 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-221/1/ii/solution

Among observed pretreatment variables, every sufficient set must contain

M_{1}

to block

Z_{1} \leftarrow M_{1} \to C \to Y

and

Z_{3}

to block

Z_{1} \leftarrow S \to Z_{3} \to Y .

Conditioning on

Z_{3}

opens the collider on

Z_{1} \leftarrow S \to Z_{3} \leftarrow M_{3} \to C \to Y,

M_{3}

must also be included. The resulting minimal sufficient adjustment set is

X_{0} = {M_{1}, M_{3}, Z_{3}} .

It blocks every path from

Z_{1}

Y

that remains after removing

A \to Y

, while the open path

Z_{1} \leftarrow S \to Z_{2} \to A

M_{2}

blocks no required relevance path, so

X_{1} = {M_{1}, M_{2}, M_{3}, Z_{3}}

is also sufficient.

There are no others. In particular, adding

Z_{2}

blocks the displayed relevance path. Without

M_{2}

, conditioning on

Z_{2}

also opens

Z_{1} \leftarrow S \to Z_{2} \leftarrow M_{2} \to C \to Y,

which violates independence; adding

M_{2}

closes that path but leaves no open path from

Z_{1}

A

. Hence the complete list is

X_{0}

and

X_{1}

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