Exploring Relationships with a Third Variable: Understanding, Controlling, and Specifying Causality

1. 11. Controlling for a 3rd Variable

2. + a b c + + a b 0 Explicating a bivariate relationship with a third variable Identifying a misspecified relationship: A spurious relationship is one type of misspecification Observe But

3. Explicating a bivariate relationship with a third variable (continued) More generally a misspecified relationship is when the magnitude or direction of the relationship you observe between a and b is not due to a causing b, but to c partly or wholly causing both a and b. When you control for c the relationship between a and b changes in magnitude or direction.

4. # of Fire trucks sent to fire Severity of Damage Initial Report Severity of damage + + # Fire trucks Severity of damage - • E.g. Should we keep the fire trucks home?

5. 1 truck > 1 truck 1 truck > 1 truck 70% 80% 30% 50% Not severe damage Not severe damage 30% 20% 70% 50% Severe damage Severe damage n= 100 n= 100 n= 10 n= 10 Taub < 0 Taub < 0 1 truck > 1 truck Not severe damage Taub > 0 Severe damage n= 110 n= 110 Not Serious Serious Initial Report

6. 30% 30% 70% 70% a 60% 40% Taub > 0 Spurious b 40% 60% 100% 100% c (3 values) low high medium a a a 70% 70% 50% 50% b b b 30% 30% 50% 50% Taub = 0 Taub = 0 Taub = 0

7. ++ a b c + + a b + Or you might find: Taub = 0.6 0 < Taub < 0.6 for each value of c Here we would have overestimated the impact of a on b. A does cause b, but controlling for c we realize the effect is less than we initially thought.

8. Controlling for a third variable thus allows us to test alternative explanations for a hypothesis. • When you cannot do a proper true experimental design that eliminates alternative explanations, you need to do statistical controls. Here we have just looked at how you do a statistical control.

9. a b 83% 70% 17% 30% Conditional Relationships: Specification is another reason to control for a third variable c Low Ed. High Ed. No Worked for Political Candidate Yes

10. 75% 90% 70% 70% 25% 10% 30% 30% Men Women Low Ed. High Ed. Low Ed. High Ed. No No Worked Worked Yes Yes Small + Taub Large + Taub Relationship between education and working for a candidate is positive for both men and women, but is stronger for women than men.

11. + a b c 40% 30% 0 60% 70% a b + Multiple Causes (Enhancement): Two variables may be causes of a third variable, while the two are unrelated to each other. a b n = 200 n = 200 c

12. 50% 30% 40% 20% 50% 70% 60% 80% a a b b n = 100 n = 100 n = 100 n = 100 Our estimate of the impact of a on b is unchanged, but by also looking at c we can better predict b. Both a and c are causes of b.

13. a a b c Race Race Income Education Using a third variable to find an intervening relationship: b A causes b. All or some of the way a causes b is through c. Income First, we observe minorities earn lower incomes than non-minorities. Then we ask, to what extent is that because they achieve lower levels of education and lower levels of education result in less income?

14. 63% 70% 50% 47% 60% 40% 37% 30% 50% 60% 53% 40% Race Min. Non-Min. Low Taub ++ Income High n = 150 n = 300 Control for Education Low Education High Education Min. Non-Min. Min. Non-Min. Low Low Income High High n = 100 n = 100 n = 50 n = 200 Taub + Taub +

15. Some, but not all, of the impact of race on income is due to education. Education partly explains the way in which race affects income. • Remember race is still the cause, we are looking at the mechanism. • If Taub = 0 with control, then all the effect of race would have worked through education.

16. a a c b a b + Using a third variable to find an antecedent cause: b + Acauses b, but we can learn more by finding a is caused by c. Here we start with: a b We ascertain: c a c With… a Then we identify a as intervening by predicting b with c and controlling for a. To the extent the relationship is attenuated by the control, c is antecedent and works through a.

17. a a b b c c a a b b Theory is key in drawing the causal arrows. , then the simple If will be misspecified. But if then c is intervening. is correct in estimating the magnitude of the effect of a on b.C become a mechanism of how a causes b. The researcher must draw the arrow correctly. Statistics can’t solve this problem.

18. a b Hint: Typically (though not always) Demographic Attitude Behavior Avoid reciprocal relationships: a b But if you think: a b You can mention that b may have a small impact on a, but the overwhelming effect is of a causing b. You can then just consider: