Exact Max-t Inference for Correlated Contrasts

Why the Maximum Matters

When several means or treatment contrasts are reported together, separate 95% intervals do not generally provide 95% coverage for the whole collection. The relevant event is that every interval covers its target. A direct way to build such intervals is to calibrate the largest absolute studentized error across the prespecified contrasts. This uses their dependence rather than treating the contrasts as independent.

The finite-sample distribution requires care. Under Gaussian sampling, the vector of contrast estimates and the sample covariance have an exact joint structure. But if each contrast is divided by its own estimated standard error, the resulting statistics do not, in general, have the conventional multivariate t distribution with one shared random denominator. The distinction determines which critical value gives exact coverage.

Setup and the Exact Studentized Maximum

Definition: Let $X_1,\ldots,X_n$ be independent observations from $N_p(\mu,\Sigma)$, where $\Sigma$ is positive definite and $n\ge 2$. Let $A$ be a prespecified $m\times p$ matrix with nonzero, linearly independent rows $a_1',\ldots,a_m'$. The targets are the contrasts $\theta_j=a_j'\mu$. Write $\bar X$ for the sample mean and $S$ for the unbiased sample covariance matrix. The separately studentized contrast statistics and their maximum are $$ T_j=\frac{a_j'\bar X-a_j'\mu}{\sqrt{a_j'Sa_j/n}}, \qquad M=\max_{1\le j\le m}|T_j|. $$

Define the contrast covariance matrix $\Gamma=A\Sigma A'$, its diagonal standard-deviation matrix $D=\operatorname{diag}(\sqrt{\Gamma_{11}},\ldots,\sqrt{\Gamma_{mm}})$, and the contrast correlation matrix $R=D^{-1}\Gamma D^{-1}$. Under the assumptions above, $R$ is positive definite. The next result gives the exact finite-sample law of $M$, when $R$ is specified.

Theorem: Put $\nu=n-1$. Let $Z\sim N_m(0,R)$ and, independently, let $W\sim\operatorname{Wishart}_m(R,\nu)$, where the Wishart matrix is the sum of $\nu$ independent outer products of $N_m(0,R)$ vectors. Then $$ (T_1,\ldots,T_m)' \ \overset{d}{=}\ \left(\frac{Z_1}{\sqrt{W_{11}/\nu}},\ldots, \frac{Z_m}{\sqrt{W_{mm}/\nu}}\right)'. $$ Thus the exact distribution of $M$ is the distribution of the maximum absolute coordinate of this Gaussian–Wishart ratio vector. Each coordinate has a univariate $t_\nu$ distribution, but the vector is not generally a conventional multivariate t vector with a common denominator.

Proof. The sample mean of a Gaussian sample is independent of the sample covariance. In particular, $\sqrt n(\bar X-\mu)\sim N_p(0,\Sigma)$ and $\nu S\sim\operatorname{Wishart}_p(\Sigma,\nu)$. Applying $A$ and standardizing each coordinate by its population contrast standard deviation gives $$ Z=D^{-1}\sqrt n\,A(\bar X-\mu)\sim N_m(0,R), \qquad W=D^{-1}A(\nu S)A'D^{-1}\sim\operatorname{Wishart}_m(R,\nu). $$ The two transformed quantities remain independent. Also, $$ \frac{a_j'Sa_j}{a_j'\Sigma a_j} =\frac{W_{jj}}{\nu}. $$ Substitution into the definition of $T_j$ gives the stated representation. The marginal numerator is standard normal, and its corresponding diagonal Wishart entry is $\chi^2_\nu$, independently, so each ratio has a $t_\nu$ distribution. $\square$

Intervals and Familywise Coverage

For a prespecified $R$ and familywise error rate $\alpha$, let $c_\alpha(R)$ be a quantile of $M$ satisfying $P_R(M\le c_\alpha(R))=1-\alpha$. Continuity of the distribution under the stated assumptions allows this equality for $0<\alpha<1$. The same critical value is used for all contrasts; different contrast standard errors still enter their respective interval widths.

Theorem: If $R$ is the true contrast correlation matrix and $c_\alpha(R)$ satisfies $P_R(M\le c_\alpha(R))=1-\alpha$, then the intervals $$ a_j'\bar X\ \pm\ c_\alpha(R)\sqrt{\frac{a_j'Sa_j}{n}}, \qquad j=1,\ldots,m, $$ have simultaneous coverage $1-\alpha$.

Proof. For any fixed $j$, the event that the $j$th interval contains $a_j'\mu$ is equivalent to $$ |a_j'\bar X-a_j'\mu| \le c_\alpha(R)\sqrt{\frac{a_j'Sa_j}{n}}, $$ which, since the standard error is positive with probability one, is equivalent to $|T_j|\le c_\alpha(R)$. All intervals cover simultaneously exactly when $\max_j|T_j|\le c_\alpha(R)$. The theorem’s definition of the critical value therefore gives simultaneous coverage $1-\alpha$. $\square$

This is exact calibration, not an independence adjustment: the distribution used for $c_\alpha(R)$ accounts for the correlations among the contrasts and among their estimated variances. It also shows an important limitation. With unrestricted unknown $\Sigma$, the true $R$ is generally unknown. Replacing it by an estimate and using the resulting $c_\alpha(\hat R)$ does not, by itself, preserve exact finite-sample coverage. Exact use of this critical value requires the contrast correlation structure to be specified or otherwise known. A conservative critical value that controls coverage uniformly over a stated set of possible correlation matrices is another option, but may produce wider intervals.

