Correlation is a bivariate analysis that measures the strengths of association between two variables. In statistics, the value of the correlation coefficient varies between +1 and -1. When the value of the correlation coefficient lies around ± 1, then it is said to be a perfect degree of association between the two variables. As the value goes towards 0, the relationship between the two variables will be weaker. Usually, in statistics, we measure three types of correlation: Pearson correlation, Kendall rank correlation and Spearman correlation.
Pearson r correlation: Pearson r correlation is widely used in statistics to measure the degree of the relationship between the linear related variables. For the Pearson r correlation, both variables should be normally distributed. For example, in the stock market, if we want to measure how two commodities are related to each other, Pearson r correlation is used to measure the degree of relationship between the two commodities. The following formula is used to calculate the Pearson r correlation:
Where:
r = Pearson r correlation coefficient
N = number of value in each data set
∑xy = sum of the products of paired scores
∑x = sum of x scores
∑y = sum of y scores
∑x2= sum of squared x scores
∑y2= sum of squared y scores
Pearson r correlation in SPSS: To perform the Pearson r correlation in SPSS, we will select “correlate” from the analysis menu and select “bivariate” from the correlate option. After selecting this option, the following window appears:
Figure 1
Figure 2
Select the variables for which we want to calculate the Pearson r correlation and drag them to the right side of the variable list. Click on “option” and select “descriptive statistics” from there. Select “Pearson test” from the window. Click on the “ok” button. The result window will show the result of the Pearson correlation. Figure 2 shows the SPSS output table for the Pearson r correlation.
Kendall rank correlation: Kendall rank correlation is a non-parametric test that does not assume any assumptions related to the distributions— like Pearson’s correlation. The following formula is used to calculate the value of Kendall rank correlation:
Where:
Nc= number of concordant
Nd= Number of discordant
=Total number of possible pairing of observations
Kendall rank correlation in SPSS: Most of the procedures for Kendall rank correlation will be the same as the Pearson correlation in SPSS. We will select “correlate” from the analysis menu and select “bivariate” from the correlate menu. As we click on the bivariate, the figure 3 window will appear. We will then select the variable and drag it into the variable list. Select “Kendall test” from this window, and click on the “ok” button. Figure 4 shows the result for the Kendall test that will appear in the result window. In this window, we will find the value of Kendall correlation coefficient and the associated significance value for the correlation coefficient.
Figure 3
Figure 4
Spearman rank correlation: Spearman rank correlation is a non parametric test that is used to measure the degree of association between the two variables. It was developed by Spearman, thus it is called the Spearman rank correlation. Spearman rank correlation test does not assume any assumptions about the distribution. Spearman rank correlation test is used when the Pearson test gives misleading results.
The following formula is used to calculate the Spearman rank correlation:
Where:
P= Spearman rank correlation
di= the difference between the ranks of corresponding values Xi and Yi
n= number of value in each data set
Spearman rank correlation in SPSS: Click on “correlate” from the analysis menu. Select “bivariate” from the correlate option. The bivariate correlation window will appear after clicking on the bivariate option. Select the correlation variable and put them on the right hand side of the variable list. Then select the “Spearman test” from the window. Click on the “ok” button. The following table will show the Spearman rank correlation from the result window:
This table will show the coefficient for Spearman rank correlation with the significance value. We will accept or reject the null hypothesis based on the significance value.
Correlation Resources
Algina, J., & Keselman, H. J. (1999). Comparing squared multiple correlation coefficients: Examination of a confidence interval and a test significance. Psychological Methods, 4(1), 76-83.
Bobko, P. (2001). Correlation and regression: Applications for industrial organizational psychology and management (2nd ed.). Thousand Oaks, CA: Sage Publications.
Bonett, D. G. (2008). Meta-analytic interval estimation for bivariate correlations. Psychological Methods, 13(3), 173-181.
Chen, P. Y., & Popovich, P. M. (2002). Correlation: Parametric and nonparametric measures. Thousand Oaks, CA: Sage Publications.
Cheung, M. W. -L., & Chan, W. (2004). Testing dependent correlation coefficients via structural equation modeling. Organizational Research Methods, 7(2), 206-223.
Coffman, D. L., Maydeu-Olivares, A., Arnau, J. (2008). Asymptotic distribution free interval estimation: For an intraclass correlation coefficient with applications to longitudinal data. Methodology, 4(1), 4-9.
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences. (3rd ed.). Mahwah, NJ: Lawrence Erlbaum Associates.
Hatch, J. P., Hearne, E. M., & Clark, G. M. (1982). A method of testing for serial correlation in univariate repeated-measures analysis of variance. Behavior Research Methods & Instrumentation, 14(5), 497-498.
Kendall, M. G., & Gibbons, J. D. (1990). Rank Correlation Methods (5th ed.). London: Edward Arnold.
Krijnen, W. P. (2004). Positive loadings and factor correlations from positive covariance matrices. Psychometrika, 69(4), 655-660.
Shieh, G. (2006). Exact interval estimation, power calculation, and sample size determination in normal correlation analysis. Psychometrika, 71(3), 529-540.
Stauffer, J. M., & Mendoza, J. L. (2001). The proper sequence for correcting correlation coefficients for range restriction and unreliability. Psychometrika, 66(1), 63-68.
