We are interested in the causal effect of a treatment (versus no treatment/control) on an outcome.
We used the counterfactual/potential outcomes to define causal effects.
| \(Y(1)\) | \(Y(0)\) | |
|---|---|---|
| John | 0.54 | 0.94 |
| Sally | 0.91 | 0.91 |
| Kate | 0.81 | 0.60 |
| Jason | 0.60 | 0.84 |
From the fundamental problem of causal inference, we generally cannot observe both \(Y(1),Y(0)\).
Also, the counterfactual outcomes differ from the observed outcomes \(Y\) in that the observed outcome is a realization for a particular value of the treatment assignment \(A\).
| \(Y\) | \(A\) | |
|---|---|---|
| John | 0.94 | 0 |
| Sally | 0.91 | 0 |
| Kate | 0.81 | 1 |
| Jason | 0.60 | 1 |
How do we learn about \(Y(1), Y(0)\) from the observable data \(Y,A\)?
First, let’s make the following assumption known as stable unit treatment value assumption (SUTVA) (Rubin (1980)) or causal consistency (page 4 of Hernán and Robins (2020)):
\[Y = AY(1) + (1-A) Y(0).\]
It’s also common to rewrite the above assumption as
\[Y = Y(A), \quad{} \text{ or if } A=a, \text{ then } Y = Y(a).\]
The latter version covers the case when the treatment \(A\) is not binary (e.g., discrete, continuous)
In words, SUTVA states the observed outcome \(Y\) is one realization of the counterfactual outcomes \(Y(a)\) based on the observed value of the treatment \(A\).
More subtly, SUTVA implies two “mini” assumptions.
It’s useful to understand this assumption by studying the case when the assumption is violated.
Let’s go back to the smoking example from the first lecture where we defined the causal effect of daily smoking (i.e., treatment) versus never smoking (i.e., control) on lung function.
Daily smoking may include different type of daily smokers.
We can define counterfactual outcomes for all types of daily smokers:
By assuming SUTVA, we eliminate these variations in the counterfactuals. Formally,
\[Y(1) = Y(2) = \ldots =Y(k).\]
In words, SUTVA implies that the lung function of a daily smoker who smokes at least one cigarette per day (i.e., \(Y(1)\)) is equal to the lung function of the same daily smoker living in the same environment except that he/she smokes at least one pack of cigarettes per day (i.e., \(Y(2)\)).
There are settings where no multiple versions of treatment is plausible. For example,
Broadly speaking, SUTVA forces you to define meaningful \(Y(a)\); see the first lecture.
It is useful to understand this assumption with a counterexample.
Suppose we want to study the causal effect of getting the varicella vaccine (i.e., chickenpox vaccine) on getting the chickenpox. Let’s define the following counterfactual outcomes:
If the chickenpox vaccine is 100% effective for everyone, it’s likely that \(Y(1) = 0\).
Now, suppose John has a sibling Sally and let’s consider John’s counterfactual universe where he is not vaccinated and Sally’s vaccination status varies.
We can redefine the counterfactual outcomes to incorporate Sally’s vaccination status.
SUTVA implies that John’s counterfactual outcome only depends on John’s vaccination status, not Sally’s vaccination status. Formally,
\[Y(0,1) = Y(0,0) = Y(0)\]
There are settings where the no interference assumption is plausible.
We can write a more general version of the no interference assumption. - Formally, let \(i=1,\ldots,n\) index \(n\) study units. - Let \(Y_i(a_i,a_{-i}) \in \mathbb{R}\) denote the counterfactual outcome of unit \(i\) if unit \(i\) gets treatment status \(a_i \in \mathbb{R}\) and unit i’s peers get treatment status \(a_{-i} \in \mathbb{R}^{n-1}\)$. - No interference for all units states that for every \(i=1,\ldots,n\),
\[ Y_i(a_i,a_{-i}) = Y_i(a_i,a_{-i}') = Y(a_i)\]
for all \(a_i \in \mathbb{R}\), \(a_{-i}, a_{-i}' \in \mathbb{R}^{n-1}\).
