Statistical power for detecting single stratum shift in a multi-strata production process.
Agarwal, Atul ; Baker, R.C.
INTRODUCTION
Many manufacturing firms still suffer from poor quality of
manufactured goods in spite of using quality control charts for over a
decade. These firms continue to face challenges in implementing quality
programs due to difficulties in correctly applying the statistical
process control (SPC) techniques to their processes. Given that a
control chart's function is to monitor a production process,
selection of an appropriate sampling method is crucial for the charts to
function effectively. It is critical that the chart detects changes in
process as soon as possible (with a high degree of sensitivity) after
they occur, especially for processes that are barely capable of meeting
the specifications [Caulcutt, 1995; Evans, 1993]. Osborn (1990) states
that an insensitive control chart may miss out in detecting small shifts
in a process and jeopardize a company's continuous improvement
efforts. Thus, a chosen sampling method can be termed appropriate if it
enables control charts to detect process shifts with greater sensitivity
without generating excessive false alarms [Osborn, 1990; Wheeler, 1983].
The problem of choosing appropriate sampling method is
straight-forward when a production process consists of just one
population (stratum). The control charts ([bar.x] and R) would be based
on a rational sample selected from successive time periods of production
[Grant, 1988; Wadsworth, 1986]. In certain chemical and pharmaceutical
applications the production process may consist of multiple fill-heads,
thus producing a mix of populations (strata). When applying quality
control techniques to monitor such process, the choice of an appropriate
sampling method is not so easy. For example, a four cavity machine could
be producing four distinctly different populations (strata) and the
choice of different sampling methods would affect the sensitivity of
detecting a shift in one or more strata. Most often, whichever is
"simplest", "most convenient", or "seemingly
logical" is used to determine the sampling method [Caulcutt, 1995;
Mayer, 1983; Osborn, 1990; Squires, 1982].
In the past, most literature has focused on studying different
aspects of process shift for production processes producing only one
population. The seminal studies by Scheffe (1949) and King (1952)
develop operating characteristic (OC) curves for [bar.x] and R charts,
when samples are rational and process standards are given. Olds (1961)
investigates power characteristics of control charts for detecting
process shifts in the context of rational sampling for both the "no
standard" and "standard given" cases. Costa (1997) shows
that the [bar.x] chart with variable sample size and sampling intervals
is more sensitive than the traditional [bar.x] charts in detecting even
moderate shifts in the process. Osborn's (1990) work emphasizes the
importance of "statistical power" to a QC practitioner's
ability in detecting a particular shift in the process average and
laments its lack of understanding among practitioners who utilize
process control techniques. He also emphasizes the role of sample size
in enhancing the statistical power of control charts. The work by Davis
et al. (1993) improvises on the explanation of the "statistical
power" of a [bar.x] control chart as used by Osborn (1990). Unlike
any previous study, Palm (1990) and Wheeler (1983) present tables of the
power function for the [bar.x] chart using multiple detection rules for
the mean process shift.
The literature focusing on process shifts for production systems
producing multiple populations is still limited. Ott and Snee (1973)
present three different methods of analysis plots of raw data, residuals
methods, and analysis of variance method--to examine fills of individual
heads in a multiple fill-head machine. Montgomery (1982) proposes use of
group control charts for detecting process shifts when the multiple
population streams are not highly correlated. The work by Mortell and
Runger (1995) suggests using a pair of control charts, Shewart and CUSUM
charts, to monitor multiple stream processes. Runger et al. (1996)
propose using multivariate techniques to detect assignable causes for
processes with multiple population streams. Lanning et al. (2002) use
adaptive fractional approach to monitor processes with a large number of
population streams. Even though the above studies present different
statistical methods to monitor a multi strata process, none of them
examine the comparative effect of alternate sampling methods on the
sensitivity of detecting a process shift.
The objective of this paper is to investigate the comparative
sensitivity of detecting a process shift in a multi-strata production
process under random and stratified sampling methods. Power curves are
used to depict the sensitivity of detecting process shifts. This paper
uses Monte Carlo simulation to develop power curves under alternate
sampling methods using both x and R control charts. The resulting power
curves can be important decision tool for a QC practitioner in selecting
the appropriate sampling method.
