 Open Access
 Total Downloads : 412
 Authors : V. B. Ghute, D. M. Zombade
 Paper ID : IJERTV3IS052054
 Volume & Issue : Volume 03, Issue 05 (May 2014)
 Published (First Online): 07062014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Nonparametric Control Charts for Monitoring Process Variability
D. M. Zombade
Department of Statistics, Walchand College of Arts and Science,
Solapur, (M.S.), India.

B. Ghute
Department of Statistics, Solapur University,
Solapur413255, (M.S.), India.
Abstract–In this paper, two nonparametric control charts are developed for monitoring the process variability. The charts are Shewharttype charts and are based on nonparametric two sample tests for testing equality of variance developed by Sukhatme and Mood. The performance of the proposed control charts is evaluated through average run length for the normal, double exponential and uniform distributions.
Keywords: Average run length, Control chart, Process variability, Statistical process control, Nonparametric tests.

INTRODUCTION
Control charts are statistical process control tools that are widely used for controlling and monitoring a process. Shewhart R and S control charts are most popular control charts for monitoring process variability. Both of these control charts are designed and evaluated under the assumption that the underlying distribution of the quality characteristic is normal. In real applications, there are many situations in which the process data come from a nonnormal distribution which need to be monitored by appropriate control charts. To monitor such type of data, development of control charts that do not depend on a particular distributional assumption is desirable. Nonparametric control charts can serve this purpose. The main advantage of a nonparametric control chart is that it does not assume any probability distribution for the characteristic of interest. A formal definition of nonparametric or distributionfree control chart is given in terms of its in control run length distribution. The number of samples that needs to be collected before the first outofcontrol signal given by a chart is a random variable called the runlength; the probability distribution of the runlength is referred to as the runlength distribution. If the incontrol run length distribution is same for every continuous distribution then the chart is called distributionfree or nonparametric (Chakraborti et al. (2004)).
In literature, several nonparametric control charts are proposed for monitoring location of a univariate process. Some of these are based on sign and/or rank statistics by assuming a known incontrol target value for process location. Chakraborti et al. (2001) presented an extensive overview of the literature on univariate nonparametric control charts. Bakir (2004) developed a distributionfree Shewhart control chart for monitoring process center based on the signedranks of
grouped observations. Bakir (2006) proposed Shewhart, CUSUM and EWMA control charts based on signedranklike statistics of grouped data for monitoring a process center when incontrol target center was not specified and studied the robustness of the charts against outliers. There exist only few articles on nonparametric charts for monitoring process variability. Lehmann (1975) suggested using nonparametric tests for the equality of two variances for use as control statistics in nonparametric control charts for variability. Control charts using tests statistics for comparing two variances would require obtaining an initial sample (of size m) when the process is considered to be incontrol. Then at each sample time i, a sample of size n is obtained from the process, and the pooled sample of size (m + n) is obtained. The observations in the pooled sample then are ranked from smallest to largest, and some statistic based on the ranks of the observations is calculated. Das and Bhattacharya (2008) proposed a nonparametric control chart for monitoring process variability based on Convers squared rank test for variance. Das (2008) developed two nonparametric control charts for monitoring process variability based on two nonparametric tests. Murakani and Matsuki (2010) proposed a nonparametric control chart for dispersion based on the rank sum statistic.
When the process distribution is normal, Shewhart R and S charts are appropriate control chart for monitoring the process variability. If underlying process distribution is nonnormal, then the need of development of nonparametric control chart based on appropriate nonparametric test arises. In this paper, we introduce two Shewharttype nonparametric control charts for monitoring process variability for the case that the location parameter is under control. The proposed nonparametric control charts are based on two sample nonparametric tests proposed by Sukhatme (1956) and Mood (1954). These are most powerful test statistics for detecting scale shifts. The performance of the proposed charts is assessed for both the in control state and outofcontrol state under different underlying distributions.

