Product moment correlation coefficient (pmcc)

Understanding Product Moment Correlation Coefficient (PMCC)

  • The Product Moment Correlation Coefficient (PMCC) is a numerical measure that describes the strength and direction of a linear relationship between two variables.

  • The PMCC is denoted by the Greek letter ρ (rho).

  • The value of PMCC can be anywhere between -1 and 1.

  • A PMCC value of 1 means a perfect positive linear correlation between the variables. This indicates as one variable increases, the other variable also increases at a fixed rate.

  • A PMCC value of -1 means a perfect negative linear correlation between the variables. This means as one variable increases, the other variable decreases at a fixed rate.

  • If PMCC is zero, then there is no linear correlation between the two variables. This means each variable behaves independently of the other.

  • The PMCC is calculated using a formula that involves the mean of the variables and their standard deviation.

Calculating the Product Moment Correlation Coefficient (PMCC)

  • The formula for calculating PMCC is r = ∑[(xi - x̅)(yi - ȳ)] / nσxσy, where:
    • ∑[(xi - x̅)(yi - ȳ)] represents the sum of products of deviations of x and y from their respective means.
    • x̅ and ȳ are the means of x and y.
    • σx and σy are the standard deviations of x and y.
    • n is the total number of observations.
  • This formula can be broken down into steps:
    1. Calculate the mean and standard deviation of both variables.
    2. Subtract the mean from each observation to get the deviations.
    3. Multiply the deviations of x and y for each observation.
    4. Sum these products to get the numerator.
    5. Multiply the standard deviations of x and y.
    6. Multiply this by number of observations to get the denominator.
    7. Divide the numerator by the denominator to get the PMCC.
  • If you do not have the standard deviations, the PMCC can also be calculated as r = [ nΣ(xy) - ΣxΣy ] / √[nΣx² - (Σx)² ][nΣy² - (Σy)² ]. Be sure to understand both formulas and when to use each.

Using the Product Moment Correlation Coefficient (PMCC)

  • The PMCC is an important tool in statistics as it can help to summarise a scatter plot and quantify relationships between variables.

  • It is often used in a preliminary analysis to investigate a possible linear relationship between two variables before using more advanced techniques.

  • Keep in mind the PMCC only assesses the strength and direction of a linear relationship. It does not tell you anything about the slope or specific model of the relationship, and it may not be accurate for non-linear relationships.

  • Be aware that a strong correlation does not establish causation - there may be other variables at play or the relationship may be due to chance.

  • Interpreting the PMCC requires careful judgement. It’s important not to overstate the importance of a weak correlation or ignore relevant context.