Once we assume SUTVA (i.e. \(Y= AY(1) + (1-A)Y(0)\)), the other assumptions for causal identification can be motivated by a connection to a missing data problem.
| \(Y(1)\) | \(Y(0)\) | \(Y\) | \(A\) | |
|---|---|---|---|---|
| John | NA | 0.94 | 0.94 | 0 |
| Sally | NA | 0.91 | 0.91 | 0 |
| Kate | 0.81 | NA | 0.81 | 1 |
| Jason | 0.60 | NA | 0.60 | 1 |
Under SUTVA, we only see one of the two counterfactual outcomes based on \(A\).
Suppose we are interested in the causal estimand \(\mathbb{E}[Y(1)]\) (i.e. the mean of the first column).
One approach to study it is to take the average of the “complete cases” (i.e., Kate and Jason’s \(Y(1)\)s).
See here for an introduction to missing data.
Here, we illustrate MCAR through a small simulation study.
Consider a data table where the first column consists of \(Y(1)\) and the second column consists of \(A\), the missingness indicator where \(A=1\) indicates that \(Y(1)\) is not missing and \(A=0\) indicates that \(Y(1)\) is missing. Each row of the table represents an individual.
set.seed(1)
library(knitr)
n = 100; mu = 10
Y1 = rnorm(n,mu);
A = as.numeric(runif(n) < 0.5)
dat = data.frame(Y1 = Y1,A=A);
dat[dat$A==0,"Y1"] = NA
knitr::kable(dat[1:10,],format="pipe",digits=2,align="l",
caption = "First 20 rows of the data",
col.names=c("Y(1)","A"))| Y(1) | A |
|---|---|
| 9.37 | 1 |
| 10.18 | 1 |
| NA | 0 |
| 11.60 | 1 |
| 10.33 | 1 |
| NA | 0 |
| NA | 0 |
| 10.74 | 1 |
| 10.58 | 1 |
| NA | 0 |
MCAR states that there are no patterns in how the NAs appear in the table; the entries of \(Y(1)\) are missing completely by chance. Then, intuitively, taking the mean of \(Y(1)\) among the observed values should be a good estimate of the mean of all \(Y(1)\).
mean(dat[dat$A == 1,"Y1"]) # mean among non-missing values[1] 10.10171mean(Y1) #mean among all $Y_i(1)$ (missing and non-missing)[1] 10.10889The two means should be similar to each other, confimring our intuition.
Now, suppose the missingness is not random. For example, in the simulation below, all values of \(Y(1)\) that are greater than 10 are not missnig and all values of \(Y(1)\) that are less than 10 are missing. In this case, \(A\), the variable that indicates whether \(Y(1)\) is missing or not, depends on the value of \(Y(1)\) and \(A \not\perp Y(1)\).
A = as.numeric(Y1 > 10) #all values of
dat = data.frame(Y1 = Y1,A=A);
dat[dat$A==0,"Y1"] = NA
knitr::kable(dat[1:10,],format="pipe",digits=2,align="l",
caption = "First 20 rows of the data",
col.names=c("Y(1)","A"))| Y(1) | A |
|---|---|
| NA | 0 |
| 10.18 | 1 |
| NA | 0 |
| 11.60 | 1 |
| 10.33 | 1 |
| NA | 0 |
| 10.49 | 1 |
| 10.74 | 1 |
| 10.58 | 1 |
| NA | 0 |
It’s useful to draw the distribution of the observed \(Y(1)\) versus the distribution of the entire (missing and non-missing) \(Y(1)\).
dY1 = density(Y1) #all of Y1
dY1_obs = density(dat$Y1,na.rm=TRUE) #observed Y1
plot(dY1,xlim=c(min(Y1),max(Y1)),ylim=c(0,max(dY1_obs$y)),
main="Estimated densities",xlab="Y(1)")
lines(dY1_obs,col=2)
legend("topright", legend = c("All of Y(1)", "Observed Y(1)"),
lty=2,col=1:2)
From the plot, it’s clear that the mean among the observed \(Y(1)\) will be a poor estimate of the mean of the entire \(Y(1)\) distribution; the mean of the observed \(Y(1)\) will be slightly higher than the mean of the entire \(Y(1)\).
mean(dat[dat$A == 1,"Y1"]) # mean among non-missing values[1] 10.76445mean(Y1) #mean among all $Y_i(1)$ (missing and non-missing)[1] 10.10889Formally, MCAR can be stated as \[A \perp Y(1) \text{ and } 0 < \mathbb{P}(A=1)\]
A natural question to ask is whether you can assess, with the observed data \(A,Y\) that \(A \perp Y(1)\). Without assumptions, this is impossible as \(Y(1)\) and \(Y\) are not linked. Even if they are linked via SUTVA, you cannot check \(A \perp Y(1)\) by simply checking \(A \perp Y\); see the next callout block.
