Questions about the levels of a single factor

We now consider a randomised experiment with a single factor that has several levels.


yij  =  µ

 + 
(explained by X)
βi

 + 
(unexplained)
εij

where β1 = 0. In this and the following few pages, we will look for structure within the different factor levels by examining whether we can make constraints on the values of the β-parameters.

Experiments with a control level

In many experiments about the effect of different interventions (e.g. applying fertiliser to a crop or antibiotic to animals), some of the experimental units are left without any intervention. This treatment is called a control level and is a baseline against which the interventions can be compared.

If there is a control level, two questions are often asked.

Is the control any different from the other treatments?

If we interpret this question as asking whether the mean response for the control treatment is equal to the average of the mean responses for the non-control treatments, it can be expressed as a question about whether the following constraint holds:

β1  =  average (β2, β3, ..., βg)

Since we define β1 = 0, this is equivalent to the sum of the other β-parameters being zero.

Are the non-control treatments different from each other?

This question can also be expressed in terms of constraints on the parameters of the model. If all non-control treatments are equivalent, the following (g - 1) constraints will hold:

β2  =  β3  =  ...  =  βg


Effect of fertiliser on the yield of tomato plants

The diagram below shows results from an experiment to assess the effect of four types of fertiliser on the yield of tomato plants. Five plants were given each fertiliser and a further five were in a control group that received no fertiliser.

Two sets of constraints can be applied with the checkboxes. Clicking All fertilisers equal applies three linear constraints (count the number of equals signs) and constrains the mean yield for all four fertiliser types to be equal.

The other checkbox constrains the mean response for the control treatment (no fertiliser) to be equal to the average of those from the four fertiliser types. When both checkboxes are selected, all five response means are equal.