CONTROL CHARTS UNDER RANDOM AND STRATIFIED SAMPLING
Consider a production process that consists of four different
fill-heads to fill a batch of four vials at a time to a specified
weight. After a batch of four vials is filled, it is replaced by another
batch of four vials. This represents a scenario where the measurements
come from four different strata. It is desired to determine if a random
or stratified sampling plan would best assist in designing a control
system for this production process. For example, in designing a process
control system utilizing [bar.x] and R charts, would it be better to
sample four vials at random from the process or sample one from each
fill-head for a total of four.
A sample for this process is "random" when the vials from
different populations are mixed together forming a "pool of mixed
product", and random samples of n vials are taken over time. Under
this plan, it is important to make sure that the vials output between
time intervals is large enough so that all combinations of possible
samples from the four fill-heads are equally likely. That is, the sample
could consist of four vials from a single fill-head or any other
combinations of fill-heads. After N such samples are taken, [bar.x] and
R charts would be determined and the question of whether or not the
process is in a state of statistical control answered. If the answer to
this question is "yes", then these control charts would be
used to monitor the process.
A sample is considered "stratified" when each succeeding
sample consists of four measurements, one from each population at
specified intervals of time [Burr, 1979; Grant, 1988]. If all strata are
identical populations, conventional control charts would act as if the
stratified sample was a random sample from that common population.
However, if the different strata have different means, conventional
control charts should indicate this stratification problem that can be
fixed using either one of the following three approaches: 1) Adjust the
bias (as proposed by Burr & Weaver, 1949; Westman & Lloyd, 1949)
which involves adjusting the data of each strata by the difference
between the grand mean and the mean of the corresponding strata to scale
down R, thus providing overlapping probability distributions for all the
strata with a common target mean; 2) Monitor each strata separately
which involves developing separate control charts for each strata; and
3) Fix strata to common mean which involves adjusting the fill heads or
the process physically to a common mean value.
The remaining sections of this article will assume that either the
different strata are identical initially or upon detecting a
stratification problem are fixed in accordance with cases 1 or 3 above.
Even if the process is concluded to be in-control, not only is a shift
in all populations possible but a shift in a single stratum very likely
to occur. It is very crucial for a QC practitioner to detect this shift
as early as possible. Thus, this paper focuses on addressing the
important question, which sampling method (random or stratified) would
be more sensitive in detecting this shift of the single stratum. The
next section develops power curves for R and x charts to compare the
relative sensitivity of each sampling plan (random and stratified) in
detecting the shift in a single stratum.
POWER CURVES
If a multi-strata in-control production process is suddenly
affected by an assignable cause, it may cause a single stratum to shift
above or below the target value. When this happens, it becomes important
to detect this shift with a high degree of sensitivity.
Power curves are useful graphical tools that represent the
statistical power (sensitivity) of detecting a process shift of a
specified magnitude by the first subgroup taken following the shift
[Osborn, 1990]. The vertical axis of the power curve provides the
probability of rejection (out of control indication) corresponding to a
given shift in the process mean of a single strata.
The following notations have been used to develop the power curves.
n = Sample size
N = Size of each strata
[bar.x] = Sample mean
R = Sample range
s = Number of strata in the process
[delta] = A real no. indicating the magnitude of mean shift as a
multiple of Standard Deviation
[[sigma].sub.x] = Standard deviation of each strata (Assumed to be
equal)
T = Target or initial mean of each strata (assuming they are equal
or have been fixed in accordance with cases 1 or 3)
[P.sub.L] = P(R or [bar.x] < LCL)
[P.sub.U] = P(R or [bar.x] > UCL)
[P.sub.m] = Probability of a sample mean greater than UCL or less
than LCL = [P.sub.L] + [P.sub.U]
[P.sub.R] = Probability of a sample range greater than UCL or less
than LCL = [P.sub.L] + [P.sub.U]
At this point it should be noted that the power curves developed in
this paper are based on one control chart for all four fill-heads.
