PMCC & Non-linear Regression (OCR A Level Maths A: Statistics): Revision Note

Product moment correlation coefficient (PMCC)

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 $r$

  • $r$ can take any value such that $-1\le r\le 1$

  • A positive value of r  describes positive correlation

  • A negative value of r describes negative correlation

  • If $r=0$ there is no correlation

  • $r=1$ means perfect positive correlation and $r=-1$ means perfect negative correlation

  • The closer to 1 or -1, the stronger the correlation

• The gradient does not change the value of $r$

Non-linear regression

Previously, 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 are non-linear regression models?

• 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 $y=a{x}^n$ or $y=k{b}^x$ 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 $y=a{x}^n$ the data should be coded from $x$- values to $y$- values using $X=\log x$ and  $Y=\log y$

    • If $y=a{x}^n$  for constants a and n, then $\log y=\log a+n\log x\mathrm{or}Y=nX+\log a$

    • Plotting $\log x$ against $\log y$ will give a linear graph

    • The y – intercept would be $\log a$ 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 $y=k{b}^x$ the data should be coded from $x$ values to $y$ values using  $X=x$ and  $Y=\log y$

    • If $y=k{b}^x$ for constants k and b , then $\log y=\log k+x\log b$ or $Y=\left(\log b\right)X+\log k$

    • Plotting $x$against $\log y$ will give a linear graph

    • The y – intercept would be $\log k$ and the gradient of the line would be $\log b$

    • This can be shown in the same way by taking logarithms of both sides

    • For example:

$y=k{b}^x$

Take logarithms of both sides

$\log y=\log \left(k{b}^x\right)$

Use the addition law for logarithms

$\log y=\log k+\log b^x$

Use the power law for logarithms

$\log y=\log k+x\log b$

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

How do I use non–linear regression models?

• 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 $\log x$ and $\log y$ back into the X and Y values in the regression line and rearranging