Tests for an Upper Percentile of a Lognormal Distribution Based on Samples with Multiple Detection Limits and Sample-Size Calculation

Abstract

The problem of determining sample size for testing an upper percentile of a lognormal distribution based on samples with multiple detection limits is considered. Two tests, the signed likelihood ratio test and another test based on a pivotal statistic, are outlined. These tests are very satisfactory in controlling type I error rates and comparable in terms of powers. Procedures and R codes for calculating sample sizes for these tests to attain a specified power are given. It is noted that for guaranteeing a given power, increased sample size is necessary due to the presence of detection limits, and the required sample size goes up as the proportion of non-detects goes up. It is also noted that in the multiple detection limit scenario, sample-size determination does not require knowledge of the proportions of non-detects that are expected to be below the individual detection limits; rather, what is required is a knowledge of the overall percentage of non-detects that is to be expected in the entire sample. Sample-size calculation is illustrated using a practical situation.