From Contrast Estimates to Exact Simultaneous Intervals
A collection of individually valid confidence intervals does not automatically provide the requested confidence level for the whole collection. When the contrasts are correlated, the joint distribution of their studentized estimates can be used directly: choose one cutoff for the maximum absolute statistic, then use that cutoff in every interval. The calculation is exact when the model supplies the contrast correlations and the statistics share a common estimated scale.
This tutorial works through that calculation under a multivariate normal model with a known covariance shape and an unknown common scale. The known shape is an important condition, not a technical convenience: it makes the correlation matrix for the contrasts available for exact calibration.
Assume \(n\geq 2\), \(\Sigma_0\) is known, and the contrast rows are nonzero and linearly independent. Write \(\nu=p(n-1)\). The common scale estimate pools variation across the \(p\) coordinates; it is not a separate variance estimate for each contrast.
Worked Example: Studentizing Two Contrasts
Worked Example: Studentizing Two Contrasts
Suppose \(n=12\) observations have three coordinates, and the model is \(X_i\sim N_3(\mu,\sigma^2I_3)\). The observed mean and sample covariance are \[ \bar X=\begin{pmatrix}5\\4\\2\end{pmatrix}, \qquad S=I_3. \] Consider the two prespecified contrasts \(c_1'\mu=\mu_1-\mu_2\) and \(c_2'\mu=\mu_2-\mu_3\), so \[ C=\begin{pmatrix}1&-1&0\\0&1&-1\end{pmatrix}. \]
The pooled scale estimate is \[ \hat{\sigma}^2=\frac{\operatorname{tr}(S)}{3} =\frac{1+1+1}{3}=1, \] and the degrees of freedom are \(\nu=3(12-1)=33\). The estimated contrasts are \[ \hat{\delta}_1=5-4=1,\qquad \hat{\delta}_2=4-2=2. \] For each contrast, \(c_j'c_j=2\), so the estimated standard error is \[ \hat{\sigma}\sqrt{\frac{c_j'c_j}{n}} =1\sqrt{\frac{2}{12}} =\frac{1}{\sqrt{6}}\approx 0.4082. \] Thus the observed studentized estimates, when testing the respective null values \(\delta_j=0\), are \(1/(1/\sqrt{6})=\sqrt{6}\approx2.449\) and \(2/(1/\sqrt{6})=2\sqrt{6}\approx4.899\).
Why the Joint Max-t Distribution Is Exact
The exact calibration follows from two features of the model: the sample mean is independent of the residual covariance matrix, and the pooled scale estimate has a chi-square distribution. The same random scale appears in every studentized contrast, so the resulting statistics are generally dependent even when some of their normal numerators are uncorrelated.
Proof. Let \(A=\Sigma_0^{-1/2}\), so \(AX_i\) has covariance \(\sigma^2I_p\). The normal sample mean and residuals are independent. In whitened coordinates, the sum of squared residuals has a chi-square distribution with \(p(n-1)=\nu\) degrees of freedom. Equivalently, \[ U=\frac{\nu\hat{\sigma}^2}{\sigma^2}\sim\chi^2_\nu, \] and \(U\) is independent of \(\bar X\). To see the scale, \(\operatorname{tr}(\Sigma_0^{-1}S)/\sigma^2\) is the sum of \(p(n-1)\) independent squared standard normal residual coordinates divided by \(n-1\); multiplying by \(\nu/p=n-1\) gives \(U\).
The vector of unstudentized contrast errors, standardized by their true standard deviations, is \[ Z_j=\frac{\sqrt{n}\,c_j'(\bar X-\mu)} {\sigma\sqrt{c_j'\Sigma_0c_j}},\qquad j=1,\ldots,q. \] It is multivariate normal with mean zero and correlation matrix \(R\). Since \(U\) is independent of \(\bar X\), it is independent of \(Z\). Direct substitution into the definition of \(T_j\) gives \[ T_j=\frac{Z_j}{\sqrt{U/\nu}}. \] This common-denominator representation is precisely the multivariate \(t\) distribution with \(\nu\) degrees of freedom and correlation matrix \(R\). \(\square\)
Worked Example: The Contrast Correlation and Critical Value
Worked Example: The Contrast Correlation and Critical Value
Continue with the three-coordinate model and contrasts above. Because \(\Sigma_0=I_3\), the unscaled contrast covariance matrix is \[ C\Sigma_0C' =CC' =\begin{pmatrix}2&-1\\-1&2\end{pmatrix}. \] Both diagonal entries are 2, so standardizing the rows gives \[ R=\begin{pmatrix}1&-1/2\\-1/2&1\end{pmatrix}. \] The contrasts are negatively correlated: increasing the middle coordinate raises the first difference in the opposite direction to the second difference.
