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### Causal reasoning in Biomedical Informatics

Chitta Baral

Professor

Department of Computer Science and Engineering

chitta@asu.edu

Causal connection versus Correlation

- Rain, Falling Barometer: They usually have a 1-1 correspondence
- Does falling barometer cause rain?
- Does rain cause falling barometer?
- Rain and Mud:
- Does rain cause mud?
- Smoking and Cancer:
- Does smoking causes cancer?
- What causes global warming?

Simpson’s Paradox (cont.)

- Summary
- 60% of males who took drug recovered
- 70% of males who did not take drug recovered
- 20% of females who took drug recovered
- 30% of females who did not take drug recovered
- 50% of people who took drug recovered
- 40% of people who did not take drug recovered
- Paradox:
- If we know a patient is male or a female then we should not give the drug!
- If we do not know the sex then we should give the drug!

Why the Simpson’s Paradox

- From the given data we can calculate the following
- P(recovery|took drug, male)
- P(recovery|took drug, female)
- P(recovery|too drug)
- We should be calculating the following
- P(recovery| do(drug), male)
- P(recovery| do(drug), female)
- P(recovery| do(drug) )

Causality: story and questions

- Story: In a group of people, 50 % were given treatment for an ailment and 50% were not. Of the 50 % in both the treated and untreated group, 50 % recovered and 50 % did not.
- Joe, a patient took the treatment and died.
- What is the probability that Joe’s death occurred due to treatment?
- Or, what is the probability that Joe, who died under treatment, would have lived had he not been treated?
- Can we answer these questions from the above story?
- No?
- Why not?

The probability distribution

- X =1

treatment was given

- X = 0

treatment was not given

- Y = 1

patient died

- Y = 0

patient recovered

Causal Model 1 of the story

- The model
- U1: a variable that decides treatment; U2: a variable that decides if someone will die
- X, Y: treatment given, patient died
- X U1 (X = U1)
- Y U2 (Y = U2)
- P(U1) = 0.5 P(U2) = 0.5. leads to the same probability table
- Observation: Joe took the treatment and died.
- X = 1 and Y = 1. Thus U1 = 1 and U 2 =1.
- Question: What is the probability that Joe would have lived if he had not taken the treatment.
- We do X= 0. Find P( Y = 0 | do(X=0))?
- Y = U2 = 1 (regardless of the value of X and U1)
- Hence P( Y = 0 | do(X=0)) = 0. (Joe would have died anyway.)

Causal Model 2 of the story

- The model
- U2: A genetic factor which if present, kills people who take the treatment and if absent kills people who do not take the treatment
- U1, X, Y: decides treatment, treatment given, patient died
- X U1 (X = U1)
- Y U2 Y X (Y = X*U2 + (1-X)(1-U2) )
- P(U1) = 0.5 P(U2) = 0.5. leads to the same probability table
- Observation: Joe took the treatment and died.
- X = 1 and Y = 1. Thus U1 = 1 and U 2 =1.
- Question: What is the probability that Joe would have lived if he had not taken the treatment.
- We do X= 0. Find P( Y = 0 | do(X=0))?
- Y = 0 * 1 + (1-0)(1-1) = 0
- Hence P( Y = 0 | do(X=0)) = 1. (Joe would not have died.)

Summary of the story

- There are at least two causal models which is consistent with the data (50% of …)
- In model 1 Joe would still have died if he had not taken the treatment.
- But in model 2 Joe would have lived if he had not taken the treatment.
- Moral:
- Causal models are the key.
- Just statistical data is not much useful.
- Multiple causal models may correspond to the same statistical data.

Causal relations in molecular biology

- Certain proteins (transcription factors) regulate the expression of genes
- One protein may inhibit or activate another protein or another biochemical molecule
- Catalysts in metabolic reactions

Sources: http://www.ornl.gov/sci/techresources/Human_Genome/graphics/slides/images/REGNET.jpg

How do we get causal models?

- Traditional approach
- Knockout genes (too slow)
- Temporal or time series data
- Not feasible for human cells
- Can we infer causal information from steady state data?
- To some extent

Suppose we have 3 variables A, B, and C obtained from the data that:

- A and B are dependent.
- B and C are dependent.
- A and C are independent.

Think of A, B and C that satisfy the above.

C

B

Example (cont.)Most likely your interpretation of A, B and C would satisfy the causal relations A B C as shown below.

Causal model

- Causal structure: a directed acyclic graph (DAG)
- Causal model: Causal structure with parameters (functions for each variables with parents, and probabilities for the variables without parents)

Conditional independence and d-separation

- X and Y are said to be conditionally independent given Z if P(x | y, z) = P(x | z) whenever P(y, z) > 0.
- d-separation: A path p is said to be d-separated by a set of nodes Z if
- p contains i m j or i m j and m is in Z or
- p contains i m j and neither m nor any of its descendant is in Z.
- Z is said to d-separate X and Y if every path between a node in X and a node in Y is d-separated by Z

Observationally equivalent

- Two directed acyclic graphs (DAGs) are observationally equivalent if they have the same set of independencies.

