What Does the Likelihood Principle Say About Statistical Process Control? Gemai Chen, University of Calgary Canada July 10, 2006

It is well know that given a random sample X 1, X 2, …, X n from, then is independent of and

Therefore, inference for  and  can be done using and separately.

From a statistical theory point of view, this supports the practice that process mean and variability are usually monitored by two different control charts.

For this presentation, let’s consider the widely used X-bar chart and the S chart.

Now let X denote a certain quality characteristic of a process, let  denote the process mean and let  denote the process standard deviation.

Suppose that X ij, i = 1, 2, … and j = 1, …, n, are measurements of X arranged in sub-groups of size n with i indexing the sub-group number.

We assume that X i1,…,X in is a random sample from a normal distribution with mean  + a  and standard deviation b , where a = 0 and b = 1 indicate that the process is in control.

Define sample mean and sample variance by

The 3-sigma X-bar chart plots the sample means against

with a Type I Error probability when the process is in control. For the S chart, we use a version with probability control limits, where

a probability is assigned to each tail so that the Type I Error probability is also when the process is in control.

The joint behaviour of the (X-bar, S) combination judged by average run length (ARL) is summarized in the following table.

a b ARL of (X-bar, S) Combination When n = 5

Some efforts have been made to use one chart by combining two existing charts, one for the mean and one for the variability.

For example, the Max chart by Chen & Cheng plots

where

Under the same in control ARL of 185.4, the Max chart has the identical ARL performance to the (X-bar,S) combination.

Can we design a control chart which by nature is meant to monitor both mean and variability simultaneously?

Here we report what we have tried. First, as any test of uniformity can be turned into a chart.

a b 0.25Inf ARL Based on Kolmogorov Test When n = 5

a b 0.25Inf ARL Based on Cramer-von Mises Test When n = 5

a b 0.25Inf Inf ARL Based on Anderson-Darling Test When n = 5

It looks that the popular tests of uniformity do not lead to efficient monitoring of the mean and variability changes, especially when the variability of the process decreases.

Next, we consider Fisher’s method of combining tests. Let

Then

a b 0.25Inf Inf ARL Based on Fisher’s Test When n = 5

We see that Fisher’s test leads to a chart that is better than the (X-bar, S) combination for monitoring mean changes and variability increases.

However, this chart can hardly detect any variability decreases.

Finally, we consider the likelihood ration test of the simple null H 0 : Mean   and SD   versus the composite alternative H 1 : Mean   and/or SD  .

a b ARL Based on Likelihood Ratio Test When n = 5

We see that the likelihood ratio test leads to a chart that has more balanced performance monitoring the mean and variability changes than the (X-bar, S) combination or any of the cases considered.

To understand better, let’s have a look at the likelihood ratio statistic

Conclusion : Mean  and standard deviation  are functionally related under the normality assumption, even though and are statistically independent.