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.
