1Class 24Social StatisticsClass Outline• The Elaboration Model—Analyses of Three or More Variables • Three Types of Elaboration Models– Interaction Effects– Models with intervening variables– Models with confounding VariablesAnalysis of Three VariablesThe 3rdvariable is an intervening variable.XYZ2.The 3rdvariable is a confounding variable.XYZ3.XYZThe 3rdvariable interacts with the independent variable.XYZ1.Note: Z is the 3rdvariable.2Elaboration Model: Step by Step1. A relationship is observed to exist between two variables.2. A third variable is held constant in the sense that the cases under study are subdivided according to the attributes of that third variable3. The original two-variable relationship is recomputed within each of the subgroups.4. Compare the original relationship with the relationships found within each subgroupE.g. Race Æ education, control variable: sex.Example of Interaction Effects:Race and Sex Differences in Education11.7512.69Female11.7511.51Black12.6913.05WhiteSubtotalMaleMean Years of Schooling by Race and SexWhite men have more education than white women. (0.36)Black men have less education than black women. (-0.24)The racial gap among men is 1.54 years.The racial gap among women is 0.84 years.Interaction Effects• We say that X and Z have interactive effects on Y when – the effect of X on Y depends on the level of Z,– or equivalently, the effect of Z on Y depends on the level of X.• If X and Y are continuous variables and Z is a dichotomous variable, interaction effects Æ non-parallel regression linesZ=1Z=0XYZ=1Z=0XYInteraction between X and Z No interaction between X and Z Example: women’s income decreases with the number of children, while men’s income does not vary by the number of children.3Causal Mechanisms: Step by Step(Intervening Variables)Three steps to show that Z is a causal mechanism of the effect of X on Y.1. Argue that X is a cause Z, and Z a cause of Y.2. Show all bivariate correlations:• X and Y • X and Z• Z and Y3. Show that the correlation between X and Y is weakened after we control for Z in the model (i.e., in subgroups of Z).Example of Intervening Variable:Race, Satisf. with Finance and HappinessSatisfaction withFinancial situationBlackHappiness+--Code race as: Black = 1White = 0Very happy = 1Pretty happy or not too happy = 0HealthMarital statusBlackHappiness-% Very happy22.5%black32.5%White4Black-10039.542.418.1Black10024.045.230.8WhiteRow totalNot at all satisfiedMore or lessSatisfiedSatisfaction withFinancial situationHappiness+48Satisfied28.8More or less16.2Not at all satisfied% very happySatisfaction withFinancial situation% very happy by level of financial satisfaction22.213.822.340.2Black32.516.929.848.7WhiteRow totalNot at all satisfiedMore or lessSatisfiedSatisfaction withFinancial situationBlackHappiness+--Note that race difference in happiness at each level of financial satisfaction is smaller than the over race difference.5Confounding and Bias:Sex, Exercise, and Heart AttackExerciseFemaleHeart attack-+-Omitted Variable Bias:Exercise increase the risk of heart attack.Leaving out the confounding variable sex in the analysis would overestimate the effect of exercise on the risk of heart attack.Confounding and Bias:Sex, Exercise, and Heart AttackHeart attacksNHeart attacksNHeart attacksN100200Female11300300Total82009Weekly exercise21003No weekly exerciseMale(Note: Not real data)For men and women combined, weekly exercise increases the risk of heart attack three times (from 0.01 to 0.03). For women, weekly exercise doubles the risk of heat attack (from 0.005 to 0.01.)For men, weekly exercise also doubles the risk of heat attack (from 0.02 to 0.04). Confounding and Bias:Age, Exercise, and Heart AttackExerciseAgeHeart attack-++Omitted Variable Bias:Leaving out the confounding variable age in the analysis would underestimatethe effect of exercise on the risk of heart attack.6Confounding and Bias:Age, Exercise, and Heart AttackHeart attacksNHeart attacksNHeart attacksN300200Middle-aged62500500Total820014Weekly exercise63008No weekly exerciseElderly(Note: Not real data)For men and women combined, weekly exercise increases the risk of heart attack ____times (from ____ to ____). For women, weekly exercise increases the risk of heat attack ____times (from ______ to ______.)For men, weekly exercise also increases the risk of heat attack ____times (from ______ to
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