1.2.1 Estimands: ATE, ATT, ATC, S-, P-

• ATE is Average Treatment Effect (all units)
• ATT is Average Treatment Effect on the Treated
• ATC is Average Treatment Effect on the Controls

1.2.2 Estimands: ATE, ATT, ATC, S-, P-

Say that units are randomly assigned to treatment in different strata (maybe just one); with fixed, though possibly different, shares assigned in each stratum. Then the key estimands and estimators are:

where \(x\) indexes strata, \(p_x\) is the share of units in each stratum that is treated, and \(w_x\) is the size of a stratum.

1.2.4 Broader classes of estimands: LATE/CATE

The CATEs are conditional average treatment effects, for example the effect for men or for women. These are straightfoward.

However we might also imagine conditioning on unobservable or counterfactual features.

• The LATE (or CACE: “complier average causal effect”) asks about the effect of a treatment (\(X\)) on an outcome (\(Y\)) for people that are responsive to an encouragement (\(Z\))

\[LATE = \frac{1}{|C|}\sum_{j\in C}(Y_j(X=1) - Y_j(X=0))\] \[C:=\{j:X_j(Z=1) > X_j(Z=0) \}\]

We will return to these in the study of instrumental variables.

1.2.7 Distribution estimands

Many inquiries are averages of individual effects, even if the groups are not known,

But they do not have to be:

• Inquiries might relate to distributional quantities such as:

  • The effect of treatment on the variance in outcomes: \(var(Y(1)) - var(Y(0))\)
  • The variance of treatment effects: \(var(Y(1) - Y(0))\)
  • Other inequality measures (e.g. Ginis; (Imbens and Rubin (2015) 20.3.2))

You might even be interested in \(\min(Y_i(1) - Y_i(0))\).

1.2.8 Spillover estimands

There are lots of interesting “spillover” estimands.

Imagine there are three individuals and each person’s outcomes depends on the assignments of all others. For instance \(Y_1(Z_1, Z_2, Z_3\), or more generally, \(Y_i(Z_i, Z_{i+1 (\text{mod }3)}, Z_{i+2 (\text{mod }3)})\).

Then three estimands might be:

• \(\frac13\left(\sum_{i}{Y_i(1,0,0) - Y_i(0,0,0)}\right)\)
• \(\frac13\left(\sum_{i}{Y_i(1,1,1) - Y_i(0,0,0)}\right)\)
• \(\frac13\left(\sum_{i}{Y_i(0,1,1) - Y_i(0,0,0)}\right)\)

Interpret these. What others might be of interest?

1.2.9 Differences in CATEs and interaction estimands

A difference in CATEs is a well defined estimand that might involve interventions on one node only:

• \(\mathbb{E}_{\{W=1\}}[Y(X=1) - Y(X=0)] - \mathbb{E}_{\{W=0\}}[Y(X=1) - Y(X=0)]\)

It captures differences in effects.

An interaction is an effect on an effect:

• \(\mathbb{E}[Y(X=1, W=1) - Y(X=0, W=1)] - \mathbb{E}[Y(X=1, W=0) - Y(X=0, W=0)]\)

Note in the latter the expectation is taken over the whole population.

1.2.10 Mediation estimands and complex counterfactuals

Say \(X\) can affect \(Y\) directly, or indirectly through \(M\). then we can write potential outcomes as:

• \(Y(X=x, M=m)\)
• \(M(X=x)\)

We can then imagine inquiries of the form:

• \(Y(X=1, M=M(X=1)) - Y(X=0, M=M(X=0))\)
• \(Y(X=1, M=1) - Y(X=0, M=1)\)
• \(Y(X=1, M=M(X=1)) - Y(X=1, M=M(X=0))\)

Interpret these. What others might be of interest?

1.4.2 Identification (Definition)

• Identifiability Let \(Q(M)\) be a query defined over a class of models \(\mathcal M\), then \(Q\) is identifiable if \(P(M_1) = P(M_2) \rightarrow Q(M_1) = Q(M_1)\).

• Identifiability with constrained data Let \(Q(M)\) be a query defined over a class of models \(\mathcal M\), then \(Q\) is identifiable from features \(F(M)\) if \(F(M_1) = F(M_2) \rightarrow Q(M_1) = Q(M_1)\).

Based on Defn 3.2.3 in Pearl.

• Essentially: Each underlying value produces a unique data distribution. When you see that distribution you recover the parameter.

1.4.3 Identification (Example without identification)

Informally a quantity is “identified” if it can be “recovered” once you have enough data.

• Say for example average wage is \(x^m\) for men and \(x^w\) for women (in some very large population).
• If I gather lots and lots of data on the wages of (male and female) couples, e.g. \(x^c_i = x^m_i + x^w_i\) then, although this will be informative, it will never be sufficient to recover \(x^m\) for men and \(x^w\).
• I can recover \(x^c\), but there are too many combinations of possible values of \(x^m\) and \(x^w\) consistent with the observed data.

