STAT 350 Lecture 2

The one way layout as a linear model

The sum of squares decomposition in one example

The data consist of blood coagulation times for 24 animals fed one of 4 different diets. Here are the data with the 4 diets being the 4 columns.

The usual ANOVA model equation is

which we can write in matrix form by stacking up the observations into a column.

Let X denote the design matrix in this formula. Usually we reparametrize the model in the form

which would lead to a design matrix which looked like X above with an extra column on the left all of whose entries are equal to 1. The parameter vector would now be

It will turn out that trying to use this parametrization the different parameters are not identifiable, that is, they cannot be separately estimated, because making bigger by a certain amount and each smaller by the same amount leaves the data unchanged. We usually solve this problem by defining

and

Now it is automatic that so we usually eliminate by replacing it in the model equation by the quantity

This leads to

and

Further analysis of this data.