
Protein signaling: single cell cytometry
4 – 6 PM - Reception: Hors d’oeuvres etc., Lobby outside A51
Dinner: On your own

6. Wednesday: Latent Variables, etc., Break-out Sessions
Morning: 9 AM – Noon, Baker Hall A51 : Giant Eagle Auditorium
Latent Variable Model Search
Mixed Data (Continuous and Discrete)
Afternoon: 1:30 PM – 3:30 PM, Baker Hall A51  breakout rooms
Break-out Sessions – 2
Dinner: On your own

7. Thursday: Research Area Overviews, Datathon
Morning: 9 AM – Noon, Baker Hall A51 : Giant Eagle Auditorium
fMRI – Brain
Cancer: Genomic Drivers
Protein Signaling Pathways: Single Cell Data
Consultation sign-up
Short Course : Ends at Noon
Datathon – Noon – 8 PM, Baker Hall A51: Giant Eagle Auditorium

8. Friday: DataThon
Morning: 9 AM – Noon, Baker Hall A51 : Giant Eagle Auditorium
Breakfast with Q&A (9 -10 AM)
Data Hacking (10-12)
Lunch, 12-1: on your own
Afternoon: 1:00 PM – 4 PM, Giant Eagle Auditorium: Datathon
1:00 – 3:00 Data Hacking & Visualization
3:00 Participant Presentations

9. Questions?

10. Overview of CCD Greg Cooper, MD, PhD

11. Causal Discovery

12. 25 Years of Advances 1992  2017: tremendous progress in:
Mathematically representing causal networks
Discovery algorithms for finding causal networks from a combination of data and knowledge.
These methods apply to biomedical data.
Statistics, Computer Science, Philosophy, etc.
America, Europe, Japan

13. Modern Theory of Statistical Causal Models
Graphical
Models
Intervention & Manipulation
Modern Theory of Statistical Causal Models
Potential Outcome Models
Testable Constraints
(e.g., Independence)
Counterfactuals 1

14. Causal Inference Requires More than Probability
Prediction from Observation ≠ Prediction from Intervention
P(Lung Cancer 1960 = yes | Tar-stained fingers = yes) ≠
P(Lung Cancer 1960 = yes | Tar-stained fingers do(1950 = yes))
In general: P(Y=y | X=x, Z=z) ≠ P(Y=y | do(X=x), Z=z))
Causal Prediction vs. Statistical Prediction:
Non-experimental data
(observational study)
Background Knowledge
P(Y,X,Z)
P(Y=y | X=x, Z=z)
Causal Structure
P(Y=y | do(X=x, Z=z)

15. P(Stained Teeth = yes) =.5

16. P(Smoker= yes) = .5

17. P(Lung Cancer = yes) = .5

18. Conditioning on Stained Teeth = no

19. P(Lung Cancer = yes | Stained Teeth = no) =
.25

20. Lung Cancer _||_ Stained Teeth

21. Manipulating White Teeth

22. Pm(Lung cancer = yes | do(Stained Teeth= no)) =.5

23. Lung Cancer _||_m Stained Teeth
P(LC = yes) = = Pm(LC= yes | do(Stained Teeth = no)) = .5
Lung Cancer _||_m Stained Teeth

25. Causal Estimation vs. Causal Search
Estimation (Potential Outcomes)
Causal Question: Effect of Zidovudine on Survival among HIV-positive men (Hernan, et al., 2000)
Problem: confounders (CD4 lymphocyte count) vary over time, and they are dependent on previous treatment with Zidovudine
Estimation method discussed: marginal structural models
Assumptions:
Treatment measured reliably
Measured covariates sufficient to capture major sources of confounding
Model of treatment given the past is accurate
Output: Effect estimate with confidence intervals
Fundamental Problem: estimation/inference is conditional on the model 1

26. Causal Estimation vs. Causal Search
Search (Causal Graphical Models)
Causal Question: which genes regulate flowering in Arbidopsis
Problem: over 25,000 potential genes.
Method: graphical model search
Assumptions:
RNA microarray measurement reasonable proxy for gene expression
Causal Markov assumption
Etc.
Output: Suggestions for follow-up experiments
Fundamental Problem: model space grows super-exponentially with the number of variables 1

27. Number of variables (nodes)
The Number of Causal Model Structures as a Function of the Number of Measured Variables*
Number of variables (nodes)
Number of Causal Model Structures 1 2 3 25 4
543 5
29,281 6
3,781,503 7
1.1 x 109 8
7.8 x 1011 9
1.2 x 1015 10
4.2 x 1018
* Assumes there are no latent variables and no directed cycles.

