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

For the pdf slides, click here

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 \(Y^a\) is the outcome we would see if treatment was set to \(A=a\)

  • Each person has potential outcome \(Y^0, Y^1\)

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 \(Y^0\)

    • If my treatment is \(A=0\), then my counterfactual outcomes is \(Y^1\)

  • Connection between potential and conterfactuals outcomes

    • Before the treatment decision is made, any outcome is a potential outcome, \(Y^0\) and \(Y^1\)
    • After the study, there is an observed outcome \(Y^A\), and counterfactual outcome \(Y^{1-A}\)

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 \(Y^1\) differs from \(Y^0\)

  • 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 \(E(Y^1 - Y^0)\)

    • 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

\[ E(Y^1 - Y^0) \neq E(Y\mid A=1) - E(Y\mid A=0) \]

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

  • In the right hand side, \(E(Y\mid A=1)\) 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

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

Visualization of real world

Other causal effects

  • \(E(Y^1 / Y^0)\): causal relative risk

  • \(E(Y^1 - Y^0 \mid A=1)\): causal effect of treatment on the treated

  • \(E(Y^1 - Y^0 \mid V=v)\): 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 \(A=a\), \(Y^a\), is equal to the observed outcome if the actual treatment received is \(A=a\)

\[ Y=Y^a \text{ if } A=a, \text{ for all } a \]

Ignorability

Ignorability assumption

  • Ignorability assumption: given pre-treatment covariates \(X\), treatment assignment is independent from the potential outcomes \[ Y^0, Y^1 \perp A \mid X \]

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

  • Example: \(Y^0\) and \(Y^1\) 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 \[ P(A=a \mid X=x) > 0, \text{ for all } a \text{ and } x \]

  • 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 \(E(Y\mid A=a, X=x)\) equals the potential outcomes \(E(Y^a \mid X=x)\)

\[\begin{align*} E(Y\mid A=a, X=x) &= E(Y^a\mid A=a, X=x) \text{ by consistency}\\ &= E(Y^a\mid X=x) \text{ by ignorability}\\ \end{align*}\]

  • If we want a marginal causal effect, we can average over \(X\) \[ E(Y^a) = \sum_x E(Y \mid A=a, X=x) P(X=x) \]

Standardization

  • Standardization involves stratifying and then averaging

    • First obtain the mean treatment effect within each stratum \(E(Y \mid A=a, X=x)\)
    • 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

\[\begin{align*} &P(\text{MACE} \mid \text{Saxa}=\text{yes}) = 350/4000 = 0.088\\ &P(\text{MACE} \mid \text{Saxa}=\text{no}) = 500/7000 = 0.071 \end{align*}\]

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

\[\begin{align*} &P(\text{MACE} \mid \text{Saxa}=\text{yes}) = 50/1000 = 0.05\\ &P(\text{MACE} \mid \text{Saxa}=\text{no}) = 200/4000 = 0.05 \end{align*}\]

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

\[\begin{align*} &P(\text{MACE} \mid \text{Saxa}=\text{yes}) = 300/3000 = 0.10\\ &P(\text{MACE} \mid \text{Saxa}=\text{no}) = 300/3000 = 0.10 \end{align*}\]

Example continued: mean potential outcome for Saxa

\[\begin{align*} & E(Y^{\text{saxa}})\\ = ~& E(Y\mid A=\text{saxa}, X = \text{prior OAD use yes}) P(\text{prior OAD use yes})\\ & + E(Y\mid A=\text{saxa}, X = \text{prior OAD use no}) P(\text{prior OAD use no})\\ = & (300/3000) (6000/11000) + (50/1000) (5000/11000) \\ = & 0.077 \end{align*}\]

  • Similarly, \(E(Y^{\text{sita}}) = 0.077\)

  • 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