Counting critical limit

In this chapter, I will examine the statistical nature of radioactivity counting. Statistics is unavoidably mathematical in nature and many equations will emerge from the discussion. However, only as much general statistical mathematics wiU be introduced as is necessary to understand the relevant matters. I wiU go on to discuss the statistical aspects of peak area measurement, background subtraction, choosing optimum counting parameters and the often superficially understood critical limits and minimum detectable activity. I end with an examination of some special counting situations. 

Critical limit (Lq) — a decision level Is the net count significant ... 

These are considered in some detail by Currie (1968) and from a different perspective by Sumerling and Darby (1981). Note that, with exception of the MDA, the limits are calculated as a count rather than as an activity or other derived quantity. Note also that critical limit and upper limit relate to a measurement just made, whereas detection limit (and the associated MDA) and determination limit pose hypothetical what if questions. 

How can we decide whether any particular measurement near to zero is truly zero or represents a true positive count There must be some level, which we can caU the critical limit, above which we can be confidenf to a degree, that a net count is valid. We might decide that if the count. A, were above a certain number of standard... 

Figure 5.9 Definition of critical limit (the vertical axis represents the frequency of observing a particular count)...

The factor would be selected to provide a predetermined degree of confidence in the conclusion. For example, we may consider that it would be acceptable that if a count happened to be at the critical limit there would only be a 1 in 20, or 5 %, chance that we would judge the count to be present when in reality it was not. This is the same as saying that at the critical limit we would be 95 % certain that the count was not statistically significant. In this case, in statistical probability terms a = 0.05 and, from one-tailed probability tables (Table 5.3), we find that would be 1.645. (We use the one-tailed tables because we are only interested in the level being exceeded on one side, the higher, of the distribution.)... 

Given that this count is not statistically significant, what is the maximum statistically reasonable count The critical limit is used to assess the statistical validity of a calculated net count. If the net count, N, is below or equal to Lq, then the activity must be declared not detected and an upper limit or less-than level quoted. So we wish to define a level which we can be confident (to an appropriate degree) exceeds the actual peak area, if any. We can relate this to the notional distribution of counts we might obtain if we were to count the particular sample a large number of times (distribution (b) in Figure 5.10)... 

In both cases, the square root term is the standard deviation of the count, or of the estimated peak area, calculated in the normal manner, rather than of the background. You may notice that if N happens to be precisely zero, then Equations (5.58) and (5.59) reduce to the critical limit expressions (Equations (5.52) and (5.56)). Quite so. If the net count were zero, we would be 95 % certain that the true count were less than Lq - which is the definition of the critical limit. In spite of this, the upper and critical limits should not be used interchangeably. 

What is the minimum number of counts I can be confident of detecting It is important to appreciate that the critical limit and upper limit are both a posteriori estimates based upon actual measured counts. They are statements of what has been achieved in the measurement. The detection limit answers the a priori question If you were to measure a sample, what would the count have to be for, say, 95 % certainty of detection it is, therefore, a statement of what might be achieved. Detection limit is often confused with the critical limit. However, if the sample activity did happen to be exactly Lq (distribution (b) in Figure 5.10), statistically we would only be able to be sure (or 95 % sure ) of detection in 50 % of cases because the counts would be distributed symmetrically about Lq. It is clear that Lq must be some way above Lq (see distribution (c) in Figure 5.10). 

Prom Figure 5.10, we can see that if the expected count due to a sample was below the critical limit we would almost certainly not detect the activity. If the expected count was above the critical Umit but below the detection limit then we might detect the activity. If it was above the detection limit, then it is more likely than not that we would detect the activity. It is important to realize that it is possible to detect a count below the Lp - the detection limit. At first, this seems perverse. Consider a gamma spectrum. Is it not reasonable to suppose that, if a peak was detectable in 95 % of cases, it would be visible within the spectrum Indeed, it is quite easy to show, by mathematically creating a continuum plus peaks, as in Figure 5.11, that if a peak contains a number of counts equivalent to Lp that it is indeed, in most cases,... 

For example, if we wish to calculate the critical limit, Lq, we must consider the distribution of counts when the... 

When counts are judged against the critical limit and found to be not significant , the upper limit should be quoted, not the MDA. 

One aspect of computer calculation of MDA is worthy of comment. Let us assume that a peak area is measured by the normal peak integration method (Chapter 5, Section 5.4), the peak background being estimated using a few channels beyond the peak limits on either side of the peak. It is a feature of some programs that if the net peak area fails the critical limit (or similar) test, the upper limit (or MDA) is then estimated from the channel counts within the peak region, using them as an estimate of the... 

CRITICAL LIMIT (Lj.) The level below which a net signal carmot reUably be delected. In radiometrics, the number of counts below which, using the conventional confidence interval, we are 95% certain the coimt is part of the background distribution. 

In the presented work an algorithm for the primary radiation filter optimization has been developed and realized in the Mathcad envelope which provides a minimal detection limit of a critical element both at the given X-ray tube power and at the given maximal acceptable count rate. 

Much of the confusion arising in the interpretation of dust counts may be traced to an inadequate appreciation of their actual significance. Many investigators have ignored the limitations of the technique, used and this has caused severe criticism of such counts in evaluating a dusty, environment, particularly in attempts to comprehend the meaning of wide variations so often obtained in results. Therefore, what are the criteria which make dust counting definitely valuable ... 

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