Multivariate Methods. Multivariate Methods: Topics of the Day

Transcription

1 Multivariate Methods LIR 832 Multivariate Methods: Topics of the Day A. Isolating Interventions in a multi-causal world B. Multivariate probability Distributions C. The Building Block: covariance D. The Next Step: Correlation 1

2 A Multivariate World Isolating Interventions in a Multi-Causal World A. Example of problem: Evaluate a program to reduce absences from a plant? Is there age discrimination? B. Types of data Experimental Quasi-experimental Non-experimental C. Need multivariate analysis to sort out causal relationships. Bi-Variate Relations: A First Run at Multivariate Methods A. Many of the issues we are interested in are essentially about the relationship between two variables. B. Bi-variate can be generalized to multivariate relationships C. We learn bi-variate formally and make more intuitive reference to multivariate. D. What do we mean by bi-variate relationship? 2

3 Bi-Variate Example Our firm, has formed teams of engineers, accountants and general managers at all plants to work on several issues that are considered important in the firm. The firm has long been committed to gender diversity and we are interested in the distribution of gender among our managerial classifications. We are particularly concerned about the distribution of gender on these teams and particularly among engineers. Consider the distribution of two statistics about these three person teams. a. gender of the team members (X: x = number of men) b. is the engineer a woman (Y: 0 = man, 1 = woman) Bi-Variate Example (cont.) 3

4 Bi-Variate Example (cont.) Bi-Variate Example (cont.) 4

5 Bi-Variate Example (cont.) Bi-Variate Example (cont.) We can also use this information to build We can also use this information to build conditional probabilities: What is the likelihood that the engineer is a woman, given that we have a man on the team? 5

6 Bi-Variate Example (cont.) What is the likelihood that the engineer is a woman, given that we have a man on the team? P(Y = 1 & X = 1 X= 1) = P(Y = 1 & X = 1)/P(X= 1) = (2/8) / (3/8) = 2/3 Note: P(Y= 1 X=2) is: the probability bilit that t Y is equal to 1 given that t X = 2" or the probability that Y = 1 conditional on X = 2" Bi-Variate Example (cont.) What is the likelihood that there is only one man, given the engineer is a woman? P(Y = 1 & X = 1 Y= 1) = P(Y = 1 & X = 1)/P(Y= 1) = (2/8)/(4/8) = 2/4 =1/2 6

7 Bi-Variate Example (cont.) What is the likelihood that the engineer is a woman? P(Y= 1) = 1/2 But if we know that there are two men, we can improve our estimate: P(Y=1 X=2) = P(Y=1 &X X=2 X=2) 2) = P(Y=1 &X=2) / P(X=2) = 1/8 / 3/8 = 1/3 What about calculating the likelihood of two men given the engineer is a woman? Example: Gender Distribution 7

8 Example: Gender Distribution Working with Conditional Probability: P(female) = 50.91% P(female LRHR) = p(female & LRHR)/P(LRHR) = 0.36/0.55 = 65% P(LRHR) = 0.55% P(LRHR Female) = p(lrhr & female)/p(female) =.36/50.91 =.70% Independence Defined Now that we know a bit about bi-variate relationships, we can define what it means, in a statistical sense, for two events to be independent. If events are independent, then Their conditional probability is equal to their Their conditional probability is equal to their unconditional probability The probability of the two independent events occurring is P(X)*P(Y) = P(X,Y). 8

9 Importance of Independence Why is independence important? If events are independent, then we are getting unique information from each data point. If events are not independent, then A practical example on running a survey on employee satisfaction within an establishment. Example: Employee Satisfaction 9

10 Covariance Covariance: Building Block of Multi- variate Analysis All very nice, but what we are looking for is a means of expressing and measuring the strength of association of two variables. How closely do they move together? Is variable A a good predictor of variable B? Move to a slightly more complex world, no more 2 and three category variables Example: Age and Income Data 10

11 Example: Age and Income Data Example: Age and Income Data 11

12 Example: Age and Income Data Descriptive Statistics: age, annual income Variable N Mean Median StDev SE Mean age annual I Variable Minimum Maximum Q1 Q3 age annual I Example: Age and Income Data 12

13 Example: Age and Income Data Adding some info to the graph Covariance and Correlation Defined Define Covariance and Correlation for a random sample of data: Let our data be composed of pairs of data (X i,y i ) where X has mean μ x and Y has mean μ y. Then the covariance, the co-movement around their means, is defined as: 13

14 Example: Covariance We observe the relationship between the number of employees at work at a plant and the output for five days in a row: Attendance Output What is the covariance of attendance and output? Example: Covariance (cont.) The covariance is positive. This suggests that when attendance is above its mean, output is also above its mean. Similarly, when attendance is below its mean, output is below its mean. 14

15 Example: Overtime Hours and Productivity 15

16 Example: Overtime Hours and Productivity Example: Overtime Hours and Productivity Covariances: prod-avg, week prod-avg week prod-avg week

17 Example: Overtime Hours and Productivity (cont.) Example: Overtime Hours and Productivity (cont.) 17

18 Example: Overtime Hours and Productivity (cont.) Covariances: prod-avg, week, week-hours prod-avg week week-hours prod-avg week week-hours Example: Overtime Hours and Productivity (cont.) 18

19 Example: Overtime Hours and Productivity (cont.) Example: Overtime Hours and Productivity (cont.) 19

20 Correlation vs. Covariance A limitation of covariance is that it is difficult to interpret. Its units are not well defined. Thus, we need a measure which is more readily interpreted and tells about the strength of association. Correlation: Population Correlation is Defined as: Correlation =

21 Correlation = 0.94 Correlation =

22 Correlation = Correlation: Previous Examples 22

23 Correlation: Previous Examples 23

24 Correlation: Previous Examples Correlation: Previous Examples Overtime-Productivity: Limit to 5 days, 10 hours: Correlations: prod-avg, week, week-hours prod-avg week week week-hours

25 Correlations: Previous Examples Example: Correlation 25

26 Example: Correlation Example: Correlation What about some real data: Relationship What about some real data: Relationship between age gender and weekly earnings among human resource managers (admin associated occupations)? 26

27 Example: Correlation Descriptive Statistics: Female, age, weekearn Variable N N* Mean Median TrMean StDev Female age weekearn Variable SE Mean Minimum Maximum Q1 Q3 Female age weekearn Example: Correlation Tabulated Statistics: Female Rows: Female weekearn weekearn Mean StDev Male Female All

28 Example: Correlation Tabulated Statistics: Female Rows: Female weekearn age weekearn age Mean Mean StDev StDev male female all Example: Correlation Covariances: age, weekearn, Female age weekearn Female age weekearn Female

29 Example: Correlation Correlations: age, Female, weekearn age Female Female weekearn Example: Correlation 29

30 Example: Non-Linearity Correlation and Covariance So covariance and correlation are measures So covariance and correlation are measures of linear association, but not measures of association in general (or of non-linear association). 30

31 Correlation and Covariance What if we do not have data on individuals but data on distributions? Example, we have plant level data but plants vary widely in employment. We want to give greater weight to plants with more employees. Correlation and Covariance 31

32 Correlation and Covariance Correlation and Covariance 32

33 Correlation and Covariance Correlation and Covariance 33