Although four different control charts for four different fill-heads
will detect a single stratum shift quicker, multiple charts tend to pose
certain disadvantages. According to Mortell and Runger (1995), multiple
charts not only increase the opportunities for false alarms but under
certain circumstances make it difficult for individual charts to detect
an assignable cause affecting a single stratum. Most practitioners
suggest using either--(a) one chart where the sample contains one
observation from each stratum or (b) one chart where the sample is
selected from the pooled production of all strata. In the next section,
power curves for both R and charts using Monte Carlo simulation are
developed under alternate sampling plans.
POWER CURVES FOR R AND [bar.x] CHARTS UNDER STRATIFIED SAMPLING
Assume that the process is in a state of statistical control and
the mean of each probability distribution of four fill-head populations
in the process is T with standard deviation [[sigma].sub.x.] The LCL and
UCL for the R-chart can be shown to be [D.sub.3][bar.R] and [D.sub.4]
[bar.R] (where [D.sub.3] and [D.sub.4] are constants depending on the
sample size) respectively and for the [bar.x]-chart to be T [+ or -]
3[[sigma].sub.x]/[square root of n] for n=ks where k represents the
number of replicated sets of one observation taken from each of s
strata. Without loss of generality, the standard normal distribution
with T=0 and ax=1 will be used for the purpose of simulation.
LCL AND UCL VALUES FOR THE R-CHART
As stated earlier, the control limits for the R-chart are given by:
LCL = [D.sub.3][bar.R] and UCL = [D.sub.4] [bar.R] (1)
On substituting
[bar.R] = [[sigma].sub.x] x [d.sub.2], in (1) we get:
LCL = [D.sub.3] x [[sigma].sub.x] x [d.sub.2] and UCL = [D.sub.4] x
[[sigma].sub.x] x [d.sub.2] (2)
where [[sigma].sub.x] = 1 for standardized normal random variate.
Since [D.sub.3] = 0, [D.sub.4] = 2.282, and [d.sub.2] = 2.059, for
n=4, from the quality parameter table in Grant et al. [5], we get the
control limits as:
LCL = 0 and UCL = 4.698 (3)
LCL AND UCL VALUES FOR THE [bar.x]-CHART
The control limits for the [bar.x]-chart are given by T [+ or -] 3
[[sigma].sub.x]/[square root of n]. It can be shown that for the
standard normal random variable (Z) with a mean of 0 and a standard
deviation of 1, the control limits are given by:
LCL = - 3/[square root of n] = -1.5 and UCL = 3/[square root of n]
= 1.5 (4)
The control limits for the R and [bar.x] charts given by equations
(3) and (4) respectively will be used to determine the probability of
rejection for developing power curves.
Now, if the mean of a single stratum shifts by (5ax) above or below
the mean (T), the analytical expressions for a new common mean and
standard deviation for the R-chart are difficult to develop since the
statistical sampling distribution for range (R) is unknown. Hence, we
propose to use Monte Carlo simulation to calculate the probability of
detecting this shift in one stratum by both R and [bar.x] charts.
To perform Monte Carlo simulation for the above scenario, a
software package, Insight.xla (Business Analysis software for Microsoft
Excel), is used. The three step approach that the software requires to
perform the Monte Carlo simulation for stratified sampling plan is as
follows:
Step 1: Build model for s=4 different strata
The fill values for each of the three strata in a state of
statistical control were generated using a random number generating
function, gen_Normal (0,1). This function randomly generated
standardized normal values (z) with a mean of 0 and a standard deviation
of 1 for each of the three strata. For the fourth stratum whose mean
shifted by [delta]ax, the corresponding function used was, gen Normal
([delta], 1), where 0.5 [less than or equal to] [delta] [less than or
equal to] 10.5. Formulae for calculating the Mean and Range values for
the four strata were incorporated in the worksheet.
Step 2: Specify simulation setting
The primary simulation setting that the software requires is the
"number of trials". The number of trials was determined by
asking the question--what sample size would enable the determination of
the probability (proportion) of rejection (P) to within 0.005 with 95%
confidence for the overlapping probability distributions of all the four
strata. Knowing that the probability of an out of control indication is
approximately 0.003 for both the [bar.x] and R control charts for an
in-control process, a sample size of 2000 would allow an estimate for
the probability of out of control indication within [+ or -] 0.0024 with
95% confidence.
