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
.