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PU8UCATI0NS Allies aa7E37

NBSIR 81-2222

Some Conservative Statistical

Approaches for Presenting

Interlaboratory N.D.A. Enrichment

Measurements

Clifford H. Spiegelman

Center for Applied Mathematics

National Engineering Laboratory

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

March 1981

Final Report

Prpared for

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81-2222

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SOME CONSERVATIVE STATISTICAL . APPROACHES FOR PRESENTING no.

INTERLABORATORY N.D.A. ENRICHMENT

MEASUREMENTS ^ >

Clifford H. Spiegelman

Center for Applied Mathematics

National Engineering Laboratory

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

March 1981

Final Report

Prpared for U.S. Nuclear Regulatory Commission

Washington, DC 20555

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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Some Conservative Statistical Approaches for Presenting

Interlaboratory N.D.A. Enrichment Measurements

By Clifford Spiegelman

SECTION 1 . Introduction.

Realistic evaluation of Interlaboratory comparisons of measurements on

prototype and primary standards is important for establishing uniformity of

reported results. The emphasis in this paper is on multichannel analyzer

count measurements, such as those obtained from gamma spectroscopy. These

measurements and other nondestructive assay (N.D.A.) measurements are used to

measure the enrichment (percentage) of special materials, such as 11303, in the

standards. The measurements are based on an estimate of the total number of

counts from the special material. This estimate is called the peak area

calculation. There is no widespread agreement on the correct formulas to

compute peak area, see [ 1 ]. Thus when different laboratories measure

prototype standards (and hence possibly different formulas are used for

computing peak area) it is difficult to distinguish differences among the

standards from superficial differences caused by variations in the peak area

formulas used by different laboratories, see [1].

The central issue to be considered is whether or not M proposed standards

have enrichments sufficiently similar to be certified with the same

enrichment level. Several statistical tests of similarity are considered. For

two of these tests, algorithms are provided for finding linear peak area

formulas which make the standards appear either most or least homogeneous.

These two tests are the usual F-statistic and the standardized range. This

report provides specific procedures for finding the linear peak area formulas

that are best and worst in the sense that they make the prototype standards

look most or least homogeneous.
2

The measurements are a vector of counts. Each entry in the vector is

called a channel. A channel contains an Integer count of the number of

particles at nearly a fixed energy level striking the counter. Elevated

counts in certain channels Indicate the presence of particular radioisotopes.

However, due to residue from past analysis and imperfect counter resolution,

adjustments for "background" must be made to the raw counts. Adjusted counts

are used to provide more accurate estimates of the amounts of radioisotopes

present. One of these adjustments is the peak area computation.

Two components contribute to the peak area computation. One component is

counts from the target radioisotope, the other is background. The peak area

calculation attempts to find the shaded area in Figure 1. This area is

estimated by subtracting an estimate of the area due to background (shown by

dots) from the estimate of total area in the peak region. The area due to

background is estimated from counts in adjacent "background region" channels.

The estimates of area we consider here are linear and are described in detail

later in this section. These are the most common type of area estimates, see

[1]. Some notation is given below.

Let Ij denote the number of counts in channel j from the target

radioisotope.

Let 3j denote the number of background counts in channel j. Let Jj denote

the set of indices of the channels in the peak region and J 2 denote the set of

indices of the channels in the purely background region. Without loss of

generality take = {1, ..., j^} and J 2 = (ji + 1, •••, jz^* (Assume these

regions are given and fixed throughout the remainder of this paper.) Typically

and J 2 are not overlapping. We assume that there are M proposed

standards measured by each laboratory. Since different multichannel analyzers
3

may have different characteristics such as different geometrical construction

it does not make sense to seek a fixed peak area formula for all laboratories.

Instead, different laboratories' measurements will be dealt with individually.

All the notation that follows is for a single laboratory's measurements.

The observables are:

j e Jl

j e J2

The peak area (PA) calculations we consider are linear. Specifically:

PA -

JeJ^

-.1

JeJ2

“ji'j

Since the weights Wj=0 for j=l, 2, ..., 22 would produce a zero peak area

for all standards a constraint must be placed on the weights. Otherwise one

may get the trivial vector = (wi, ...» w-; ) = (0, 0, ..., 0) for the

J2

weights that make the standards appear most homogeneous. The constraint

used here is

I"j = 1

which is standard in numerical quadrature. Since the statistics used in the

next section are scale invariant the value, 1, serves only to rule out the

vector W = 0. In addition it should be noted that nearly all peak area rules

can be written as [W(y)]ty where the vector W(y) is some (often nonlinear)

function of (yj^, ..., yj^) =

yt, see [1].