For 95% simultaneous coverage, the required cutoff solves \[ P(-c\leq T_1\leq c,\ -c\leq T_2\leq c)=0.95, \] where \(T\) has a bivariate \(t\) distribution with \(\nu=33\) and correlation \(-1/2\). This is an exact finite-sample probability, not a normal approximation. It can be evaluated by integrating over the shared chi-square scale: \[ \int_0^\infty P\!\left(-c\sqrt{u/33}\leq Z_1\leq c\sqrt{u/33},\ -c\sqrt{u/33}\leq Z_2\leq c\sqrt{u/33}\right) f_{\chi^2_{33}}(u)\,du=0.95, \] where \((Z_1,Z_2)'\sim N_2(0,R)\). Numerical integration followed by root-finding gives \(c_{0.05}\approx2.29\) (rounded to two decimal places). The cutoff is determined by the joint maximum distribution, not by treating the two statistics as independent.
Worked Example: Forming and Reading the Intervals
Worked Example: Forming and Reading the Intervals
Use the estimates, standard errors, and critical value from the previous calculations. The common simultaneous margin is \[ 2.29\left(\frac{1}{\sqrt{6}}\right)\approx 2.29(0.4082)\approx0.935. \] The first interval is therefore \[ 1\pm0.935=[0.065,\ 1.935], \] and the second is \[ 2\pm0.935=[1.065,\ 2.935]. \] The endpoints are rounded to three decimal places.
The pair of intervals has 95% simultaneous coverage under the stated model: the probability that both contain their respective population contrasts is 0.95. This is a joint statement about the family, not a claim that each interval has 95% coverage independently of the other. The intervals are also based on the contrasts chosen in advance; changing the family changes the maximum distribution and may change the critical value.
Coverage Guarantee and Its Conditions
The coverage statement follows directly from the max-\(t\) definition. For each \(j\), the interval contains \(\delta_j\) exactly when \[ |\hat{\delta}_j-\delta_j| \leq c_\alpha\hat{\sigma}\sqrt{\frac{c_j'\Sigma_0c_j}{n}}, \] which is equivalent to \(|T_j|\leq c_\alpha\). Consequently, all intervals cover their targets exactly when \(\max_j|T_j|\leq c_\alpha\). The theorem gives the probability of this event.
Proof. By the equivalence just established, the event inside the probability is \(\{\max_j|T_j|\leq c_\alpha\}\). By the definition of \(c_\alpha\), this event has probability \(1-\alpha\). The multivariate \(t\) theorem ensures that the distribution used to define the cutoff is the sampling distribution under the stated model. \(\square\)
Common Mistake: Treating an Estimated Correlation as Known
A frequent error is to calculate a sample correlation matrix for the contrasts and then describe the resulting max-\(t\) intervals as exact. Exactness here relies on \(R\) being known from the specified covariance shape \(\Sigma_0\). If the covariance shape is unrestricted and \(R\) is replaced by a random estimate, the cutoff is no longer calibrated from the exact sampling distribution established above. The intervals may still be useful as an approximation under suitable methods, but this derivation does not establish their exact coverage.
A related error is to use separate univariate \(t\) cutoffs and assume their coverage probabilities multiply. The statistics share the random scale \(U\), and their numerators may also be correlated. Their joint probability is therefore not obtained by multiplying marginal probabilities. The max-\(t\) procedure accounts for that joint distribution directly.
Check Your Understanding
Use the definitions and derivations above to answer these questions.
- Why does the multivariate \(t\) representation use one common chi-square scale variable rather than a separate scale for each contrast?
- For \(\Sigma_0=I_3\) and contrasts \(\mu_1-\mu_2\) and \(\mu_2-\mu_3\), calculate their correlation.
- What event in terms of the studentized statistics is equivalent to all \(q\) simultaneous intervals covering their targets?
- Why does replacing the known contrast-correlation matrix by an estimated one require additional justification before claiming exact coverage?
- In the worked model with three coordinates and twelve observations, how many degrees of freedom does the pooled scale estimate have, and why?