Dear Allstat,
Here are the responses to my technical SPC question - all gratefully receive, and reproduced below in no particular order.
A rich collection of ideas which I will test out.
My main focus will be to keep it simple and take action - i.e., the Shewhart approach.
Regards
Martin
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Scenario: We have approximately 200 data points collected over a long period of time. Data is right skewed and there is no rational sub-grouping.
Purpose: To establish if process is stable.
Analysis: To plot data on an Individuals and Moving Range chart. (I am aware that data does not need to be normally distributed to use control
charts, ref; Shewhart's original work and current work by Don Wheeler).
Traditionally we would estimate sigma from moving average or median ranges from first 25 data points and then calculate 3 sigma control limits. If any
points were out of control then we would remove these points and then re-estimate sigma and the limits, and then use the control chart in process.
An alternative view is to estimate sigma from all the data and calculate the limits. The resulting control chart shows the process is out of control, or
is this a false warning? Nest step is to use a suitable Box-Cox lambda transform to achieve normality and recalculate sigma, draw the charts and
now the process appears stable.
My question relates to how many sub-groups (values) in this case are required to estimate sigma?
For x-bar / range charts it is commonly assumed to be at least 20 sub-groups of size 5 or at least enough shifts / days of operation to reflect "normal"
process variation. For individuals and moving range charts the traditional approach is 25 values and then proceed as above.
Are there any formal rules? Indeed does it really matter so long as we are improving the process by taking actions on out of control points? Or, do we
ignore skewness by taking a transformation and then assume everything is stable?
Has any research been done in this area please?
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Why treat this as an SPC problem. If you take equal samples of say 70 from either end, order them and pair by order you should obtain a plot that reveals any significant change in shape between these periods. No change would be the 45 degree line. There are variants on this, but see what happens first.
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I think your key question was at the end: "does it really matter so long as we are improving the process by taking actions on out of control points?"
Short answer: No.
Your concern with the distribution (Normal, skewed, or what) applies mostly when you wish to interpret those 'out of control' points. IN a
standard x-bar-R chart with n typically = 3-5, you say anything beyond 3 sigma is an outlier. What you really mean is that you have one chance
in 500 (roughly) of being wrong, when you say it is unusual. Hard to bet against that.
Shewhart wanted the chart to be right when it said 'outlier,' so he picked a very small alpha risk.
If you are willing to give the troops a little slack, you can pick another alpha risk. If you want to focus on tracking down 'assignable
causes' and are willing to accept that sometimes you won't find any, then you don't have to get detailed about your 'true' distribution or an
appropriate limit.
I would suggest that you use all 200 points to set up your initial control limits, removing outliers and transforming as desired to obtain
a 'Normal' dist. Back transform for placing those limits on the chart, make it up large enough for all to see (a meter across works for about
90 points), then see what everyone thinks about the chart. So a point is close to the limit, but not over it. Let's argue a while, point some
fingers. Might be good or poor entertainment. Then get down to what we can do to reduce the value tomorrow. Every time we have an outlier, we
mark the reason on the chart. reminds everyone of what not to do again.
Every time we have a bright thought that works, or shows on the chart that it doesn't work, we mark it on the chart. Try to collect & plot
the data reasonably frequently. 1/month is a no-no. 1/day is much better, if you can.
Long answer: yes.
You care if the process is stable (you say. I don't think so, but whatever :) Take all your data, transform it, clean it, discover the mean and stdev of the transformed data. Back transform for the 3s control limits. Make your 'long time' plot, see how many points are out of control. Look most closely at the R chart. If you use a x- moving R chart pair, be _very_ careful about normality. Do the laboratory charts in the transformed space. Correct the moving range chart for
auto-correlation, which it is. You might want to compare with subgroups of 3 and see if the results are much different.
Net conclusion: the process is stable, at a confidence level of 95% (lousy phrasing, but you know what I really mean). So what do you do
now? Live with it?
OR: The process is not stable. There is less than a 0.005 chance that this statement is false. So what do you do now? You take your
careful chart, recast it to 'real' measurement numbers, and jump whenever a point falls outside the limit line. Eventually everyone will
learn not to report wild numbers (hopefully that won't happen. they will eventually learn what not to do.) However, common cause variation
will remain. Is that OK by you?
Also, sources of variation that put you 2.5 sigma out will continue. You need to break down the limit lines and use the "Westinghouse Rules"
to really detect those. And you will need to do those special rules in the transformed, Normal space. Then back transform them so everyone
will understand what happened. What a mess! all because you insist on a Shewhart defined, dichotomous boundary between 'assignable' and 'common'
cause event.
Better to try for improvement, all the time. What causes that 'common cause' variation? Let's go after that, too. Just because it is defined
as 'common' doesn't mean we can't control and get rid of it.
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We have lots of right skewed control charts here, mainly counts of defects. Here's what we do:
1. Collect lots of data, 200 or more points
2. Screen outliers often 4 pseudo sigma at the 90th percentile, sometimes 5 ps
3. Set limits at 99th or 99.5th percentile, extrapolate on a normal probability plot if there is not much data at the extremes.
4. Apply limits to existing data and see if out of controls are the type of thing we want to respond to. If not go back to 2.
The basic idea is to set limits you can respond to.
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This a partly a question of philosophy. SPC practitioners tend to fall into two camps: the statisticians and the pragmatists. The latter group includes Shewhart, Deming and Wheeler. Wheeler regard statisticians as having done much damage by "adding-on" to Shewhart. In my 10 years of practising SPC in ICI I started by being with the statisticians, but after a few years of doing it on real plants and processes I shifted camp.
As you say, Shewart's 3-sigma rule was a purely empirical rule which works reasonably well for the vast majority of distributions. SPC does not impose any model on the data. The silliest thing I've seen is the replacement of 3s by 3.09s in the British Standard.
You have answered your own question below: Indeed does it really matter so long as we are improving the process by taking actions on out of control points?
Yes indeed. The objective of SPC is not to characterise the process, but to bring it into a state of control. In Wheeler's book 'Advanced Topics in Statistical Process Control' he states 'The purpose of analysis is insight rather than numbers. The objective is not to compute the "right limits", but rather to take the right action on the process.' If you have to calculate approximate limits from a small amount of data, just be aware that the limits will be fuzzy. You'll probably have plenty of alarms to be going on with. Don Wheeler once told me that the smallest amount of data he'd ever used to set up an SPC chart was 2.
As your application is retrospective analysis I'd use all the data points. I occasionally used a log transform for variables such as impurity concentrations where the distribution can be expected to be lognormal. I don't think I ever used a Box-Cox. I'm wary of transforms generally because often a right skew is caused by the presence of out of control points. However of all the applications of SPC, the "report card" is the weakest.
You don't say what type of process you're dealing with, but autocorrelation is common in continuous processes and if that's a serious problem with yours then you do need to adjust your limits.
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I'm not an expert in SPC but from your description I suspect the main problem is the non-normality of your data. A possible solution is to use density estimation to establish the control limits, rather than calculating the "3-sigma"s.
If your data is low-dimensional, traditional kernel density estimation might be ok; or if high-dimensional, you may wish to try Gaussian mixture models.
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