Step 3: Run simulation and examine results
It is assumed that the mean of the fourth stratum shifts by
[delta][[sigma].sub.x] such that 0.5 [less than or equal to] [delta]
[less than or equal to] 10.5. The shift values ([delta]) are considered
in increments of 0.5. A simulation run for 2000 trials was performed for
each shift ([delta]) value and the resulting values for R and [bar.x]
recorded.
After comparing the R values obtained in Step 3 with the control
limits for the R-chart ([LCL.sub.R] = 0, [UCL.sub.R] = 4.698) and
[bar.x] values with the control limits for the [bar.x]-chart
([LCL.sub.[bar.x]] = -1.5, [UCL.sub.[bar.x]] = 1.5), the proportion of
values falling below LCL ([P.sub.L)] and above UCL ([P.sub.U)] were
determined for each chart.
Table 1 shows the [P.sub.m,] [P.sub.R,] and P values for [bar.x]
and R charts under stratified sampling for different values of 5. Figure
1 shows the power curves for both R and [bar.x] charts under stratified
sampling.
POWER CURVES FOR R AND [bar.x] CHARTS UNDER RANDOM SAMPLING
Assume the same initial conditions of the process as given in
previous section for stratified sampling. Hence, the initial mean and
standard deviation of each of the four fill-head populations in the
process is given by T and [[sigma].sub.x.] Monte Carlo simulation is
used to determine the sensitivity of both the R and [bar.x] charts in
detecting the shift of one stratum mean. The three step approach to run
Monte Carlo simulation for random sampling plan is presented next.
Step 1: Build model for s=4 different strata
Recall that the random sample involves selection of items in such a
way that all the items in the population of interest have the same
probability of being selected. As a result, the probability of a
measurement being selected from each of the 4-strata is 0.25. Since the
distribution of 3-strata constitutes a common population, it can be
shown that for a random sample, the probability of a measurement being
selected from the common population is 0.75 and that from the
distribution of the shifted stratum is 0.25. Hence, a random sample of
size 4 was generated by incorporating a logical random number generating
function, IF Rand 0 > 0.25, gen_Normal (0,1), gen_Normal
([delta],1),in 4 different cells of the worksheet.
This would ensure that all the 4 measurements in the random sample
come from one or any combination of the 4-strata. The formulae to keep
track of the mean and range of the random samples were also incorporated
in the worksheet.
Steps 2 and 3: These steps were same as given under stratified
sampling.
The proportion of R and [bar.x] values obtained using Monte Carlo
simulation which fall outside the LCL ([P.sub.L]) and UCL ([P.sub.U])
for R and [bar.x] charts respectively were determined.
Table 2 shows the [P.sub.L], [P.sub.U], and P values for R and
[bar.x] charts under random sampling plans for different values of
[delta]. Figure 1 also shows the power curve for [bar.x] and R charts
under random sampling plan.
DISCUSSION
The power curves in Figure 1 show the relative sensitivity of
detecting a shift in the mean of a single stratum by R and [bar.x]
charts under stratified and random sampling methods.
POWER CURVES FOR R-CHART
For both stratified and random sampling methods, the R-chart is
found to be more sensitive than the [bar.x]-chart in detecting the
shifts of different magnitude in the mean of a single stratum. For
example, under stratified sampling method, if a stratum mean shifts by
3[[sigma].sub.x], its probability of detection is 26.15% by the R-chart
as compared to 7.50% by the [bar.x]-chart. However, under random
sampling method, both R and [bar.x] charts are equally sensitive in
detecting single stratum shifts up to 3[[sigma].sub.x]. For shift values
(5) > 3[[sigma].sub.x], the R-chart is relatively more sensitive than
the [bar.x]-chart.
The R-chart shows higher sensitivity in detecting shifts in a
single stratum mean under stratified sampling when compared to random
sampling. For example, Figure 1 shows the probability of detecting a
3.5[[sigma].sub.x] shift to be 38.20% and 29.25% under stratified and
random sampling methods respectively. It is interesting to note that
even though the power of R-chart for detecting a single stratum shift
increases with increasing shift (5) values, this detection power
stabilizes for shift levels greater than 7[[sigma].sub.x] under both the
sampling methods. For example, the sensitivity of R-chart for detecting
shifts ([delta]) [greater than or equal to] 7[[sigma].sub.x] stabilizes
around 1 under stratified sampling whereas it stabilizes around 68%
under random sampling. In summary, if R-chart is the only tool used by a
practitioner to monitor a multi-strata production process, then the
stratified sampling method should be preferred over the random sampling
method.
