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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 847-868

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2009.239

\vspace{3pc} \normalsize \centerline{\large\bf METHODS FOR
IDENTIFYING INFLUENTIAL VARIABLES} \vspace{2pt}
\centerline{\large\bf  IN AN OUT-OF-CONTROL MULTIVARIATE}
\vspace{2pt} \centerline{\large\bf   NORMAL PROCESS} \vspace{.25cm}
\author{Chia-Ling Yen and Jen Tang}
\vspace{.4cm} \centerline{\it National Tsing-Hua Unversity and
Purdue University} \vspace{.55cm} 
\begin{quotation}
\noindent {\it 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.\par \vspace{9pt} \noindent {\it Key
words and phrases:} Hotelling's  $T^2$ statistic, hypothesis
testing, influential variables, likelihood, mean vector,
multivariate process control, out-of-control. \par
\end{quotation}

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