Hotelling's T-squared Test

Hotelling's T-squared test is the multivariate generalization of the two-sample t test, testing whether two groups have equal mean vectors on several correlated outcome variables.

Hotelling T² 검정은 두 표본 t 검정을 여러 변수로 확장한 것으로, 서로 상관된 여러 결과변수에서 두 집단의 평균 벡터가 같은지 검정하는 방법이다.

How do I compare two groups on several related outcomes at once instead of running many t tests?

여러 결과변수를 두 집단 사이에서 t 검정을 여러 번 하지 않고 한 번에 비교하려면 어떻게 하나요?

Hotelling's T² weights the vector of mean differences by the inverse pooled covariance matrix, so correlations among the outcomes are taken into account and the familywise error rate is held at α. It converts exactly to an F statistic with p and n₁ + n₂ − p − 1 degrees of freedom.

When to use Hotelling's T-squared Test

• Two groups are compared on two or more continuous outcomes.
• The outcomes are correlated with each other.
• You want one overall test before looking at individual outcomes.

When to use something else

• More than two groups are compared; use MANOVA instead.
• Outcomes are severely non-normal or have outliers; use a permutation-based multivariate test instead.

Assumptions and requirements

• Observations are independent.
• Outcomes are multivariate normal within each group.
• Both groups share the same covariance matrix.
• The sample size exceeds the number of outcomes.

What the analysis reports

• T² statistic
• F approximation with df and p-value
• Mahalanobis distance between group means
• Univariate mean differences per outcome

How to report it

Hotelling's T² test showed that the mean vectors of [group 1] and [group 2] [differed/did not differ] across [p] outcomes, T² = [value], F([df1], [df2]) = [value], p = [value].

Related methods

• MANOVA
• Box's M Test
• Independent Samples t test

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