Simultaneous Inference for Transition Coefficients

Why simultaneous intervals are needed

A transition-regression model may estimate a separate covariate effect for several transitions. Reporting a 95% confidence interval for each coefficient does not generally provide 95% confidence that all the intervals cover their respective coefficients. When many intervals are examined together, the probability that at least one misses can exceed 5%. Simultaneous inference addresses the family as a whole: it controls the probability of one or more coverage failures across the specified set.

This matters when the scientific question compares effects across transitions, or when conclusions are drawn from a set of transition-specific estimates. The intervals must reflect not just each estimate's standard error but also the estimated covariance among estimates. Transitions observed on the same individuals can yield correlated coefficient estimates, even when their regression terms have different labels.

Definition: Let $\beta$ be a vector of regression coefficients for a multi-state transition model, and let $\widehat{\beta}$ estimate it with estimated covariance matrix $\widehat V$. A family of $m$ targets can be written as $\theta_j=l_j^\mathsf{T}\beta$, $j=1,\ldots,m$, where each $l_j$ is a specified contrast vector. Simultaneous $100(1-\alpha)\%$ confidence intervals have joint coverage at least $1-\alpha$ (or approximately $1-\alpha$ for an asymptotic method): the probability that every interval in the family covers its target is at least $1-\alpha$.

A coefficient for a transition-specific covariate effect is one such target. For example, in a proportional-hazards transition model, $\beta_r$ may be the log hazard ratio for a one-unit covariate increase on transition $r$. A contrast such as $\beta_1-\beta_2$ compares two transition-specific log hazard ratios. Its exponential, $\exp(\beta_1-\beta_2)$, is the ratio of those hazard ratios, not an overall risk ratio.

Covariance-aware intervals

Suppose the fitted model provides $\widehat{\beta}$ and $\widehat V$, with $\widehat V$ estimating the covariance matrix of $\widehat{\beta}$. For a contrast $l^\mathsf{T}\beta$, the estimate and its estimated standard error are

\widehat{\theta}=l^\mathsf{T}\widehat{\beta}, \qquad SE(\widehat{\theta})=\sqrt{l^\mathsf{T}\widehat V l}.

The off-diagonal entries of $\widehat V$ are essential whenever a contrast combines multiple coefficients. Treating the coefficients as independent replaces the correct variance with a different quantity and can produce intervals that are too narrow or too wide.

Under regularity conditions for the fitted likelihood, identifiable coefficients, and a correctly specified model, the estimator is approximately multivariate normal:

\widehat{\beta}\ \dot{\sim}\ N(\beta,V).

The covariance estimate must be appropriate for that model and estimation procedure. This tutorial uses the model-based covariance from the fitted model; it does not address replacing it with a sandwich covariance estimator. If the likelihood model or its covariance assumptions are unsuitable, a multiplicity adjustment alone cannot repair the resulting uncertainty estimates.

Result: For $m$ prespecified contrasts, Bonferroni simultaneous intervals are $$ l_j^\mathsf{T}\widehat{\beta}\ \pm\ z_{1-\alpha/(2m)} \sqrt{l_j^\mathsf{T}\widehat V l_j},\qquad j=1,\ldots,m, $$ where $z_q$ is the $q$ quantile of the standard normal distribution. They provide at least $1-\alpha$ joint coverage when the individual intervals have their stated coverage. With estimated covariance and asymptotic normality, this guarantee is an asymptotic one.

Bonferroni does not require the estimates to be independent. For each contrast, the two-sided interval has error probability at most $\alpha/m$; the probability that any of the $m$ intervals fails is at most the sum of those error probabilities, hence at most $\alpha$. The method can be conservative, particularly for a large family or highly correlated estimates.

Worked Example: Bonferroni Intervals for Two Transition Effects

Suppose a hypothetical fitted transition model estimates two log hazard ratios for the same covariate: $\widehat{\beta}_1=0.40$ and $\widehat{\beta}_2=-0.10$. Their estimated covariance matrix is

\widehat V= \begin{pmatrix} 0.0400 & 0.0120\\ 0.0120 & 0.0225 \end{pmatrix}.

