Question & Answer
Question
Will SPSS compute Hotelling's T^2 ("T-squared") for multivariate tests of mean differences in the two-sample case?
Answer
The MANOVA and GLM procedures compute a multivariate statistic named Hotelling's Trace (sometimes called the Lawley-Hotelling Trace and sometimes the Hotelling-Lawley Trace).
Multiplying Hotelling's Trace by (N - L), where N is the sample size across all groups and L is the number of groups, gives a generalized version of Hotelling's T^2. When L=2, the product is Hotelling's
two-sample T^2. In practice, T^2 and Hotelling's Trace are transformed to equal values of F with the same degrees of freedom. The Exact F and its significance level for Hotelling's Trace, as found in the MANOVA and GLM output, are the appropriate F and significance for T^2.
See S. Kotz & N.L Johnson, (1983), "Encyclopedia of Statistical Sciences: Vol. 3" (pp. 669-677) for discussions of the derivation of Hotelling's T^2 and Hotelling's Trace and the relationship between
them. The article on Hotelling's Trace (pp. 673-677) provides the most information on the relationship, particularly on pages 673-674. Focus on the information for Hypothesis II (as labelled on p. 673).
The following example syntax command runs a multivariate Analysis of Variance with the GLM command. The variables are from the data file bankloan.sav, which is included in the Samples subdirectory of the directory where IBM SPSS Statistics is installed. The three dependent variables are DEBTINC ("Debt to income ratio (x100)"), CREDDEBT ("Credit card debt in thousands") and INCOME ("Household income in thousands") and the factor is DEFAULT ("Previously defaulted"). The GLM command can be run in the menu system from Analyze->General Linear Model->Multivariate. The multivariate tests, including the Hotelling's Trace, are printed by default. The Advanced Statistics module must be installed to run the GLM command (or the MANOVA command that is mentioned in the Question).
GLM debtinc creddebt income BY default
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/PRINT=DESCRIPTIVE PARAMETER
/CRITERIA=ALPHA(.05)
/DESIGN= default.
The Hotelling's Trace for DEFAULT is printed in the "Multivariate Tests" table in the General Linear Model output. For this analysis, based on 700 cases and two values for DEFAULT, the Hotelling's Trace is .209, which is converted to an F of 48.537 with 3 and 696 degrees of freedom. The significance level is printed as .000 (i.e., p < .0005).
Therefore, Hotellings T^2 = .209 * (700 - 2) = 145.882. As noted above, the significance for T^2 is already printed in the "Multivariate Tests" table as the significance for the transformed Hotelling's Trace.
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Modified date:
16 June 2018
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