4.2.2 Correlation Coefficients

What is Pearson’s product-moment correlation coefficient?

• Pearson’s product-moment correlation coefficient (PMCC) is a way of giving a numerical value to a linear relationship of bivariate data
• The PMCC of a sample is denoted by the letter 
  • r can take any value such that 
  • A positive value of r describes positive correlation
  • A negative value of r describes negative correlation
  • r = 0 means there is no linear correlation
  • r = 1 means perfect positive linear correlation
  • r = -1 means perfect negative linear correlation
  • The closer to 1 or -1 the stronger the correlation

How do I calculate Pearson’s product-moment correlation coefficient (PMCC)?

• You will be expected to use the statistics mode on your GDC to calculate the PMCC
• The formula can be useful to deepen your understanding

• • • is linked to the covariance
    •  and  are linked to the variances
  • You do not need to learn this as using your GDC will be expected

When does the PMCC suggest there is a linear relationship?

• Critical values of r indicate when the PMCC would suggest there is a linear relationship
  • In your exam you will be given critical values where appropriate
  • Critical values will depend on the size of the sample
• If the absolute value of the PMCC is bigger than the critical value then this suggests a linear model is appropriate

What is Spearman’s rank correlation coefficient?

• Spearman's rank correlation coefficient is a measure of how well the relationship between two variables can be described using a monotonic function
  • Monotonic means the points are either always increasing or always decreasing
  • This can be used as a way to measure correlation in linear models
  • Though Spearman's Rank correlation coefficient can also be used to assess a non-linear relationship
• Each data is ranked, from biggest to smallest or from smallest to biggest
  • For n data values, they are ranked from 1 to n
  • It doesn't matter whether variables are ranked from biggest to smallest or smallest to biggest, but they must be ranked in the same order for both variables
• Spearman’s rank of a sample is denoted by 
  • r_s can take any value such that 
  • A positive value of r_s describes a degree of agreement between the rankings
  • A negative value of r_s describes a degree of disagreement between the rankings
  • r_s = 0 means the data shows no monotonic behaviour
  • r_s = 1 means the rankings are in complete agreement: the data is strictly increasing
    • An increase in one variable means an increase in the other
  • r_s = -1 means the rankings are in complete disagreement: the data is strictly decreasing
    • An increase in one variable means a decrease in the other
  • The closer to 1 or -1 the stronger the correlation of the rankings

How do I calculate Spearman’s rank correlation coefficient (PMCC)?

• Rank each set of data independently
  • 1 to n for the x-values
  • 1 to n for the y-values
• If some values are equal then give each the average of the ranks they would occupy
  • For example: if the 3^rd, 4^th and 5^th highest values are equal then give each the ranking of 4
    • 
• Calculate the PMCC of the rankings using your GDC
  • This value is Spearman's rank correlation coefficient

Which correlation coefficient should I use?

• Pearson’s PMCC tests for a linear relationship between two variables
  • It will not tell you if the variables have a non-linear relationship
    • Such as exponential growth
  • Use this if you are interested in a linear relationship
• Spearman’s rank tests for a monotonic relationship (always increasing or always decreasing) between two variables
  • It will not tell you what function can be used to model the relationship
    • Both linear relationships and exponential relationships can be monotonic
  • Use this if you think there is a non-linear monotonic relationship

How are Pearson’s and Spearman’s correlation coefficients connected?

• If there is linear correlation then the relationship is also monotonic
  • 
  • 
  • However the converse is not true

• It is possible for Spearman’s rank to be 1 (or -1) but for the PMCC to be different
  • For example: data that follows an exponential growth model
    • as the points are always increasing
    • as the points do not lie on a straight line

Are Pearson’s and Spearman’s correlation coefficients affected by outliers?

• Pearson’s PMCC is affected by outliers
  • as it uses the numerical value of each data point
• Spearman’s rank is not usually affected by outliers
  • as it only uses the ranks of each data point