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UNIVERSITY OF TORONTO SCARBOROUGH Department of Computer and Mathematical Sciences STAC50H3 Final Exam December 15, 2021, 8:45am - 11am Last name: LinboLinboLinboLinboLinboLinbo First name: LinboLinboLinboLinboLinboLinbo Student number: LinboLinboLinboLinboLinboLinbo Instructions • ATTEMPT ALL QUESTIONS, including answering the academic integrity questions as instructed. • You may not use any aid, except for a non-programmable calculator. You may NOT use R . • Please note that academic integrity is fundamental to learning and scholarship. You may NOT discuss questions with other students. • Please round the final results to the nearest second decimal. • You may use notations in the lecture notes without defining them. Good Luck!!! Question: 1 2 3 4 5 6 7 8 Total Points: 15 10 16 33 10 16 100 Score 1

1. Academic integrity is a fundamental value of learning and scholarship at the UofT. Participating hon- estly, respectfully, responsibly, and fairly in this academic community ensures that your UofT degree is valued and respected as a true signifier of your individual academic achievement. The University of Toronto’s Code of Behaviour on Academic Matters outlines the behaviours that constitute academic misconduct, the processes for addressing academic offences, and the penalties that may be imposed. You are expected to be familiar with the contents of this document. Potential offences include, but are not limited to: • Working together to answer questions. • Looking at someone else’s answers. • Letting someone else look at your answers. • Sharing or posting the assignment questions. • Discussing answers or the assignment questions with anyone else in or outside the course. • Misrepresenting your identity or having someone else complete your assignment. Prior to beginning this assignment, you must attest that you will follow the Code of Behaviour on Aca- demic Matters and will not commit academic misconduct in the completion of this online assignment. Affirm your agreement to this by completing the following Statement: Please write down the following as the answer to the first question I, [name] (type your full name here), [stnum] (type your student number here), agree to fully abide to the Code of Behaviour on Academic Matters. I will not commit academic misconduct, and am aware of the penalties that may be imposed if I commit an academic offence. 2

2. (15pts) Which of the following is a DAG? Find all that apply. For those graphs that are not a DAG, explain why. Egg Chick Hen (a) Chicken-egg graph Isabella Dustie Tony Andrea Lisa (b) High school dating (c) Smoking and cardiac arrest 3

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3. (10pts) A baseball batter Tim has a better batting average than his teammate Frank. However, someone notices that Frank has a better batting average than Tim against both right-handed and left-handed pitchers. How can this happen? (Limit your response to no more than two sentences.) Come up with a table with hypothetical numbers with your explanation. 4

4. (16pts) It has often been found out that (i) mother’s smoking during pregnancy is associated with higher infant mortality, but (ii) if the baby happens to be born underweight, then the babies of smoking mothers have a better survival rate than those of non-smokers. It is also the case that (iii) the babies of smokers are lighter on average than the babies of nonsmokers, and (iv) this would lead to poorer survival. (a) (8pts) Suppose that Dr. F from a tobacco company wants to use this study to argue that (v) mother’s smoking has a direct protective effect on baby’s survival. Draw a DAG that represents the causal relations between smoking during pregnancy ( A ), birth weight ( W ), and infant sur- vival ( S ). Use your DAG to support Dr. F’s claim. Your explanation needs to be compatible with (i) - (v). Limit your explanation into four lines. (b) (8pts) Scientists nowadays have shown that Dr. F’s claim is not true: (vi) mother’s smoking is in fact detrimental to baby’s survival. Add a four variable birth defect ( D ) onto your graph and use the new graph to explain the birth weight paradox. Your explanation needs to be compatible with (i) - (iv) and (vi). Limit your explanation into four lines. 5

5. (33pts) For the DAG in the Figure 1 X W 1 Z 1 L 1 U 1 L 2 W 2 Z 2 Y U 2 L 3 W 3 Z 3 Figure 1: A causal graph. (a) (3pts) Find the parents of Y (b) (3pts) Find the ancestors of Z 2 . (Recall that, by convention, every node is its own ancestor.) (c) (3pts) Find the descendants of X . (Recall that, by convention, every node is its own descendant.) (d) (3pts) Consider the path: X → W 1 → Z 1 ← U 1 → W 2 ← L 2 → Y . Which nodes are colliders and which are non-colliders on this path? (e) (3pts) Find a path between X and Y on which W 2 is a collider. (f) (3pts) Find a path between X and Y on which W 2 is a non-collider. (g) (3pts) Is X d-separated from Y given { Z 1 , Z 2 , Z 3 } ? Explain. (h) (3pts) Is X d-separated from Y given { W 1 , W 2 , W 3 } ? Explain. (i) (3pts) Is X d-separated from Y given { L 1 , Z 1 , L 2 , Z 2 , L 3 , Z 3 } ? Explain. (j) (3pts) Identify all of the nodes which are d-connected to Y given ∅ (k) (3pts) Is there a subset of { W 1 , W 2 , W 3 , Z 1 , Z 2 , Z 3 } which d-separates X and Y . Explain why or why not (in four or fewer sentences). 6

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6. (10pts) For each of questions below, use the template to draw a DAG that contains four variables A, B, C , and D . Each DAG should imply the (conditional) independencies listed in the corresponding question, and only these independencies. (a) (5pts) A | = D, A | = B, B | = C, B | = D A C B D (b) (5pts) A | = C and B | = D | A, C A C B D 7

7. (16pts) Refer to the following DAG: A 0 L 1 A 1 Y U Here A 0 refers to HIV treatment at time 0, L 1 refers to CD4 count at time 1, A 2 refers to HIV treatment at time 1, and Y refers to CD4 count at time 2. U refers to unmeasured common causes of L 1 and Y (a) (4pt) Can we use a regression model to estimate the causal effect of A 0 on Y ? If so, what regression model would you use? What additional assumptions would you need to make to make sure that your estimate represents the causal effect of interest? (b) (4pt) Can we use a regression model to estimate the causal effect of A 1 on Y ? If so, what regression model would you use? If not, explain. (c) (4pt) Can we use a regression model to estimate the causal effect of A 0 and A 1 on Y simultane- ously? If so, what regression model would you use? If not, explain. (d) (4pt) If U were observed, can we use a regression model to estimate the causal effect of A 0 and A 1 on Y simultaneously? If so, what regression model would you use? If not, explain. 8

8. All suspected cases of academic dishonesty will be investigated following the procedures outlined in the Code of Behaviour on Academic Matters. Please sign the Statement below by filling in your name and student number before you submit your exam answers. Please write down the following as the answer to this last question: I am attesting to the fact that I, [name] (type your full name here), [stnum] (type your student number here), have abided fully to the Code of Behaviour on Academic Matters. I have not committed academic misconduct, and am aware of the penalties that may be imposed if I have committed an academic offence. 9

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