Chapter 6 Measures of Bivariate Association 1

Transcription

1 Chapter 6 Measures of Bivariate Association 1 A bivariate relationship involves relationship between two variables. Examples: Relationship between GPA and SAT score Relationship between height and weight Usually we are interested in either a test of statistical significance about the relationship, or a measure of association which indicates the strength of relationship. 2 Chapter 6 Measures of Bivariate Association 1

2 The type of statistic used to measure relationship between two variables depends on the type of variables. The most commonly used statistics are as follows: Predictor Variables(X) Criterion Quantitative Ordinal Nominal Variables Variables (Y) Quantitative Variables Pearson Correlation or Spearman Correlation Spearman Correlation ANOVA Ordinal Spearman Correlation Kruskal Wllias Test Nominal c 2 test 3 Note:Certain assumptions need to hold in each case for the statistic to be valid. Examples: Chi-Square test of independence o Tests independence of two nominal variables. Variable1: Place of residence East,West, Midwest Variable2: Political Party Democrat, Republican, Other Hypothesis: People in Midwest are more likely to belong to the Republican party 4 Chapter 6 Measures of Bivariate Association 2

3 Democrat Republican Other East * * * West * * * Midwest * * * A Sample of registered voters from each region in taken and their political party affiliation is recorded. Spearman Correlation Coefficient Ranking of 100 largest universities -- Intellectual Environment -- Football Program Pearson Correlation -- Income -- Age 5 The symbol for Pearson Correlation is r and is computed as r = ( x i - x)( y i - y ) 2 2 Ø ( y - y ) ( x i º - x) i øß Where x i and y i are observations on two variables. Properties of r: 1) -1 r 1 2) r is unit-less 6 Chapter 6 Measures of Bivariate Association 3

4 When to use r? Both variables must be quantitative Both variables must assume a large number of values Both variables must come from a normally distributed population 7 Interpreting r r can be interpreted only if there is an approximate (or exact) linear relationship between the two variables involved. Sign of r r > 0 r < 0 x and y increase or decrease together x and y increase or decrease in opposite directions. 8 Chapter 6 Measures of Bivariate Association 4

5 Size of r: The closer the absolute value of r to 1, the stronger the linear relationship is (if there is a linear relationship). An r close to zero indicates lack of linear relationship. Rule of Thumb: ± 1 = perfect linear relation ±.8 = strong correlation ±.5 = moderate correlation.2 = weak correlation 0 = no correlation 9 To determine whether there is a linear relationship, it is best to look at the scatter plot of variables: PROC PLOT DATA= data-set-name; PLOT criterion variable* predictor variable;; RUN; Y X 10 Chapter 6 Measures of Bivariate Association 5

6 Summary to determine interpretability of 1) Determine whether each variable is normally distributed, using PROC UNIVARIATE 2) Inspect the scatter plot of variables to see if the relationship is linear, using PROC PLOT 1) If normal and linear, the r is interpretable. LINK: Example1.SAS 11 /* 1.School: Contains the name of each school 2.School_Type: Coded 'LibArts' for liberal arts and 'Univ' for university 3.SAT: Median combined Math and Verbal SAT score of students 4.Acceptance: % of applicants accepted 5.$/Student: Money spent per student in dollars 6.Top 10%: % of students in the top 10% of their h.s. graduating class 7.%PhD: % of faculty at the institution that have PhD degrees 8.Grad%: % of students at institution who eventually graduate */ Options NODATE; DATA EDUCATION; INFILE 'c:/classes/sta5206/notes/chapter6/sas_files/colleges.dat'; INPUT (NAME) (TYPE) (SAT) (ACCEPT) (SPENT) (Top_HS) (PHD) (GRADS) (2.); PROC UNIVARIATE NORMAL; PROC PLOT HPCT=50 VPCT=75; plot SAT*ACCEPT; plot SAT*Top_HS; plot ACCEPT*TOP_HS; Run; 12 Chapter 6 Measures of Bivariate Association 6

7 Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D > Cramer-von Mises W-Sq Pr > W-Sq > Anderson-Darling A-Sq Pr > A-Sq > Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D > Cramer-von Mises W-Sq Pr > W-Sq Anderson-Darling A-Sq Pr > A-Sq Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W < Kolmogorov-Smirnov D Pr > D < Cramer-von Mises W-Sq Pr > W-Sq < Anderson-Darling A-Sq Pr > A-Sq < Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D > Cramer-von Mises W-Sq Pr > W-Sq > Anderson-Darling A-Sq Pr > A-Sq > Chapter 6 Measures of Bivariate Association 7