Replicate measurements are usually made for each standard. The resulting peak

areas are then compared by an analysis of variance procedure, see [2]. Let
4

denote the mean peak area for the k-th standard. Let y denote the

average value of over all the M standards. One possible measure of

M _

homogeneity for the M standards is A2=^(yj^-y)2, if a is small, the standards

1 can be considered homogeneous.

One Important pair of hypotheses is:

Hq: a < 6

vs.

Hi: A > 6

for some appropriate 6.

Other measures may also be considered, such as one based on the range.

Let: R = Max yj^ - Min yj^ . k k

Then another pair of hypotheses is

H'q: R < 6"

vs.

H'l: R >

6'

It is not likely that any set of M standards can contain identical

enrichments. Therefore, it may not be necessary to consider

hypotheses that specify Identical enrichments. We take 6 and 6'

to be strictly positive.

The choice of 6 and 6' cannot be given by any a priori rule. The

laboratory making the measurements will have to choose 6 and 6". If the

standards do satisfy Hq or they must be similar enough to fulfill their

intended purpose, see Section 4.
5

SECTION 2. Some test statistics for the hypotheses.

Let PA(m,n) denote the peak area of the n-th

replicate of the m-th standard, m = 1, ...» M, n = 1, ...» N. Let

PA = )[ PA(m,n)/M*N and PA(m) =

'I PA(m,n)/N

m,n n

Then a good test (the uniformly most powerful invariant unbiased test if peak

areas are assumed to be independent and normal) of Hq

vs. is given by using the noncentral F statistic

F(A/o2, M-1, M(N-D)

nJ] (l^(m) - PA)

2

' m

I (PA(m,n) - PA(m))2

m,n

This statistic has noncentrality parameter A/o2, and M-1 and M(N-l) degrees

of freedom.

Any test of the form

Reject Hq when F > k

do not Reject Hq when F < k

is a good test. The constant k is chosen to provide a desired level of

significance. In applications M(N-l) 6/^(PA(m,n)-PA)2 can be used as m,n

an estimate of the noncentrality parameter. The constant k can be chosen

from the tables in [3].
6

A statistic for testing H^q vs. is based on the range of the

averages PA(m). Let M* and M* denote the indexes corresponding to the

largest and smallest values of PA(m). If there are two or more indexes

corresponding to the largest or smallest value an arbitrary choice may be made

A among them. Let the range of the statistics PA(m) be denoted by R, i.e.,

A R = PA(M*) - PA(M*). We take as an estimate for the variance of R,

2 ^(PA(m,n) - PA(m))2

^2 m,n

a =

M(N-l)

By taking this choice we Ignore the possibility that M* and M* do not

correspond to the populations with the largest and smallest values of y,

respectively. No universally better choice is known by the author.

Then a statistic for testing H"q vs. is:

(R-6^)

T = -

In many applications there are hundreds of counts in each channel. Therefore

the probability distribution of these counts can be well approximated by a

normal distribution, see [4]. In these cases T has approximately the

distribution of student's t with M(N-l) degrees of freedom, see also [5].

The test rejects H'q if

T > K'

and does not reject H^q if

T < K
7

The constant is obtained from any t table. It should be noticed that the

indexes M* and M* may not correspond to the standards with the largest and

smallest theoretical mean level y. Therefore, the constant K'’ may have to be

modified, see [6].

Notice that if either hypothesis Hq or is rejected then multiple

comparison methods should be applied to find the "bad actors" among the group

of standards, see [2]. An example Illustrating the use of these tests is

given in Section 4.

The next section deals with the interaction between the weights W and the

test statistics.

SECTION 3. Finding extreme peak areas.

It is clear that the choice of weights, W, used in defining peak area

affects the values of F and T. It would be wonderful if we could plot F and T

as a function of W. However, 22 * dimension of W, is usually much larger

than 2, see Section 4. This implies that these plots are not practical.

Instead we settle for the comparison of a few choices of weights. The

most important choices are those that the measurement laboratory believes

produce meaningful and accurate peak areas, see [1]. In addition it may be

useful to compute the set of weights which make the statistics F and T take on

their largest and smallest values. Procedures for calculating these weights

are given below.

