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Struggling
with Independence
by Paul A. Keller,
CQE, CQA
An important but frequently
overlooked assumption of statistical process control charts is that observations
are independent. Unfortunately, the real world operates in ignorance of
many statistical assumptions, which can lead to problems in analysis.
Fortunately, these problems may be easily overcome, so long as they are
recognized.
The Problem
with Autocorrelation
Independence of observations
implies that the current data value provides no indication of the next
data value. For independent processes in a state of statistical control,
the best predictor of the next observation is the average. For dependent
(or autocorrelated) processes, the best predictor of the next observation
is based upon some function of the current observation (or a prior observation).
For example, the outdoor temperature at 12:33 is likely to be highly correlated
with the temperature at 12:32, explaining in part why the television meteorologist
can accurately predict that it is in fact hot outside, and will remain
hot for most of the day.
Why is the independence
assumption so important? Consider how X-bar charts are constructed. Each
subgroup is used to estimate the "short term" average and variation
of the process. The average short term variation (R-bar) is then used
to estimate the control limits on both the X-bar and Range charts. If
the process is "in control", or stable, then the aver- age short
term variation provides a good indication of long term variation. Therefore,
in forming subgroups, a convenient rule to remember is that the short
term (or within subgroup) variation must be comparable to the long term
(or between subgroup) variation. In practical terms, the potential causes
of within subgroup variation (machines, materials, methods, manpower,
measurement, and environment) should be comparable to those causes that
exist between subgroups.
Similarly, the Individual-X
/ Moving Range chart is also prone to these errors. Here, since the subgroup
size is one, the variation between consecutive subgroups is used to calculate
the short term variation.
A process that violates
the independence requirement would frequently cause the short term variation
to be a poor indicator of long term variation. Consider the wait time
for a bank teller station. It is likely that each person's wait time would
be influenced by (i.e. dependent on) the wait time of the person immediately
in front of them. If we form subgroups of several consecutive customers'
wait time, the variation within the subgroup would most likely be much
lower than the variation between subgroups.
When product is manufactured
in distinct batches, and these batches are well mixed (homogeneous), the
variation from sample to sample in a given batch is likely to be small
compared to the variation between batches. Consider, for example, the
brewing of a 5000 barrel batch of beer. Samples of this beer taken at
various locations in the vessel will show subtle differences of any parameter,
such as pH, within the vessel. These differences may be due to many factors
such as measurement error, temperature gradients, and mixing inefficiencies.
Now consider the parameters influencing pH variation between batches.
The raw ingredients, grain and water, have properties of their own which
affect pH. These batch-to- batch variables will more strongly influence
pH than within batch factors: the between subgroup variation is much greater
than the within subgroup variation.
A subgroup created from
a sample of each head of a six-head machining operation (or six doctors
performing the same procedure) would show correlation between every sixth
observation. Differences between the heads cause the subgroup range to
be large, resulting in excessively wide X-bar control limits.
These examples point out
how control limits could be either too large or too small, resulting in
failure to look for special causes when they really do exist, or searching
for special causes that don't exist. The important point to note here
is that these errors are not caused by the methodology itself but rather
by ignoring a key requirement of the methodology: independence.
Detecting
Autocorrelation
The scatter diagram in
Figure 1 shows the correlation,
or in this case the autocorrelation, between each observation and the
one observed immediately (one period or lag) following it. Each period
is one minute, or in other words one sample was taken every 60 seconds.
The scatter diagram in
Figure 2 shows the autocorrelation
using observations made two periods apart, or 2 minutes between samples.
Figure 3 and Figure
4 , respectively, show 5 and 10 minutes between samples. As seen
by the plots, the influence of an observed temperature on the temperature
one minute later is stronger than on temperature readings made 10 minutes
later.
While Scatter Diagrams
offer a familiar approach to the problem, they are a bit cumbersome to
use for this purpose since you must have separate Scatter Diagrams for
each lag period. A more convenient tool for this test is the Autocorrelation
Function (ACF). It will directly indicate departures from the assumption
of independence.
The ACF plots directly
the autocorrelation at each lag, as shown in Figure
5.
For example, if observations
46.2, 46.6, 46.5, 47.2, 47.9,... were recorded from a process with an
overall average of 47, the autocorrelation at lag one and lag two is calculated
as follows:
(46.2 -47)(46.6-47) + (46.6-47)(46.5-47) + (46.5-47)(47.2-47)+...
r1= -----------------------------------------------------------------
(46.2-47)2 + (46.6-47)2 + (46.5-47)2 + (47.2-47)2+...
(46.2 -47)(46.5-47) + (46.6-47)(47.2-47) + (46.5-47)(47.9-47)+...
r2= -----------------------------------------------------------------
(46.2-47)2 + (46.6-47)2 + (46.5-47)2 + (47.2-47)2+...
As you can see, this calculation
is similar to those shown in statistical texts for the correlation between
two factors. The ACF will lie between plus and minus one, with values
closer to these limits signifying stronger correlation. In the figure,
we have shown 95% confidence limits. We evaluate the ACF from lag one
to lag n/4 (where n is the number of data points), since ACFs greater
than that have been shown statistically unreliable (see Box, Stephen E.
P. and Gwilym M. Jenkins, Time Series Analysis: Forecasting & Control.
San Francisco, Holden- Day, 1970).
Dealing
with Autocorrelation
Such problems with autocorrelation
can be accommodated in a number of ways. The simplest technique is to
change the way we take samples, so that the effects of process autocorrelation
are negligible. To do this, we consider the reason for the autocorrelation.
If the autocorrelation
is purely time-based, we can set the time between samples large enough
to make the effects of autocorrelation negligible. (This idea is discussed
in detail in Pyzdek's Guide to SPC, Volume Two. Applications and Special
Topics, by Thomas Pyzdek, 1992, ASQC, Quality Press, Milwaukee, and Quality
Publishing, Tucson, page 140.) In the example above, by increasing the
sampling period to greater than 10 minutes, autocorrelation becomes insignificant.
We can then apply standard X-bar or Individual-X charts.
A disadvantage of this
approach is that it may force the time between samples to be so large
that process shifts are not detected in a reasonable (economical) time
frame. Alternatively, we could model the process based on its past behavior,
including the effects of autocorrelation, and use this process model as
a predictor of the process. Changes in the process (relative to this model)
can then be detected as special causes. SPC-PC IV's EWMA chart with moving
center line has been designed for these autocorrelated processes.
When the autocorrelation
is due to homogeneous batches, we may consider taking sub- groups of size
one using an Individual-X chart. In this case, each plotted point represents
a single sample from each batch, with only one sample per batch. Now the
subgroup to sub- group variation is calculated using the Moving Range
statistic, which is the absolute value of the difference between consecutive
samples. An enhancement to this method is to take multiple samples per
batch, then average these samples and plot the average as a single data
point on an Individual-X chart. This is sometimes referred to as a Batch
Means Chart. Each plotted point will better reflect the characteristics
of the batch, since an average is used. The calculation of the batch average
is easily done using SPC-PC IV's spreadsheet formulas.
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