Home / Statistical Tools / Control Charts / T² and Generalized Variance

T² and Generalized Variance¶

A control chart for several correlated variables watched together. Every other chart in this section watches one measurement at a time. This one takes two or more variable columns and plots two statistics computed from all of them at once: T², which watches the joint mean, and Generalized Variance, which watches the joint spread.

Three ribbon commands, one tool¶

QXL Stat Tools > Control Charts > Multivariate

Command	What you get	Worksheet title
T² and Generalized Variance	both panels, stacked	T² and Generalized Variance Chart
Hotelling T²	the T² panel alone	Hotelling T² Chart
Generalized Variance	the Generalized Variance panel alone	Generalized Variance Chart

The three are the same analysis with a different amount of it drawn. The engine computes both panels from a single pass whichever command you pick, so the numbers on a Hotelling T² sheet are the numbers the combined sheet would have shown. The choice is made on the ribbon and there is no panel control in the dialog.

Why one chart instead of several¶

Run a separate chart for each of five correlated measurements and you get five chances to raise a false alarm where you wanted one. Set each at a 0.27 per cent false-alarm rate and the chance that at least one of the five signals on an in-control process is far higher than 0.27 per cent.

Separate charts also cannot see a relationship break. Two dimensions that normally move together can each stay comfortably inside their own limits while the pair moves somewhere the process has never been. T² measures distance from the centre in the shape the data's own covariance defines, so a point that is ordinary on each axis and unusual as a pair is far from the centre in that metric.

What each panel plots¶

T², the joint mean. One point per subgroup, or one per row when the subgroup size is 1. It is a squared distance, so it is never negative and its lower limit is 0: the chart signals a joint mean that has moved, in any direction.

Generalized Variance, the joint spread. The determinant of the covariance matrix, one value per subgroup. It collapses the whole spread and the whole correlation structure into a single number, and it has both an upper and a lower limit: the spread can grow, and it can also shrink in a way worth knowing about.

What it needs¶

Requirement	Detail
At least two variable columns	one column is not multivariate. The dialog will not finish with fewer, and the engine refuses with TSGV requires at least 2 variable columns.
A subgroup size	1 means individuals, one point per row. Anything larger groups consecutive rows
A row count divisible by the subgroup size	a partial trailing subgroup is rejected rather than plotted short
Enough rows	at least two subgroups

Missing cells are handled rather than refused, and the rules differ by panel and by layout. They are on Math Details.

What lands on the worksheet¶

A worksheet named from your data with the prefix QXL_T2, carrying:

Part	Where	What it holds
User Input	columns A and B, at the top	the selections the run was made from: Variable columns:, GroupBy columns:, Stage column:, X-Label column:
The panels	below it	one or two charts, stacked, with the plotted points, the centre line and the limits
Summary table	to the right of the panels	the same summary every variable chart writes, plus a T² Alpha row when the T² panel is drawn
Out-of-Control Decomposition (T²)	below the summary table	one row per out-of-control point per variable: Point, Variable, Contribution, P-Value
Mean Vector and Covariance Matrix	its own band at column AC	the parameters the calculation actually used, one block per stage

The decomposition table is the part worth going to this chart for. A T² point above the limit tells you the group has moved; it does not tell you which measurement moved it. The decomposition answers that, and it is described on Math Details.

The table appears only when the T² panel is drawn and there is at least one point to list. A Generalized Variance sheet has no decomposition table, and neither does a T² sheet with nothing out of control.

The parameter table says whether the numbers were estimated or supplied. Its blocks are headed Mean (estimated) and Covariance (estimated), or Mean (historical) and Covariance (historical) when you typed them in. If Box-Cox was on it also carries Values are in Box-Cox transformed units, which is the reminder that the numbers in that table are not in your measurement's units.

Phase I and Phase II¶

Leave the Historical Parameters tab empty and the chart estimates its centre and covariance from your data. That is Phase I: the limits describe the data they were computed from.

Type a historical mean and covariance in and the chart is judged against them instead. That is Phase II, and the limits change form rather than just changing value. Both forms are on Math Details.