The weights for the F statistic are provided first. Let denote the

counts in the j~th channel of the n-th replicate of the m-th standard,

j e Ji+J2» l<m<M, l<n<N. The usual * notation is used to denote an
8

average. The average over any set of subscripts is denoted by replacing

the subscript(s) with a doty e.g.j ~ ^

* nm nm

Let the matrix A^, where a superscript t denotes transpose, have its jm-th

entry, aj^, given by the equation aj„j = * This matrix has

dimensions 22 x M. In addition let the matrix (j 2 x NM) have entries whose

( (m-1 )N+n)-th column is:

^Imn “

• • •

Tj^mn “ ^12“*

Recall that = (wj^, ..., Wj 2

)» The F statistic can now be rewritten as:

F = N W^AtAW

The weights which maximize and minimize F subject to ^w-; = 1 are

characterized below. If the elements of B are noirmal and n > 2 then B^B

has full rank j 2

* (The assumption of a normal distribution is only an

approximation to the truth. There remains a very small chance that B^B is not

of full rank. If the rank of B^B is not j 2 , then the rest of the analysis in

this section cannot be done.) It is known that B^B can be rewritten as the

square of a full rank symmetric matrix, Q, i.e., B^B = Q^, see [7].

Let u “ Q W (notice that p does not denote any mean value). Then F

satisfies the following equations:
9

F » NWtQ(Q”lAtAQ”l)QW/uty

I = NM*^(Q“^A^AQ”^)li/litM .

The constraint =1 can be rewritten as

jeJl

Q”^ d = 1 where d is a vector having ones in the first and zeroes in the

last (j2”ji) places, i.e.,

dt = (1 . . . 1 0 . . . 0)

Jl J2

It is known that the p vectors which maximize and minimize

M Q~ ^ A*^ AQ“ 1 ) p/ p

t y

are the eigenvectors corresponding to the largest and smallest eigenvalues

of Q”^A^AQ“1 respectively, (see [7]).

Let v(l), VO2 ) denote an orthonormal set of characteristic

vectors (eigenvectors) for this matrix (see [7]).

The eigenvectors V^j) corresponding to largest and smallest

eigenvalues satisfy

v(i)Q-ld t 0

vO)Q”ld * 0

with probability one and are the solution, i.e., they maximize and

minimize F. (We note that the Independent eigenvectors need only be

orthogonal, and need not be orthonormal. The F statistic is invariant to
10

scale changes.) The untransformed vectors v(i), vO) are only of intermediate

interest. The final choice of weights is obtained from

W = Q"lv .

The study of these eigenvectors (weights) may be of independent interest.

This is particularly true for the weights which minimize the F statistic.

Our analysis is similar to principal components analysis, see [7]. In

particular if identical standards are analyzed by the above analysis then the

eigenvectors help to characterize the measurement process.

Next the weights for the T statistic are found. We use the matrix Q

defined in the last derivation. Consider the M(M-l) vectors Dmm"* =

^-1

Ylm- - Yim'.

^

12

“’ ~ ^^

2“

‘

for all m and m^. Let p = Q W as before. Notice that T

is proportional to max mm Then the vector y which maximizes T is

(y^y)

the vector of greatest length. (Recall that the constraint

= 1 serves only to rule out the null vector.)

J

1

The weights that minimize T satisfy the following quadratic

programming problem:

Min Z W

such that for all m and ra'

^^mm^ ^ Z and

y*^y = 1 where

y = QW.
11

Discussion.

The largest and smallest values of a set of F and T statistics do not have

F and t distributions. The distributions for the F statistic case are given

in [7J under Hq with 6=0. The distributions for the T statistic case have

yet to be worked out, and according to M.A.H. Dempster L 8 J distributions for

stochastic programming problems are complicated.

SECTION 4. An F and T Statistic Example.

The National Bureau of Standards (NBS) is participating in a certification

program for 0303 low enrichment (<3X) standards. At the time of this writing

NBS has only analyzed prototype standards. There are three replicate

measurements for each of twelve prototype standards. The three measurements

for one peak and its backround region, for one of the samples, are shown in

Figure 2. The peak area was computed using the sum rule, i.e., * W2

=* ...

= Wj^= .1 and = Wj ^+2 ® ••• * ^^

2“ where 22 ~ The value of

the F statistic for this peak area formula is shown in Table 1 along with

the individual means PA(m) for each of the prototype standards. The value of

the T statistic is 63.7. These statistics correspond to real differences

among the standards of about (10”^)% enrichment. On the basis of both

statistics we can say the prototype standards are different but good enough

for their intended use.

