Methods for identifying influential variables in an out-of-control multivariate normal process

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Hotelling's T 2 is a well-known statistic for testing the mean vector of a multivariate normal distribution. Control charts based on T 2 have been widely used in statistical process control for monitoring a multivariate process. Although it is a powerful tool, the T 2 statistic has a practical problem, namely, that a significant T 2 -value that normally signals an overall out-of-control condition in the process mean vector does not provide direct information about which variable or group of variables may have caused this out-of-control condition. We propose a diagnostic method to identify the influential variable(s) for cases with and without a speci- fied out-of-control mean vector. Our approach, based on the likelihood principle, computes the conditional likelihood of a variable or sub-group of variables causing or not causing the overall out-of-control condition. Unlike many existing meth- ods, our method assumes that an out-of-control condition already exists; hence, all conditional likelihoods in this paper are based on non-central distributions of the monitoring/testing statistics. By comparing these conditional likelihoods, we iden- tify the influential variable(s). We use an example from the literature to illustrate our method and to demonstrate its effectiveness.