[FIGURE 1 OMITTED]
POWER CURVES FOR [bar.x]-CHART
Figure 1 also shows that for stratum shifts ([delta]) [less than or
equal to] 6[[sigma].sub.x], random sampling is more sensitive than
stratified sampling in detecting these shifts. For example, the
probability of detecting 3[[sigma].sub.x] shift in the mean of a single
stratum under random and stratified sampling is 18.45% and 7.5%
respectively. However, for stratum shifts ([delta]) >
6[[sigma].sub.x], stratified sampling is found to be more sensitive than
random sampling in detecting a single stratum shift. Thus, for a given
sample size there exists a threshold shift level below which the random
sampling method and above which the stratified sampling method is
superior in detecting a single stratum shift. In the above example where
sample size is n=4, the threshold shift level occurs at [delta] =
6[[sigma].sub.x].
POWER CURVES FOR R AND [bar.x] CHARTS COMBINED
The last column in Tables 1 and 2 represent the total probability
of detecting a single stratum shift by R and [bar.x] charts combined
under stratified and random sampling respectively. The corresponding
power curves are presented in Figure 2.
[FIGURE 2 OMITTED]
On considering both the control charts together, random sampling is
found to be marginally superior to stratified sampling in detecting
single stratum shifts up to 3.5[[sigma].sub.x]. This can be attributed
to the relative superiority of R chart under stratified sampling when
compared to random sampling (see Figure 1) being off-set by that of
[bar.x] chart under random sampling. For example, for a 2[[sigma].sub.x]
shift in the mean of a single stratum, the probability of detection is
12.78% under random sampling and 8.64% under stratified sampling.
Figure 2 also shows that for the shift level in the single stratum
mean such that [delta]=3.5[[sigma].sub.x], both the sampling methods are
equally sensitive. We define it as the threshold shift level
([[delta].sub.threshold] = 3.5[[sigma].sub.x]) where the relative
superiority of R-chart under stratified sampling gets balanced by that
of [bar.x]-chart under random sampling.
For shift values greater than [[delta].sub.threshold] ([delta] >
3.5[[sigma].sub.x]), stratified sampling is found to be more sensitive
than random sampling. It is due to the relative superiority of R-chart
for stratified sampling plan far exceeding that of [bar.x]-chart for
random sampling when 3.5 < 5 [less than or equal to] 6. Further, for
[delta] > 6, both R and [bar.x] control charts show relative
superiority in detecting a single stratum shift under stratified
sampling as compared to random sampling.
Theoretically, it can be argued that if both R and [bar.x] charts
are used to monitor quality for a stratified production process, then
for processes where mean shifts up to 3.5[[sigma].sub.x] do not cause
significant quality problems (i.e. processes with higher capability
index), a stratified sampling plan should be preferred. For processes
with smaller capability index, a random sampling plan would be most
desirable.
CONCLUSION
This article has examined the relative superiority of detecting a
single stratum shift in a multi-strata process under alternate sampling
methods. The results find R-chart to be more sensitive than the
[bar.x]-chart in detecting a single stratum shift under both stratified
and random sampling plans. In addition, the R chart shows higher
sensitivity in detecting all shift levels in a single stratum mean under
stratified sampling than under random sampling method. This research
also defines the threshold shift level and discusses the impact of
process capability index on the relative superiority of the alternate
sampling methods in detecting shifts above or below this threshold
level.
This research is not the final authority on detecting quality
problems underlying a multistrata production process. It is intended to
help QC practitioners gain better insights for selecting appropriate
sampling methods that can have differentiating impact in detecting
process shifts in a single stratum. While our analysis was based on
[bar.x] and R control charts, future extensions of this work may wish to
investigate appropriate sampling method under special purpose charts
like CUSUM or geometric moving average control charts.
