Regression All about going back (regressing) from the points to the function from which they came. We are looking at linear regression.

Regression Finding, using mathematics, the line of best fit. Overview of GCSE Fit by eye Equal numbers of points above and below Going through the mean

A mathematical approach Residuals Aim: to minimise Σei2

An anchor point

An anchor point

An anchor point

An anchor point

An anchor point

An anchor point

Least Squares Minimising Σ ei does not tend to help. We seek instead to minimise Σ ei2 i.e. the sum of the squares of the ei’s. This gives what is called the least squares regression line of y on x.

Least Squares Regression Line It can be shown that minimising Σ ei2 happens when

b is known as the regression coefficient of y on x

Regression line of y on x

Regression line of y on x

Height and Weight

Summary Statistics

Line of regression of y on x

Height and Weight This is the point formed by the means 50 60 70 80 90 100 110 150 160 170 180 190 200 210 220 230 240 Height (cm) Weight (kg) y = 0.760x - 56.293 This is the point formed by the means Notice that this is not the y-intercept of –56.923 since the axis does not begin at 0

Use of the y on x line To predict y values from known x values by substituting for x values. Safe to use if interpolating. When interpolating, results are usually reliable. Dangerous to use if extrapolating. When extrapolating, results are usually NOT reliable.

There is another line

There is another line This time the aim is to minimise the sum of the squares of the horizontal distances from the points to the line. The is called the line of regression of x on y.

The regression coefficient of x on y Regression line of x on y

Line of regression of x on y

Line of regression of x on y

Height and Weight x on y y on x 50 60 70 80 90 100 110 150 160 170 180 190 200 210 220 230 240 Height (cm) Weight (kg) y = 0.760x - 56.293 x = 1.227y + 81.405 x on y y on x

Use of the x on y line To predict x values from known y values by substituting for y values. Safe to use if interpolating. When interpolating, results are usually reliable. Dangerous to use if extrapolating. When extrapolating, results are usually NOT reliable.

Observation

Observation

Observation

Observation

Dependent and Independent Variables If one variable changes as another one changes the former is called the dependent variable and the latter is called the independent variable. In a given situation which is the dependent variable and which is the independent variable will depend on which variable is being used to predict which variable. If you are using a regression line of y on x then x is the independent variable and y the dependent one. If you are using a regression line of x on y then y is the independent variable and x the dependent one!

Controlled Variables Be aware that sometimes one of the variables is controlled e.g, suppose measurements of a variable y are taken every time the x variable has increased by 5. The x variable is a controlled variable. In this case a regression line of x on y would make no sense since you could not minimise the horizontal distances of points from the line since these are fixed. In this case whether you were predicting y from x, or predicting x from y, you should use the regression line of y on x in both cases.

Link between b, b' and r

Link between b, b' and r

Link between b, b' and r

Link between b, b' and r

Link between b, b' and r

Link between b, b' and r

When are the regression lines the same? y on x line

When are the regression lines the same? x on y line

When are the regression lines the same? x on y line

When are the regression lines the same? y on x line x on y line

Final thoughts Be ready to talk in context about the meaning of a and b in the regression equation of y on x, namely y = a + bx. a is the value of y when x = 0. b is the amount by which y increases every time x increases by 1. Remember IN CONTEXT.