4 PETER SPRENT
Pearson ( 1901) extended Adcock's (equal error variance) solution to the
p-
variate situation, the direction cosines of the normal to a
(p-
1 )-dimensional
hyperplane of best fit now being determined by the smallest principal com-
ponent. These basic solutions for the bivariate or p-variate case with inde-
pendent errors, identically normally distributed with zero means, were redis-
covered by many workers in the 1920s and 1930s; references were given in
Anderson (1984).
Dent ( 1935) appears to be the first to propose what is now often called the
geometric mean functional relationship
estimator of slope, viz.
(2)
(
m )
I/2
b
=
sgn(m
XY)
___r.r_
mxx
where
myy, m x x
are the sample variances and
m
XY
is the sample covari-
ance. This estimator has been widely used, especially in fisheries research.
When no assessment of the error variances, or even their ratio, is possible,
this estimator has intuitive appeal but is generally not consistent.
It
only ig-
nores the problem of identifiability inherent in the problem when we assume
normality but know nothing about the error variances.
During this period a diversity of terminology evolved.
Regression with er-
rors in x, errors-in-variables models, and measurement error models
are still
in common usage. While factor analysis was being developed, at the same
time, as a tool for psychologists, its relation to what we now call structural
and functional relationships (terms themselves to be clearly defined only in
the 1950s; see Section 3) was not then evident. There is a logical distinction
between
regression with errors in x,
where the
y
variable(s) have a "depen-
dent"
status and functional and structural relationships, where all variables
are on an equal footing. However, Anderson ( 1984) discussed the mathemat-
ical equivalence between the
regression with error
models and the
functional
and structural
models.
2. More systematic studies
Reservations about some of the assumptions made by early workers led to
a more systematic study of linear relationships. Roos ( 1937) was concerned
about lack of invariance under scale changes for the Adcock (bivariate) or
Pearson ( p-variate)
principal component
type of solution. Roos proposed an
arbitrarily weighted least squares solution to overcome this difficulty. Lindley
( 1947) pointed out that this led to a generally inconsistent slope estimator.
Wald (
1940) proposed a method of grouping data and, subject to certain
conditions, showed that the slope of the line joining the group means pro-
vided a consistent estimator of
PI . If
there are
n
=
2m
observations and
the groups are of equal size and
u1
,
e1
have finite variance, then it suffices
to take groups
(xi , x
2
, ••• ,
xm)
and
(xm+I , xm+2
, ••. ,
x 2m)
such that
(1)
J(xi +x2 · ·+xm)/m-(xm+I +xm+2 · ·+x2m)/ml
0 as
n-
oo,
and
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