More generally, assumptions involving counterfactuals like \(A \perp Y(1)\) must be verified by the study design or the process in which the data was generated. For example, in addition to obtaining the data \(A,Y\), if someone told us that \(A\) was generated from an independent, random flip of a coin, then we know \(A \perp Y(1)\). To put it differently, it is impossible to check whether \(A\) came from an independent, random flip of a coin based on the numerical values of \(A,Y\) alone as the values themselves do not tell you information about how \(A\) was generated.
Note that \(A \perp Y(1)\) (i.e. missingness indicator \(A\) for the \(Y(1)\) column is completely random) is not equivalent to \(A \perp Y\) (i.e. the missingness indicator \(A\) is not associated with \(Y\)), with or without SUTVA.
Without SUTVA, \(Y\) and \(Y(1)\) can be two completely different variables and thus, the two independence assumptions are generally not equivalent to each other. In other words, \(A \perp Y(1)\) makes an assumption about the counterfactual outcome whereas \(A \perp Y\) makes an assumption about the observed outcome.
With SUTVA, \(Y = AY(1) + (1-A)Y(0)\). But, \(A \perp Y(1)\) still does not necessarily imply that \(Y\), which is a linear combination of \(Y(1)\) and \(Y(0)\) are independent of \(A\). The easiest way to see this is by going back to the data table with \(Y(1), Y(0), Y,A\).
One special case where \(A \perp Y(1)\) implies \(A \perp Y\) is when \(Y(1) = Y(0)\), or that there is no causal effect of treatment for everyone.
Suppose SUTVA and MCAR hold:
Then, we can identify the causal estimand \(\mathbb{E}[Y(1)]\) by writing it as the following function of the observed data \(\mathbb{E}[Y | A=1]\): \[\begin{align*} \mathbb{E}[Y \mid A=1] &= \mathbb{E}[AY(1) + (1-A)Y(0) \mid A=1] && \text{(A1)} \\ &= \mathbb{E}[Y(1) \mid A=1] && \text{Algebra} \\ &= \mathbb{E}[Y(1)] && \text{(A2)} \end{align*}\] (A3) is used to ensure that \(\mathbb{E}[Y | A=1]\) is a well-defined quantity.
Technically speaking, to establish \(\mathbb{E}[Y | A=1] = \mathbb{E}[Y(1)]\), we only need mean independence:
\[\mathbb{E}[Y(1) | A=1] = \mathbb{E}[Y(1)] \]
Note that \(A \perp Y(1)\) implies mean independence above, but the converse is not true.
In a similar vein, to identify the ATE \(\mathbb{E}[Y(1)-Y(0)]\), a natural approach would be to use \(\mathbb{E}[Y | A=1] - \mathbb{E}[Y | A=0]\).
This approach would be valid under the following variation of the MCAR assumption: \[A \perp Y(0),Y(1), \quad{} 0 < \mathbb{P}(A=1) < 1\]
Suppose SUTVA and MCAR hold:
Then, we can identify the ATE from the observed data via: \[\begin{align*} &\mathbb{E}[Y|A=1] - \mathbb{E}[Y | A=0] \\ =& \mathbb{E}[AY(1) + (1-A)Y(0) | A=1] \\ & \quad{} - \mathbb{E}[AY(1) + (1-A)Y(0) | A=0] && \text{(A1)} \\ =& \mathbb{E}[Y(1)|A=1] - \mathbb{E}[Y(0) | A=0] && \text{Algebra} \\ =& \mathbb{E}[Y(1)] - \mathbb{E}[Y(0)] && \text{(A2)} \end{align*}\]
(A3) ensures that the conditioning events in \(\mathbb{E}[\cdot |A=0]\) and \(\mathbb{E}[\cdot |A=1]\) are well-defined.
\[\underbrace{\mathbb{E}[Y|A=1] - \mathbb{E}[Y | A=0]}_{\text{Measure of Association}} = \underbrace{\mathbb{E}[Y(1)] - \mathbb{E}[Y(0)]}_{\text{Measure of Causation}}\]
This equality implies that under (A1,SUTVA), (A2, Ignorability), and (A3, Positivity), a measure of association between \(A\) and \(Y\) based on difference in means (i.e., the left-hand-side ) is equal to a measure of causation based on difference in counterfactual means (i.e., the right-hand side).