8 Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W < Kolmogorov-Smirnov D Pr > D < Cramer-von Mises W-Sq Pr > W-Sq < Anderson-Darling A-Sq Pr > A-Sq < Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D Cramer-von Mises W-Sq Pr > W-Sq Anderson-Darling A-Sq Pr > A-Sq The SAS System 13 Plot of SAT*ACCEPT. A=1, B=2, etc. Plot of SAT*Top_HS. A=1, B=2, etc. SAT SAT 1400 ˆ A 1400 ˆ A B AA A A A A A A A A A A AAA A B A A 1300 ˆ AA A A 1300 ˆ AA A A A A B AA A A AAA A A A A A B A AA A A A A A A A A A A A B A A A A A A A A B B A A A A 1200 ˆ A AAA 1200 ˆ A A A A A A A A A A A A A A 1100 ˆ 1100 ˆ Šˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆ Šˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒ ACCEPT Top_HS 16 Chapter 6 Measures of Bivariate Association 8

9 The SAS System 14 Plot of ACCEPT*Top_HS. A=1, B=2, etc. ACCEPT 70 ˆ A A 60 ˆ A A A A A AA A ˆ A 50 A A A A A A A ˆ 40 A A A A A A A A A AA A A 30 ˆ A A A A A A A CA AA A A 20 ˆ A AAA 10 ˆ Šƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒ Top_HS 17 Note: r is not interpretable if the relation between x and y is nonlinear. r 0 Perfect relation r.8 nonlinear relation 18 Chapter 6 Measures of Bivariate Association 9

10 Computing Pearson Correlations PROC CORR DATA=data-set-name Options; VAR variable_1 variable_2 ; RUN; This will produce some standard summary statistics, correlation between variables, and a p-value testing Ho: There is significant correlation between x&y Ha: There is NOT a significant correlation between x&y LINK: Example. SAS 19 /* 1.School: Contains the name of each school 2.School_Type: Coded 'LibArts' for liberal arts and 'Univ' for university 3.SAT: Median combined Math and Verbal SAT score of students 4.Acceptance: % of applicants accepted 5.$/Student: Money spent per student in dollars 6.Top 10%: % of students in the top 10% of their h.s. graduating class 7.%PhD: % of faculty at the institution that have PhD degrees 8.Grad%: % of students at institution who eventually graduate */ Options NODATE; DATA EDUCATION; INFILE 'c:/classes/sta5206/notes/chapter6/sas_files/colleges.dat'; INPUT (NAME) (TYPE) (SAT) (ACCEPT) (SPENT) (Top_HS) (PHD) (GRADS) (2.); PROC CORR; VAR SAT ACCEPT; Run; PROC CORR; VAR SAT ACCEPT TOP_HS; RUN; PROC CORR; VAR SAT; WITH ACCEPT TOP_HS; RUN; 20 Chapter 6 Measures of Bivariate Association 10

11 The SAS System 1 The CORR Procedure 2 Variables: SAT ACCEPT Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum SAT ACCEPT Pearson Correlation Coefficients, N = 50 Prob > r under H0: Rho=0 SAT ACCEPT SAT <.0001 ACCEPT < The SAS System 2 The CORR Procedure 3 Variables: SAT ACCEPT Top_HS Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum SAT ACCEPT Top_HS Pearson Correlation Coefficients, N = 50 Prob > r under H0: Rho=0 SAT ACCEPT Top_HS SAT < ACCEPT <.0001 <.0001 Top_HS < Chapter 6 Measures of Bivariate Association 11

12 The SAS System 3 The CORR Procedure 2 With Variables: ACCEPT Top_HS 1 Variables: SAT Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum ACCEPT Top_HS SAT Pearson Correlation Coefficients, N = 50 Prob > r under H0: Rho=0 SAT ACCEPT <.0001 Top_HS VAR V1 V2 V3; Produces correlations between all pairs VAR V1 V2; With V3 V4 V5 V6 ; Produces correlations between V1 and V2 with V3, V4, V5, V6 (total of8 correlations) The following options may be useful COV: prints covariance between variables NOMISS: Drops all cases with missing values SPEARMAN: Prints spearman correlations 24 Chapter 6 Measures of Bivariate Association 12

13 Spearman Correlations It s often used when at least one variable is ordinal and the other variable is ordinalor quantitative. It s denoted by r s When both variables are quantitative and at least one shows departure from normality, it is better to use r s rather than r. It s a distribution free test. i.e. Does not assume distribution for variables. 25 r s is the ordinary correlation between the ranked values of two variables. In PROC CORR use the option Spearman to produce r s. LINK: Example3.SAS 26 Chapter 6 Measures of Bivariate Association 13

14 /* 1.School: Contains the name of each school 2.School_Type: Coded 'LibArts' for liberal arts and 'Univ' for university 3.SAT: Median combined Math and Verbal SAT score of students 4.Acceptance: % of applicants accepted 5.$/Student: Money spent per student in dollars 6.Top 10%: % of students in the top 10% of their h.s. graduating class 7.%PhD: % of faculty at the institution that have PhD degrees 8.Grad%: % of students at institution who eventually graduate */ Options NODATE; DATA EDUCATION; INFILE 'c:/classes/sta5206/notes/chapter6/sas_files/colleges.dat'; INPUT (NAME) (TYPE) (SAT) (ACCEPT) (SPENT) (Top_HS) (PHD) (GRADS) (2.); PROC CORR SPEARMAN; Run; VAR SPENT PHD; PROC CORR; Run; VAR SPENT PHD; 27 The SAS System 1 The CORR Procedure 2 Variables: SPENT PHD Simple Statistics Variable N Mean Std Dev Median Minimum Maximum SPENT PHD Spearman Correlation Coefficients, N = 50 Prob > r under H0: Rho=0 SPENT PHD SPENT PHD Chapter 6 Measures of Bivariate Association 14

