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

6.5.4.3.1.

T2 Chart for Subgroup Averages -- Phase I

Estimate with

Since is generally unknown, it is necessary to estimate analogous to the way that is estimated when an chart is used. Specifically, when there are rational subgroups, is estimated by , with

Obtaining the _i

Each _i, i = 1, 2, ..., p, is obtained the same way as with an chart, namely, by taking k subgroups of size n and computing

.

Here is used to denote the average for the lth subgroup of the ith variable. That is,

with x_ilr denoting the rth observation (out of n) for the ith variable in the lth subgroup.

Estimating the variances and covariances

The variances and covariances are similarly averaged over the subgroups. Specifically, the s_ij elements of the variance-covariance matrix S are obtained as

with s_ijl for i j denoting the sample covariance between variables X_i and X_j for the lth subgroup, and s_ij for i = j denotes the sample variance of X_i. The variances (= s_iil) for subgroup l and for variables i = 1, 2, ..., p are computed as

.

Similarly, the covariances s_ijl between variables X_i and X_j for subgroup l are computed as

.

Compare T² against control values

As with an chart (or any other chart), the k subgroups would be tested for control by computing k values of T² and comparing each against the UCL. If any value falls above the UCL (there is no lower control limit), the corresponding subgroup would be investigated.

Formula for plotted T² values

Thus, one would plot

for the jth subgroup (j = 1, 2, ..., k), with denoting a vector with p elements that contains the subgroup averages for each of the p characteristics for the jth subgroup. ( is the inverse matrix of the "pooled" variance-covariance matrix, , which is obtained by averaging the subgroup variance-covariance matrices over the k subgroups.)

Formula for the upper control limit

Each of the k values of given in the equation above would be compared with

Lower control limits

A lower control limit is generally not used in multivariate control chart applications, although some control chart methods do utilize a LCL. Although a small value for might seem desirable, a value that is very small would likely indicate a problem of some type as we would not expect every element of to be virtually equal to every element in .

Delete out-of-control points once cause discovered and corrected

As with any Phase I control chart procedure, if there are any points that plot above the UCL and can be identified as corresponding to out-of-control conditions that have been corrected, the point(s) should be deleted and the UCL recomputed. The remaining points would then be compared with the new UCL and the process continued as long as necessary, remembering that points should be deleted only if their correspondence with out-of-control conditions can be identified and the cause(s) of the condition(s) were removed.