Current and past literature discusses critical levels, detection limits, and their use in analyzing data collected for numerous analytical processes.  The use of the modified Bessel functions is a straightforward approach to readily arrive at both quantities.  One can readily Use this approach to validate the Brodsky formulation for paired counting.  Using Modified Bessel functions, Neyman-Pearson confidence intervals are also determined. In analyzing data from counting systems, confidence intervals help quantify results with negative net counts and lack precision. Described in this article is the use of a computer code to generate tables for p-values, critical levels, detection limits, and confidence intervals.

INTRODUCTION

A personal computer is used to generate exact results for p-values, critical levels, detection limits, and confidence intervals using the fact that if x and y are Poisson distributed random variables with expectations a and b respectively, then the probability of the difference of x and y is given by:

P(x-y = k) = exp(-a-b) sqrt((a/b)^k) I_|k|(2sqrt(ab)), Eq. (1)

where I_k(2sqrt(ab)) is the modified Bessel function of the real variable [2sqrt(ab)] and of integral order k.^[1] The function I_k(2sqrt(ab)) is defined by a power series^[1] and ïkï denotes the absolute value of k. Only for paired counting is it true that the difference between the gross count and the background count is the difference between two Poisson distributed random variables.

It is necessary to have a routine, such as the one described by Potter,^[2] capable of generating accurate numbers with sufficient precision for the modified Bessel functions.

P-VALUES
 

The literature in fields such as medicine, epidemiology, biostatistics, and radiobiology frequently uses

p-values to describe the results of various scientific investigations. The p-value of a statistical test is the probability of obtaining an observed value or a more extreme observed value when the null hypothesis is true.^[3] The p-value is calculated after the experiment has been conducted. The smaller the p-value, the less likely the null hypothesis is true and the more likely the alternative hypothesis is true.

When the null hypothesis is true a = b in eqn (1). Therefore, P(x-y = k) is given by

P(x-y = k) = exp(-2b) I_|k|(2b). Eq. (2)

Summing the above formula yields the p-value p(M), the probability of observing a net count of M or more when the null hypothesis is true. As a result p(M) is:
 

p(M) = P(x-y = k) = exp(-2b) S I_|k|(2b). Eq. (3)

^{k=M k=M}

In a similar manner the p-value q(N), the probability of observing a net count of N or less when the null hypothesis is true, is determined. Consequently q(N), is as follows:

q(N) = P(x-y = k) = exp(-2b) S I_|k|(2b). Eq. (4)

^{k=N k=N}
 

It is observed that if N = -M in Equation 4, then q(-M) = p(M). So it is only necessary to calculate p(M) for nonnegative values of M. Using addition, one may determine the p-value for a two-sided test. The quantity b is estimated from the background count.

The paper by Potter^[2] gives a function in C++ for calculating the modified Bessel functions of integral order. It is then a trivial matter to calculate the p-values given in Table 1, using a personal computer (a Pentium II® running Windows 98®) and double precision arithmetic. The values entered into Table 1, are those values with four decimal places closest in absolute value to the computed value.

In particular, for the counting of radioactive samples, activity is said to be detected at a level of significance a, if the p-value is equal to or less than a.

Table 1. "p-values" for extreme low-level, paired counting.
 

M	B=0	B=0.1	B=0.3	B=0.5	B=1	B=2	B=3	B=4	B=5
7	.0000	.0000	.0000	.0000	.0000	.0010	.0046	.0112	.0200
6	.0000	.0000	.0000	.0000	.0003	.0038	.0129	.0259	.0404
5	.0000	.0000	.0000	.0001	.0016	.0131	.0327	.0546	.0757
4	.0000	.0000	.0002	.0011	.0084	.0390	.0739	.1051	.1314
3	.0000	.0001	.0027	.0093	.0372	.1001	.1486	.1842	.2112
2	.0000	.0042	.0282	.0592	.1305	.2177	.2646	.2941	.3148
1	.0000	.0865	.2003	.2671	.3457	.3965	.4167	.4283	.4361
0	1.0000	.9135	.7997	.7329	.6543	.6035	.5833	.5717	.5639

CRITICAL LEVELS, DETECTION LIMITS

The critical value is the value of the net count used to make the decision between activity detected and activity not detected. If the net count is greater than the critical level it is concluded that activity was detected. The probability of concluding there is activity when there is no activity is called the error of the first type or the level of significance. Usually the error of the first type is given the symbol a. For discrete random variables, it is not possible, in general, to determine critical values yielding errors of the first kind equal to a specified value for a. Brodsky^[4] and Currie^[5] choose the actual error of the first kind to be equal to or less than a; this paper uses the same approach. Using Equation 2, together with the function for the modified Bessel function^[2] and a personal computer, the critical level for a specified a can be readily determined by summing the right tail of the probability density function.

The error of the second kind is the probability of concluding there is no activity on a sample when there is activity present and is usually denoted by the symbol b. Because the expected value for the gross count is restricted to nonnegative values, it is possible to have the error of the second kind equal to a predetermined value for b. For the computations of this article, the detection limit is that value, with two decimal places, that yields an error of the second kind closest in absolute value to b.

Utilizing the above discussion, critical levels and detection limits for a specified background B, a specified a = 0.5, and a specified b = 0.5 can be determined in a straightforward manner. Using a personal computer and double precision arithmetic Table 2, is constructed for a = 0.010 and b = 0.05. In Table 2, L_C is the critical level, L_D is the detection limit, Err1 is the true error of the first kind, and Err2 is the true error of the second kind.