MATERIALS AND METHODS

Sukhatme test based nonparametric control chart
Sukhamte (1956) proposed a nonparametric test for two independent samples dispersion problem. Suppose we want to compare two independent random samples
X (X1 , X2 ,…, Xm ) and Y (Y1 , Y2 ,…, Yn ) which
m (N 2 1)
are drawn from absolute continuous distributions and differ only in the scale parameters. Let X and Y be the arbitrary
E (M)
12
m n (N 1) (N 2 4)
measures of dispersion of X and Y respectively then problem of testing of hypothesis is H0 : X Y against
H1 : X Y . The Sukhamte test statistic for testing null hypothesis is defined as,
and Var (M)
180
For N greater than or equal to 30, we may consider the normalized random variable W
M E (M)
1 m n
W
Var (M)
(4)
T m n D (Xi ,Yj ) ,
(1)
and perform the test on the basis of tabulated values of the
i1 j1
where D (X , Y) 1 if
0
either 0 X Y or Y X 0 otherwise
standard normal distribution.
For N less than 30, it is not advisable to use directly the normal approximation. In that case, Laubscher recommended
We reject hypothesis if T is too large or too small. The mean
1
the use of a correction for continuity yielding the following
test statistic:
and variance of the statistic T is given by,
Var ( T ) (m n 7)
48 m n
For a large sample,
E ( T ) and
4
W M E (M)
Var (M) 1
2 Var (M)
(5)
T E (T)
Z (2)
Vat (T)
has a standard normal distribution and the test is performed on the basis of tabulated values of the standard normal distribution and the test is performed on the basis of tabulated values of the standard normal distribution.
We consider Z as the control chart statistic for the nonparametric control chart for monitoring process variability and the chart is referred as NPS chart. We consider X (X1, X2 ,…, Xm ) , as reference sample of size m from an incontrol process and that Y (Y1, Y2 ,…, Yn ) be an arbitrary test sample of size n. The sample statistics Z computed from independent observations from the process are plotted against an upper control limit UCL = 3 and LCL = 3. The process is considered outofcontrol when a plotted point lies above UCL or below LCL.

Mood test based nonparametric control chart
Mood (1954) developed a nonparametric test for equality of variances.
Suppose, we have two independent random
samples X (X , X ,…, X ) and Y (Y , Y ,…, Y ) .
and perform the test agai based on the tabulated values of the standard normal distribution.
We consider W as the control chart statistic for the nonparametric control chart for monitoring process variability and the chart is referred as NPM chart. We consider X (X1, X2 ,…, Xm ) , as reference sample of size m from an incontrol process and that Y (Y1, Y2 ,…, Yn ) be an arbitrary test sample of size n. The sample statistics W computed from independent observations from the process are plotted against an upper control limit UCL = 3 and LCL = 3. The process is considered outofcontrol when a plotted point lies above UCL or below LCL.


