Censoring: a new approach for detection limits in total-reflection X-ray fluorescence

It is shown that the detection limits in the total-reflection X-ray fluorescence (TXRF), which restrict quantification of very low concentrations of trace elements in the samples, can be accounted for using the statistical concept of censoring. We demonstrate that the incomplete TXRF measurements containing the so-called “nondetects”, i.e. the non-measured concentrations falling below the detection limits and represented by the estimated detection limit values, can be viewed as the left random-censored data, which can be further analyzed using the Kaplan–Meier (KM) method correcting for nondetects. Within this approach, which uses the Kaplan–Meier product-limit estimator to obtain the cumulative distribution function corrected for the nondetects, the mean value and median of the detection limit censored concentrations can be estimated in a non-parametric way. The Monte Carlo simulations performed show that the Kaplan–Meier approach yields highly accurate estimates for the mean and median concentrations, being within a few percent with respect to the simulated, uncensored data. This means that the uncertainties of KM estimated mean value and median are limited in fact only by the number of studied samples and not by the applied correction procedure for nondetects itself. On the other hand, it is observed that, in case when the concentration of a given element is not measured in all the samples, simple approaches to estimate a mean concentration value from the data yield erroneous, systematically biased results. The discussed random-left censoring approach was applied to analyze the TXRF detection-limit-censored concentration measurements of trace elements in biomedical samples. We emphasize that the Kaplan–Meier approach allows one to estimate the mean concentrations being substantially below the mean level of detection limits. Consequently, this approach gives a new access to lower the effective detection limits for TXRF method, which is of prime interest for investigation of metallic impurities on the silicon wafers.