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,

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 = ₀, we would then have

so that

Generalize to p variables

When t² is generalized to p variables it becomes

with

         

S^-1 is the inverse of the sample variance-covariance matrix, S, and n is the sample size upon which each _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 = ₀

with F_(p,n-p) representing the F distribution with p degrees of freedom for the numerator and n - p for the denominator. Thus, if were specified to be ₀, this could be tested by taking a single p-variate sample of size n, then computing T² and comparing it with

for a suitably chosen .

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 ₀), 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.