topic Multivariate Control Chart Run Rules in Discussions

Multivariate Control Chart Run Rules

gfirmstone — Wed, 02 Aug 2023 20:48:14 GMT

Is it possible in JMP (or even appropriate statistically) when using multivariate (T2) control charts to use run rules other than run rule #1 (result above upper control limit)? For example, run rule #2 for nine consecutive results above the mean/median. Thank you.

Re: Multivariate Control Chart Run Rules

DrewLuebe — Fri, 04 Aug 2023 15:59:45 GMT

Hi @gfirmstone

I am certainly not an expert but in an effort to get some conversation going, I will share my thoughts.

I think one of the primary challenges is that typical univariate control charts have an assumption of normality. Hotelling's T^2 distribution is more or less an F distribution. This difference would therefore mean that rules requiring X standard deviations for n iterations etc would be different simply because the probabilities of being those number of standard deviations away from the mean would differ due to different distributions.

The second difference is that a T^2 control chart is one sided where many of the run rule control charts look at responses above and below. There really isn't a below for a T^2 control chart.

All that said, many of these run rules are developed based on experience and are better described as best practices. I suppose it would be possible to mathematically calculate analogs of each of the run rules to make the frequency of random occurrence in a F distribution type variable like T^2 be equivalent to a normal distribution variable. In general though, you are likely safe to apply the rules (at least the ones that can be applied to a one sided chart). If T^2 is elevated for 9 continuous samples, there is likely something worth investigating etc. The run rules merely flag an observation as something that needs to be investigated for cause. So, the risk is rather low. If you find that applying the same run rules to T^2 is causing too frequent or too infrequent investigation, you could certainly adjust to something that fits.

Re: Multivariate Control Chart Run Rules

gfirmstone — Mon, 07 Aug 2023 15:09:50 GMT

Thanks very much DrewLuebe for your reply. That all makes total sense, appreciate your thoughts very much.

Re: Multivariate Control Chart Run Rules

ih — Tue, 08 Aug 2023 14:19:09 GMT

Great review @DrewLuebe.

There seems to be a lot of discussion around picking limits for T2 and DModX/SPE, but not so much about what to do when a limit is crossed, so I think this is a great topic. My experience is that, just like in univariate control charting, in order to pick a limit that is useful, it tends to be close enough to the data that some data points will cross that boundary by random chance.

Here are some other techniques to maximize the relevance of SPE and T2 deviations:

Smoothing - Reducing noise from inputs means limits means the T2 limit does not need to expand to compensate for that noise.

Transformations - With non-linear data, one side of a cluster of data might have a very definite edge while the opposite side has a lot of noise. The T2 and DModX boundries tend to be defined by the noisy side, which means deviations are missed on the well-defined side.