28. Causal Search Causal Search:
Find/compute all the causal models that are indistinguishable given background knowledge and data
Represent features common to all such models
Multiple Regression is often the wrong tool for Causal Search:
Example: Foreign Investment & Democracy

29. Foreign Investment
Does Foreign Investment in 3rd World Countries inhibit Democracy?
Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49,
N = 72
PO degree of political exclusivity
CV lack of civil liberties
EN energy consumption per capita (economic development)
FI level of foreign investment 1

30. Foreign Investment Correlations po fi en cv po 1.0 fi -.175 1.0

31. Case Study: Foreign Investment
Regression Results
po = *fi *en *cv
SE (.058) (.059) (.060) t P
Interpretation: foreign investment increases political repression 1

32. Case Study: Foreign Investment Alternative Models
There is no model with testable constraints (df > 0) that is not rejected by the data, in which FI has a positive effect on PO.

33. Outline Representing/Modeling Causal Systems Causal Graphs
Parametric Models
Bayes Nets
Structural Equation Models
Generalized SEMs

34. 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
Years of Education
Income
1. don’t define causality - but will introduce axioms to connect probability to causality
2. many fields proceed without agreement on definition - probability, “force” in mechanics, interpretation of quantum mechanics, etc.
3. a number of different kinds of graphs represent probability distributions and independence - advantage of directed graphs is also represents causal relations
4. will introduce several extensions
Income
Skills and Knowledge
Years of Education

35. Omitteed Common Causes
Causal Graphs
Omitteed Causes
Not Cause Complete
Income
Skills and Knowledge
Years of Education
Common Cause Complete
Omitteed Common Causes
1. don’t define causality - but will introduce axioms to connect probability to causality
2. many fields proceed without agreement on definition - probability, “force” in mechanics, interpretation of quantum mechanics, etc.
3. a number of different kinds of graphs represent probability distributions and independence - advantage of directed graphs is also represents causal relations
4. will introduce several extensions
Income
Skills and Knowledge
Years of Education

36. Tetrad: Complete Causal Modeling Tool

37. Tetrad Demo & Hands-On Smoking Stained_Teeth LC
Build and Save two acyclic causal graphs:
Build the Smoking graph picture above
Build your own graph with 4 variables
Build 2 causal graphs - one for smoking, yf, lc

38. Modeling Ideal Interventions
Interventions on the Effect
Post
Pre-experimental System
Room Temperature
Sweaters On

39. Modeling Ideal Interventions
Interventions on the Cause
Post
Pre-experimental System
Sweaters On
Room
Temperature

40. Interventions & Causal Graphs
Model an ideal intervention by adding an “intervention” variable outside the original system as a direct cause of its target.
Pre-intervention graph
Intervene on Income
“Hard” Intervention
Fat Hand - intervention - cholesterol drug -- arythmia
“Soft” Intervention

41. Interventions & Causal GraphsX5 X2 X1
Pre-intervention
Graph X4 X3 X6
Intervention:
hard intervention on both X1, X4
Soft intervention on X3 X1 X2 X3 X4 X6 X5 I S
Fat Hand - intervention - cholesterol drug -- arythmia
Post-Intervention Graph?

42. Interventions & Causal GraphsX5 X2 X1
Pre-intervention
Graph X4 X3 X6
Intervention:
hard intervention on both X1, X4
Soft intervention on X3 X1 X2 X3 X4 X6 X5 I S
Fat Hand - intervention - cholesterol drug -- arythmia
Post-Intervention Graph?

43. Interventions & Causal GraphsX5 X2 X1
Pre-intervention
Graph X4 X3 X6
Intervention:
hard intervention on X3
Soft interventions on X6, X4 I S X1 X2 X3 X4 X6 X5
Fat Hand - intervention - cholesterol drug -- arythmia
Post-Intervention Graph?

44. Interventions & Causal Graphs
Smoking
Pre-intervention Graph
Stained_Teeth LC
Trek between Stained_Teeth and LC
In Pre-Intervention Graph?
Treks  Association
Yes  Stained_Teeth _||_ LC
Smoking
Paint Teeth White
Stained_Teeth LC
Fat Hand - intervention - cholesterol drug -- arythmia
Trek between Stained_Teeth and LC
In Post-Intervention Graph?
Treks  Association
No  Stained_Teeth _||_m LC

45. Parametric Models

46. Instantiated Models

47. Causal Bayes Networks The Joint Distribution Factors
According to the Causal Graph,
P(S,YF, L) =
P(S)
P(YF | S)
P(LC | S)

