Defining Causality

STAT 20: Introduction to Probability and Statistics

Agenda

• Concept Questions
• Lab 6.1

Concept Questions

Suppose that a prisoner is about to be executed by a firing squad. A certain chain of events must occur for this to happen. First, the judge orders the execution. The order goes to a captain, who signals the two soldiers of the firing squad (soldier 1 and soldier 2) to fire. They are obedient and expert marksmen, so they only fire on command, and if either one of them shoots, the prisoner dies.

Using the conditional counterfactual definition, who caused the death of the prisoner?

A. The judge
B. The captain
C. Soldier 1
D. Soldier 2

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The goal of this example is to reinforce the definition of cause and effect by applying it to a fairly complicated causal graph. The correct answers are A and B, because had the judge not ordered, or had the captain not signaled, the prisoner would not have died. Each of the soldiers individually fail this definition because if one of them doesn’t fire (and they’re acting independently), the prisoner will still die.

You can debrief this one by drawing a causal diagram with judge -> captain -> -> two arrows to separate solider 1 and soldier 2 -> -> two arrows back to prisoner. You can start at the top, drawing 2 by 3 data frames of first the judge, then captain, then soldiers, showing one world in the first row and the other world in the second row and showing that the potential outcomes would differ based on the action of the first two.

You can also use the graph method of: if you break the causal chain after a given node, is there still a connected path to the prisoner? The answer is no for judge and captain but yes for each solider.

Students may bring up good points about ambiguity in the wording of the problem or a different interpretation of the “rules of the game” around how the causal structure works. That’s good! If students suggest different interpretations, you could draw that graph or draw that data frame, and analyze that one to identify causes.

As a postscript, this is likely one reason why firing squads are have multiple shooters. Each one has plausible deniability that “the prisoner would have died anyways”, so they didn’t cause the death. Apparently, it’s common for marksmen to occasionally neglect to fire, so this also has redundancy. The judge and captain, though they might be insolated from guilt because they are higher up the causal chain, they are still a cause of the death by this definition.

In the next slide, you will see the first few rows of a dataset containing demographic information on California counties. Scroll to see all of the rows.

We are interested in determining whether a difference in median_edu has a causal effect on homeownership. With this in mind, which county serves as the best counterfactual match to Fresno County?

1. Kern County
2. Marin County
3. Contra Costa County
4. Shasta County
5. Del Norte County

The answer is (A), since all categorical columns except homeownership have the same entries.

You may want to mention that this was a final question last year.

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name	homeownership	median_edu	metro	smoking_ban
Fresno County	55.0	some_college	yes	none
Colusa County	64.4	hs_diploma	no	none
Del Norte County	60.9	hs_diploma	no	none
Alameda County	55.1	some_college	yes	none
Contra Costa County	69.5	some_college	yes	partial
Glenn County	67.5	hs_diploma	no	none
Shasta County	66.0	some_college	yes	none
Kern County	61.4	hs_diploma	yes	none
San Luis Obispo County	61.4	some_college	yes	none
Marin County	64.0	bachelors	yes	partial

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