1.4.4 Identification : Goal

Our goal in causal inference is to estimate quantities such as:

\[\Pr(Y|\hat{x})\]

where \(\hat{x}\) is interpreted as \(X\) set to \(x\) by “external” control. Equivalently: \(do(X=x)\) or sometimes \(X \leftarrow x\).

• If this quantity is identifiable then we can recover it with infinite data.

• If it is not identifiable, then, even in the best case, we are not guaranteed to get the right answer.

Are there general rules for determining whether this quantity can be identified? Yes.

1.5.1 When to condition? What to condition on?

The key idea is that you want to find a set of variables such that when you condition on these you get what you would get if you used a do operation.

Intuition:

• You could imagine creating a “mutilated” graph by removing all the arrows leading out of X
• Then select a set of variables, \(Z\), such that \(X\) and \(Y\) are d-separated by \(Z\) on the the mutilated graph
• When you condition on these you are making sure that any covariation between \(X\) and \(Y\) is covariation that is due to the effects of \(X\)

1.5.7 Backdoor Criterion: (Pearl 1995)

The backdoor criterion is satisfied by \(Z\) (relative to \(X\), \(Y\)) if:

1. No node in \(Z\) is a descendant of \(X\)
2. \(Z\) blocks every backdoor path from \(X\) to \(Y\) (i.e. every path that contains an arrow into \(X\))

In that case you can identify the effect of \(X\) on \(Y\) by conditioning on \(Z\):

\[P(Y=y | \hat{x}) = \sum_z P(Y=y| X = x, Z=z)P(z)\] (This is eqn 3.19 in Pearl (2000))

1.5.9 Backdoor Proof

Following Pearl (2009), Chapter 11. Let \(T\) denote the set of parents of \(X\): \(T := pa(X)\), with (possibly vector valued) realizations \(t\). These might not all be observed.

If the backdoor criterion is satisfied, we have:

1. \(Y\) is independent of \(T\), given \(X\) and observed data, \(Z\) (since \(Z\) blocks backdoor paths)
2. \(X\) is independent of \(Z\) given \(T\). (Since \(Z\) includes only nondescendents)

• The intervention level relates to the observational level as follows: \[p(y|\hat{x}) = \sum_{t\in T} p(t)p(y|x, t)\]
• Think of this as fully accounting for the (possibly unobserved) causes of \(X\), \(T\)

1.5.10 Backdoor Proof

• But of course we do not observe \(T\), rather we observe \(Z\). So we now need to write everything in terms of \(Z\) rather than \(T\).

We bring \(Z\) into the picture by writing: \[p(y|\hat{x}) = \sum_{t\in T} p(t) \sum_z p(y|x, t, z)p(z|x, t)\]

• Then using the two conditions above:

  1. replace \(p(y|x, t, z)\) with \(p(y | x, z)\)
  2. replace \(p(z|x, t)\) with \(p(z|t)\)

This gives: \[p(y|\hat x) = \sum_{t \in T} p(t) \sum_z p(y|x, z)p(z|t) \]

1.5.11 Backdoor Proof

So, cleaning up, we can get rid of \(T\):

\[p(y|\hat{x}) = \sum_z p(y|x, z)\sum_{t\in T} p(z|t)p(t) = \sum_z p(y| x, z)p(z)\]

1.5.12 Backdoor proof figure

For intuition:

We would be happy if we could condition on the parent \(T\), but \(T\) is not observed. However we can use \(Z\) instead making use of the fact that:

1. \(p(y|x, t, z) = p(y | x, z)\) (since \(Z\) blocks)
2. \(p(z|x, t) = p(z|t)\) (since \(Z\) is upstream and blocked by parents, \(T\))

1.5.13 Adjustment criterion

See Shpitser, VanderWeele, and Robins (2012)

The adjustment criterion is satisfied by \(Z\) (relative to \(X\), \(Y\)) if:

1. no element of \(Z\) is a descendant in the mutilated graph of any variable \(W\not\in X\) which lies on a proper causal path from \(X\) to \(Y\)
2. \(Z\) blocks all noncausal paths from \(X\) to \(Y\)

Note:

• mutilated graph: remove arrows pointing into \(X\)
• proper pathway: A proper causal pathway from \(X\) to \(Y\) only intersects \(X\) at the endpoint

1.5.15 Frontdoor criterion (Pearl)

Consider this DAG:

• The relationship between \(X\) and \(Y\) is confounded by \(U\).
• However the \(X\rightarrow Y\) effect is the product of the \(X\rightarrow M\) effect and the \(M\rightarrow Y\) effect

Why?

