Correlation

Correlation coefficients give a numerical value to the strength and direction of a relation between variables. This chapter covers correlation techniques used with interval, ratio scale and ordinal scale variables. The chi square test of independence covered in Chapter 15 can test the association between two nominal variables. The current chapter covers Pearson’s r, Spearman’s ρ and Kendall’s τ. The correlation statistics that measure the strength and direction of a relationship are called correlation coefficients and their values range from −1.0 to +1.0. The sign of the correlation coefficient (+ or −) tells us whether any relation is positive or negative. Figure 8.1 uses scatter plots to show some examples of relationships between two arbitrary variables, x and y. A scatter plot contains points representing individuals by their values for the variables x and y. These points are called co-ordinates. A positive correlation, such as Figure 8.1(a), means that as one variable, x, increases we see an increase in the other variable, y. A negative correlation, such as Figure 8.1(b), means that as one variable, x, increases we see a decrease in the other variable, y. The absolute value of a correlation, which is when the sign is removed, is between 0.0 and 1.0. Figure 8.1(c) shows two variables that have no correlation; this would typically have a correlation coefficient of about 0.0. Figure 8.1(d) is an example of a perfect correlation where a straight line could be drawn through all the co-ordinates. It is not necessary for a straight line of best fit to pass through the origin of the graph for there to be a perfect correlation. Furthermore, perfect correlations can be positive, as is the case in Figure 8.1(d), or negative. Where a perfect correlation exists, the correlation coefficient will be −1.0 if it is a negative correlation or +1.0 if it is a positive correlation. Figures 8.1(a) and 8.1(b) are examples of correlations that are strong but not perfect. The coefficients of correlation here would have absolute values higher than 0.85.