Standard parameterisation of one-factor model

Before explaining fully how the two-factor model is described with parameters, it is necessary to further discuss the parameters of the one-factor model.

The one-factor model for an experiment with g levels was initially written in the following form. If there are ni replicates at factor level i, we use the notation yij to denote the j'th of the ni response values at level i of the factor.

yij  =  µi   +   εij       for i = 1 to g and j = 1 to ni

where εij is the normally distributed error. With this parameterisation, µi denotes the mean response at factor level i.

Alternative parameterisations

An alternative but equivalent way to express this model is

yij  =  µ   +   βi   +   εij       for i = 1 to g and j = 1 to ni

where β1 = 0. With this parameterisation,

Level 1 of the factor is called the baseline level and there are (g - 1) of the β-parameters describing differences between the mean response at the other factor levels and the baseline level.

Alternative baseline levels

Restricting β2 = 0, β3 = 0, etc, in the model instead of β1 = 0 sets a different baseline level but is equivalent in the sense that the model remains flexible enough to allow arbitrary means for all g factor levels.

Antibiotic effectiveness

A percentage of any antibiotic binds to blood serum proteins, reducing the effectiveness of the medication. This is of considerable pharmacological importance since increased binding reduces the systemic uptake of the drug. An experiment was conducted using bovine serum to determine the binding percentage for five common antibiotics.

The experiment was repeated four times for each antibiotic.

The following jittered dot plot shows the data. Initially we have not chosen a baseline antibiotic, so the five β parameters are the mean responses for the different factor levels (antibiotics).

Drag the red arrows to adjust the five parameters of the model and note that the model has the flexibility to allow any mean responses for the five antibiotics.

Click the checkbox Baseline and offsets to choose Penicillin as the baseline antibiotic. The parameter µ denotes its mean binding percentage. The β parameters now denote the difference between the mean binding percentages of the other antibiotics and Penicillin. Again note that the model has the same flexibility — the red arrows can be used to give any mean responses for the five antibiotics.

Click Least squares to set the parameters to their least squares estimates. The fitted values are now all equal to their group mean.

Finally, use the pop-up menu to change the baseline category. Observe that although the values of the parameters change, the fitted values from the model remain the same.