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