9.3.4. Hotelling’s TSquared Analysis
This is also known as the multivariate control analysis where multiple variables can be controlled in a single chart using the Hotelling’s TSquared statistic. The Hotelling’s TSquared statistic is the multivariate equivalent of the tstatistic (see 6.1.1. t and FTests and 6.1.4. Hotelling’s TSquared Test). In this case, the Hotelling TSquared statistic is calculated using the average within sample covariance matrix as an estimate of the population covariance matrix, rather than using each sample covariance matrix to calculate TSquared for that sample only.
To run this procedure, you will need to select two or more variables by clicking on [Variable] and one [Sample] column for subgroups.
9.3.4.1. Hotelling’s TSquared Inputs
The parameter input dialogue asks for the following:
UCL Probability: This is critical probability of the test and is defined as the probability of declaring the process out of control when it is not out of control. This is equivalent to the Type I error probability.
Use Average N: If this is 1 then the average sample size will be used to determine the critical value for each sample. This will mean that the UCL will be a straight line. Otherwise, if the samples have different sample sizes, then the UCL will vary from sample to sample.
Variable n: You can enter a target level for each variable selected. The default value suggested is the sample mean of each variable.
9.3.4.2. Hotelling’s TSquared Output Options
The Output Options Dialogue offers the following choices:
Summary Information: For each sample, the sample size, Hotelling’s TSquared statistic, the estimated F and its associated probability are displayed in a table. The estimated Fvalue is a transformation from the TSquared statistic to a variable which follows the F distribution.
Chart Summary: For each sample, the sample size, Hotelling’s TSquared statistic and the UCL are displayed in a table.
Table of Means: For each sample, the sample means of all variables are displayed in a table.
Average Withinsample Covariance Matrix: This is the average of each sample covariance matrix. It is used in calculating the Hotelling’s TSquared statistic, instead of taking a different sample covariance matrix for each sample.
Hotelling’s TSquared Chart: TSquared and UCL values are plotted against the group numbers. Clicking the [Opt] button situated to the left of the Hotelling’s TSquared Chart option will place the plot in UNISTAT’s Graphics Editor. The plot can be further customised and annotated using the tools available under the graphics window’s Edit menu.
9.3.4.3. Hotelling’s TSquared Example
Open ANOTESTS and select Statistics 2 → Quality Control → Hotelling’s TSquared Analysis. Select Age and Capacity (C10 – C11) as [Variable]s and Age Group (C9) as [Sample]. In the next dialogue accept the default values.
Hotelling’s TSquared Analysis
Summary Information
Sample: Age Group
FStatistic with (2, N2) Degrees of Freedom
Sample 
N 
TSquared 
FStatistic 
Probability 
1 
12 
6.1748 
2.5728 
0.1255 
2 
28 
6.0366 
2.8027 
0.0790 
3 
44 
3.1140 
1.4862 
0.2379 
Chart Summary
Sample 
N 
TSquared 
UCL 
1 
12 
6.1748 
9.8468 
2 
28 
6.0366 
7.2563 
3 
44 
3.1140 
6.7465 
Beta Probability = 
0.0500 
Table of Means
Sample 
Age 
Capacity 
1 
49.7500 
3.9492 
2 
37.7857 
4.4718 
3 
39.7955 
4.4620 
Overall 
42.4437 
4.2943 
Average WithinSample Covariance Matrix

Age 
Capacity 
Age 
103.8656 
5.1348 
Capacity 
5.1348 
0.6704 