What is the product moment correlation coefficient?

• The product moment correlation coefficient (PMCC) is a way of giving a numerical value to?linear correlation?of bivariate data
• The PMCC of a sample is denoted by the letter
  • can take any value such that
  • A positive value of ?describes positive correlation
  • A negative value of ?describes negative correlation
  • If ?there is no correlation
  • means perfect positive correlation and ?means perfect negative correlation
  • The closer to 1 or -1, the stronger the correlation
• The gradient does not change the value of

How is the product moment correlation coefficient calculated?

• You must learn how to use your calculator to calculate value of the PMCC, ?for the relationship between two variables
• All calculators are different and you should make sure you can calculate the PMCC on your personal calculator
  • Make sure you know how put your calculator into the statistics mode
    • You will be given the option to turn the frequency on or off, choose off for most calculations of the PMCC
  • With the statistics mode switched on on your calculator, there will be a ‘statistics’ option, followed by a regression option in the form A + BX
    • Your calculator will give you two columns into which you can input the and ?data values
  • Once the data has been entered into your calculator, choose the ‘r’ value from the ‘STAT’ options

At AS Level you learned how to use linear regression models to describe a relationship between two variables. However, it is possible for two variables to have a relationship that does not fit a linear model, but still shows a pattern based on exponential growth or decay. A linear regression model is only appropriate if the PMCC is close to 1 or -1.

What forms can non – linear regression models take?

• If a?bivariate data set?appears to have a non – linear relationship it could fit an exponential model
  • A non – linear regression model could take the form or ?where?a, n, k?and?b?are constants
• It is possible to use logarithms to rearrange the non – linear form of the model to obtain a?linear regression model?which can then be used to examine trends in the data
  • If the regression model takes the form the data should be coded from - values to - values using and
    • If ? for constants a and n, then
    • Plotting against ?will give a linear graph
    • The y – intercept would be ?and the gradient of the line would be?n
    • This can be shown by taking logarithms of both sides
  • If the regression model takes the form the data should be coded from values to values using and
    • If for constants?k?and?b ,?then
    • Plotting against ?will give a linear graph
    • The y – intercept would be and the gradient of the line would be
    • This can be shown in the same way by taking logarithms of both sides
    • For example:

Take logarithms of both sides

Use the addition law for logarithms

Use the power law for logarithms

• Using logarithms to code the data in this way is called?changing the variables

How can non – linear regression models be used?

• Non – linear regression models can be used in much the same way as?linear regression models
• By coding the original data using logarithms (changing the variables) a regression line of?Y?on?X?can be found
  • This can be used to make predictions for data values that are within the range of the given data (interpolation)
  • Making a prediction outside of the range of the given data is called?extrapolation?and should not be done
• The non – linear regression model can then be found by substituting and ?back into?the?X?and?Y?values in the regression line and rearranging

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