A Multivariate Subgroup Rationality Test

Abstract

In one of our recent articles (Holmes, D. S., Mergen, A. E. (2003). Testing rationality of subgroups in multivariate control charts. In: The 2003 Northeast Decision Sciences Institute Meeting Proceedings. Providence, Rhode Island, March 27–29, Fariborz Partovi, 286–288.) we proposed an aggregate measure for checking rationality of the subgroups in multivariate control charts, such as T ² charts. The purpose of this article is to build on that measure and propose a similar multivariate test to check the rationality of each subgroup used in the chart. The proposed test uses the statistical process control (SPC) approach; i.e., maintains the time sequence of the data. The method is based on a test for equality of two covariance matrices (Kramer, C. Y., Jensen, D. R. (1969). Fundamentals of multivariate analysis—part II: inference about two treatments. Journal of Quality Technology 1(3):189–204.). The first covariance matrix is the one normally described in statistical literature (for example Johnson and Wichern (Johnson, R. A., Wichern, D. W. (1992). Applied Multivariate Statistical Analysis. New Jersey: Prentice Hall 92–122.) The second matrix used in the test is the multivariate mean square successive difference (MSSD) covariance matrix (Holmes, D. S., Mergen, A. E. (1993). Improving the performance of T-square control chart. Quality Engineering 5(4):619–625.). If the test indicates a significant difference of these two matrices, this is an indication of the lack of rationality of the corresponding subgroup(s) being used on the multivariate control chart. The test result for each subgroup can then be plotted on a control chart to explore which subgroup(s) may be the cause of the nonrationality. Two examples of the use of the proposed method are given.

Keywords:

• Subgroup rationality
• Multivariate statistical process control
• T ² control charts
• Mean square successive difference (MSSD)
• MSSD covariance matrix