Stats II

Question 1

Pearl - Intro

Language of queries

Answer 1

The description of an intervention or treatment, written symbolically.

eg. Effect of a drug (D) on lifespan (L)

P(L | do(D))

Question 2

Pearl - Intro

What is a counterfactual?

Answer 2

“What would have happened if we had acted differently (not taken the treatment)?”

Allows us to emulate in retrospect, reflect on past actions and envision alternative scenarios.

Question 3

Session 2

Individual Treatment Effect

Answer 3

The difference of outcomes between a group under treatment and the control group.

ITE = Y1i − Y0i

Question 4

Session 2

Fundamental Problem of Causal Inference

Answer 4

“It is impossible to observe two states of the world for the same person at the same time (e.g. vaccine effectiveness).”

Question 5

Session 2

Average Treatment Effect

Answer 5

An average of the individual level effect.

ATE = Avg[Y1i − Y0i]

Also known as the Estimand.

Question 6

Session 2

True ATE

Answer 6

AVG[ITE1 + ITE2 + … ITE X]

Question 7

Session 2

Difference in Means

Answer 7

**difference in means = ATE + selection bias
**

naive difference in means = Y1i-Y0i

We can interpret the ATE as a difference in means when there is no selection bias.

Experiments ensure that the selection bias term is 0.

Question 8

Session 2

Randomization

Answer 8

Random assignment guarantees that:

**E[Y0i|Di = 1] = E[Y0i|Di = 0]
**
Which means the selection bias term equals zero.

Question 9

Session 2

Law of Large Numbers - LLN

Answer 9

Sample average can be brought as close as we want to the AVG in population by increasing the sample.

Question 10

Session 2

Constant Treatment Effects

Answer 10

When the treatment affects groups in the same way.

In a hypothetical example, an outcome of Treatment ITE = 6 and and Control ITE 4 means different effects. If both had 4 and 4, there would be a constant treatment effect.

Question 11

Session 2

Randomization considerations

Answer 11

• Feasibility: Not everything that counts should be randomized, and not everything that can be randomized counts.
• Ethics: Concerning the denial of services we know to be good, the effect of bad treatment, the fair allocation of incentives, the repercussions of using human subjects.

Question 12

Session 3

DAG

Answer 12

Directed Acyclic Graphs (DAG) represent the beliefs, relationships and assumptions of a causal model.

Question 13

Session 3

Confounder

Answer 13

A variable that has a mixing effect associated with our causal path. Occurs when treatment and outcome have a shared common cause not controllef for.

May be observable or unobservable.

Can lead to OVB and selection bias.

https://jamanetwork.com/journals/jama/fullarticle/2790247

Question 14

Session 3

Backdoor Paths

Answer 14

Backdoor paths are non-causal, open paths that create associations even in the absence of a causal effect.

D <- X -> Y

Question 15

Session 3

Mediator

Answer 15

A mechanism that mediates the causal relationship of a set of variables. Must be added one wants the total effect of the treatment.

D -> X -> Y

Question 16

Session 3

Collider

Answer 16

A third variable that has been influenced by both the treatment and the outcome.

It is a closed path unless controlled for, which induces statistical association between the variables (collider bias).

D -> O <- A -> Y with O being the collider.

https://jamanetwork.com/journals/jama/fullarticle/2790247

Question 17

Session 3

Collider Bias

Answer 17

Threatens the internal validity of a study and the accurate estimation of causal relationships.

Question 18

Session 3

Conditioning

Answer 18

Means holding the confounder (X) fixed at some value.

eg. “adjusting or controlling for”.

Question 19

Session 3

Backdoor Criterion

Answer 19

• Conditioning for a confounder closes an open backdoor path (eliminates selection bias)
• Conditioning for a collider opens a closed path (and we get collider bias).
• Conditioning for a mediator is fine, depending on what the effect to measure (to get the total effect, we would need the to add the direct + mediated treatment effects).

Question 20

Session 3

Overcontrolling

Answer 20

“Sometimes you end up controlling for the thing you are trying to measure” (Pearl on Ezra Klein’s example).

Question 21

Session 4

Regression

Answer 21

Predicts the value of an outcome variable based on one or more input explanatory variables.

What is our best guess of y given an observed x?

Bivariate regression: yi = α+βxi+ei
*Modeling variable y as a function of one variable x

Question 22

Session 4

Loss function

Answer 22

Sum of the squared residuals

Question 22

Session 4

Conditional expectation function

Answer 22

E[Yi|Xi = X] = α+βxi

give years of schooling = expected income

α+βxi + vi + s1 * 1500 + s2 * log(100,000)
- v is associated with the dummies, eg. SAT score
- we can change Pi to compare treatment / control

**On average and holding all else constant, a one-unit change in P (eg. from 0 to 1) is associated with a B-unit change in log(y).
**

Question 23

Session 4

Bias-variance tradeoff

Answer 23

The difference between being systematically off, but consistent (low bias, high variance), or accepting bias but having low variance.

Question 24

Session 4

Regression of conditional means

Answer 24

When we know there is selection bias, the naive difference does not equal ATE. We have to:
- look at naive difference between groups themselves
- get mean or weighted AVG

Question 25

Session 4

Overall average effect

Answer 25

log() can help us get a normal distribution for eg. income, education

Yi = α+βxi+ vAi + ei

Question 26

Pearl - Intro

Why did the causal revolution arise?

Answer 26

It has been possible, unlike other historical periods, thanks to the vocabulary that allowed us to capture:

1- causal diagrams (what we know)
2- symbolic language (what we want to know).

Question 27

Session 4

Regression for causal effects

Answer 27

We can include Estimated B by following our DAG.

If we control for a confounder, given a treatment, we include all controls in the X and keep them constant.

E[Yi | Di = 1, X] - E[Yi | Di = 0, X]