Worked Example: Two Correlated Contrasts

Suppose two prespecified contrasts have standard errors $0.50$ and $0.80$, and their true sampling correlation is $0.60$. Assume $n=10$, so $\nu=9$, and that the correlation is known. The exact maximum statistic has the representation $$ M=\max\left\{ \frac{|Z_1|}{\sqrt{W_{11}/9}}, \frac{|Z_2|}{\sqrt{W_{22}/9}} \right\}, \quad Z\sim N_2\left(0,\begin{pmatrix}1&0.60\\0.60&1\end{pmatrix}\right), \quad W\sim\operatorname{Wishart}_2\left( \begin{pmatrix}1&0.60\\0.60&1\end{pmatrix},9\right), $$ with $Z$ and $W$ independent.

For a 95% simultaneous procedure, define $c_{0.05}(R)$ by $P_R(M\le c_{0.05}(R))=0.95$. If the observed contrast estimates are $1.20$ and $-0.40$, the resulting intervals are $$ 1.20\ \pm\ 0.50c_{0.05}(R), \qquad -0.40\ \pm\ 0.80c_{0.05}(R). $$ For example, their lower endpoints are $1.20-0.50c_{0.05}(R)$ and $-0.40-0.80c_{0.05}(R)$; their upper endpoints are $1.20+0.50c_{0.05}(R)$ and $-0.40+0.80c_{0.05}(R)$. The same critical value protects both intervals, while the standard errors determine their different widths. The quantile must come from the correlated max-t law above, not from an independence formula.

When Is the Conventional Multivariate t Law Appropriate?

A conventional multivariate t vector has a different construction: it is a correlated normal vector divided in every coordinate by the same independent random scale, such as $\sqrt{U/\nu}$ for $U\sim\chi^2_\nu$. This is the correct model when the statistics truly share a common variance estimate. It is not generally the law of contrast statistics whose denominators are their separate sample standard deviations.

The distinction can be seen in the Wishart representation. The separately studentized statistic $T_j$ uses $W_{jj}$ in its denominator, so different coordinates use different random quantities. A common-scale multivariate t instead uses one random quantity for every coordinate. Each marginal distribution may still be $t_\nu$; matching the marginals does not establish the joint distribution. In particular, using the common-scale multivariate t quantile for separately studentized contrasts need not give the claimed familywise coverage.

Worked Example: Separate Sample Variances Are Not One Shared Scale

Consider $n=3$ observations on two coordinates, with the two coordinate means as the contrasts: $$ (0,0),\qquad(2,0),\qquad(0,1). $$ The sample mean is $(2/3,1/3)'$. The first coordinate’s centered values are $-2/3,4/3,-2/3$, so its sample variance is $$ S_{11}=\frac{(-2/3)^2+(4/3)^2+(-2/3)^2}{2} =\frac{4/9+16/9+4/9}{2}=\frac{4}{3}. $$ The second coordinate’s centered values are $-1/3,-1/3,2/3$, giving $$ S_{22}=\frac{(-1/3)^2+(-1/3)^2+(2/3)^2}{2} =\frac{1/9+1/9+4/9}{2}=\frac{1}{3}. $$ Thus the two standard errors of the sample means are $\sqrt{4/9}=2/3$ and $\sqrt{1/9}=1/3$. The observed standard errors differ because each contrast uses its own sample variance. In repeated samples, those variances are random and are not represented by one shared chi-square scale. The conventional common-denominator multivariate t distribution therefore cannot be assumed for these separately studentized statistics.

Computing and Reporting an Exact Procedure

The exact max-t critical value is a property of the full joint law, not a universal number determined only by $m$, $n$, and $\alpha$. It depends on $\nu$ and $R$. For a given $R$, its defining probability is $$ P_R(M\le c) = P\left( |Z_j|\le c\sqrt{W_{jj}/\nu}\ \text{for every }j \right). $$ The probability can be evaluated by integration or by simulation from the independent Gaussian and Wishart representations. The separate tutorial on computing exact max-t critical values treats that numerical task; here the key inferential point is that the calibration must match the statistic actually used.

For a fully specified correlation structure, reporting the method should identify the contrast family, the degrees of freedom, the correlation information used for calibration, the chosen familywise level, and the resulting common critical value. If the correlation structure was estimated from the same sample, describe the procedure as plug-in calibration rather than claiming exact finite-sample coverage. The difference is substantive: exactness is a probability statement under a specified sampling distribution, not a consequence of using a statistic with t-distributed marginals.

Common Mistake

Wrong reasoning: “Each contrast statistic has a $t_{n-1}$ marginal distribution, so the vector has a multivariate t distribution with the estimated contrast correlations.”
Correction: Marginal t distributions do not determine a joint distribution. With separate sample variances, the exact vector has the Gaussian–Wishart ratio representation, with a distinct Wishart diagonal term for each contrast. A common-scale multivariate t law applies only when the statistics genuinely share that random denominator. Also, if the true contrast correlation matrix is unknown, substituting an estimate does not make the resulting critical value exact.

Key takeaway: Maximum absolute studentization gives a natural simultaneous procedure, but exact coverage depends on its joint finite-sample law. Under Gaussian sampling, derive that law from the contrast correlation structure and the covariance estimator; do not infer it from marginal t distributions alone.

Check Your Understanding

Use the definitions and results above to answer these questions.

What event in terms of $M$ is equivalent to all $m$ simultaneous intervals covering their contrast targets?
Why does the Gaussian–Wishart representation give a univariate $t_{n-1}$ distribution for each coordinate?
What feature distinguishes a conventional common-scale multivariate t vector from separately studentized contrast statistics?
Which correlation information determines the exact max-t critical value when it is known?
Why does replacing the true contrast correlation matrix by an estimate not automatically yield exact finite-sample coverage?