ENBIS: European Network for Business and Industrial Statistics

Sample size determination for testing the variance compound in a one-way random effects model

23 September 2009, 10:25 – 10:45

Abstract

Submitted by

Michael Meyners

Authors

Michael Meyners

Affiliation

Nestlé Research Center, 1000 Lausanne 26, Switzerland

Abstract

In a food industrial mixing process, it is important to control and minimize heterogeneity of the distribution of supplements added at low concentrations like vitamins. To assure high product quality, the variability σ^2_H of the concentration must not exceed a certain value δ. Measuring the concentrations of interest is subject to analytical error as expressed by σ^2_A. Therefore, n samples are taken and analyzed k-times. Assuming normality, we consider a one-way random effects model. The purpose of this paper is to determine combinations of n and k minimizing the product n*k while assuring sufficient power for testing H_0: σ^2_H > δ.
One approach is to use the classical upper confidence limit to test H_0. As an alternative, we adapt the more recently proposed test for the reverse hypothesis based on generalized p values to suit our setting.
For both approaches, we analytically simplify the description of the null distribution. Under mild conditions, we show that the power of the two alternative tests does not depend on the mean concentration, and that it depends on δ and σ^2_A only through δ / σ^2_H and σ^2_A / σ^2_H, respectively. We therefore confine ourselves to a limited number of parameter settings. For these we search the combinations of n and k that give a minimal number of measurements n*k while warranting a power of a least 80%.
The comparison between the two approaches indicates that the approach based on generalized p values is more powerful and therefore yields smaller sample sizes. In general, the reduction in number of samples is between 5% and 10%, while it goes up to 20% in exceptional cases. For sufficiently small ratios σ^2_A / σ^2_H, waiving replications (i.e. k=1) and using the classical chi-square test for a single variance is found to be even more powerful.

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