Course Notes: A Crash Course on Causality -- Week 1: Intro to Causal Effects

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Notations and Terminologies

Notations

  • We are interested in the causal effect of some treatment A on some outcome Y

  • Treatment: A, binary

    • A=1 if receive treatment; and A=0 if receive control

    • Example: A=1 if receive active drug; and A=0 if receive placebo

  • Outcome: Y, can be binary or continuous

    • Example: Y=1 if dead; Y=0 otherwise
    • Example: Y can be time until death

  • Pre-treatment covariates: X

Potential outcomes and conterfactuals

Potential outcomes

  • Potential outcome Ya is the outcome we would see if treatment was set to A=a

  • Each person has potential outcome Y0,Y1

Counterfactuals

  • Conterfactual outcomes: the outcomes would have been observed, had the treatment been different

    • If my treatment is A=1, then my counterfactual outcomes is Y0

    • If my treatment is A=0, then my counterfactual outcomes is Y1

  • Connection between potential and conterfactuals outcomes

    • Before the treatment decision is made, any outcome is a potential outcome, Y0 and Y1
    • After the study, there is an observed outcome YA, and counterfactual outcome Y1−A

Immutable variables

  • Variables that we cannot control (or change), such as race, gender, age, are immutable variables

  • For immutable variables, causal effects are not well defined

  • In this course, we focus on treatments that could be thought of as interventions

What are Causal Effects?

Causal effects

  • Definition: treatment A has a causal effect on the outcome Y, if Y1 differs from Y0

  • Example
    • Y: headache gone one hour from now (yes=1, no=0)
    • A: take ibuprofen (A=1) or not (A=0)

Fundamental problem of causal inference

Fundamental problem of causal inference

  • Fundamental problem of causal inference: we can only observe one potential outcome for each person

  • However, with certain assumptions, we can estimate population level (average) causal effects E(Y1−Y0)

    • Average value of Y if everyone was treated with A=1 minus average value of Y if everyone was treated with A=0
  • Headache example:
    • Hopeless: What would have happened to me had I not taken ibuprofen? (Unit level causal effect)
    • Hopeful: What would the rate of headache remission be if everyone took ibuprofen when they had a headache versus if no one did? (Population level causal effect)

Visualization of population average causal effect

Population average causal effect versus conditioning on treatment/control

E(Y1−Y0)≠E(Y∣A=1)−E(Y∣A=0)

  • In the left hand side, E(Y1) is the mean of Y if the whole population was treated with A=1

  • In the right hand side, E(Y∣A=1) is restricting to the subpopulation of people who actually had A=1

    • This subpopulation may differ from the whole population in important ways
    • For example, people at higher risk for flu are more likely to choose to get a flu shot

  • E(Y∣A=1)−E(Y∣A=0) is not a causal effect, because it is comparing two different populations of people

Visualization of real world

Other causal effects

  • E(Y1/Y0): causal relative risk

  • E(Y1−Y0∣A=1): causal effect of treatment on the treated

  • E(Y1−Y0∣V=v): average causal effect in the subpopulation with covariate V=v

Visualization of causal effect of treatment on the treated

Causal Assumptions

Most common causal assumptions

  • Stable Unit Treatment Value Assumption (SUTVA)

  • Consistency

  • Ignorability

  • Positivity

Stable Unit Treatment Value Assumption (SUTVA)

Stable Unit Treatment Value Assumption (SUTVA)

  • SUTVA involves two assumptions

  1. No interference

    • Units do not interfere with each other
    • Treatment assignement of one unit does not affect that outcome of another unit
    • Spillover or contagion are also terms for interference

  2. One version of treatment

  • SUTVA allows us to write potential outcome for a person in terms of only that person’s treatments

Consistency

Consistency assumption

  • Consistency assumption: the potential outcome under treatment A=a, Ya, is equal to the observed outcome if the actual treatment received is A=a

Y=Ya if A=a, for all a

Ignorability

Ignorability assumption

  • Ignorability assumption: given pre-treatment covariates X, treatment assignment is independent from the potential outcomes Y0,Y1⊥A∣X

  • Among people with the same values of X, we can think of treatment A as being randomly assigned

  • Example: Y0 and Y1 are not independent from A marginally, but within levels of X, treatment might be randomly assigned

    • X: age; can take values ‘younger’ or ‘older’
    • Y: hip fracture
    • Older people are more likely to get treatment A=1
    • Older people are also more likely to have the outcome, regardless of treatment

Positivity

Positivity assumption

  • Positivity assumption: for every set of values of X, treatment assignment was not deterministic P(A=a∣X=x)>0, for all a and x

  • If for some values of X, treatment was deterministic, then we would have no observed values of Y for one of the treatment groups for those values of X

Standardization and Stratification

Observed data and potential outcomes

  • Under all above assumptions, the observed data average outcome E(Y∣A=a,X=x) equals the potential outcomes E(Ya∣X=x)

E(Y∣A=a,X=x)=E(Ya∣A=a,X=x) by consistency=E(Ya∣X=x) by ignorability

  • If we want a marginal causal effect, we can average over X E(Ya)=∑xE(Y∣A=a,X=x)P(X=x)

Standardization

  • Standardization involves stratifying and then averaging

    • First obtain the mean treatment effect within each stratum E(Y∣A=a,X=x)
    • Then pool across stratum, weighing by the probability (size) of each stratum P(X=x)

Standardization example: two diabetes treatments

  • Treatments: saxagliptin (new medicine) vs sitagliptin

  • Outcome: major adverse cardiac event (MACE)

  • Covariate: past use of oral antidiabetic (OAD) drug

  • Challenge

    • Saxa users were more likely to have past use of OAD drug
    • Patients with past use of OAD drugs are at higher risk of MACE

  • Stratify parents in two subpopulations by whether having prior OAD use

    • Within levels of the prior OAD use variable, treatment can be thought of as randomized (ignorability)

Example continued: unstratified

Unstratified raw data
MACE=yes MACE=no Total
Saxa=yes 350 3650 4000
Saxa=no 500 6500 7000
Total 850 10150 11000

P(MACE∣Saxa=yes)=350/4000=0.088P(MACE∣Saxa=no)=500/7000=0.071

Example continued: subpopulation without prior OAD use

Prior OAD use = no
MACE=yes MACE=no Total
Saxa=yes 50 950 1000
Saxa=no 200 3800 4000
Total 250 4750 5000

P(MACE∣Saxa=yes)=50/1000=0.05P(MACE∣Saxa=no)=200/4000=0.05

Example continued: subpopulation with prior OAD use

Prior OAD use = yes
MACE=yes MACE=no Total
Saxa=yes 300 2700 3000
Saxa=no 300 2700 3000
Total 600 5400 6000

P(MACE∣Saxa=yes)=300/3000=0.10P(MACE∣Saxa=no)=300/3000=0.10

Example continued: mean potential outcome for Saxa

E(Ysaxa)= E(Y∣A=saxa,X=prior OAD use yes)P(prior OAD use yes)+E(Y∣A=saxa,X=prior OAD use no)P(prior OAD use no)=(300/3000)(6000/11000)+(50/1000)(5000/11000)=0.077

  • Similarly, E(Ysita)=0.077

  • Hence, the treatment Saxa or not has no causal effects on the MACE outcome

Problems with standardization

  • There will be many X variables needed to achieve ignorability

  • Stratification would lead to many empy cells

  • Alternative to standardization: matching inverse probability of treatment weighting (IPTW), etc

References