8
PETER SPRENT
While the distinction between the models is certainly logically sound and
theoretically important, modern developments suggest that the practical con-
sequences of the distinction may not always be great from an estimation
viewpoint. Ehrenberg ( 1982) suggested the term
law-like relationships
for
true relationships when observables may be subject to errors, and this em-
braces all underlying structural or functional (and even ultra-structural) re-
lationships. In all cases the validity of assumptions about error structure is
important.
In the absence of replication for each true x there may be difficulties
in justifying assumptions, although in many physical or biological situations
there may be external information about the likely error structure. In par-
ticular,
measurement
errors may be estimable by repeated sampling of the
x
(i.e., by observing several
X
1
corresponding to a given (unknown) x
1
)
even
if there is only one measurement of
r; .
A good example is given by Fuller
( 1987, Example 1.2.1 ), though this is a situation where there are clearly error
components other than measurement error (in the above sense) associated
with the true functional or structural relationship. (See Section 9 below.)
4. Alternative error structures and more general models
The 1960s and 1970s saw a rapid extension to models with a variety of
error structures for both single relationships in the 2, and
p
2 , variate
case. Many anomalies associated with different assumptions about errors
were resolved, and difficulties over confidence intervals were also clarified.
Two reviews that are strongly recommended as reflecting the state of the art
at the times they were written are those by Madansky (1959) and Moran
(1971). More recently, the whole subject area has been reviewed in depth by
Anderson (1984). The discussion appended to Sprent (1966) is wide rang-
ing, and contributions by M. J. R. Healy and by the late E. M.
L.
Beale shed
light on why some of the confidence intervals suggested by earlier workers
for slope proved unsatisfactory. For example, one of these intervals stem-
ming from work by Williams ( 1955) was not distinguishing between a test for
adequacy of a linear model and estimation of slope for an adequate model.
The situation then was analogous to lumping degrees of freedom and sums
of squares for deviations from linearity with those for error in a standard
regression analysis of variance. Both procedures sometimes result in bizarre
conclusions. Gieser and Hwang ( 1987) illuminated the basic difficulty with
confidence sets for errors-in-variables models and gave references to a num-
ber of papers relevant to what they describe as "the long and controversial
history" of this topic.
There was also interest in the 1960s and 1970s in extension to several (lin-
ear) relationships between
p
variates subject to error. This occurred both in
economics, where simultaneous relationships (often involving large measure-
ment errors) had long been studied, and also in the physical sciences. Gieser
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