# Answer to Question #10989 Submitted to "Ask the Experts"

*Category: Instrumentation and Measurements — Surveys and Measurements (SM)*

The following question was answered by an expert in the appropriate field:

What is the preferred method to deal with a result that is less than the MDA (minimum detectable activity) but meets or exceeds the L_{c }(critical level)?

Before answering your specific question, I will offer some thoughts on what the critical level, detection limit, and minimum detectable activity represent, how they are obtained, and how they are used. If you don't need the introductory discussion, you may want to skip to the last two paragraphs.

By definition, the critical level, L_{c}, is that net count rate above which net radioactivity is assumed to be present in a measured sample. In many facilities, when establishing the L_{c} value, the a priori assumption is made that the critical level should be such that 0.05 of the time when no net radioactivity is present the analyst will make the false conclusion that radioactivity is present—i.e., when the net counting rate is zero, the critical level will be set at 1.645 standard deviations above the zero level, assuming a normal distribution about the zero count rate value. This fraction, 0.05, is often referred to as the alpha value, a, associated with the false positive count rate. Of course, one may specify an alternative value of a as desired. The count rate can naturally be converted to activity by dividing by the appropriate counting efficiency. Accepting an a value of 0.05 also implies that if a sample contains real net radioactivity at a level close to that, which yields a count rate close to the background level, one would expect that one might falsely conclude 95 percent of the time that no net radioactivity is present. To reduce this false negative rate, it is common practice to use the detection limit concept.

The detection limit, L_{d}, is a net count rate higher than the critical level, often set at a value such that the false negative conclusion will prevail less frequently, the most commonly assumed value being 0.05 of the measurements. In terms of the shape of the assumed normal distribution, the assumption is that the detection limit is set high enough above the critical level so that the tip of the left tail of the distribution curve will extend below the critical level for 0.05 of the measurements—i.e., when the net counting rate is at the value of L_{d}, the separation between the L_{c} value and the L_{d} value will be 1.645 standard deviations of the L_{d} value. This implies that when net radioactivity is actually present at the L_{d} level, one would conclude that no net radioactivity is present 5 percent of the time. The selected value of 0.05 here represents the so-called beta value, ß, which represents the probability of making a false negative conclusion that no net radioactivity is present when it actually is. The minimum detectable activity, MDA, is obtained by dividing the L_{d} value by the appropriate counting efficiency.

From normal counting statistics it can be shown that the detection limit is related to the critical level by:

L_{d} = k^{2}/T_{S } + 2L_{c},

where k represents the number of standard deviations that separate the L_{c} value and the zero value in the zero net count rate distribution, and the (same) number of standard deviations that separate the L_{c} value from the L_{d} value in the L_{d} distribution. The above expression would be different if different values of k were selected for the two distributions—i.e., if a and ß had different values. The gross sample counting time is given by T_{S}. When k^{2}/T_{S} in the above expression is very small, as it often is for longer counting times, the value of L_{d} is twice that of L_{c}.

Now, to finally attempt to answer your question, we would have to conclude that any result that shows a net count rate above the L_{c} value does contain net radioactivity. In my opinion, all such values should be reported as positive. This does not mean, necessarily, that such results should be recorded as significant. This is where the L_{d} and associated MDA concept has been invoked. It has become common practice at many facilities to report activity values that are less than the MDA as simply

Beyond this, we could also make an argument for recording all values, those below as well as those above the L_{c} value. In such a case, when no samples contained any net radioactivity, we would expect, over an extended period, to record as many negative values as positive values (assuming a normal distribution around zero). Such information can be helpful in evaluating the performance of the analytical/counting system by observing perturbations from the expected distribution. Even if the official reporting record includes only values above the critical level, or even exclusively those above the MDA, maintaining a second data log book that includes all results can prove very useful and I believe is well worth the rather small additional effort required.

I hope this is useful to you.

George Chabot, PhD