Statistical Methods in Medical Research 4ed

This discussion is rather more mathematical than many other parts of the book,
although it is hoped that appreciation of the more detailed parts of the argument
is not necessary to gain a general understanding of this topic.
It is most convenient for the following discussion to assume that the data have
been written in the succinct form of (12.39), the longitudinal aspect of the data
being represented by the block diagonal nature of the dispersion matrix of the
residual term. The true dispersion matrix which is, of course, unknown, is
denoted by VT. If the data are analysed by ordinary least squares, that is, the
longitudinal aspect is ignored, then the estimate of the parameters of interest, b,is
^b O ˆ…X T X† 1 X T y, …12:48†
where the subscript O indicates that the estimator uses ordinary least squares.
Despite the implicit misspecification of the dispersion matrix, this estimator is
unbiased and its variance is
…X T X† 1 X T VTX…X T X† 1 : …12:49†
If ordinary least squares were valid and VT ˆ s 2 I, then (12.49) would reduce to
the familiar formula (11.51). However, in the general case, application of (11.51)
would be incorrect, but (12.49) could be used directly if a suitable estimator of
VT were available. Provided that the mean, Xb, is correctly specified, then a
suitable estimator of the ith block within VT is simply:
…y i Xi ^ b O†…y i Xi ^ b O† T , …12:50†
and, if these N estimators are collected together in ^ VT and this is used in place of
VT in (12.49), then a valid estimator of the variance of ^ b O is obtained. The
estimator is valid because, regardless of the true dispersion matrix, the variance
of y is correctly estimated by the collection of matrices in (12.50). This estimator
of variance is occasionally called a robust estimator because it is valid independently
of model assumptions; the estimator of the variance of ^ b O is referred to as a
`sandwich estimator' because the estimator of the variance of y is `sandwiched'
between other matrices in (12.49) (see Royall, 1986).
Although using (12.49) in conjunction with (12.50) yields a valid estimate of
error, the estimates may be inefficient in the sense that the variances of elements
of ^ b O are larger than they would have been using an analysis which used the
correct variance matrix. The unbiasedness of (12.48) does not depend on using
ordinary least squares and an alternative would be to use
^b G ˆ…X T V 1
W X† 1 X T V 1
W y, …12:51†
with the associated estimator of variance being
…X T V 1
W X† 1 X T V 1 1
W VTV W X…XT V 1
W X† 1 :
12.6 Longitudinal data 441