PMCC & Non-linear Regression | OCR A Level Maths: Statistics Revision Notes 2018

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 $- 1 \leq r \leq 1$
- 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 $r$

2-5-1-pmcc-diagram-1

Worked example

2-5-1-pmcc-we-diagram-1

Three scatter diagrams, showing observations from different bivariate data sets, are shown above.

(i)

Match each of the three scatter diagrams show above to one of the values of

r

given below. You should use each given value of

r

no more than once.

r = - 0.7134 r = 0.1652 r = 0.8134 r = - 0.9993

(ii)

Sketch a scatter diagram for the remaining value of

r

listed above.

2-5-1-pmcc-we-solution

Non-linear Regression

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 $y = a x^{n}$ 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 $\log y = \log a + n \log x or Y = n X + \log a$
  - 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 $Y = (\log b) X + \log y$
  - 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

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

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

Worked example

The graph below shows the distribution of the height, $h$ m, of a group of children and the amount of time, $t$ hours, they spend napping in the day. It is believed the data can be modelled using the form $t = k h^{n}$ .

2-5-1-non-linear-regression-we-diagram

The data are coded using the changes of variables $X = \log h$ and $Y = \log t$ . The regression line of $Y$ on $X$ is found to be $Y = - 3.5 X$ .

(i)

Find the values of

X

and

Y

for a child that is 75 cm tall and naps for 4 hours per day, giving your answers to four decimal places.

(ii)

Using the regression line, show that a child of height 0.9 metres would be expected to nap for approximately 1.45 hours per day.

(iii)

State an assumption that was made in order to justify the use of the regression line in part (ii).

(iv)

By first substituting

\log h

for

X

and

\log t

for

Y

in the equation of the regression line given, show that the relationship between the height of a child and the time they spend sleeping can be modelled by

t = h^{- 3.5}

2-5-1-non-linear-regression-we-solution-2-part-1

Uo6sLrf6_2-5-1-non-linear-regression-we-solution-2-part-21

Examiner Tip

Be careful when using original and coded data interchangeably, it is easy to forget which one you are working with. Remember that if your regression line was calculated using coded data then you will need to reverse this if finding predictions. Make sure that you are familiar with using logarithms, indices and their laws. Be careful to check which base logarithms were used for coding the data, if $\log x$ was used then it is reversed using $10^{\log x}$ , but is $\ln x$ was used then it should be reversed using $e^{\ln x}$ .

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

Revision Note

What is the product moment correlation coefficient?

What forms can non – linear regression models take?

How can non – linear regression models be used?

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1. Statistical Sampling

1.1 Sampling & Data Collection

1.1.1 Sampling & Data Collection

2. Data Presentation & Interpretation

2.1 Statistical Measures

2.1.1 Basic Statistical Measures

2.1.2 Frequency Tables

2.1.3 Standard Deviation & Variance

2.2 Data Presentation

2.2.1 Data Presentation

2.2.2 Box Plots & Cumulative Frequency

2.2.3 Histograms

2.3 Working with Data

2.3.1 Outliers & Cleaning Data

2.3.2 Intrepreting Data

2.4 Correlation & Regression

2.4.1 Correlation & Regression

2.5 Further Correlation & Regression (A Level only)

2.5.1 PMCC & Non-linear Regression

2.5.2 Hypothesis Testing for Correlation

3. Probability

3.1 Basic Probability

3.1.1 Calculating Probabilities & Events

3.1.2 Venn Diagrams

3.1.3 Tree Diagrams

3.2 Further Probability (A Level only)

3.2.1 Set Notation & Conditional Probability

3.2.2 Further Venn Diagrams

3.2.3 Further Tree Diagrams

3.2.4 Probability Formulae

4. Statistical Distributions

4.1 Probability Distributions

4.1.1 Discrete Probability Distributions

4.2 Binomial Distribution

4.2.1 The Binomial Distribution

4.2.2 Calculating Binomial Probabilities

4.3 Normal Distribution (A Level only)

4.3.1 The Normal Distribution

4.3.2 Normal Distribution - Calculations

4.3.3 Standard Normal Distribution

4.4 Choosing Distributions (A Level only)

4.4.1 Modelling with Distributions

4.4.2 Normal Approximation of Binomial

5. Hypothesis Testing

5.1 Hypothesis Testing

5.1.1 Hypothesis Testing

5.2 Hypothesis Testing (Binomial Distribution)

5.2.1 Binomial Hypothesis Testing

5.3 Hypothesis Testing (Normal Distribution) (A Level only)

5.3.1 Sample Mean Distribution

5.3.2 Normal Hypothesis Testing

6. Large Data Set

6.1 Large Data Set

6.1 Large Data Set

Author: Amber