Introduction to Multivariate Relationships

Transcription

1 Introduction to Multivariate Relationships Jamie Monogan University of Georgia Introduction to Data Analysis Jamie Monogan (UGA) Multivariate Relationships POLS / 13

2 Why do we need multivariate statistics? Jamie Monogan (UGA) Multivariate Relationships POLS / 13

3 Objectives By the end of this meeting, participants should be able to: Define the necessary conditions for a relationship to be causal. Define the concept of control, both experimental and statistical. Illustrate and substantively distinguish among a variety of potential multivariate relationships. Jamie Monogan (UGA) Multivariate Relationships POLS / 13

4 Testing Causal Relationships Three criteria are necessary conditions for a causal relationship: 1 Association between the variables. 2 An appropriate time order. (Early plug: Time Series Analysis.) 3 The elimination of alternative explanations. Side Note When many people talk about causal inference, they are referring to the Neyman-Rubin causal inference framework. Much work in matching methods and experimental research follows this theory. The way we lay-out causal inference today is more in line with the work of Bollen or Pearl. Jamie Monogan (UGA) Multivariate Relationships POLS / 13

5 Elimination of Alternative Explanations How would you define the concept of control? Experimental control. Statistical control. Jamie Monogan (UGA) Multivariate Relationships POLS / 13

6 Spurious Association between X 1 and Y X 1 X 2 Y If you estimated the effect of X 1 on Y and completely ignored X 2, you would wrongly conclude that there is a relationship between X 1 and Y. If you modeled Y as a function of both X 1 and X 2, the spurious association between X 1 and Y would disappear. (Which is good!) Jamie Monogan (UGA) Multivariate Relationships POLS / 13

7 Chain Relationship X 1 X 2 Y Here, we say X 2 is a mediator or intervening variable. X 1 has an indirect effect on Y. Analogy: if you set-up a line of dominoes and push the first over, pushing the first indirectly knocked-over the last. Jamie Monogan (UGA) Multivariate Relationships POLS / 13

8 Interaction X 2 X 1 Y Here, we say X 2 is a moderator variable. X 1 has a conditional effect on Y. Jamie Monogan (UGA) Multivariate Relationships POLS / 13

9 Multiple Causes X 1 Y X 2 If X 1 and X 2 are truly independent, then controlling for one will not change your assessment of the other s effect. Jamie Monogan (UGA) Multivariate Relationships POLS / 13

10 Both Direct and Indirect X 1 Y X 2 In this case, ignoring X 2 would lead you to overestimate the direct effect of X 1 on Y. By estimating a model of X 2 and a model of Y, you can determine both the direct and indirect effects of X 1 on Y. A special case of this kind of relationship would be a suppressor relationship. What kind of circumstances would produce this? Jamie Monogan (UGA) Multivariate Relationships POLS / 13

11 Statistical Control in Practice: A Control Table (1) #INITIAL BIVARIATE ANALYSIS #Load library for CrossTable command library(gmodels) #Load library with data and load dataset: library(car) data(greene) #Create a binary variable for judge="desjardins" Greene$bin.judge<-as.numeric(Greene$judge== Desjardins ) #Bivariate analysis on whether Desjardins judges differently: CrossTable(y=Greene$bin.judge, x=greene$decision, prop.r=false, prop.t=false, chisq=true) Jamie Monogan (UGA) Multivariate Relationships POLS / 13

12 Statistical Control in Practice: A Control Table (2) #CONTROL FOR MERITS OF CASE #Subset based on whether an independent rater thinks the case has merit. merit.yes<-subset(greene, rater== yes ) merit.no<-subset(greene, rater== no ) #Does Desjardins differ when there is merit? CrossTable(y=merit.yes$bin.judge, x=merit.yes$decision, prop.r=false, prop.t=false, chisq=true) #Does Desjardins differ when there is not merit? CrossTable(y=merit.no$bin.judge, x=merit.no$decision, prop.r=false, prop.t=false, chisq=true) #What kind of relationship is this? Jamie Monogan (UGA) Multivariate Relationships POLS / 13

13 For Next Time Read Agresti & Finlay, Chapter 11, Multiple Regression and Correlation. Answer questions 10.6 & Analyze these data on professors salaries: library(car); data(salaries) What is the bivariate correlation between years since Ph.D. (yrs.since.phd) and professors salaries (salary)? Create three subsets of the data based on rank of the professor (rank). Report the bivariate correlation for years since Ph.D. and professors salaries for each of subsets (Assistant, Associate, and Full). What kind of bivariate relationship is there between years since Ph.D. and professors salaries? Why is that? Jamie Monogan (UGA) Multivariate Relationships POLS / 13