1.5.16 Identification through the front door

If:

• \(M\) (possibly a set) blocks all directed paths from \(X\) to \(Y\)
• there is no backdoor path \(X\) to \(M\)
• \(X\) blocks all backdoor paths from \(M\) to \(Y\) and
• all (\(m,z\)) combinations arise with positive probability

Then \(\Pr(y| \hat x)\) is identifiable and given by:

\[\Pr(y| \hat x) = \sum_m\Pr(m|x)\sum_{x'}\left(\Pr(y|m,x')\Pr(x')\right)\]

1.5.17 Frontdoor criterion (Proof)

We want to get \(\Pr(y | \hat x)\)

From the graph the joint distribution of variables is:

\[\Pr(x,m,y,u) = \Pr(u)\Pr(x|u)\Pr(m|x)\Pr(y|m,u)\] If we intervened on \(X\) we would have (\(\Pr(X = x |u)=1\)):

\[\Pr(m,y,u | \hat x) = \Pr(u)\Pr(m|x)\Pr(y|m,u)\] If we sum up over \(u\) and \(m\) we get:

\[\Pr(m,y| \hat x) = \Pr(m|x)\sum_u\left(\Pr(y|m,u)\Pr(u)\right)\] \[\Pr(y| \hat x) = \sum_m\Pr(m|x)\sum_u\left(\Pr(y|m,u)\Pr(u)\right)\]

The first part is fine; the second part however involves \(u\) which is unobserved. So we need to get the \(u\) out of \(\sum_u\left(\Pr(y|m,u)\Pr(u)\right)\).

1.5.18 Frontdoor criterion

Now, from the graph:

1. \(M\) is d-separated from \(U\) by \(X\):

\[\Pr(u|m, x) = \Pr(u|x)\] 2. \(X\) is d-separated from \(Y\) by \(M\), \(U\)

\[\Pr(y|x, m, u) = \Pr(y|m,u)\] That’s enough to get \(u\) out of \(\sum_u\left(\Pr(y|m,u)\Pr(u)\right)\)

1.5.19 Frontdoor criterion

\[\sum_u\left(\Pr(y|m,u)\Pr(u)\right) = \sum_x\sum_u\left(\Pr(y|m,u)\Pr(u|x)\Pr(x)\right)\]

Using the 2 equalities we got from the graph:

\[\sum_u\left(\Pr(y|m,u)\Pr(u)\right) = \sum_x\sum_u\left(\Pr(y|x,m,u)\Pr(u|x,m)\Pr(x)\right)\]

So:

\[\sum_u\left(\Pr(y|m,u)\Pr(u)\right) = \sum_x\left(\Pr(y|m,x)\Pr(x)\right)\]

Intuitively: \(X\) blocks the back door between \(Z\) and \(Y\) just as well as \(U\) does

1.5.20 Frontdoor criterion

Substituting we are left with:

\[\Pr(y| \hat x) = \sum_m\Pr(m|x)\sum_{x'}\left(\Pr(y|m,x')\Pr(x')\right)\]

(The \('\) is to distinguish the \(x\) in the summation from the value of \(x\) of interest)

It’s interesting that \(x\) remains in the right hand side in the calculation of the \(m \rightarrow y\) effect, but this is because \(x\) blocks a backdoor from \(m\) to \(y\)

1.5.21 Front foor

Bringing all this together into a claim we have:

If:

• \(M\) (possibly a set) blocks all directed paths from \(X\) to \(Y\)
• there is no backdoor path \(X\) to \(M\)
• \(X\) blocks all backdoor paths from \(M\) to \(Y\) and
• all (\(m,z\)) combinations arise with positive probability

Then \(\Pr(y| \hat x)\) is identifiable and given by:

\[\Pr(y| \hat x) = \sum_m\Pr(m|x)\sum_{x'}\left(\Pr(y|m,x')\Pr(x')\right)\]

1.6.6 Illustration of identification failure from conditioning on a collider

U1 <- rnorm(10000);  U2 <- rnorm(10000)
Z  <- U1+U2
X  <- U2 + rnorm(10000)/2
Y  <- U1*2 + X

lm_robust(Y ~ X) |> tidy() |> kable(digits = 2)

lm_robust(Y ~ X + Z) |> tidy() |> kable(digits = 2)

1.6.9 Let’s look at that in dagitty

g <- dagitty("dag{U1 -> Z  ; U1 -> y ; 
             U2 -> Z ; U2 -> x  -> y; 
             Z -> x}")
adjustmentSets(g, exposure = "x", outcome = "y")

{ U1 }
{ U2, Z }

which means you have to adjust on an unobservable. Here we double check that including or not including “Z” is enough:

isAdjustmentSet(g, "Z", exposure = "x", outcome = "y")

[1] FALSE

isAdjustmentSet(g, NULL, exposure = "x", outcome = "y")

[1] FALSE