The weights corresponding to the extreme values of the F and T statistics

were not computed. However it is conjectured that the weights corresponding

to the maximum of these statistics will give large weight to anomalies in the

data. It is also conjectured that if the prototype standards are nearly equal
12

then the weights which minimize the statistics should give appropriate peak

area rules.

SECTION 5. Additional Considerations.

In addition to the T and F other statistics may be worth computing. These

include rank tests, see [9]. For testing Hq'* vs. Hi' the maximum likelihood

ratio statistic is known to be good when N is large.

The minimizing and maximizing weights for the T statistic may not be the

only weights of interest. For exploratory purposes it will be useful to

choose weights orthogonal to the previously chosen ones, which maximize the T

statistic. Finally it may be worthwhile to track the optimum weights over

time, since they are directly related to the necessary background correction.

SECTION 6. Conclusion and Summary.

Two important statistics, F and T, were given for testing the sets of

hypotheses (Hq vs. H]^) and (Hq' vs. These hypotheses are related

directly to two possible measures of the homogeneity of the standards. For

these statistics procedures were given for finding weights which make the

standards look most and least homogeneous. The F and T statistics from these

weights can be used as a basis for conservative decisions about the standards.

For example, the minimum value found for the T statistic is at least as small

as the value of T that would be found using a "proper" choice of peak area

weights. Thus if the minimum value of T is large, one can be sure that no

linear peak area formula could reduce an apparent inhomogeneity of standards.

Acknowledgement: I thank J. R. Rosenblatt for many helpful suggestions.

The data in Section 4 were jointly analyzed with D. Hogben.
13

References

[1] Helmer, R. G. , Lee, M. A. (1980). Analytical Functions for Fitting Pears

from GE Semiconductor Detectors, Nuclear Instruments and Methods 178,

499-512.

[2] Scheffe", H. (1959). The Analysis of Variance, John Wiley, New York.

[3] Tiku, M.L. (1967). Tables of the Power of the F-Test, JASA 62, 525-39.

[4] Feller, W. (1968). An Introduction to Probability Theory and Its

Applications-3rd Edition, Wiley, New York.

[5] Beyer, W. H. (1968). Handbook of Tables for Probability and Statistics,

Second Edition, Part VIII, The Chemical Rubber Co. (CRC), Cleveland.

[6] Gibbons, J.D. , Olkin, I., and Sobel, M. (1977). Selecting and Ordering

Populations, John Wiley, New York.

[7] Anderson, T.W. (1958). An Introduction to Multivariate Statistical

Analysis, John Wiley, New York.

[8] Dempster, M.A.H. Ed. (1974). International Conference on Stochastic

Programming, I. Introduction, Oxford, New York.

[9] Hajek, J. and Sidak, Z. (1967). Theory of Rank Tests, Academic Press,

New York.
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NBS-n4A tREV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET CSee instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 81-2222

2. Performing Organ. Report No.

714

3. Publication Date

4. TITLE AND SUBTITLE

Title: Some Effective Statistical Approaches for Presenting Interlaboratory N.D.A.

Enrichment Measurements

Subtitle: Presenting N.D.A. Enrichment Measurements

5. AUTHOR(S)

Clifford Spiegelman

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

national bureau of STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. NRC AT(49-25)-9009

8. Type of Report & Period Covered

NBSIR

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

NRC

10.

SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Soft>ware Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

SECTION 1. Introduction.

It is important to realistically evaluate interlaboratory measurements on prototype and primary standards. The emphasis in this paper is on nondestructive

assay (NDA) measurements. These measurements are based on an estimate of the total

number of counts from a target source. This estimate is called the peak area calculation. It is necessary to distinguish real differences between laboratory

findings from artificial differences such as those caused by artifacts of individual

peak area calculations. This report provides guidance for assessing the magnitude

of these peak area artifacts. A wide variety of statistical tests for homogeneity

of the peak areas for these standards is considered. For these tests, algorithms are provided for finding linear peak area computations which either make the standards

appear most or least homogeneous. These tests Include the usual F statistics as well as the standardized range.

12. I^Y WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Principal components; multiple comparisons; multivariate analysis, peak areas,

minmax, maxfciln

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

1 ^1 Unlimited

1 1 For Official Distribution. Do Not Release to NTIS 18

r 1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402. 15. Price

Q3] Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $5.00

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