STATISTICS IN THE LABORATORY continued
1000; or (b) the process isn’t behaving now as it has behaved in the past— something is different. Because the probability of (a) is so small, our gut feeling is that possibility (b) is the more likely explanation.
This is called a three-sigma rule violation, where the rule states: A lack of control is indicated when a point falls outside a control limit. Okay, the subgroup mean at subgroup number 20 is a three-sigma rule violation. What should be done?
We should look for an “assignable cause” and try to remove it. One place to start is in the run chart. Note that the last six points in the run chart (not just the last four) are noticeably and consistently higher in value than most of the previous data points. It looks like the bias of the chromato- graphic method has shifted—the integrated peak areas have become a bit larger than they’ve been in the past. Why would that be? Well, statisticians can tell you a lot about these charts, but they probably don’t know very much about why things happen in these charts. This is where “profound knowledge” or “domain knowledge” comes in. Translation: we need to talk to a chromatographer to try to find the assignable cause. I’ve done some chromatography, but I don’t consider myself to be a real chroma- tographer (like Brian Bidlingmeyer,1
for example), so I can only speculate that maybe the lamp in the UV detector has shifted its position because
someone bumped the chromatograph…or maybe the instrument manu- facturer decided to push a new version of integrator software into the chromatograph, a version they thought was better but was only decidedly different…or maybe…I dunno. At this point I’d call Brian.
It should be said that an assignable cause isn’t always found. Sometimes— about three times in 1000—a “statistical event” occurs, one of those random, “apparently-but-not-really” out-of-control events that happens with a risk of 0.00270. Still, when a rule violation occurs, it’s a good idea to see if you can find an assignable cause, fix it, and try to prevent it from happening again.
Figure 2 shows a run-of-ten rule violation in the R chart, where the rule states: A lack of control is indicated when 10 successive values fall on the same side of the central line. Starting at subgroup number 5, there are 10 ranges that are greater than the central line.
This might happen if a routine analytical laboratory suffers a loss of per- sonnel, and the remaining analysts are overworked and don’t have time to be as careful as before. This is not good—if the variability is getting to be too large, remedial training might be in order. To be fair, sometimes the rule violation happens the other way—the ranges get smaller. This might happen in a routine analytical laboratory as the analysts become more familiar with the method and their variability decreases. This is good. In either case, if the process has changed and becomes stable in its new state, and the new state is acceptable, you might want to redraw the control limits based on the more current information.
A few statisticians use a run-of-seven rule; probably most statisticians use a run-of-eight rule; some are more conservative and use a run-of-nine rule; I’m even more conservative and use the run-of-ten rule. Why? Because the theoretical risk of triggering a false positive with the run-of-ten rule (0.001953125) is more in keeping with the risk of the three-sigma rule (0.00270) than is the original run-of-eight rule (0.0078125) or the run- of-nine rule (0.00390625). That said, as Wheeler2
points out, Shewhart3
didn’t treat control limits as strict probability limits and split (tiny) hairs like this. Shewhart knew the distributions weren’t always Gaussian (or χ2
),
weren’t always symmetrical, etc. He didn’t care if the risks were exactly 0.00270—they just needed to be small, so the three-sigma rule and the run-of-eight rule are probably good enough.
One last thing: don’t use too many rules! Beginners soon discover hun- dreds, perhaps thousands, of specialized rules in the literature. Two familiar examples are found in the Western Electric Statistical Quality Control Handbook:4
Detection Rule Two: A lack of control is indicated when Figure 2 – A run-of-ten rule violation. AMERICAN LABORATORY 44
two out of three successive values are: (a) on the same side of the central line and (b) more than two standard deviations away from the central line. Detection Rule Three: A lack of control is indicated when four out of five successive values are: (a) on the same side of the central line and (b) more than one standard deviation away from the central line. Beginners think that if a few rules are good, then more rules must be better. Actually… no. When making multiple, independent statistical decisions, each with its own risk α of giving a wrong answer, the overall risk αEW
that at least APRIL 2016
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