Read this definition of a propensity score in ordinary language.

Propensity Score: For a unit with covariates X=x, the probability of being assigned treatment:

 

PS(x) = prob(T = 1|X=x)

Basic Concepts of Statistical Inference for Causal Effects in Experiments and Observational Studies Donald B. Rubin. P 18

ICS212 Fall 2022

The propensity score of a unit with characteristics x is the probability of being assigned to the treatment group given that its covariates are x.

ICS212 Fall 2022

Translate these assertions about the assignment mechanisms for randomized experiments

= "is independent of"
= "given that" or "conditional on"

The probability that any given individual is assigned to treatment or control is independent of potential outcomes.

The probability that any given individual is assigned to treatment or control is independent of potential outcomes and is a known function of covariates (for example, we might decide to have more women than men in our treatment group).

ICS212 Fall 2022

Translate these assertions about the assignment mechanisms for randomized experiments

= "is independent of"
= "given that" or "conditional on"

ICS212 Fall 2022

means the probability of X given a, b, and c is the same as the probability of X given a.

pr(X|a,b,c) = pr(X|a)

And that means the X does not depend on b, and c.

  1. What do each of the underlined parts of this mathematical description of an assignment mechanism mean?
     
  2. What does it mean altogether?

probability of being in treatment group

given unit i's characteristics

and what would happen under treatment and no treatment

is the same as

pr(T) given i's characteristics and the observed outcome

  1. What do each of the underlined parts of this mathematical description of an assignment mechanism mean?
     
  2. What does it mean altogether?

ICS212 Fall 2022

means the probability of X given a, b, and c is the same as the probability of X given a.

pr(X|a,b,c) = pr(X|a)

And that means the X does not depend on b, and c.

ICS212 Fall 2022

means the probability of X given a, b, and c is the same as the probability of X given a.

pr(X|a,b,c) = pr(X|a)

And that means the X does not depend on b, and c.

  1. What do each of the underlined parts of this mathematical description of an assignment mechanism mean?
     
  2. What does it mean altogether?

probability of being in treatment group

given unit i's characteristics

and what would happen under treatment and no treatment

is the same as

pr(T) given i's characteristics

  1. What do each of the underlined parts of this mathematical description of an assignment mechanism mean?

ICS212 Fall 2022

means the probability of X given a, b, and c is the same as the probability of X given a.

pr(X|a,b,c) = pr(X|a)

And that means the X does not depend on b, and c.

  1. What do each of the underlined parts of this mathematical description of an assignment mechanism mean?
     
  2. What does it mean altogether?