UNION INTERSECTION TEST IN INTERPRETING SIGNAL FROM MULTIVARIATE CONTROL CHART

Statistical Process Control (SPC) has been a very important discipline in quality control study since pioneered by Walter A. Shewhart in 1920s. Control charting is one of the important tools in SPC and has received wide attention from researchers as well as practitioners. The complexity and the impracticality in monitoring several univariate control charts for a multivariate process has made many practitioners use a multivariate control chart instead. Its usage gives a better control of the overall Type I error and the interdependency among variables is retained. Unfortunately, a multivariate control chart is not able to pinpoint the responsible variable(s) once an out-of-control (OOC) signal is triggered. Many diagnostic methods have been proposed to overcome this problem but all of them have their own limitations and drawbacks. The applicability of a diagnostic method for a limited number of variables, lack of physical interpretation, the complexity of the computation procedure and lack of location invariance are among the factors that have inhibited the implementation of multivariate charts. Lack of comparative studies for various diagnostic methods also makes it difficult for practitioners to choose an appropriate diagnostic method.
This study highlights some problems that might arise in a comparison of diagnostic methods and makes suggestions to overcome them, hence, making the results of a comparative study more relevant and reliable. The effects of several factors such as the size of the deviation in a mean vector, the combination of various sizes of shifts in a mean vector and the inter-correlation among the variables on the performance of diagnostic methods are studied and a summary of the suitability of certain diagnostic methods for certain situations is given. This study presents a new comparison involving two diagnostic methods adapted from the methods proposed by Doganaksoy, Faltin and Tucker (1991) and Maravelakis et al. (2000). A problem related to the usage of eigenvectors with similar eigenvalues is revealed in this study and suggestions from previous studies regarding this matter are presented.
Due to lack of multivariate approaches in dealing with the interpretation of a multivariate control chart signal, this study proposes a new method which embraces the principles of Union Intersection Test (UIT) in diagnosing an OOC signal. A thorough discussion of the UIT principle, the hypotheses, the test statistic and the application of the union intersection technique in the diagnosis problem is presented. An extension of the first comparison study is which includes the proposed method is carried out. The performance of the new diagnostic method is studied and its strengths and weaknesses are discussed. A simplified version for the new method, involving application of spectral decomposition, is also proposed. By using this simplified approach, the common practice of considering multiple types of covariance matrices in a comparison study of diagnostic methods can be avoided to some extent. This study is concluded with a few suggestions of potential further work.