PERFORMANCE OF THE PROPOSED CONTROL CHARTS
To examine the ability of proposed NPS and NPM charts to detect variability shift in a process, we consider underlying process distributions as normal, double exponential and uniform with mean zero and variance one. The uniform distribution is considered as process distribution to see the effect of a light tailed distribution and double exponential distribution is considered to see the effect of heavy tailed distribution on the performance of proposed nonparametric
1 2 m
1 2 n
control charts. Consider a process where quality characteristic
We wish to test H0 : X Y against H1 : X Y . Let R1 R2 …. Rm be the combined samples ranks of the Xvalues in increasing order of magnitude. The Mood test
of interest X is distributed with mean and standard
deviation . Let 0 and 0 be the incontrol values of and
o respectively. When a shift in process standard deviation
statistic for testing null hypothesis is defined as,
occurs, we have change from the incontrol value 0
to the
n N 12
o ( 0 1)
M R i 2
, where N m n
(3)
outofcontrol value 1 0
. Therefore,
i 1
The mean and variance of the statistic M is given as
when control chart for variability is employed, the process shifts are measured through
1
0
. When
1, the process is considered to be in
of the proposed control charts are computed using 10000 simulations for sample size of n = 10, 15, 20 and 25.
control . For 1 an increase in occurs and for 1, decrease in occurs. Computer programs written in C
language are used to study the performance of the proposed control charts. The incontrol and outofcontrol ARL values
Table 1 to Table 4 provide the ARL values of the proposed nonparametric control charts when the underlying process data actually follows normal, double exponential and uniform distributions with sample sizes n =10, 15, 20 and 25 respectively.
Table1: ARL values of NPS and NPM charts when n = 10
Shift
NPS Chart
NPM Chart
Normal
Double Exponential
Uniform
Normal
Double Exponential
Uniform
1.0
332.45
340.16
339.58
328.93
332.94
335.48
1.2
158.54
189.13
116.89
135.24
174.71
85.03
1.4
85.85
133.77
53.03
62.06
92.96
30.26
1.6
51.65
75.38
29.67
33.09
54.86
14.83
1.8
34.50
52.40
19.90
20.13
36.10
9.16
2.0
24.85
40.09
14.01
13.66
24.69
6.43
2.2
18.59
30.24
10.71
9.91
18.15
5.01
2.4
15.00
24.80
8.70
6.82
12.63
3.71
2.6
12.24
20.71
7.44
6.13
11.67
3.44
2.8
10.04
17.16
6.38
5.13
9.51
2.99
3.0
8.84
14.98
5.60
4.39
7.96
2.69
3.5
6.40
10.73
4.45
3.34
5.79
2.25
4.0
5.24
8.57
3.71
2.71
4.45
1.95
4.5
4.36
7.05
3.26
2.34
3.67
1.79
5.0
3.82
5.88
2.92
2.10
3.16
1.67
Table2: ARL values of NPS and NPM charts when n = 15
Shift
NPS Chart
NPM Chart
Normal
Double
Exponential
Uniform
Normal
Double
Exponential
Uniform
1.0
414.70
419.74
415.25
420.16
415.82
411.57
1.2
164.85
213.42
105.95
137.21
184.70
66.75
1.4
68.36
103.25
35.08
46.02
78.83
17.62
1.6
34.13
58.25
16.18
20.50
39.69
7.49
1.8
19.71
34.98
9.46
11.25
23.02
4.53
2.0
12.61
23.20
6.41
7.10
14.85
3.19
2.2
8.94
16.49
4.80
5.00
10.45
2.44
2.4
6.56
12.65
3.76
3.59
6.75
1.91
2.6
5.29
9.76
3.14
3.05
5.95
1.80
2.8
4.30
8.03
2.69
2.59
4.83
1.64
3.0
3.72
6.65
2.41
2.23
4.05
1.53
3.5
2.67
4.57
1.93
1.74
2.85
1.32
4.0
2.20
3.53
1.67
1.49
2.26
1.23
4.5
1.86
2.83
1.52
1.36
1.92
1.18
5.0
1.66
2.43
1.42
1.28
1.71
1.16
Table3: ARL values of NPS and NPM chars when n = 20
Shift
NPS Chart
NPM Chart
Normal
Double
Exponential
Uniform
Normal
Double
Exponential
Uniform
1.0
332.45
340.16
339.58
418.01
414.41
416.31
1.2
158.54
189.13
116.89
109.29
166.97
45.11
1.4
85.85
133.77
53.03
31.19
58.02
10.23
1.6
51.65
75.38
29.67
12.73
26.67
4.42
1.8
34.50
52.40
19.90
6.61
14.51
2.65
2.0
24.85
40.09
14.01
4.09
8.92
1.95
2.2
18.59
30.24
10.71
2.93
6.11
1.58
2.4
15.00
24.80
8.70
2.28
4.50
1.42
2.6
12.24
20.71
7.44
1.87
3.46
1.28
2.8
10.04
17.16
6.38
1.66
2.87
1.20
3.0
8.84
14.98
5.60
1.47
2.44
1.16
3.5
6.40
10.73
4.45
1.25
1.81
1.09
4.0
5.24
8.57
3.71
1.15
1.51
1.05
4.5
4.36
7.05
3.26
1.09
1.34
1.04
5.0
3.82
5.88
2.92
1.07
1.24
1.03
Table4: ARL values of NPS and NPM charts when n = 25
Shift
NPS Chart
NPM Chart
Normal
Double
Exponential
Uniform
Normal
Double
Exponential
Uniform
1.0
416.96
420.62
420.04
418.44
424.64
426.27
1.2
108.62
154.37
55.64
90.68
139.41
33.15
1.4
31.47
55.39
13.23
22.46
44.64
6.83
1.6
12.92
24.67
5.44
8.39
19.05
2.96
1.8
6.82
13.41
3.19
4.42
9.86
1.88
2.0
4.14
8.32
2.25
2.83
5.81
1.46
2.2
2.96
5.61
1.78
2.05
4.10
1.27
2.4
2.27
4.15
1.51
1.65
3.05
1.16
2.6
1.88
3.26
1.36
1.44
2.41
1.11
2.8
1.64
2.62
1.26
1.29
2.02
1.07
3.0
1.48
2.28
1.19
1.20
1.76
1.05
3.5
1.24
1.69
1.10
1.09
1.39
1.02
4.0
1.14
1.42
1.05
1.04
1.21
1.01
4.5
1.08
1.26
1.03
1.02
1.13
1.01
5.0
1.05
1.17
1.02
1.02
1.08
1.00
Examinations of Table 1 to Table 4 lead to the following findings:

Incontrol ARL values of the proposed NPS and NPM control charts for different process distributions are approximately same.

Outofcontrol ARL values of NPM chart are smaller than that of the NPS chart. Therefore, NPM chart is more efficient than NPS chart for normal, light tailed uniform and heavy tailed double exponential distributions.

For normally distributed data, both NPS and NPM charts performs better than double exponential data.

For uniformly distributed data, both NPS and NPM charts perform better than normally and doubly exponential data.


CONCLUSIONS

In this paper, two nonparametric control charts are developed for monitoring process variability. The performance of the proposed control charts is studied by simulation under normal, light tailed and heavy tailed
distributions. Our simulation study indicates that the NPM control chart is more efficient than NPS control chart for detecting shifts in process variability for different process distributions. Both NPM and NPS control charts perform better when underlying process distribution is light tailed.
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