HISTORY OF RELATIONSHIPS II
unit; thus X
1
is determined a posteriori and its precise value is out of our
control. In the controlled variable model X
1
is fixed a priori at a value cho-
sen by the experimenter, and this choice has a role in determining the true
but still unknown
x
1

A typical situation is that in which a meter is set to
record, say, a current flow of 1, 2, 3,... amps (these are the
X
1
),
but the
meter has an associated measurement error and so the true
x1
are deter-
mined by our choice of
X
1

The distinction may appear subtle, and there is
perhaps a temptation to believe that one has a controlled variable situation
when such an assumption is of dubious validity. For example, a controlled
variable model would not be appropriate if one increased the current flowing
in a circuit until some observed response (e.g., a fuse blowing) occurred and
recorded the metered current at which that response took place. Here X
1
(when the fuse blows) is not fixed a priori. A useful way to view the differ-
ence between the Berkson model and the usual error-in-variables regression
model is to note that Berkson assumed that the true predictor variable
x ,
given observed
X,
has conditional mean
X;
in the usual errors-in-variables
model we assume that X , given
x ,
has conditional mean
x .
9. Errors in equations
A common type of error in lawlike relationships is often referred to as an
error in the equation. This may arise if relevant variables are left out of the
equation. The effect may sometimes be obvious from a simple analysis of
residuals (e.g., if a linear relationship is fitted when the data obviously de-
mands at least a second-degree term). In other cases a multitude of potential
variables may not have been recorded, and the combined effect of omitting
these may be much like an additive error that is simply swallowed in the e1
term of the model. Sometimes practical knowledge about the physics, chem-
istry, biology, or whatever of the real-world situation will indicate whether
we are likely to have errors in equations as well as errors in variables over
and above measurement errors. Fuller ( 1987) incorporated additive errors in
equations as a special aspect of measurement error; however, it would seem
that in the broadest context errors in equations may be either errors arising
from failure to measure some relevant variable or may reflect inadequate (or
inappropriate) model specification.
The situation is well illustrated by the data used in Example 1.2.1 in Fuller
( 1987) where a linear relationship between soil nitrogen and corn yield was
postulated. Fuller used the example to show how quantitative estimates of
measurement error of soil nitrogen (based on replicated soil samples and
chemical analyses) could be used to get a more appropriate estimate of slope
than that obtained ignoring such errors. However, any agrnomist familiar
with plant/nitrogen relations would suspect (as the data in the example sug-
gest) that there is much more to this sort of measurement error to disturb the
linear relationship. There is room for argument as to which factors should be
regarded as errors in variables and which as errors in equations. We outline
a few of the relevant factors.
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