We can take this result a bit further by considering the setting where there is no association between \(A\) and \(Y\).
More broadly, as illustrated by both examples, association can imply certain causal claims if additional assumptions hold (e.g., (A1,SUTVA), (A2,Ignorability), and (A3, Positivity)).
Consider an ideal, completely randomized trial/experiment (RCT) to assess the causal effect of a new drug (versus a control/placebo) on an outcome of interest.
RCTs have been referred to as the gold standard to study causal effects of a treatment on an outcome of interest. But why?
This was the “big” idea from Fisher in 1935 where he used randomization as the “reasoned basis’’ for causal inference. Paul Rosenbaum explains this more beautifully than I do in Chapter 2.3 of Rosenbaum (2020).
Consider the following data table.
| \(Y(1)\) | \(Y(0)\) | \(Y\) | \(A\) | X (Measured; age) | U (Unmeasured; environment) | |
|---|---|---|---|---|---|---|
| John | NA | 0.94 | 0.94 | 0 | 23 | \(U_{\rm John}\) |
| Sally | NA | 0.91 | 0.91 | 0 | 27 | \(U_{\rm Sally}\) |
| Kate | 0.81 | NA | 0.81 | 1 | 32 | \(U_{\rm Kate}\) |
| Jason | 0.60 | NA | 0.60 | 1 | 30 | \(U_{\rm Jason}\) |
If the treatment \(A\) is completely randomized (as in an RCT), we would also have \(A \perp X, U\). More generally, we have
\[\text{(A2, Ignorability) } A \perp Y(1), Y(0), X, U\].
Also, because there is at least one control unit and treated unit in an RCT, we have
\[\text{(A3, Positivity) } 0 < \mathbb{P}(A=1) < 1\].
Even with the change in (A2, Ignorability) from before, the proof to identify the ATE in an RCT remains the same as before, i.e., \(\mathbb{E}[Y(1)] - \mathbb{E}[Y(0)] = \mathbb{E}[Y|A=1] - \mathbb{E}[Y | A=0]\). This is because only need \(A \perp Y(1), Y(0)\) for identification.
An important implication from randomization of treatment assignment is covariate balance.
Roughly speaking, we say that covariate \(X\) (measured or unmeasured) is ``balanced’’ between treated and control groups if
\[\mathbb{P}(X |A=1) = \mathbb{P}(X | A=0)\]
From the RCT motivation, it’s very obvious that covariate balance holds for both \(X\) and \(U\), i.e.
\[\mathbb{P}(X,U |A=1) = \mathbb{P}(X,U | A=0)\].
In general, covariates should be balanced between treated and control groups to make causal claims about the relationship between the outcome and the treatment.
In Chapter 9.1 of Rosenbaum (2020), Rosenbaum recommends using the pooled variance when computing the difference in means of a covariate between the treated group and the control group. Specifically, let \({\rm SD}X_{A=1}\) be the standard deviation of the covariate in the treated group and \({\rm SD}X_{A=0}\) be the standard deviation of the covariate in the control group. Then, Rosenbaum suggests the statistic
\[ \text{Standardized difference in means} = \frac{\bar{X}_{A=1}-\bar{X}_{A=0}}{\sqrt{ ({\rm SD}(X)_{A=1}^2 + {\rm SD}(X)_{A=0}^2)/2}} \]
We briefly mentioned that covariates \(X\) must precede treatment assignment, i.e.
If they are post-treatment covariates, then the treatment can have a causal effect on both the outcome \(Y\) and the covariates \(X\).
In this case, it’s not unclear whether \(Y\) has a causal effect because of a causal effect in \(X\). Studying this type of question is called causal mediation analysis.
In general, we don’t want to condition on post-treatment covariates \(X\) when the goal is to estimate the average treatment effect of \(A\) on \(Y\).