Correlation Coefficients (DP IB Applications & Interpretation (AI)): Revision Note

PMCC

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

Spearman’s Rank

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

Appropriateness & Limitations

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