10. Introduction to Multivariate RelationshipsSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Simpson’s paradoxTypes of Multivariate RelationshipsSlide 10Slide 11Slide 1210. Introduction to Multivariate RelationshipsBivariate analyses are informative, but we usually need to take into account many variables.•Many explanatory variables have an influence on any particular response variable.•The effect of an explanatory variable on a response variable may change when we take into account other variables.Example: Y = whether admitted into grad school at U. California, Berkeley (for the 6 largest departments) X = gender Whether admittedGender Yes No Total %yes Female 550 1285 1835 Male 1184 1507 2691 Difference of sample proportions = … There is very strong evidence of a higher probability of admission for men than for women.•Now let X1 = gender and X2 = department to which the person applied. e.g., for Department A, Whether admittedGender Yes No Total %yes Female 89 19 108 82% Male 511 314 825 62%Now, 2 = (df = 1), but difference is …. The strong evidence is that there is a higher probability of being admitted for than .What happens with other departments?Female Male Difference of Dept. Total %admitted Total %admitted proportions 2A 108 82% 825 62% B 25 68% 560 63% C 593 34% 325 37% D 375 35% 417 33% E 393 24% 191 28% F 341 7% 273 6% Total 1835 30% 2691 44% There are 6 “partial tables,” which summed give the original “bivariate” table. How can the partial table results be so different from the bivariate table?•Partial tables – display association between two variables at fixed levels of a “control variable.”Example: Previous page shows results from partial tables relating gender to whether admitted, controlling for (keeping constant) departmentWhen control variable X2 is kept constant, association between Y and X1 is not due to the association of each of them with X2.Note: When each pair of variables is associated, then a bivariate association for two variables may differ from its partial association, controlling for the other variable.Example: Y = whether admitted is associated with X1 = gender, but each of these itself associated with X2 = department.Department associated with gender: Department associated with whether admitted: Moral: Association does not imply causation! This is true for quantitative and categorical variables.With observational data, effect of X on Y may be partly due to association of X and Y with lurking variables. (e.g., association between X = whether use marijuana and Y = GPA; other variables?)Causation difficult to assess with observational studies, unlike experimental studies that can control potential lurking variables (by randomization). When X1 and X2 both have effects on Y but are also associated with each other, there is said to be confounding. It’s difficult to determine whether either truly causes Y, because a variable’s effect could be partly due to its association with the other variable.Simpson’s paradox•It is possible for the (bivariate) association between two variables to be positive, yet be negative at each fixed level of a third variable. (see scatterplot)Example: Florida countywide data There is a positive correlation between crime rate and education!There is a negative correlation between crime rate and education at each level of urbanizationTypes of Multivariate Relationships•Spurious association: Y and X1 both depend on X2 and association disappears after controlling X2 Example:•Multiple causes – A variety of factors have influences on the response (most common in practice)In observational studies, usually all (or nearly all) explanatory variables have associations among themselves as well as with response var. Effect of any one changes depending on which other var’s are controlled (statistically), often because it has a direct effect and also indirect effects through other variables.Example: What causes Y = juvenile delinquency?•Statistical interaction – Effect of X1 on Y changes as the level of X2 changes.Example: Effect of whether a smoker (yes, no) on whether have lung cancer (yes, no) changes as value of age changes Example: U.S. median annual income by race and gender RaceGender Black WhiteFemale Male•The difference in median income between whites and blacks is: $ for females, $ for males•i.e., the effect of race on income depends on gender (and the effect of gender on income depends on race), so there is interaction between race and gender in their effects on income.Example (p. 311): X = number of years of education Y = annual income (1000’s of dollars)Suppose E(Y) = for men E(Y) = for