Statistica Sinica 22 (2012), 847-868

doi:http://dx.doi.org/10.5705/ss.2009.239

METHODS FOR IDENTIFYING INFLUENTIAL VARIABLES

IN AN OUT-OF-CONTROL MULTIVARIATE

NORMAL PROCESS

Chia-Ling Yen and Jen Tang

National Tsing-Hua Unversity and Purdue University

Abstract: 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 specified 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 methods, 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 identify the influential variable(s). We use an example from the literature to illustrate our method and to demonstrate its effectiveness.

Key words and phrases: Hotelling's T² statistic, hypothesis testing, influential variables, likelihood, mean vector, multivariate process control, out-of-control.