Transcript Design of Control Charts to Monitor Mean and Variance

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 X1, X2, …,
2
N(

,

) , then X
Xn from
2
is independent of S and
n
1
2
X   X j ~ N (  ,  / n ),
n j 1
(n  1) S

2
2
n
 ( X j  X ) / ~  .
2
j 1
2
2
n 1
Therefore, inference for 
and  can be done using X
2
and S 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 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
n
1
X i   X ij ,
n j 1
n
1
2
( X ij  X i ) .
S 

n  1 j 1
2
i
The 3-sigma X-bar chart
plots the sample means
against

UCL    3
n
CL  
LCL    3

n
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.
ARL of (X-bar, S) Combination
When n = 5
a
b
0.0
0.5
1.0
2.0
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
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
Mi  maxUi , Vi ,
where
Xi  
Ui 
,
/ n
2




(
n

1
)
S
1
i

Vi    pchisq
;
n

1
.

2





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
 Xi   


 / n 
are iid U (0,1),
any test of uniformity can
be turned into a chart.
ARL Based on Kolmogorov
Test When n = 5
a
0.0
b
1.0
2.0
0.25 Inf
54.5 1.2
1.0
0.50
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
8333.3
0.5
ARL Based on Cramer-von
Mises Test When n = 5
a
0.0
b
0.5
1.0
2.0
0.25 Inf
2083.3
1.3
1.0
0.50
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
50000
ARL Based on AndersonDarling Test When n = 5
a
b
0.0
0.5
1.0
2.0
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
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
 Xi   
Ui  
,
 / n 
 ( n  1) S


Vi  pchisq
;
n

1
.
2

 

2
i
Then
Fi  2 log(1  Ui )(1  Vi )
~ .
2
4
ARL Based on Fisher’s Test
When n = 5
a
b
0.0
0.5
1.0
2.0
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
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
H0: Mean   and SD  
versus the composite
alternative
H1: Mean   and/or SD  .
ARL Based on Likelihood
Ratio Test When n = 5
a
b
0.0
0.5
1.0
2.0
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
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
 1 (n  1) S
  
2
n 
2
i



n/2
 1 X 
i

exp  

 2 / n 

2
 1 (n  1) S
exp 
2
 2 
 n / 2
n
exp(n / 2).


2
i

 

Conclusion:
Mean  and standard
deviation  are functionally
related under the normality
assumption, even though X
2
and S are statistically
independent.