Model for two factors

The reason for expressing the one-factor model in the form

yij  =  µ   +   βi   +   εij       where β1 = 0

is that it can be extended easily to give a model without interaction for two factors.

Consider an experiment in which factor X has gX levels and factor Z has gZ levels. We use the notation yijk to denote the k'th of the replicates for which factor X is at level i and factor Z is at level j. The model with no interaction can be expressed as:

yijk  =  µ   +   βi   +   γj   +   εijk       where β1 = 0 and γ1 = 0

The parameters are interpreted as follows:

The β-parameters therefore capture all the differences between the levels of X and there are (gX - 1) of them that are non-zero. Similarly there are (gZ - 1) non-zero γ-parameters that capture the differences between the levels of Z.


Strength of asphaltic concrete

A civil engineer conducted an experiment to evaluate how different compaction methods and types of aggregate affect the strength of asphaltic concrete. Two types of aggegate and four levels of compaction were used in the experiment and three specimens were tested at each combination of levels for the two factors. The tensile strength of each specimen (psi) was recorded and will be modelled in terms of the two categorical explanatory variables Aggregate and Compaction.

  Compaction method
Aggregate type Static Regular
kneading
Low
kneading
Very low
kneading
Basalt 68
63
65
126
128
 133
 93
101
98
56
59
57
Silicious 71
66
66
107
110
116
63
60
59
40
 41
44

We will treat Static compression and Basalt as the baseline levels for the two factors. (Any other factor levels could have been used with equivalent results.)

Graphical display of model

The diagram below shows the actual data against two categorical axes (for the two factors). The means from the best model without interaction are shown by the coloured grid.

The blue value under the table is µ in the equation earlier in the page. The four red values are the parameters βi and the two green values are the parameters γj.

Drag the red arrow for Silicious aggregate to change the parameter γ2 and observe that its value is the difference between the mean response for this aggregate and the baseline aggregate (Basalt).

Similarly drag the three red arrows for Regular, Low and Very low compaction and observe that the values for the corresponding parameters are differences from the baseline compaction (Static).

Finally, click Least squares to show the least squares estimates of the parameters. Observe that the model estimates that: