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Common Calibration Errors in One-Sample Hotelling Tests

Use the exact one-sample Hotelling calibration correctly, and identify when a change in convention or a singular covariance invalidates the usual calculation.

Intermediate 12 min read

What You'll Learn

  • State the Gaussian and invertibility conditions for the exact one-sample Hotelling test.
  • Convert Hotelling’s \(T^2\) statistic to its \(F\) calibration with the correct multiplier and degrees of freedom.
  • Translate statistics computed with a covariance divisor of \(n\) to the standard divisor of \(n-1\).
  • Explain why the p-value is an upper-tail probability.
  • Diagnose when the sample covariance must be singular and when it is nonsingular almost surely.

Why Calibration Details Matter

A one-sample Hotelling test combines the discrepancy between a sample mean vector and a hypothesized mean vector with an estimate of covariance. Its exact calibration is not obtained by comparing the statistic directly with an ordinary \(F\) distribution: a particular multiplier and a particular pair of degrees of freedom are required. Reversing that multiplier, using the wrong covariance divisor, or looking in the wrong tail can change the conclusion even when the underlying data summary is unchanged.

Let \(X_1,\ldots,X_n\) be independent observations from a \(p\)-dimensional normal distribution \(N_p(\mu,\Sigma)\), where \(\Sigma\) is positive definite. Write

$$ \bar X=\frac{1}{n}\sum_{i=1}^n X_i,\qquad S=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)(X_i-\bar X)'. $$

For testing \(H_0:\mu=\mu_0\), the one-sample Hotelling statistic is

$$ T^2=n(\bar X-\mu_0)'S^{-1}(\bar X-\mu_0). $$

The statistic is defined in this form when \(S\) is invertible. The exact finite-sample calibration below additionally requires \(n>p\) and the stated independent multivariate-normal sampling model.

Theorem: Under \(H_0\), if \(X_1,\ldots,X_n\) are independent \(N_p(\mu_0,\Sigma)\) observations with positive-definite \(\Sigma\) and \(n>p\), then $$ \frac{n-p}{p(n-1)}T^2\sim F_{p,n-p}. $$ Thus the exact p-value for the two-sided-in-mean-vector alternative \(H_1:\mu\ne\mu_0\) is the upper-tail probability of this calibrated statistic.

Deriving the Multiplier and Degrees of Freedom

The scale factor is easy to misremember, so it is useful to see where each part comes from. Under the null, define

$$ Z=\sqrt{n}\,\Sigma^{-1/2}(\bar X-\mu_0),\qquad W=(n-1)\Sigma^{-1/2}S\Sigma^{-1/2}. $$

Normal-sample theory gives \(Z\sim N_p(0,I_p)\), \(W\sim\operatorname{Wishart}_p(I_p,n-1)\), and independence of \(Z\) and \(W\). Substitution into \(T^2\) yields \(T^2=(n-1)Z'W^{-1}Z\).

For completeness, the distribution of this quadratic form can be obtained from the Wishart representation. Put \(m=n-1\). Conditional on a nonzero value of \(Z\), rotational invariance of \(W\) means that the distribution of \(Z'W^{-1}Z\) is the distribution of \(\|Z\|^2(W^{-1})_{11}\). This conditional distribution does not depend on the direction of \(Z\), so the factor \((W^{-1})_{11}\) in this representation is independent of \(\|Z\|^2\). Also, \(\|Z\|^2\sim\chi^2_p\).

To identify the other factor, represent \(W=Y'Y\), where \(Y\) is an \(m\)-by-\(p\) matrix of independent standard normal entries. Partition its first column \(y_1\) from the remaining columns \(Y_2\). The block inverse formula gives

$$ \frac{1}{(W^{-1})_{11}} =y_1'\left(I-Y_2(Y_2'Y_2)^{-1}Y_2'\right)y_1. $$

The matrix in parentheses is an orthogonal projection of rank \(m-p+1\), with probability one. Conditional on \(Y_2\), \(y_1\) is an independent standard normal vector, so the displayed quadratic form has a \(\chi^2_{m-p+1}\) distribution. Its conditional distribution does not depend on \(Y_2\); therefore \((W^{-1})_{11}\) has the distribution of the reciprocal of an independent \(\chi^2_{m-p+1}\) variable. Consequently \(Z'W^{-1}Z\) has the distribution of a ratio of independent chi-square variables, with their degrees of freedom divided by those degrees of freedom:

$$ Z'W^{-1}Z \ \sim\ \frac{\chi^2_p/p}{\chi^2_{m-p+1}/(m-p+1)} \cdot\frac{p}{m-p+1} =\frac{p}{m-p+1}F_{p,m-p+1}. $$

Since \(m-p+1=n-p\), multiplying \(T^2=(n-1)Z'W^{-1}Z\) by \((n-p)/(p(n-1))\) gives the theorem. The numerator degrees of freedom are the dimension \(p\); the denominator degrees of freedom are \(n-p\). \(\square\)

Worked Example: Applying the Exact Scale and Upper Tail

Suppose \(n=12\), \(p=3\), and the observed statistic computed using the divisor-\((n-1)\) covariance matrix is \(T^2=18\). Find the calibrated statistic, its reference degrees of freedom, and the exact form of the p-value.

The multiplier is $$ \frac{n-p}{p(n-1)}=\frac{12-3}{3(12-1)}=\frac{9}{33}=\frac{3}{11}. $$ Therefore $$ F_{\mathrm{obs}}=\frac{3}{11}(18)=\frac{54}{11}\approx4.909. $$

The reference distribution is \(F_{3,9}\), not \(F_{3,11}\) or \(F_{9,3}\). Since larger values of \(T^2\) represent greater discrepancy from \(H_0\), the exact p-value is $$ P(F_{3,9}\ge 54/11)=1-F_{3,9}(54/11), $$ where \(F_{3,9}(x)\) denotes the \(F_{3,9}\) cumulative distribution function. Using a lower-tail probability would report the probability of a statistic no larger than the observed one, not the p-value for this test.

Changing the Covariance Divisor

Some software or hand calculations use the centered sum of products divided by \(n\), rather than by \(n-1\). Denote this matrix by

$$ S_n=\frac{1}{n}\sum_{i=1}^n(X_i-\bar X)(X_i-\bar X)'. $$

It is not the covariance matrix in the standard definition of \(T^2\) above. The relationship between the two matrices is exact:

Conversion rule: Since \(S_n=\frac{n-1}{n}S\), the quadratic statistic computed with \(S_n\), $$ T_n^2=n(\bar X-\mu_0)'S_n^{-1}(\bar X-\mu_0), $$ satisfies \(T_n^2=\frac{n}{n-1}T^2\). Its matching exact calibration is $$ \frac{n-p}{pn}T_n^2\sim F_{p,n-p} $$ under the same Gaussian null conditions.

Indeed, \(S_n^{-1}=\frac{n}{n-1}S^{-1}\), which gives \(T_n^2=\frac{n}{n-1}T^2\). Substituting \(T^2=\frac{n-1}{n}T_n^2\) into the standard calibration proves the second formula:

$$ \frac{n-p}{p(n-1)}T^2 =\frac{n-p}{p(n-1)}\frac{n-1}{n}T_n^2 =\frac{n-p}{pn}T_n^2. $$

This conversion is not an optional small-sample refinement. Applying the usual multiplier to a statistic computed with \(S_n\) makes the calibrated value too large by a factor of \(n/(n-1)\).

Worked Example: Correcting a Divisor-\(n\) Calculation

Let \(n=8\), \(p=2\). Suppose the usual divisor-\((n-1)\) calculation gives \(T^2=14\). Then the statistic calculated with \(S_n\) is $$ T_n^2=\frac{8}{7}(14)=16. $$

The standard calibration gives $$ \frac{8-2}{2(8-1)}(14)=\frac{6}{14}(14)=6, $$ with reference distribution \(F_{2,6}\). Starting instead from \(T_n^2=16\), the corresponding multiplier is $$ \frac{8-2}{2(8)}=\frac{6}{16}=\frac{3}{8}, $$ so the calibrated value is \(\frac{3}{8}(16)=6\), as required.

A common incorrect calculation applies the standard multiplier \(6/14\) directly to \(T_n^2=16\), producing \(48/7\approx6.857\). That number does not have the claimed \(F_{2,6}\) calibration: the statistic and multiplier use inconsistent covariance conventions.

Degrees of Freedom, Tail Direction, and Invertibility

Three further checks prevent frequent errors. First, the \(F\) degrees of freedom are \(p\) and \(n-p\), in that order. The numerator degrees of freedom reflect the \(p\)-dimensional mean, while estimating the covariance from \(n\) observations leaves \(n-p\) in the denominator. Second, the test rejects for large \(T^2\), so the p-value and critical-value rule use the upper tail. The statistic is nonnegative and measures a squared, covariance-adjusted distance; there is no negative direction to test separately.

Third, \(S^{-1}\) exists only if \(S\) is nonsingular. For any centered data matrix, its rank is at most \(n-1\), because its \(n\) centered rows sum to zero. Its rank is also at most \(p\). Thus, if \(n\le p\), the rank is at most \(n-1<p\), and \(S\) must be singular. If \(n>p\), singularity is no longer forced by dimension. Under independent sampling from a multivariate normal distribution with positive-definite \(\Sigma\), \(S\) is nonsingular with probability one. For particular observed data, however, the matrix should still be checked; duplicated or linearly constrained observations can make it singular.

Worked Example: Detecting a Singular Sample Covariance

A researcher has \(p=4\) outcomes but only \(n=4\) independent subjects. Can the ordinary one-sample Hotelling statistic be computed?

No. The centered data matrix has four rows, but those rows sum to zero, so its rank is at most \(n-1=3\). It has four columns, and therefore cannot have column rank four. Equivalently, the sample covariance matrix has rank at most three and is singular. Its inverse in the ordinary \(T^2\) formula does not exist.

This is not fixed by using an \(F_{4,0}\) reference distribution: the denominator degrees of freedom \(n-p\) are zero, and the required statistic itself is undefined. The standard exact one-sample calibration requires \(n>p\). A different inferential method or a genuinely lower-dimensional analysis would be needed; silently substituting a generalized inverse does not produce the stated Hotelling test.

A Calibration Checklist

For a reported one-sample Hotelling result, check the ingredients in order. Confirm the dimension \(p\), sample size \(n\), null mean vector, and covariance divisor. Verify that \(n>p\) and that the observed covariance matrix is invertible. Then compute \(T^2\) with the divisor-\((n-1)\) covariance, multiply by \((n-p)/\{p(n-1)\}\), and use \(F_{p,n-p}\). Report the upper-tail probability, or reject when the calibrated statistic exceeds the upper \(\alpha\) critical value.

These conditions also mark the limits of the exact result. The \(F\) calibration is exact for independent multivariate-normal observations under the null. A covariance estimate alone does not make the result exact for nonnormal data, dependent observations, or a singular covariance matrix.

Common pitfall: A large \(T^2\) should lead to a large calibrated \(F\) and a small upper-tail p-value. Do not use the reciprocal multiplier, swap the \(F\) degrees of freedom, use the lower tail, or pair a divisor-\(n\) covariance statistic with the divisor-\((n-1)\) multiplier. Each error changes the calibration rather than merely changing notation.

Check Your Understanding

For each question, identify the calibration issue and state the correction.

  1. For \(n=15\) and \(p=5\), what are the multiplier for the standard \(T^2\) statistic and the reference \(F\) degrees of freedom?
  2. A statistic was computed using a covariance matrix with divisor \(n\). How does it relate to the standard \(T^2\), and what multiplier should be used with it?
  3. Why is the exact p-value an upper-tail probability even though the alternative is written \(H_1:\mu\ne\mu_0\)?
  4. Can the ordinary one-sample Hotelling statistic be defined when \(n=p\)? Explain using the rank of the centered data matrix.
  5. Under what sampling assumptions does the stated \(F\) calibration hold exactly?

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