C++ Computer Code for Exact Decision Levels And Errors of type I When The Sample Count Time Is An Integer Times Greater Than The Background Count Time.
W. E. Potter (Consultant, Sacramento, CA) and J. Strzelczyk (University of Colorado Hospital, Aurora, CO)
In the past there have been papers where the blank (background) is counted for the same amount of time or longer than the sample. In emergency and cleanup operations situations may arise where the sample is counted longer than the blank. In particular, it is well known that the optimum ratio of blank count time to sample count time equals the square root of the ratio of the respective count rates. The presented approach assumes that both the blank count and the sample contribution to the gross count are Poisson distributed. Also it is assumed that the expected blank count is known. The net count is transformed into an integer. A code in C++ computes the exact probability density function for the transformed net count when there is no activity in the sample. The validity of the computations is verified by checking that the sum of probabilities is close to 1.0 and that the expected value of the net count is close to zero. The decision level is determined by summation of the right tail of the probability density function when there is no activity in the sample. Activity is said to have been detected if the observed net count is greater than the decision level. The code runs on the current Microsoft operating systems. The entire C++ code for decision levels is given and a code fragment is given for the probability distribution for the net count when there is activity in the sample. This fragment can be utilized to extend the code to computing confidence intervals and detection limits. The computed results are compared with the usual Poisson-normal approximation for the decision level. Minimally the computed results are valid for expected blank counts in the sample count time <= 300.0, ratios of the sample count time to the blank count time <= 20, and errors of type I >= 0.001. Uncertainty in the expected blank count can be readily examined by utilizing confidence intervals for the expected value of a Poisson distribution in conjunction with the code.