Notation
If two factors, X and Z, are varied in an experiments, a model in which there is no interaction between their effects has the following form.
yi = |
(explained by X) f (xi) |
+ |
(explained by Z) g (zi) |
+ |
(unexplained) εi |
There are various ways to specified the functions f (xi) and g (zi) and these depend on whether X and Z are numerical or categorical.
To express these models more easily, a different notation is needed using multiple subscripts for the observations. If the factor X has a levels and Z has b levels, and there are r replicates for each treatment combination, ...
Notation for response | In words: | Possible subscripts |
---|---|---|
yijk | k 'th of the replicate response values for the treatment combination X = i and Z = j | i = 1 to a j = 1 to b k = 1 to r |
Model for two categorical factors
For categorical factors, we allow each factor level to have a parameter that allows it to have an arbitrary effect. This model can be expressed in the form,
yijk = |
(explained by X) µi |
+ |
(explained by Z) γj |
+ |
(unexplained) εijk |
As in our models for a single categorical factor, there are a parameters, µ1, µ2, ..., µa, for factor X. However we cannot estimate all b of the parameters γ1, γ2, ...,γb, for factor Z. We therefore set γ1 = 0 since this does not restrict the flexibility of the model.
In order to treat both factors in a more symmetric way (and to make it easier to extend the models later), we write the two-factor model in the form:
yijk = µ + |
(explained by X) βi |
+ |
(explained by Z) γj |
+ |
(unexplained) εijk |
where β1 = 0 and γ1 = 0.
The model therefore has (a - 1) parameters describing differences between the factor levels for X and (b - 1) parameters describing differences between the factor levels for Z.
The model with two categorical factors X (a levels) and Z (b levels) has (a + b - 1) degrees of freedom.
Interpretation of parameters
The parameters are interpreted as follows:
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: