Linear model

If the controlled factor, x, is numerical, we would normally expect that the mean response would change fairly smoothly with x.

the simplest model is for the mean response to increase linearly with it,



yij  =  
(explained
by factor
)
β0 + β1 xi


 + 

(unexplained)
εij
 

for i = 1 to g and j = 1 to ni

where εij  ∼  normal (0, σ)

This model has two parameters, β0 and β1 that give flexibility to our model for the explained variation. The model is therefore said to have two degrees of freedom.

Of these parameters, the slope parameter, β1, is most important. If the slope is zero, the explanatory variable does not affect the response at all.

Quadratic model

Models for experimental data vary in how the mean response depends on the explanatory variables. A different model for a single numerical explanatory variable that allows for some curvature in the relationship is the quadratic model,



yij  =  
(explained
by factor
)
β0 + β1 xi + β2 xi2


  +  

(unexplained)
εij
 

for i = 1 to g and j = 1 to ni

where εij  ∼  normal (0, σ)

This model has more flexibility in matching how the explanatory variable affects the response — the mean involves three parameters, that can be adjusted to get a good fit to the data, so the model has three degrees of freedom.

Generalising from the observed factor levels

Linear and quadratic models are valid for any value of x. It would be a mistake to extrapolate far beyond the x-values that were used in the experiment since we have no information about the curvature of the relationship outside the range of the data. However:

The models can be used to predict the response at intermediate values of x that were not used in the experiment.


Hardness of ball bearings

The hardness of steel ball bearings is related to the rate X at which they were cooled after being made. Data were obtained from an experiment where two ball bearings were cooled at each of several rates of cooling.

  Hardness Index  
Y
  Rate of cooling  
(°C/min), X
    Hardness Index  
Y
  Rate of cooling  
(°C/min), X
48.60
47.80
47.60
46.70
46.20
45.70
46.55
46.57
46.49
41.82
41.40
42.10
42.01
41.67
38.96
40.97
10
10
15
15
20
20
25
25
30
30
35
35
40
40
45
45
  38.71
37.00
35.88
36.25
39.23
34.18
34.59
37.56
33.49
33.93
31.02
31.57
26.99
28.38
50
50
55
55
60
60
65
65
70
70
75
75
80
80

The diagram below shows the data with a straight line representing a potential linear model.

It only takes two points to define the position of a straight line, so there are two red arrows that can be dragged to adjust the model (i.e. to adjust the values of the two parameters, β0 and β1). Adjust the line 'by eye' to make the model means close to the data points (and give small residuals).

Quadratic model for antibiotic effectiveness data

A quadratic curve can be defined by the position of any three points on it and the diagram below allows three such points to be dragged. (Hence three degrees of freedom.)

Drag the three arrows to match the curve as closely as possible to the data. (Again remember the target of small residuals.)