多變量品質管制中用來決定製程是否失控最常用統計量是侯特齡T平方,但是侯特齡T平方統計量仍然有些缺點,如當製程失控時因為變數之間不是獨立所以它不能偵測出來是那個變數出問題.所以有許多學者針對在T平方失控下,想找出是那個變數最有可能引起它出問題這方面作研究.在我所題出論文中證明出由變數之間所算出來T平方值等於因子得分點所算出來T平方值.所以我所題出方法是在T平方失控下想找出是由那個因子所導致.

Most multivariate quality control charts for determining whether the process mean vector is in-control or out-of-control are based on aggregate statistics, such as Hotelling’s . When no correlation is present among the characteristics, monitoring with a statistic reduces to running independent Shewhart charts on the individual variables. When correlations do exist, the is a powerful tool; but has a major drawback, that is, it does not gives any direct information as to what has caused the out-of-control condition, when the statistic indicates an out-of-control process.
When the value of a Hotelling’s signals an out-of-control condition, an investigation must be initiated to find out possible causes of the problem by studying not only the individual variables but also the correlations among them. Principal components and factor analysis are two useful tools for studying the correlations. It is proved in this paper that the value of the Hotelling’s based on the original variables is the same as that based on the factor scores, and hence it is reasonable to use a method that is based on the factor scores for determining whether a process is out-of-control or not, and, if any, the factor(s) that had caused the out-of-control condition. We consider the cases with known and unknown population covariance matrix S. The proposed method is easy to implement using S-PLUS package on a personal computer. Finally, we compare our method with several existing methods to illustrate the usefulness of the proposed method.

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