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Atul Agarwal, University of Illinois Springfield
R C Baker, The University of Texas at Arlington
Table 1: Rejection Probabilities for and R charts under Stratified
Sampling Plan
[sup. ~]chart
. [P.sub.L] [P.sub.u] [P.sub.m] = Pu+[P.sub.L]
0.5 0.0010 0.0010 0.0020
1 0.0005 0.0055 0.0060
1.5 0 0.0120 0.0120
2 0 0.0245 0.0245
2.5 0 0.0360 0.0360
3 0 0.0750 0.0750
3.5 0 0.1105 0.1105
4 0 0.1690 0.1690
4.5 0 0.2375 0.2375
5 0 0.2970 0.2970
5.5 0 0.4000 0.4000
6 0 0.4930 0.4930
6.5 0 0.6100 0.6100
7 0 0.6910 0.6910
7.5 0 0.7765 0.7765
8 0 0.8370 0.8370
8.5 0 0.9005 0.9005
9 0 0.9395 0.9395
9.5 0 0.9610 0.9610
10 0 0.9775 0.9775
10.5 0 0.9850 0.9850
R-chart Total
. [P.sub.L] [P.sub.u] [P.sub.R] = Pu+[P.sub.L] P
0.5 0 0.0070 0.0070 0.0090
1 0 0.0115 0.0115 0.0174
1.5 0 0.0365 0.0365 0.0481
2 0 0.0635 0.0635 0.0864
2.5 0 0.1370 0.1370 0.1681
3 0 0.2615 0.2615 0.3169
3.5 0 0.3820 0.3820 0.4503
4 0 0.5295 0.5295 0.6090
4.5 0 0.7030 0.7030 0.7735
5 0 0.8225 0.8225 0.8752
5.5 0 0.9090 0.9090 0.9454
6 0 0.9545 0.9545 0.9763
6.5 0 0.9825 0.9825 0.9932
7 0 0.9940 0.9940 0.9981
7.5 0 0.9965 0.9965 0.9992
8 0 1.0000 1.0000 1.0000
8.5 0 1.0000 1.0000 1.0000
9 0 1.0000 1.0000 1.0000
9.5 0 1.0000 1.0000 1.0000
10 0 1.0000 1.0000 1.0000
10.5 0 1.0000 1.0000 1.0000
Table 2: Rejection Probabilities for and R charts under Random
Sampling Plan
[sup.~]chart R-chart Total
. PL Pu Pm = PL+Pu PL Pu PR = PL+Pu P
0.5 0.0015 0.0035 0.0050 0 0.0060 0.0060 0.0110
1 0.0005 0.0125 0.0130 0 0.0110 0.0110 0.0239
1.5 0.0000 0.0395 0.0395 0 0.0260 0.0260 0.0645
2 0.0010 0.0760 0.0770 0 0.0550 0.0550 0.1278
2.5 0.0000 0.1240 0.1240 0 0.1200 0.1200 0.2291
3 0.0005 0.1840 0.1845 0 0.1915 0.1915 0.3407
3.5 0.0005 0.2506 0.2511 0 0.2925 0.2925 0.4702
4 0.0005 0.3026 0.3031 0 0.3923 0.3923 0.5765
4.5 0.0005 0.3440 0.3445 0 0.4937 0.4937 0.6681
5 0.0000 0.3794 0.3794 0 0.5720 0.5720 0.7344
5.5 0.0008 0.4297 0.4305 0 0.6263 0.6263 0.7872
6 0.0008 0.4748 0.4756 0 0.6568 0.6568 0.8200
6.5 0.0002 0.5108 0.5110 0 0.6588 0.6588 0.8332
7 0.0000 0.5586 0.5586 0 0.6820 0.6820 0.8596
7.5 0.0005 0.5985 0.5990 0 0.6747 0.6747 0.8696
8 0.0000 0.6190 0.6190 0 0.6750 0.6750 0.8762
8.5 0.0015 0.6295 0.6310 0 0.6770 0.6770 0.8808
9 0.0000 0.6650 0.6650 0 0.6790 0.6790 0.8925
9.5 0.0000 0.6660 0.6660 0 0.6800 0.6800 0.8931
10 0.0005 0.6845 0.6850 0 0.6815 0.6815 0.8997
10.5 0.0005 0.6950 0.6955 0 0.6830 0.6830 0.9035
