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The purpose of this article is to study the link between mean shift and inflation coefficient when the underlying null hypothesis is rejected in the analysis of variance (ANOVA) application after the preliminary test on the model specification.

A new approach is proposed to study the link between mean shift and inflation coefficient when the underlying null hypothesis is rejected in the ANOVA application. First, we determine this relationship from the general perspective of Six Sigma methodology under the normality assumption. Then, the approach is extended to a balanced two-stage nested design with a random effects model in which a preliminary test is used to fix the main test statistic.

The features of mean-shifted and inflated (but centred) processes with the same specification limits from the perspective of Six Sigma are studied. The shift and inflation coefficients are derived for the two-stage balanced ANOVA model. We obtained good predictions for the process shift, given the inflation coefficient, which has been demonstrated using numerical results and applied to case studies. It is understood that the proposed method may be used as a tool to obtain an efficient variance estimator under mean shift.

In this work, as a new research approach, we studied the link between mean shift and inflation coefficients when the underlying null hypothesis is rejected in the ANOVA. Derivations for these coefficients are presented. The results when the null hypothesis is accepted are also studied. This needs the help of preliminary tests to decide on the model assumptions, and hence the researchers are expected to be familiar with the application of preliminary tests.

After studying the proposed approach with extensive numerical results, we have provided two practical examples that demonstrate the significance of the approach for real-time practitioners. The practitioners are expected to take additional care before deciding on the model assumptions by applying preliminary tests.

The proposed approach is original in the sense that there have been no similar approaches existing in the literature that combine Six Sigma and preliminary tests in ANOVA applications.