Table 2. Critical Levels and Detection Limits for Paired

Counting when a = 0.010 and b = 0.05
 

B	LC	Err1	LD	Err2
0	0	0	3.00	0.05000
0.1	1	0.004248	4.88	0.04991
0.2	2	0.000950	6.51	0.04993
0.3	2	0.002726	6.61	0.04995
0.4	2	0.005525	6.71	0.04986
0.5	2	0.009271	6.80	0.05001
0.6	3	0.001977	8.27	0.05012
0.7	3	0.003141	8.35	0.05010
0.8	3	0.004614	8.43	0.05003
0.9	3	0.006388	8.51	0.04992
1.0	3	0.008446	8.58	0.05005
1.1	4	0.002195	9.97	0.05001
1.2	4	0.002933	10.04	0.04991
1.3	4	0.003796	10.10	0.05005
1.4	4	0.004783	10.17	0.04991
1.5	4	0.005891	10.23	0.04999
1.6	4	0.007116	10.29	0.05005
1.7	4	0.008452	10.35	0.05009
1.8	4	0.009893	10.41	0.05011

CONFIDENCE INTERVALS

Confidence intervals provide a way to interpret both measurements that lack precision and measurements with a negative net count. Neyman-Pearson principles, as described by Pearson and Hartley,^[6] are used in conjunction with Equation 1, to compute confidence intervals of the form [0, ##.##] for an arbitrary confidence level, t. Frequently, the confidence level is expressed as a percent by multiplying t by 100.

B denotes the observed background count. Analytically, the results of the measurement process are depicted in the upper half plane with the observed net count, OC, as abscissa and the expected net count as ordinate. The results of activity measurements on a particular sample will lie along a line of fixed, but unknown, expected net count. Where possible, for each measured pair of numbers, B and OC, a corresponding nonnegative value for the expected value of OC is determined such that the probability of realizing OC or fewer counts is (1-t). For each value of B > 0, a smooth curve can be drawn through the points. Whatever be the state of nature, the probability of a measurement lying on or to the right of the smooth curve is at least t.

The computer code presented in Potter^[2] can be readily modified to calculate confidence intervals for an arbitrary level of confidence t. Table 3, was constructed using a personal computer and double precision arithmetic.

Table 3. Confidence Intervals (99%) of the Form [0, ##.##] for

the Expected Net Count for Extreme Low-level, Paired Counting
 

OC	B=0	B=0.1	B= 0.5	B= .75	B=1.0
-3	NA	0	0	0.95	1.69
-2	NA	0	2.1	2.88	3.48
-1	NA	2.27	4.1	4.69	5.18
0	4.61	4.94	5.92	6.39	6.80
1	6.64	6.86	7.61	8.00	8.35
2	8.41	8.58	9.20	9.54	9.86
3	10.05	10.19	10.73	11.04	11.32
4	11.60	11.73	12.22	12.50	12.76

Zero is entered in the table because at the 99% confidence level it is impossible to get a positive value for the expected value of the net count; at a specified higher level of confidence, a positive value for the expected net count may exist. NA is entered in the first column because when B = 0 it is impossible to get a negative net count.

CONCLUSION

Many diverse fields such as health physics, medicine, epidemiology, biostatistics, and radiobiology frequently use p-values to describe the results of various scientific investigations. Decisions on whether or not to take action based on data analysis rely on exact results for p-values, critical levels, detection limits, and confidence intervals. It is necessary to utilize a computer code capable of generating accurate numbers with sufficient precision for the modified Bessel functions. This article describes the use of one such computer code to generate tables for p-values, critical levels, detection limits, and confidence intervals.

    About the Author

    Bill Potter has a bachelor's degree in Engineering-Science and master's degrees in Engineering Science-Nuclear Engineering and Mathematics from the University of California at Berkeley.  He has additional graduate work at Berkeley in nuclear engineering, statistics, medical physics, mechanical engineering, industrial engineering and operations research together with a wide range of technical interests.  He has been a guest with the Health Physics Department at Lawrence Berkeley National Laboratory. Bill is a member of the Health Physics Society, the Central Rocky Mountain Chapter of the Health Physics Society, and the Colorado Section of the American Nuclear Society.

REFERENCES

1. Feller, W., "An Introduction to Probability Theory and its Applications," Vol. II., New York: John Wiley & Sons, 1966.

2. Potter, W. E., "Neyman-Pearson Confidence Intervals for Extreme Low-level, Paired Counting," Health Phys, 76:186-190, 1999.

3. Woolson, R. F., "Statistical Methods for the Analysis of Biomedical Data," New York: John Wiley & Sons, 1987.

4. Brodsky, A., "Exact Calculation of Probabilities of False Positives and False Negatives for Low Background Counting," Health Phys., 63(2): 198-204, 1992.

5. Currie, L. A., "Lower Limit of Detection: Definition and Elaboration of a Proposed Position for Radiological Effluent and Environmental Measurements," Washington, DC: U.S. Nuclear Regulatory Commission, Report No. NUREG/CR-4007, 1984.

6. Pearson, E. S., Hartley, H. O., "Biometrika Tables for Statisticians," Vol. I. Cambridge: University Press, 1966.

Trademarks

Pentiun II-450 processor, Intel Corporatiion, 3065 Bowers Avenue, Santa Clara, CA 95051.

Windows 98, Microsoft Corporation, One Microsoft Way, Redmond, WA 98052-6399.

Visual C++6.0(Standard Edition), Microsoft Corporation, One Microsoft Way, Redmond, WA 98052-6399.