48. Causal Bayes Networks The Joint Distribution Factors
According to the Causal Graph,
P(S) P(YF | S) P(LC | S) = f()
All variables binary [0,1]:  =
{1, 2,3,4,5, }
P(S = 0) = 1
P(S = 1) = 1 - 1
P(YF = 0 | S = 0) = 2 P(LC = 0 | S = 0) = 4
P(YF = 1 | S = 0) = 1- 2 P(LC = 1 | S = 0) = 1- 4
P(YF = 0 | S = 1) = 3 P(LC = 0 | S = 1) = 5
P(YF = 1 | S = 1) = 1- 3 P(LC = 1 | S = 1) = 1- 5

49. Causal Bayes Networks The Joint Distribution Factors
According to the Causal Graph,
P(S,YF, LC) = P(S) P(YF | S) P(LC | S) = f()
All variables binary [0,1]:  = {1, 2,3,4,5, }
P(S,YF, LC) = P(S) P(YF | S) P(LC | YF, S) = f()
All variables binary [0,1]:  =
{1, 2,3,4,5, 6,7, }

50. Causal Bayes Networks The Joint Distribution Factors
According to the Causal Graph,
P(S,YF, L) = P(S) P(YF | S) P(LC | S)
P(S = 0) = .7
P(S = 1) = .3
P(YF = 0 | S = 0) = .99 P(LC = 0 | S = 0) = .95
P(YF = 1 | S = 0) = .01 P(LC = 1 | S = 0) = .05
P(YF = 0 | S = 1) = .20 P(LC = 0 | S = 1) = .80
P(YF = 1 | S = 1) = .80 P(LC = 1 | S = 1) = .20
P(S=1,YF=1, LC=1) = ?

51. Causal Bayes Networks The Joint Distribution Factors
According to the Causal Graph,
P(S,YF, L) = P(S) P(YF | S) P(LC | S)
P(S = 0) = .7
P(S = 1) = .3
P(YF = 0 | S = 0) = .99 P(LC = 0 | S = 0) = .95
P(YF = 1 | S = 0) = .01 P(LC = 1 | S = 0) = .05
P(YF = 0 | S = 1) = .20 P(LC = 0 | S = 1) = .80
P(YF = 1 | S = 1) = .80 P(LC = 1 | S = 1) = .20
P(S=1,YF=1, LC=1) =
P(S=1)
P(YF=1 | S=1)
P(LC = 1 | S=1)
P(S=1,YF=1, LC=1) = * *
.20
= .048

52. Calculating the effect of a hard interventions
P(YF,S,L) = P(S) P(YF|S) P(L|S)
Pm (YF,S,L) =
P(S) P(L|S)
P(YF| I)

53. Calculating the effect of a hard intervention
P(S,YF, L) = P(S) P(YF | S) P(LC | S)
P(S=1,YF=1, LC=1) = * * = .048
P(YF =1 | I ) = .5
Pm (S=1,do(YF=1), LC=1) = ?
Pm (S=1, do(YF=1), LC=1) = P(S) P(YF | I) P(LC | S)
Pm (S=1, do(YF=1), LC=1) =
.3 * * = .03

54. Calculating the effect of a soft intervention
P(YF,S,L) = P(S) P(YF|S) P(L|S)
Pm (YF,S,L) =
P(S) P(L|S)
P(YF| S, Soft)

55. Tetrad Demo & Hands-On Use the DAG you built for Smoking, YF, and LC
Define the Bayes PM (# and values of categories for each variable)
Attach a Bayes IM to the Bayes PM
Fill in the Conditional Probability Tables (make the values plausible).
Build 2 causal graphs - one for smoking, yf, lc

56. Updating

57. Tetrad Demo Use the IM just built of Smoking, YF, LC
Update LC on evidence: YF = yes
Update LC on evidence: do(YF = yes)

58. Structural Equation Models
Causal Graph
Structural Equations
For each variable X  V, an assignment equation:
X := fX(immediate-causes(X), eX)
1. example of recursive structural equation model without correlated errors
2. can show that assumption of independence of errors guarantees correctness of probabilitic interpretation
3. this represents both probability and causality
Exogenous Distribution: Joint distribution over the exogenous vars : P(e)

59. Linear Structural Equation Models
Path diagram
Causal Graph
Equations:
Education := Education
Income :=Educationincome
Longevity :=EducationLongevity
Exogenous Distribution:
P(ed, Income,Income )
- i≠j ei  ej (pairwise independence)
- no variance is zero
1. example of recursive structural equation model without correlated errors
2. can show that assumption of independence of errors guarantees correctness of probabilitic interpretation
3. this represents both probability and causality
Structural Equation Model:
V = BV + E
E.g.
(ed, Income,Income ) ~N(0,2)
2 diagonal,
- no variance is zero

60. Hands On Parameterize a DAG as a SEM. Add an IM
Make all coefficients positive
Examine “Implied Covariance Matrix” and “Implied Correlation Matrix”
Attach another IM  Standardized SEM

61. Lunch