Comparison of biased and unbiased estimators of variances of qualitative and semi-quantitative results of testing

Unbiased estimators of within-laboratory and between-laboratory (or within reference material unit and between-unit) variances of results of qualitative and semi-quantitative testing are formulated and discussed. Qualitative and semi-quantitative test results were treated as binary nominal and ordinal values, respectively, in framework of the newly developed ordinal analysis of variance (ORDANOVA). It is shown that the difference of the unbiased and the biased estimators of a within-laboratory variance does not exceed 5 %, when the number of replicate tests in a laboratory is larger than 20. Such a difference is increasing when the replicate number is decreasing, not depending on the number of laboratories and the between-laboratory variation, since both the unbiased and the biased estimators are based on the averaged within-laboratory variances. The difference of the unbiased and the biased estimators of the between-laboratory variance depends not only on the number of replicates, but also on the number of laboratories and on the ratio of the contributions to the total variance (the between-laboratory variance and the averaged within-laboratory variance). This difference does not exceed 5 %, when the number of replicates and the number of laboratories are larger than 20 and the ratio of the between-laboratory to the averaged within-laboratory variances does not yield 1. For a limited size of experiment (smaller numbers of replicates and laboratories), the difference is increasing with the size decreasing and can be significant.