Alternative Definition:

- Two DAGs are observationally equivalent if they have the same skeleton and the same set of v-structures
- V-structures are structures of the form a x b such that there is no arrow between a and b.

Observationally equivalent networks

- Two networks that are observationally equivalent can not be distinguished without resorting to manipulative experimentation or temporal information.

Preference

- Ordering between DAGs: G1 is preferable to G2, if every distribution that can be simulated using G1 (and some parameter) can also be simulated using G2 (and some parameter).
- In the absence of hidden variables, tests of preference and (observational) equivalence can be reduced to tests of induced dependencies, which can be determined directly from the topology of the DAG without considering about the parameters.

Stability/faithfulness

- Stability/faithfulness: A DAG and distribution are faithful to each other if they exhibit the same set of independencies. A distribution is said to be faithful if it is faithful to some DAG.
- With the added assumption of stability, every distribution has a unique minimal causal structure (up to d-separation equivalence), as long as there are no hidden variables.

IC algorithm and faithfulness

- Given a faithful distribution the IC and IC* algorithms can find the set of DAGs that are faithful to this distribution, in absence and in presence of hidden variables, respectively

a

b

Sab

a

╨

b

a

b

Not Sab

IC Algorithm: Step 1- For each pair of variables a and b in V, search for a set Sab such that (a╨b | Sab) holds in P – in other words, a and b should be independent in P, conditioned on Sab .
- Construct an undirected graph G such that vertices a and b are connected with an edge if and only if no set Sab can be found.

a

a

╨

b

C

c

a

No

b

c

b

IC Algorithm: Step 2- For each pair of nonadjacent variables a and b with a common neighbor c, check if c Sab.
- If it is, then continue;
- Else add arrowheads at c
- i.e a→ c ← b

Our work on learning causal models

- We developed an algorithm for learning causal relationship with knowledge of topological ordering information;
- Uses conditional dependencies and independencies among variables;
- Incorporates topological information; and
- Learns mutual information among genes.

Steps of learning gene causal relationships: mIC algorithm and its evaluation

- Step1: obtain the probability distribution, data sampling and the topological order of the gene;
- Step2: apply algorithms to find causal relations;
- Step3: compare the original and obtained networks based on the two notions of precision and recall;
- Step4: repeat step 1-3 for every random network;

We applied the learning algorithm in Melanoma Dataset

- melanoma -- malignant tumor occurring most commonly in skin;

Knowledge we have

- The 10 genes involved in this study chosen from 587 genes from the melanoma data;
- Previous studies show that WNT5A has been identified as a gene of interest involved in melanoma;
- Controlling the influence of WNT5A in the regulation can reduce the chance of melanoma metastasizing;

Partial biological prior knowledge:

MMP3 is expected to be at the end of the pathway

Pirin causatively influences WNT5A –

“In order to maintain the level of

WNT5A we need to directly control

WNT5A or through pirin”.

WNT5A

Causal connection between WNT5A and

MART-1

“WNT5A directly causes MART-1”

Important Information we discoveredB

f

C

C=f(A,B)

Modeling and simulation of a causal Boolean network (BN)Proper function: The function that reflects the influence of the operators.

Example:

c = f(a,b) = (ab) (a b)

= a is not a proper function.

- Boolean network:

- Simulation process in our study:
- Generate M BNs with up to 3 causal parents for each node;
- For each BN, generate a random proper function for each node;
- Assign random probabilities for the root gene(s);
- Given one configuration, get probability distribution;
- Collect 200 data points for each network;
- Repeat above steps 3-5 for all M networks.

Comparing original and obtained networks

- Original graph is a DAG, while obtained graph has both directed and undirected edges;

Recall = ATP/(AFN+ATP), Precision = ATP/(ATP + AFP)

Conclusion

- Causality differs from correlation.
- P(X|Y) differs from P(X| do(Y)).
- While P(X|Y) can be answered using joint probability distributions and other representations of it (such as Bayes nets), to answer P(X|do(Y)) one needs a causal model.
- We have worked on various causal model representations and how to reason with them.
- Causal models can be learned by knock out experiments and from temporal and time series data.
- Recent algorithms have been proposed to learn causal models from steady state data.
- IC algorithm.
- We have improved on the IC algorithm.

References

- Judea Pearl: Reasoning with cause and effect. http://singapore.cs.ucla.edu/IJCAI99/index.html
- An algorithm to learn causal connection between genes from steady state data: simulation and its application to melanoma dataset. Xin Zhang, Chitta Baral, Seungchan Kim. Proc. of 10th Conference on Artificial Intelligence in Medicine (AIME 05) 23 - 27 July 2005 Aberdeen, Scotland. pages 524-534. http://www.public.asu.edu/~cbaral/papers/AIME05_final.pdf

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