This example produces a correlation analysis with descriptive statistics and four measures of association: the Pearson productmoment correlation, the Spearman rankorder correlation, Kendall’s taub coefficients, and Hoeffding’s measure of dependence, .
The Fitness
data set created in the section Getting Started: CORR Procedure contains measurements from a study of physical fitness of 31 participants. The following statements request all four measures
of association for the variables Weight
, Oxygen
, and Runtime
:
ods graphics on; title 'Measures of Association for a Physical Fitness Study'; proc corr data=Fitness pearson spearman kendall hoeffding plots=matrix(histogram); var Weight Oxygen RunTime; run; ods graphics off;
Note that Pearson correlations are computed by default only if all three nonparametric correlations (SPEARMAN, KENDALL, and HOEFFDING) are not specified. Otherwise, you need to specify the PEARSON option explicitly to compute Pearson correlations.
The “Simple Statistics” table in Output 2.1.1 displays univariate descriptive statistics for analysis variables. By default, observations with nonmissing values for each variable are used to derive the univariate statistics for that variable. When nonparametric measures of association are specified, the procedure displays the median instead of the sum as an additional descriptive measure.
Output 2.1.1: Simple Statistics
Measures of Association for a Physical Fitness Study 
3 Variables:  Weight Oxygen RunTime 

Simple Statistics  

Variable  N  Mean  Std Dev  Median  Minimum  Maximum 
Weight  31  77.44452  8.32857  77.45000  59.08000  91.63000 
Oxygen  29  47.22721  5.47718  46.67200  37.38800  60.05500 
RunTime  29  10.67414  1.39194  10.50000  8.17000  14.03000 
The “Pearson Correlation Coefficients” table in Output 2.1.2 displays Pearson correlation statistics for pairs of analysis variables. The Pearson correlation is a parametric measure of association for two continuous random variables. When there are missing data, the number of observations used to calculate the correlation can vary.
Output 2.1.2: Pearson Correlation Coefficients
Pearson Correlation Coefficients Prob > r under H0: Rho=0 Number of Observations 


Weight  Oxygen  RunTime  
Weight 




Oxygen 




RunTime 



The table shows that the Pearson correlation between Runtime
and Oxygen
is 0.86843, which is significant with a value less than 0.0001. This indicates a strong negative linear relationship between these two variables. As Runtime
increases, Oxygen
decreases linearly.
The Spearman rankorder correlation is a nonparametric measure of association based on the ranks of the data values. The “Spearman Correlation Coefficients” table in Output 2.1.3 displays results similar to those of the “Pearson Correlation Coefficients” table in Output 2.1.2.
Output 2.1.3: Spearman Correlation Coefficients
Spearman Correlation Coefficients Prob > r under H0: Rho=0 Number of Observations 


Weight  Oxygen  RunTime  
Weight 




Oxygen 




RunTime 



Kendall’s taub is a nonparametric measure of association based on the number of concordances and discordances in paired observations. The “Kendall Tau b Correlation Coefficients” table in Output 2.1.4 displays results similar to those of the “Pearson Correlation Coefficients” table in Output 2.1.2.
Output 2.1.4: Kendall’s Taub Correlation Coefficients
Kendall Tau b Correlation Coefficients Prob > tau under H0: Tau=0 Number of Observations 


Weight  Oxygen  RunTime  
Weight 




Oxygen 




RunTime 



Hoeffding’s measure of dependence, , is a nonparametric measure of association that detects more general departures from independence. Without ties in the variables,
the values of the statistic can vary between and , with indicating complete dependence. Otherwise, the statistic can result in a smaller value. The “Hoeffding Dependence Coefficients” table in Output 2.1.5 displays Hoeffding dependence coefficients. Since ties occur in the variable Weight
, the statistic for the Weight
variable is less than .
Output 2.1.5: Hoeffding’s Dependence Coefficients
Hoeffding Dependence Coefficients Prob > D under H0: D=0 Number of Observations 


Weight  Oxygen  RunTime  
Weight 




Oxygen 




RunTime 



When you use the PLOTS=MATRIX(HISTOGRAM) option, the CORR procedure displays a symmetric matrix plot for the analysis variables listed in the VAR statement (Output 2.1.6).
The strong negative linear relationship between Oxygen
and Runtime
is evident in Output 2.1.6.
Note that this graphical display is requested by enabling ODS Graphics and by specifying the PLOTS= option. For more information about ODS Graphics, see Chapter 21: Statistical Graphics Using ODS in SAS/STAT User's Guide.