Control Charts for Dependent and Multivariate Measurements

Abstract

The control chart, first developed by W. A. Shewhart, is a useful tool in statistical process control. It is an on-line process control technique used to detect the occurrence of any significant process change, and to call for a corrective action. The construction of a control chart is basically equivalent to the plotting of the acceptance regions of a sequence of hypothesis testing over time. For example, \( \overline X \)-chart is a control chart used to monitor the process mean μ. It plots the sample means \( \overline X \) of subgroups of the observed {X 1,X 2,…} and is equivalent to testing the hypotheses H 0: μ = μ0 vs. H a: μ ≠ μ 0 (for some μ 0 required by the engineers) conducted over time using \( \overline X \) as the test statistic. Here we assume that {X 1, X 2,…} are the sample measurements of a particular quality characteristic from the distribution F whose mean is μ and standard deviation σ. When there is not enough evidence to reject H 0, the process is said to be in control. Otherwise it is said to be out of control. The decision rule to accept or to reject H 0 is based on the value of \( \overline X \), the sample mean of observations taken at each time. These decision rules are graphically displayed in the control chart as the upper and the lower control limits (UCL and LCL). The region between the control limits is the acceptance region of H 0. The process is considered out of control when an observed sample mean falls outside the limits. When this occurs it suggests that the process may have been affected by some assignable causes. Investigation of these causes should then be initiated. As in hypothesis testing, to obtain the control limits, we need to find the sampling distribution of \( \overline X \) — μ when H 0 is true. More precisely, for a given α, we need to locate two values, U and Z, such that, under H 0, $$ P\left( {L < \bar{X} - {\mu_0} < U} \right) = 1 - \alpha $$ (1.1)