Model for explained and unexplained variation
For most experimental data, the response, yi, for the i'th of the experimental units is modelled using a statistical distribution and we assume that all of these distributions are independent of each other.
µi = f (xi, zi, ..., )
Unknown parameters
The form of the function describing the explained variation depends on the characteristics of the response and factors, but it usually involves one or more unknown parameters. The following are examples of possible functions for the explained variation.
• | In an experiment with a single factor that has g levels, the response mean might be modelled as |
µi = βj if the i'th experimental unit gets factor level j | |
This model has one unknown parameter for each factor level, β1, β2, ..., βg. | |
• | In an experiment with a numerical factor, the response mean might be modelled as |
µi = exp( β0 + β1 xi ) if the i'th experimental unit has factor value xi | |
This model has two unknown parameters, β0 and β1. |
Whatever the form of the function, the experimental data are used to estimate the unknown parameters. The parameter estimates reflect how strongly the controlled factors and known structure of the experimental units affect the respones.
Distribution describing unexplained variation
Unexplained variation in the data from the experiment is modelled with a standard statistical distribution. The distribution that should be used depends on the type of response measurement. The following examples indicate some possibilities.
In practice, most experimental data are modelled with normal distributions, even when there is a better alternative. This is partly because normal distributions are a reasonable approximation for many types of data, but also because the analysis and interpretation of the models are much easier.
In this e-book, we will only deal with normal models