Tolerance Bounds and C_pk Confidence Bounds Under Batch Effects

Abstract

The capability index C _pk for a process, that produces parts with normally distributed characteristic X, is defined as C _pk = min(U − µ, µ − L)/(3σ) = (T - ∣µ − v∣)/(3σ), where U and L are upper and lower specification limits for X, µ and σ are process mean and standard deviation, and v = (U + L)/2, T = (U − L)/2. Using a sample X ₁,…, X _n of independent observations from N(µ, σ ²) Chou et al. (1990) [with clarification by Kushler and Hurley (1992)] showed how to get lower confidence bounds for C _pk . Here we extend this methodology to cover the situation where samples come in batches and the intra batch correlation reduces the amount of independent information. In parallel we also apply this extension to the closely related tolerance bounds or confidence bounds for quantiles. Introducing the simple trick of effective sample size these problems are linked quite successfully to existing tables for tolerance bounds or C _pk confidence bounds. The basic idea is to “approximate” the complicated data situation with an i.i.d. scenario with reduced overall sample size. The approximation is anchored by analysis to the two extreme situations where the within batch correlation is zero or one. For the in-between cases the effective sample size is chosen on a simple heuristic basis, namely by matching the variances of the sample mean under the batch effect model and its i.i.d. approximation. The coverage properties of the resulting method, examined by simulation, were found to be reasonably accurate near the extreme cases and mildly conservative in-between.