6.1.4. Hotelling’s TSquared Test
This is the multidimensional equivalent of One Sample tTest. The null hypothesis “the population mean vector is equal to the given mean vector” is tested. Hotelling’s TSquared statistic is computed as follows:
where:
· is the sample mean vector.
· µ is the expected mean vector (target level).
· S is the sample covariance matrix.
The test statistic (which is Fdistributed) is found as:
df numerator = p
df denominator = n – p
where p is the number of variables and n is the number of valid cases.
Select two or more columns by clicking on [Variable]. The next dialogue prompts for the given target levels, where the mean value of each variable is offered by default. Any rows containing at least one missing value are omitted. The output includes the sample covariance matrix, observed means, target levels, Hotelling’s TSquared statistic and its tail probability.
Also see the related quality control procedure 9.3.4. Hotelling’s TSquared Analysis.
Example
Example 13.3 on p. 474 Armitage & Berry (2002). Measurements are made on babies when they were 25 and 50 days old. The null hypothesis “there is no significant difference between measurements on 25 and 50 days” is tested.
Open PARTESTS and select Statistics 1 → Parametric Tests → Hotelling’s TSquared Test. Select Haemoglobin, Platelets, log Leucocytes and Systolic BP (C10 to C13) as variables and all target levels as zero. The following results are obtained.
Hotelling’s TSquared Test

Target Values 
Mean 
Difference 
Haemoglobin 
0.0000 
0.5300 
0.5300 
Platelets 
0.0000 
0.0300 
0.0300 
Log Leucocytes 
0.0000 
0.5900 
0.5900 
Systolic BP 
0.0000 
3.1000 
3.1000 
Hotelling’s TSquared Statistic = 
7.4391 
F(4,6) = 
1.2398 
RightTail Probability = 
0.3869 
The result is not significant at 10% level. Thus do not reject the null hypothesis.