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Introductory slide deck for my course on Causal Inference
Definition of Causal Effect
The role of counterfactuals and estimation of casual effects
Patrick F. McArdle, Ph.D.
Two aspects of Causality
What is causality? Are there causal laws?
How are causes identified? How can we test a causal hypothesis?
Two different causal hypotheses
Smoking causes lung cancer
Smoking caused my lung cancer
The distinction between General and Specific may not be precise enough.
There are two different causal queries, even when discussing a specific individual.
Effects of Causes
Causes of Effects
I have a headache.
Will it help to take aspirin?
My headache is gone.
Is it because I took aspirin?
Note both questions are “specific” in nature, that is pertaining to a single observational unit.
From Dawid: J Amer Stat Assoc Vol. 95, No. 450, Jun., 2000
Identification of causes of effects is often a mental exercise.
In the law, the cause of an effect is often of interest.
The harm has already occurred.
Was it because of the actions of some party?
The But-For Legal Test:
The harm would not have occurred but for the actions of the defendant.
Y is the outcome
X is the exposure
YX=x(u) is the value that Y would obtain in unit u had X been x.
YX=1(u) is the outcome for unit u when exposure value X=1
YX=0(u) is the outcome for unit u when exposure value X=0
If YX=1(u) ≠ YX=0(u) then X can be said to cause Y for unit, u .
But we can never observe both YX=1(u) and YX=0(u) for a single unit.
Sharp Causal Null Hypotheses
YX=1(u) = YX=0(u) for all units in the population.
Individual causal effects are defined as contrasts between counterfactual states.
Thus some claim we can never truly identify individual causality.
Yet we do make such inferences in everyday life.
If I am hit by a bus and break a leg, can we not know that the bus caused my broken leg?
“It is a paradigm of philosophical analysis, lucid, concise, rigorous, and informed throughout by a luminous clarity of vision. The book concerns itself with a single problem of fundamental philosophical interest and importance: what do counterfactual conditionals mean, and when are they true? And such is the author's consummate brilliance that he manages to solve this problem, in its essentials, in less than a hundred and fifty pages.”
S. Guha, Amazon.com Reviewer
“I wrote my doctoral dissertation on Lewis's theory of possible worlds, part of which is contained in Counterfactuals. I regret every second of it.”
A. Apter, Amazon.com reviewer
Not all counterfactual statements have corresponding causal answers
Is the following statement true or false?
If the Chesapeake Bay were in California, it would flow into the Pacific Ocean
One man’s cause is another man’s cure
I propose that we subscribe to satellite TV
My wife claims that will cause our expenses to increase.
I claim that will cause our expenses to decrease.
What happens without Satellite TV?
I listen to the game on the radio
I go to a bar and run up a big tab
Expenses without Satellite TV
Expenses with Satellite TV
Lewis’ Closest World
We will only observe one possible world, the world we actually live in when we either get satellite TV or not.
There are infinite other possible worlds that could exist.
We get satellite TV and the company forgets to bill us.
We don’t get satellite TV and the players go on strike and cancel the season.
Lewis suggests that we align these worlds on some sort of continuum and make causal inferences based upon comparing our real world against the closest of the possible worlds.
Epidemiology is a population science and is therefore concerned with population causal effects (effects of causes).
If we can not identify individual causal effects, can we identify population causal effects?
Population Casual Effect Notation
is the proportion of subjects that would develop outcome Y if all are exposed to X=1.
Also commonly referred to as the counterfacutal risk.
There is a population causal effect if
Pr[YX=1] ≠ Pr[YX=0]
Average Causal Effect
What is the “true” causal effect?
Pr[YX=1] - Pr[YX=0]
Pr[YX=1] / Pr[YX=0]
Pr[YX=1] / (1- Pr[YX=1] )
Pr[YX=0] / (1- Pr[YX=0])
1. How is it measured?
The causal effect of an exposure can be different in populations with varying prevalence of component causes.
2. In what population?
3. At what time?
E.g.: The causal effect of some occupation hazard could be quite different before or after labor safety laws.
What role does statistics play?
Role 1: The fundamental problem of causal inference
We need to estimate the missing counterfactual.
Note, this is the fundamental problem because it exists even if we can observe everyone on the population.
Role 2: We can not observe everyone in the population
Even in the ideal situation (perfect exchangeability) where Pr[YX=1] = Pr[Y|X=1] we usually can only estimate Pr[Y|X=1] from the individuals in our sample, not the entire population.
The group of people about which our scientific question is relevant.
Assume we believe that individuals with a genetic variant metabolize drugs differently and we believe that patient genotype should be taken into consideration when prescribing. We can perform a randomized clinical trial with two arms:
Standard of Care vs Genotype Directed Prescribing
We can ask the question: Does genotype directed prescribing have an effect on outcome.
The true causal effect of genotype directed prescribing will depend on the prevalence of the genetic mutation in the population.
A causal contrast requires two parameters, at least one of which but often both, are unobserved and must be estimated.
These are some causal contrasts:
risk difference, risk ratio, odds ratio
These are not causal contrasts:
p-values, correlation coefficients, chi-square statistics
The goal of causal inference: make “good” estimates of the causal contrast for our inferential target.
Questions posed by Dawid
Are counterfactuals to be regarded as genuine features of the external world, or are they purely theoretical terms?
Can any inference base on counterfactuals be allowed, or should they be restricted to those that could in principle be formulated without mention of counterfactuals?
Do counterfactual terms in a model have a clear relationship with meaningful aspects of the problem addressed? Can counterfactual constructions and arguments help to clarify understanding?
Can use of counterfactuals streamline thinking and assist analyses, or do they promote misleading lines of argument and false conclusions?
CIWC Dawid 2000 J Amer Stat Assoc
Go thru course example before starting slides.
If we can not identify individual level causality, why are so many people in prison?
Example of why choosing a referent is important. Ultimately the very identification of a cause is based upon the counterfactual comparison.
Maybe go with the epi joke: How’s the wife? Compared to what?
is the proportion of sub