The effects on the variance of Rp when Y and/or X are not normally
distributed are not fully understood, except for a few exceptions
where some exact results are known (Mielke and Flueck, 1976).
One
of these exceptions that is potentially applicable to radionuclide
data is where Y and X are distributed as bivariate lognormal variables. With Y and X both lognormally distributed, the ratio ¥/X is
also lognormally distributed (Aitchison and Brown, 1969, p. 11) so
that the quantity
Ru
i
>
"ei
=
evo]
log
Ys
Z|
ey
is an estimate of the median ratio in the original (untransformed)
scale.
An approximate confidence interval on the true median ratio
is given by the antilogarithms of the lower and upper confidence
limits UL and uy for
1
n
case 23,
It estimates the median value that would be obtained if all
possible observations had been taken. An advantage of Rs is that confidence limits are easily calculated from tables of the binomial
distribution, as illustrated e.g. by Snedecor and Cochran (1967, p.
124) or Conover (1971, p. 111).
At this point, we can now make a few general recommendations on how
to proceed when estimating an average ratio,
First, view the problem
as a regression of Y on X and plot Y versus X.
If a straight line
relationship through the origin appears reasonable, then the average
ratio given by Y¥/X should be statistically unbiased.
If the relationship is a straight line, but not through the origin, then the ratio
is not constant for all X.
In this case an average ratio does not
adequately summarize the data and it is better to estimate the
regression of Y on X with a non-zero intercept.
We note that the bias discussed in this paper refers to only statistical
bias, f.e. the bias in the estimate arising due to the data not completely satisfying the assumptions underlying the statistical treatment of
the data. We ignore here any systematic biases that may be introduced
due to such things as inappropriate field sampling or sample preparation techniques and analytical bias in the laboratory.
n & 18. x,
NUMERICAL EXAMPLES
ue and Uy are easily obtained since
We have chosen three sets of data from NAEG studies at safety-shot
sites on NTS and TTR to illustrate and compare the various methods
of calculating estimates of an average ratio.
ee
2
> log,
i=l
>=
Fal
1
a
is normally distributed when Y,/X, {£s lognormal.
Thus, an approximate
confidence interval for Ry, is given by the limits
R, - exp {¥.}
and
Ry, + exp {us} .
(6)
Another estimate of the average ratio is the median of the observed
Rs is nonparametric in the sense that no
ratios denoted here as Rs.
assumptions on the form of the distributions of Y, X, or Y/X are made.
As an example of the computation of R5 suppose we have a sample of 5
observations ranked from smallest to largest, say 18, 19, 23, 28 and
The sample median (Rs) is the middle observation, or in this
35,
610
The first set of data consists of 24!aM and ?3972"9pPu concentrations
on the same aliquot from a soil sample collected at 15 random locations
from stratum 3 of the Double Track site, TTR (Gilbert, et al, 1975).
The ratio of interest here is Pu to Am.
A plot of the data is given
in Figure 6, where the points are labeled by the value of their ratio.
A straight line through the origin seems to fit the data very well.
The ratios of the 15 data pairs are ordered from smallest to largest
at the top of the figure.
The ratio estimates, their standard errors
and 95% confidence intervals (CI) are given in Table 1.
The first
method is R, where X is considered to be a constant and Var(Y) is
assumed constantfor all X. Methods 2, 3 and 4 all use the same
estimate Rp = Y/X but with three different estimates of variance,
V(R2), V'(Ro), and V"(R2), respectively. Method 2 is based on the
assumption that X has no varlability while methods 3 and 4 assume X
is a variable.
Method 2 assumes Var(Y) is proportional to X.
Method
5 is R3, the average of the individual ratios.
It assumes that X
is not a variable and that Var(Y) is proportional to x2,
611