SUMMARISING AND PRESENTING DATA

PRESENTING DATA FOR TWO CONTINUOUS MEASUREMENTS

Regression

Example - Effects of storage conditions on shelf life

To examine the effects of storage conditions on shelf life of a product several experiments were conducted. One involved measuring the moisture content of samples of the product at different levels of relative humidity. The results were as follows:

Relative Humidity (%) x: 30   35   40   45   50   55
Moisture Content  (%) y:  7   10   11   12   15   16

Java applet "Regression"

Points on the scatter plot lie almost on a straight line so we can represent the relationship between moisture content and relative humidity by an equation

1. We want to estimate the constants a and b in this equation
2. We want to predict the moisture content for a specific relative humidity

Fit a line to points "by eye". From the graph

at x = 35, y = a + 35b = 9

at x = 50, y = a + 50b = 14

Solve for a and b:

35b + a = 9
50b + a = 14

Subtract the first equation

so the equation of the straight line is approximately

y = - 2.67 + 0.33x

In general we need a systematic way of finding a straight line

y = a + bx

It is not satisfactory to fit a line "by eye" or arbitrarily take two points on the line the calculate a and b because different people would obtain different equations.

Method of Least Squares

The most commonly used systematic method for estimating a and b is called the method of least squares.

The vertical distance from point (x_i, y_i) to the line y = a + bx is y_i - (a + b.x_i)

The sum of the squared distance is

The method of least squares consists of finding values of a and b which minimise S. To find the minimum, you differentiate S with respect to a and b, set the derivatives and to zero, and solve for a and b. (The following derivation does not need to be remembered)

Hence the required values for a and b are given by the solution of the equations

The solution is obtained by solving for a in equation (1)

For moisture data, calculation of the least squares line is as follows:

So the line, estimated by the method of least squares, is

y = -2.98 + 0.35x

(compared with the approximation

y = -2.67 + 0.33x

obtained "by eye")

• The equation summarises the relationship between the x and y variables so it can be used instead of a table of values.
• To estimate the average moisture content for a relative humidity of 37% substitute x = 37 into the equation.

y = -2.98 + 0.35(37) = 9.97, i.e. approx 10%

• Usually predict for x values within the scope of the model i.e. between 30% and 55% relative humidity.

Also, the points do not lie exactly in a line - there is some random variation. Therefore the estimates and and predictions of y values are also subject to some variation.