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6. Process or Product Monitoring and Control
6.5. Tutorials
6.5.4. Elements of Multivariate Analysis

6.5.4.3.

Hotelling's T squared

Hotelling's T2 distribution A multivariate method that is the multivariate counterpart of Student's t and which also forms the basis for certain multivariate control charts is based on Hotelling's T2 distribution, which was introduced by Hotelling (1947).
Univariate t-test for mean Recall, from Section 1.3.5.2, t=x¯μs/n has a t distribution provided that X is normally distributed, and can be used as long as X doesn't differ greatly from a normal distribution. If we wanted to test the hypothesis that μ=μ0, we would then have t=x¯μ0s/n so that t2=(x¯μ0)2s2/n=n(x¯μ0)(s2)1(x¯μ0).
Generalize to p variables When T2 is generalized to p variables it becomes T2=n(x¯μ0)S1(x¯μ0), with x¯=[x¯1x¯2x¯p]μ0=[μ10μ20μp0]. S1 is the inverse of the sample variance-covariance matrix, S, and n is the sample size upon which each x¯i,i=1,2,,p, is based. (The diagonal elements of S are the variances and the off-diagonal elements are the covariances for the p variables. This is discussed further in Section 6.5.4.3.1.)
Distribution of T2 It is well known that when μ=μ0 T2p(n1)npF(p,np), with F(p,np) representing the F distribution with p degrees of freedom for the numerator and np for the denominator. Thus, if μ were specified to be μ0, this could be tested by taking a single p-variate sample of size n, then computing T2 and comparing it with p(n1)npFα(p,np) for a suitably chosen α.
Result does not apply directly to multivariate Shewhart-type charts Although this result applies to hypothesis testing, it does not apply directly to multivariate Shewhart-type charts (for which there is no μ0, although the result might be used as an approximation when a large sample is used and data are in subgroups, with the upper control limit (UCL) of a chart based on the approximation.
Three-sigma limits from univariate control chart When a univariate control chart is used for Phase I (analysis of historical data), and subsequently for Phase II (real-time process monitoring), the general form of the control limits is the same for each phase, although this need not be the case. Specifically, three-sigma limits are used in the univariate case, which skirts the relevant distribution theory for each Phase.
Selection of different control limit forms for each Phase Three-sigma units are generally not used with multivariate charts, however, which makes the selection of different control limit forms for each Phase (based on the relevant distribution theory), a natural choice.
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