Xuhua Xia Slide 1 Correlation Simple correlation –between two variables Multiple and Partial correlations –between one variable and a set of other variables Canonical Correlation –between two sets of variables each containing more than one variable. Simple and multiple correlations are special cases of canonical correlation. Multiple: x 1 on x 2 and x 3 Partial: between X and Y with Z being controlled for

Xuhua Xia Slide 2 Review of correlation XZY Compute Pearson correlation coefficients between X and Z, X and Y and Z and Y. Compute partial correlation coefficient between X and Y, controlling for Z (i.e., the correlation coefficient between X and Y when Z is held constant), by using the equation in the previous slide. Run SAS to verify your calculation: proc corr pearson; var X Y; partial Z; run;

Xuhua Xia Slide 3 Many Possible Correlations With multiple DV’s and IV’s, there could be many correlation patterns: –Variable A in the DV set could be correlated to variables a, b, c in the IV set –Variable B in the DV set could be correlated to variables c, d in the IV set –Variable C in the DV set could be correlated to variables a, c, e in the IV set With these plethora of possible correlated relationships, what is the best way of summarizing them?

Xuhua Xia Slide 4 Dealing with Two Sets of Variables The simple correlation approach: –For N DV’s and M IV’s, calculate the simple correlation coefficient between each of N DV’s and each of M IV’s, yielding a total of N*M correlation coefficients The multiple correlation approach: –For N DV’s and M IV’s, calculate multiple or partial correlation coefficients between each of N DV’s and the set of M IV’s, yielding a total of N correlation coefficients The canonical correlation Note: All these deal with linear correlations

Xuhua Xia Slide 5 Fitness Data /* First three variables: physical Last three variables: exercise Middle-aged men */ data fit; input weight waist pulse chins situps jumps cards; ;

Xuhua Xia Slide 6 SAS Program proc cancorr data=fit vdep wdep smc stb t probt vprefix=PHYS vname='Physical Measurements' wprefix=EXER wname='Exercises'; var weight waist pulse; with chins situps jumps; title2 'Middle-aged Men in a Health Fitness Club'; title3 'Data Courtesy of Dr. A. C. Linnerud, NC State Univ.'; run; What’s the meaning of these cryptic terms? Next slide

Xuhua Xia Slide 7 SAS Program proc cancorr data=fit short vdep wdep smc stb t probt SHORT - suppresses all default output except the tables of Canonical correlations and multivariate statistics. VDEP - requests multiple regression analyses with the VAR variable as dependent variables and the WITH variables as regressors. WDEP does the opposite SMC - prints squared multiple correlations and F tests for the regression analyses The STB option requests standardized regression coefficients. VPREFIX - specify a variable prefix for canonical variables instead of using the default V1, V2, and so on. WPREFIX does the same.

Xuhua Xia Slide 8 Multiple Correlations DV: the Physical Measurements IV: Exercises Squared Multiple Correlations and F Tests 3 numerator df 16 denominator df 95% CI for R 2 R 2 R 2.adj Lower Upper F Pr > F weight waist <.0001 pulse Weight and WAIST are significantly associated with the exercise variables.

Xuhua Xia Slide 9 Regression of Phys. on Exer. Standardized Regression Coefficients weight waist pulse chins situps jumps t Values for the Regression Coefficients weight waist pulse chins situps jumps Prob > |t| for the Regression Coefficients weight waist pulse chins situps jumps

Xuhua Xia Slide 10 Multiple Correlations DV: Exercises IV: the Physical Measurements Squared Multiple Correlations and F Tests 3 numerator df 16 denominator df 95% CI for R 2 R 2 R 2.adj Lower Upper F Pr> F chins situps jumps

Xuhua Xia Slide 11 Regression of Exer. on Phys. Standardized Regression Coefficients chins situps jumps weight waist pulse t Values for the Regression Coefficients chins situps jumps weight waist pulse Prob > |t| for the Regression Coefficients chins situps jumps weight waist pulse

Xuhua Xia Slide 12 Canonical Correlation Adjusted Approx Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation Eigenvalue Difference Proportion Cumulative Significance test: EigenvalueLikelihood Approximate Ratio F Value Num DF Den DF Pr > F

Xuhua Xia Slide 13 Standardized Canonical Coefficients for the Physical Measurements PHYS1 PHYS2 PHYS3 weight waist pulse for the exercises EXER1 EXER2 EXER3 chins situps jumps Because the variables are not measured in the same units, the standardized coefficients rather than the raw coefficients should be interpreted.

Xuhua Xia Slide 14 Canonical Structure: correlations Between Phys. and their canonical var.: PHYS1 PHYS2 PHYS3 weight waist pulse Between Exer. and their canonical var.: EXER1 EXER2 EXER3 chins situps jumps Between Phys. and the canonical var. of Exer.: EXER1 EXER2 EXER3 weight waist pulse Between Exer. and the canonical var. of Phys.: PHYS1 PHYS2 PHYS3 chins situps jumps

Xuhua Xia Slide 15 Ecology data data candata; input Sp1 Sp2 Sp3 Sp4 Chem1 Chem2 Chem3 Chem4; cards; ;

Xuhua Xia Slide 16 SAS Program (cont.) proc cancorr vdep wdep smc stb t probt vprefix=BIO vname='Species' wprefix=ENV wname='Environment'; var Sp1 Sp2 Sp3 Sp4; with Chem1 Chem2 Chem3 Chem4; run; Run and explain