Multivariate Quality Control Based on Regression-Adiusted Variables

Abstract

When performing quality control in a situation in which measures are made of several possibly related variables, it is desirable to use methods that capitalize on the relationship between the variables to provide controls more sensitive than those that may be made on the variables individually. The most common methods of multivariate quality control that assess the vector of variables as a whole are those based on the Hotelling T ² between the variables and the specification vector. Although T ² is the optimal single-test statistic for a general multivariate shift in the mean vector, it is not optimal for more structured mean shifts-for example, shifts in only some of the variables. Measures based on quadratic forms (like T ²) also confound mean shifts with variance shifts and require quite extensive analysis following a signal to determine the nature of the shift. This article proposes Shewhart and cumulative sum (CUSUM) controls based on the vector Z of scaled residuals from the regression of each variable on all others. Each component of Z is the (Neyman-Pearson) optimal single-test statistic for testing whether that variable is shifted in mean. The Shewhart charts plot the components of Z, and the CUSUM charts are based on accumulation of components of Z, leading one to anticipate good performance by the charts. This is verified by some average run length calculations. The vector Z also has the valuable interpretive property that signals given are for shifts in the mean, or shifts in the variance, of particular variables rather than global signals indicating some unspecified departure from control.

KEY WORDS:

• Cumulative sums
• Hotelling T²
• Regression residuals