Radioanalytical data interpretation when the ratio reading/median is lognormally distributed

The purpose of this paper is to assist those who might be confronted by non-normal and non-homoscedastic error distributions representable by continuous probability density functions. Methods are presented to demonstrate how mathematical algorithms can be developed to obtain a "best fit" calibration line and how uncertainty ranges in interpretations of unknowns can be obtained from the calibration. The data used to demonstrate these methods were obtained from Brookhaven National Laboratory fission track analysis data for plutonium in urine. Examination of the variability in the fission track analysis data, during the period of time that the demonstration data were collected, revealed that the deviations from the mean were neither normal nor lognormal, but the ratios of tracks divided by the median at each plutonium level were lognormally distributed. Consequently, the differences between the logarithms of observed tracks and the median were normally distributed. The new "best fit" line was obtained by minimizing a reduced chi-square statistic made up of the squared differences in logarithms, divided by the variance in logarithms and degrees of freedom. Thus, to detect a worker urine sample to be above the 58-person "control" population 95 percentile [about 3.2 microBq (85 aCi)] at the 95% probability level (0.05 Type H error) would now require an average of about 11 microBq (300 aCi) per sample, compared to 5 microBq per sample (132 aCi per sample) in a previous paper. This paper presents the algorithms used to obtain the new calibration line and the uncertainty distributions of interpretations at various analyte levels. The importance of maintaining process control over the statistical interpretation of bioassay data as well as for the radiochemical procedures for achieving the lowest feasible level of detection is demonstrated.