15 The SAS System 2 The CORR Procedure 2 Variables: SPENT PHD Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum SPENT PHD Pearson Correlation Coefficients, N = 50 Prob > r under H0: Rho=0 SPENT PHD SPENT PHD The Chi-square Test of Independence Used to determine whether there is dependence between two classification variables (i.e. Both variables should be assessed on nominal scale) Subjects contributing data should represent a random sample from the population Each subject should fall in the intersection of only one set of variables. 30 Chapter 6 Measures of Bivariate Association 15

16 Data are often summarized in classification tables: Two-way classification table Column Variables CAT 1 CAT 2 CAT c Row Variables CAT 1 Cell (1,1) Cell (1,2) CAT (1,c) CAT 2 Cell (2,1) Cell (2,2) Cell (2,c) CAT r Cell (r,1) Cell (r.2) Cell (r,c) 31 Example: 478 students in grades 4-6 from 3 school districts in Michigan were given a questionnaire to determine which of the following is important to the student: Good Grades Athletic Ability Popularity Among other things, the questionnaire also asks for gender, grade level, race, and other demographic information LINK: Example4.SAS 32 Chapter 6 Measures of Bivariate Association 16

17 data pop_kids; infile 'c:/classes/sta5206/notes/chapter6/sas_files/popkids. dat'; input Gender$ Grade Age Race$ Urban_Rural$ School$ Goals$ Grades Sports Looks Money; proc print; *proc freq; * table Gender*Goals; run; * table Race*Goals; 33 The SAS System 00:13 Wednesday, October 4, Urban_ Obs Gender Grade Age Race Rural School Goals Grades Sports Looks Money 1 boy 5 11 White Rural Elm Sports boy 5 10 White Rural Elm Popular girl 5 11 White Rural Elm Popular girl 5 11 White Rural Elm Popular girl 5 10 White Rural Elm Popular girl 5 11 White Rural Elm Popular girl 5 10 White Rural Elm Popular girl 5 10 White Rural Elm Grades girl 5 10 White Rural Elm Sports girl 5 10 White Rural Elm Sports girl 5 11 White Rural Elm Sports girl 4 10 White Rural Elm Grades boy 4 9 White Rural Elm Popular boy 4 9 White Rural Elm Popular boy 4 9 Other Rural Elm Popular girl 4 9 White Rural Elm Grades girl 4 9 White Rural Elm Sports girl 4 9 White Rural Elm Popular girl 4 9 White Rural Elm Grades girl 4 9 White Rural Elm Sports girl 4 9 White Rural Elm Popular girl 4 9 White Suburban Brentwoo Grades girl 4 9 White Suburban Brentwoo Popular Chapter 6 Measures of Bivariate Association 17

18 Using this data the following classification tables were obtained: GOALS GENDER Grades Popular Sports Boy Girl The Chi-square test can be used to test whether there is a dependence between variable Goals and Gender. Inspecting each column of the table there seems to be differences. The question is whether these differences are statistically significant. 36 Chapter 6 Measures of Bivariate Association 18

19 USING SAS: PROC FREQ DATA data-set-name ; TABLES row-variable-name* column-variable-name options; WEIGHT number-variable-name; RUN; Example: LINK: Example5.SAS 37 data pop_kids; input Gender$ Goals$ Frequency; cards; Boy Grades 117 Boy Popular 50 Boy Sports 60 Girl Grades 130 Girl Popular 91 Girl Sports 30 ; proc freq; tables Gender*Goals /chisq expected exact; Weight frequency; run; 38 Chapter 6 Measures of Bivariate Association 19

20 The SAS System 00:13 Wednesday, October 4, The FREQ Procedure Table of Gender by Goals Gender Goals Frequency Expected Should be larger than 5 Percent Row Pct Col Pct Grades Popular Sports Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ # of boys Boy ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Girl # of girls ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total Expected frequency should be larger than 5 39 Statistics for Table of Gender by Goals Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square <.0001 Likelihood Ratio Chi-Square <.0001 Mantel-Haenszel Chi-Square Phi Coefficient Contingency Coefficient Cramer's V Fisher's Exact Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Table Probability (P) 1.625E-07 Pr <= P 2.088E-05 Sample Size = 478 Small p-value indicates dependence Fisher s exact test is used when the sample size is small 40 Chapter 6 Measures of Bivariate Association 20

21 Raw data can also be used on input. LINK: Example6.SAS 41 Chapter 6 Measures of Bivariate Association 21