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.