Pearson's Correlation Coefficient
This is one in a series of tutorials using examples from WINKS SDA.
Definition: Measures the strength of the linear relationship between two variables.
Assumptions: Both variables (often called X and Y) are interval/ratio and approximately normally distributed, and their joint distribution is bivariate normal.
Characteristics: Pearson's Correlation Coefficient is usually signified by r (rho), and can take on the values from -1.0 to 1.0. Where -1.0 is a perfect negative (inverse) correlation, 0.0 is no correlation, and 1.0 is a perfect positive correlation.
Related statistics: R2 (called the coefficient of determination or r squared) can be interpreted as the proportion of variance in Y that is contained in X.
Tests: The statistical significance of r is tested using a t-test. The hypotheses for this test are:
H0: rho = 0
Ha: rho <> 0
A low p-value for this test (less than 0.05 for example) means that there is evidence to reject the null hypothesis in favor of the alternative hypothesis, or that there is a statistically significant relationship between the two variables.
Note: This test is equivalent to the test of no slope in the simple linear regression procedure.
Location in WINKS: Pearson's correlation coefficient is found in the following locations:
1. Regression and Correlation - The Correlation procedure produces both Pearson and Spearman Correlation coefficients. The t-test for statistical significance of r is calculated. R2 is also reported.
2. Regression and Correlation - The Simple linear regression reports the Pearson correlation coefficient and the t-test. R2 is also reported.
3. Regression and Correlation - The Correlation Matrix procedure produces a matrix of correlations for a number of pairs of variables at a time, and includes the p-value for the test or significance of r.
Graphs: An important part of interpreting r is to observe a scatterplot of the data. Scatterplots are available from the Graphs option, as a part of Simple Linear Regression and in the Graphical Correlation Matrix option in Regression and Correlation.