ABSTRACT

Least squares estimation

In this book we have generally been concerned with inference about

eects associated with treatments, such as direct treatment and carry-

over eects. On those occasions when we need explicit expressions for

estimators of these eects to be able to compare designs, for example,

we want to obtain them by the simplest route. In this appendix we

show how the correct expressions can be derived in several ways using

dierent models for the expectations of (continuous) cross-over data. In

this context the use of dierent models is an artice to obtain simple

derivations of estimators of eects; it does not imply that analyses based

on dierent models would be equivalent. In particular, it is easier to

manipulate models with sequence (group) eects rather than subject

eects, and we show below how each leads to equivalent estimators, but

analyses based on the two types of model would not lead to the same

estimates of error. We demonstrate equivalences for the following three

cases:

1. A model that contains a dierent parameter for each subject (xed

subject eects)

2. A model with sequence eects but no subject eects

3. A model for the contrasts between the means for each period in each

sequence group

We will be deriving expressions for Generalized Least Squares

(GLS) estimators under a general covariance matrix for the repeated

measurements from a subject. This general framework contains as a spe-

cial case,Ordinary Least Squares (OLS) estimation and includes the

analyses described in Chapter 5. It is assumed that we have a cross-over

design with s sequence groups, p periods and n

i

subjects in each group

and let n =

P

n

i

.