The mathematical assumption underlying the calculation of an average
ratio between two quantities Y and X is that the true ratio is constant at each value of X and that if X = 0, then Y = 0.
This implies
that a plot of Y against X is a straight line through the origin
(illustrated by line A, Figure 2}.
If this is true then the ratio
Y/X which is the slope of the line (in this case 0.5), is constant
for all values of X.
If the relationship between Y and X is a
straight line, but not through the origin (line B, Figure 2), then
the ratio Y/X changes for different values of X
(from 1.5 to
.7}
although the slope of the line is the same (0.5) as that of line A.
In transuranic field studies both the numerator and denominator are
often highly variable. However, the classical approach to ratio
estimation is one of regression through the origin applied to an
experimental situation which assumes that the numerator has variability but that the denominator, which is under the control of the
experimenter, is constant except for possible measurement errors.
In this latter situation the optimal estimator of the ratio depends
on the relationship of the variance of Y to the value of X at which
it is measured.
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Consider first the case where the denominator has very little variability relative to the numerator so that the classical approach is
justified. We assume the regression goes through the origin so that
the slope and the average ratio are synonymous.
Suppose the variance
of Y is the same for all values of X (Figure 3).
This relationship
is illustrated by dotted lines representing roughly + one standard
deviation from the regression line.
In this situation the slope of
the regression Line (estimate of the average ratio) is given by
0.7
R,
1
=
n
{=1
1s
METHODS OF COMPUTING AN AVERAGE RATIO
X,Y /
ia
i
1
Xy
2
where n is the number of data pairs (YX).
0.5
V(R.)
=
> Y,? -
i=l
> X,Y,
}?
y x ;
,
Its variance is given by
(n-1) ¥ X,*
,
If the distribution of Y about the regression line is normal for a
given X, then the 100(l-a) percent confidence interval for Ry is
estimated by
0.5
0
X
FIGURE 2.
Relationship of Ratios to Regression
where t, is the value of the t distribution with (n-1) degrees of
freedom and confidence probability l-a.
Commonly chosen values of
a are ,05 or .01 giving 95% or 99% confidence intervals, respectively.
If the variance of Y increases proportional to X as X increases
(illustrated by dotted lines in Figure 4), then the optimal estimator
of the average ratio (slope) is Ry = Y/X, where
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