The standard errors are $\sqrt{0.0400}=0.200$ and $\sqrt{0.0225}=0.150$. For a 95% simultaneous family of two intervals, $\alpha=0.05$, $m=2$, and the Bonferroni critical value is $z_{0.9875}=2.241$.

For transition 1, the interval is $0.40\pm2.241(0.200)=0.40\pm0.4482$, or $[-0.0482,\,0.8482]$. For transition 2, it is $-0.10\pm2.241(0.150)=-0.10\pm0.3362$, or $[-0.4362,\,0.2362]$. Together, these intervals have at least approximately 95% joint coverage under the stated model-based large-sample assumptions.

On the hazard-ratio scale, exponentiating the endpoints gives approximately $[0.953,\,2.335]$ for transition 1 and $[0.646,\,1.266]$ for transition 2. The intervals are simultaneous for the two specified effects; they do not establish that either effect differs from zero.

Contrasts and the full covariance matrix

A comparison between transitions is a different target from the two separate coefficients. For $l=(1,-1)^\mathsf{T}$, the contrast is $\beta_1-\beta_2$. Its variance contains a covariance term:

\widehat{\operatorname{Var}}(\widehat{\beta}_1-\widehat{\beta}_2) =\widehat V_{11}+\widehat V_{22}-2\widehat V_{12}.

The same principle applies to any linear contrast: calculate $l^\mathsf{T}\widehat V l$ using the entire covariance matrix, not just its diagonal. To obtain simultaneous inference, the contrast must be included in the family before selecting the critical value.

Worked Example: A Simultaneous Interval for a Transition Contrast

Use the hypothetical estimates and covariance matrix above. Suppose the prespecified family consists of two targets: $\beta_1$ and the contrast $\beta_1-\beta_2$. The estimated contrast is $0.40-(-0.10)=0.50$. Its estimated variance is

0.0400+0.0225-2(0.0120)=0.0385, \qquad SE=\sqrt{0.0385}=0.1962.

There are again $m=2$ targets, so the Bonferroni critical value is 2.241. The contrast interval is $0.50\pm2.241(0.1962)=0.50\pm0.4397$, or $[0.0603,\,0.9397]$. Because zero is outside this simultaneous interval, the data support a nonzero difference between the two log hazard ratios at the familywise 5% level, conditional on the model and approximation.

Exponentiating the contrast interval gives approximately $[1.062,\,2.559]$. This estimates the ratio $\exp(\beta_1)/\exp(\beta_2)$ of the two hazard ratios. The covariance $0.0120$ reduces the contrast variance here; omitting it would give $0.0400+0.0225=0.0625$, a different standard error and interval.

Alternatives and their scope

Bonferroni is transparent and valid regardless of dependence among the targets, but it can sacrifice precision. For a finite, prespecified family, a max-$|Z|$ or max-$|t|$ procedure can use the joint distribution of the standardized estimates. Its critical value is the quantile of the maximum absolute standardized error, calculated using their correlation matrix. Because it accounts for dependence, it may be smaller than the Bonferroni critical value. The correlation must be estimated reliably, and the relevant multivariate reference distribution and degrees-of-freedom approximation must be justified; one should not use an ordinary single-interval critical value for the maximum.

A different option is a simultaneous Wald ellipsoid for all $p$ coefficients in $\beta$. Under the same large-sample normal approximation, it is centered at $\widehat{\beta}$ and uses the chi-square quantile with $p$ degrees of freedom. Any linear contrast can then be projected from the ellipsoid to obtain a simultaneous interval. This protects a broader collection of contrasts than a finite family may require, so it is often more conservative when only a few contrasts matter.

These methods answer related but distinct questions. Pointwise intervals target one coefficient at a time and do not control familywise error. Bonferroni and max-$|Z|$ intervals protect a specified finite set. A Wald ellipsoid protects all linear contrasts in the coefficient space under its assumptions. The earlier material on likelihood-ratio tests for transition-specific terms addresses joint hypothesis tests; intervals add direction and scale information, but do not make the test and interval interchangeable in every finite-sample setting.

Worked Example: An Ellipsoid-Based Interval for Any Contrast

Continue with the two-coefficient estimate and covariance matrix above. A 95% asymptotic Wald ellipsoid for both coefficients uses $\chi^2_{2,0.95}=5.991$. Its projection gives a simultaneous interval for any linear contrast $l^\mathsf{T}\beta$, with half-width $\sqrt{5.991}\sqrt{l^\mathsf{T}\widehat V l}$.

