What is double sampling?

Answer 1

Sample size calculations for a continuous outcome require specification of the anticipated variance; inaccurate specification can result in an underpowered or overpowered study. For this reason, adaptive methods whereby sample size is recalculated using the variance of a subsample have become increasingly popular. The first proposal of this type (Stein, 1945, Annals of Mathematical Statistics 16, 243-258) used all of the data to estimate the mean difference but only the first stage data to estimate the variance. Stein's procedure is not commonly used because many people perceive it as ignoring relevant data. This is especially problematic when the first stage sample size is small, as would be the case if the anticipated total sample size were small. A more naive approach uses in the denominator of the final test statistic the variance estimate based on all of the data. Applying the Helmert transformation, we show why this naive approach underestimates the true variance and how to construct an unbiased estimate that uses all of the data. We prove that the type I error rate of our procedure cannot exceed alpha.

Double and multiple sampling plans were invented to give a questionable lot another chance. For example, if in double sampling the results of the first sample are not conclusive with regard to accepting or rejecting, a second sample is taken. Application of double sampling requires that a first sample of size n1 is taken at random from the (large) lot. The number of defectives is then counted and compared to the first sample's acceptance number a1 and rejection number r1. Denote the number of defectives in sample 1 by d1 and in sample 2 by d2, then:

If d1 a1, the lot is accepted.

If d1 r1, the lot is rejected.

If a1 < d1 < r1, a second sample is taken.

If a second sample of size n2 is taken, the number of defectives, d2, is counted. The total number of defectives is D2 = d1 + d2. Now this is compared to the acceptance number a2 and the rejection number r2 of sample 2. In double sampling, r2 = a2 + 1 to ensure a decision on the sample.

If D2 a2, the lot is accepted.

If D2 r2, the lot is rejected.