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.


Bait Acceptability by Feral Pigs

In this experiment, trapped feral pigs were given a wheat diet on Day 1 then five different feeds on Day 2. Ten male and ten female pigs were used so each feed was given to two pigs of each gender. The response analysed was the difference in feed intake from Day 1 to Day 2.

We will treat Male as the baseline gender and Wheat only as the baseline treatment. The treatment Wheat only is a control treatment so it makes sense to compare the others with it. However Female could equally have been used as the baseline gender. The diagram below shows the means from the best model without interaction as a coloured grid.

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

Drag the red arrow for Female pigs to change the parameter β1and observe that its value is the difference between the mean response for females and the baseline category, males.

Similarly drag the four red arrows for the four non-baseline feeds (Wheat&water, Wheat,water&dye, Wheat,water&1080 and Wheat,water,dye&1080) and observe that the values for the corresponding parameters are differences from the baseline feed (Wheat only).

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