# 11. Controlling for a 3 rd Variable. Explicating a bivariate relationship with a third variable Identifying a misspecified relationship: A spurious relationship.

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11. Controlling for a 3 rd Variable

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

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.

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

66%53% 34%47% 1 truck> 1 truck Not severe damage Severe damage n= 110 Tau b > 0 20%30% 80%70% 50%70% 50%30% 1 truck > 1 truck Not severe damage Severe damage n= 100n= 10 n= 100 Tau b < 0 Not SeriousSerious Initial Report

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

Or you might find: ab ++ c ab ++ + Tau b = 0.6 0 < Tau b < 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.

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.

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

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

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

40%50% 60% 20%30% 70%80% aa bb 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.

Using a third variable to find an intervening relationship: abac b A causes b. All or some of the way a causes b is through c. RaceIncome RaceEducationIncome 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?

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

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 Tau b = 0 with control, then all the effect of race would have worked through education.

Using a third variable to find an antecedent cause: ab + ca b + A causes b, but we can learn more by finding a is caused by c. Here we start with: ab b a We ascertain: ca With… c 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.

Theory is key in drawing the causal arrows. c ab If, then the simple ab will be misspecified. But if c ab then c is intervening. ab 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.

Hint: Typically (though not always) DemographicAttitudeBehavior Avoid reciprocal relationships: But if you think: ab ab 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: ab

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