Model

As mentioned earlier, most models that are used in experimental design have the form


yi  =  
(explained)
µi

 + 
(unexplained)
εi

where the unexplained variation is modelled with a normal distribution,

εi   ∼   normal (0, σ)

Since we are assuming that there is no structure to experimental units in this chapter — they either appear identical or we are ignoring any information about their differences — all explained variation results from the treatments that we have applied to them. If the i'th experimental unit gets treatment treatmenti, the mean response is therefore

µi   =   (treatmentsi)

In experiments with two factors, treatments are combinations of the levels of the two factors. If the i'th experimental unit gets level xi of factor X and level zi of factor Z, the mean response is some function of xi and zi ,

µi   =   (xi, zi)

Importance of models

In simple experiments with a single factor, most of the data analysis can be explained without explicit use of models. However formally writing the models that are being used for more complex experiments is more important. The analysis of complex experiments is strongly related to the models that are used.

This section may seem a little abstract but it is important — its importance will not be apparent until later in this e-book.


Example of model

The diagram below shows a possible model for an experiment in which Factor A and Factor B can each take 3 levels. For each treatment (i.e. each combination of factor levels), the response has the normal distribution shown in the diagram.

Rotate the diagram to get a better feel for the model. Note in particular that the normal distributions all have the same standard deviation, σ = 1, but that their means depend on the treatment.

Click Take sample a few times to generate typical data that might arise from this model if each treatment is used for two experimental units — i.e. if there are two replicates.