What Does the Likelihood Principle Say About Statistical Process Control?

Gemai Chen, University of Calgary

CanadaJuly 10, 2006

It is well know that given a random sample X1, X2, …, Xn from , then is independent of and

.~/)()1(

),/ ,(~1

1

21

222

2

2

1

n

jnj

n

jj

XXSn

nNXn

X

X2S

) ,N( 2

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

X2S

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 Xij, 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 Xi1,…,Xin 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

.)(1

1

,1

1

22

1

n

jiiji

n

jiji

XXn

S

Xn

X

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

nLCL

CLn

UCL

3

3

with a Type I Error probability 0.0027 when the process is in control.

For the S chart, we use a version with probability control limits, where

a probability 0.00135 is assigned to each tail so that the Type I Error probability is also 0.0027 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

0.0 0.5 1.0 2.0

b

0.25 4.8 4.8 4.8 1.1

0.50 51.4 51.2 12.3 1.3

1.00 185.4 30.7 4.5 1.6

1.50 7.3 5.2 2.7 1.6

2.00 2.3 2.1 1.7 1.4

ARL of (X-bar, S) CombinationWhen 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

,V,UmaxM iii

where

.1;)1(

,/

2

21

nSn

pchisqV

n

XU

ii

ii

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.

),1,0(/

Uiidaren

X i

a

0.0 0.5 1.0 2.0

b

0.25 Inf 54.5 1.2 1.0

0.50 8333.3 41.5 2.6 1.0

1.00 185.2 29.9 5.6 1.1

1.50 44.5 19.8 6.6 1.6

2.00 21.1 13.9 6.7 2.1

ARL Based on Kolmogorov Test When n = 5

a

0.0 0.5 1.0 2.0

b

0.25 Inf 2083.3 1.3 1.0

0.50 50000 84.0 2.5 1.0

1.00 185.2 26.0 4.6 1.1

1.50 43.4 18.3 5.8 1.6

2.00 22.3 14.1 6.5 2.1

ARL Based on Cramer-von Mises Test When n = 5

a

0.0 0.5 1.0 2.0

b

0.25 Inf Inf 3.3 1.0

0.50 Inf 408.2 4.2 1.0

1.00 185.2 22.3 3.7 1.1

1.50 13.3 7.0 2.9 1.2

2.00 3.8 3.1 2.1 1.2

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

.1;)1(

,/

2

2

nSn

pchisqV

n

XU

ii

ii

Then

.~

)1)(1(log224

iii VUF

a

0.0 0.5 1.0 2.0

b

0.25 Inf Inf 50000 1.0

0.50 Inf 50000 35.3 1.0

1.00 185.2 20.1 3.9 1.1

1.50 6.5 3.5 2.0 1.1

2.00 2.3 1.8 1.5 1.1

ARL Based on Fisher’s TestWhen 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 nullH0: Mean and SD versus the composite alternativeH1: Mean and/or SD .

a

0.0 0.5 1.0 2.0

b

0.25 3.2 2.3 1.2 1.0

0.50 27.6 16.1 4.0 1.0

1.00 185.2 47.9 7.3 1.2

1.50 15.8 9.1 3.7 1.3

2.00 3.3 2.9 2.1 1.3

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

).2/exp(/2

1exp

)1(

2

1exp

)1(1

2/

2

2

22/

2

2

nnn

X

SnSn

n

ni

i

n

i

Conclusion:

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

2SX