For $l=(1,-1)^\mathsf{T}$, the estimated contrast is 0.50 and its standard error is 0.1962, as calculated above. Since $\sqrt{5.991}=2.448$, the half-width is $2.448(0.1962)=0.4803$, giving $[0.0197,\,0.9803]$. This interval covers the contrast as part of a procedure that protects all linear contrasts of the two coefficients, rather than only the two targets in the previous example. Its width is consequently not required to match the interval tailored to that finite family.

Why the ellipsoid gives simultaneous coverage

The key argument is a geometric inequality. Write $\delta=\widehat{\beta}-\beta$. On the event that the Wald ellipsoid contains $\beta$, the covariance-scaled squared distance satisfies $\delta^\mathsf{T}\widehat V^{-1}\delta\leq\chi^2_{p,1-\alpha}$. For any contrast $l$, Cauchy–Schwarz gives

|l^\mathsf{T}\delta| = |( \widehat V^{1/2}l)^\mathsf{T}(\widehat V^{-1/2}\delta)| \leq \sqrt{l^\mathsf{T}\widehat V l} \sqrt{\delta^\mathsf{T}\widehat V^{-1}\delta}.

Therefore, whenever $\beta$ lies in the ellipsoid, every contrast satisfies

|l^\mathsf{T}\widehat{\beta}-l^\mathsf{T}\beta| \leq \sqrt{\chi^2_{p,1-\alpha}}\sqrt{l^\mathsf{T}\widehat V l}.

The event of ellipsoid coverage thus implies coverage of all its projected contrast intervals at once. The asymptotic chi-square reference is justified when the coefficient estimator is approximately multivariate normal and the covariance estimate is consistent and nonsingular. This derivation also explains why the protection can be more extensive than needed for a short list of targets.

Assumptions, diagnostics, and limitations

Before reporting simultaneous intervals, verify that the targets and family are scientifically meaningful and specified clearly. The model must identify the coefficients in question, its fitted likelihood and transition structure must be appropriate, and the covariance matrix must correspond to that fit. The usual Wald intervals further rely on an adequate large-sample normal approximation. These conditions are especially consequential when some transitions have few events or a coefficient is weakly identified.

Check that the estimated covariance matrix is symmetric and positive definite (or, where appropriate, positive semidefinite), and that the reported standard errors agree with its diagonal. For each contrast, independently verify $l^\mathsf{T}\widehat V l\geq0$. Large correlations, very large standard errors, unstable estimates, or a nearly singular covariance matrix can indicate weak information or numerical problems. Transition-specific residual diagnostics and model checks help assess the fitted model, but passing a diagnostic does not prove that the model or large-sample approximation is correct.

The intervals also have a defined scope. They cover the coefficients or contrasts in the chosen family, not every possible post hoc comparison. Selecting transitions after inspecting results and then presenting intervals as if that selection had been prespecified can invalidate the stated familywise interpretation. Bonferroni remains valid for a fixed family without independence, but may be wide; max-$|Z|$ depends on a usable joint approximation; and the ellipsoid may be unnecessarily broad for a small, targeted family. None of these approaches resolves bias from model misspecification, unmeasured confounding, or a poorly defined estimand.

Common mistake

Wrong reasoning: “Each of five transition coefficients has a 95% interval, so there is 95% confidence that all five intervals cover.”
Correction: Each interval's 95% coverage is pointwise. The chance of at least one miss across five intervals can exceed 5%. Use an explicitly simultaneous procedure, such as Bonferroni with $m=5$, or a justified joint max-$|Z|$ method.
Wrong reasoning: “For $\beta_1-\beta_2$, add the two variances and ignore their covariance.”
Correction: Use $\widehat V_{11}+\widehat V_{22}-2\widehat V_{12}$. Shared individuals and model fitting can make the coefficient estimates dependent.

Key takeaway: Simultaneous inference starts by defining the entire family of transition coefficients or contrasts. Compute each contrast's standard error from the full estimated covariance matrix, then use a critical value calibrated for joint coverage—not the ordinary pointwise critical value by default.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Two coefficients have estimated variances 0.04 and 0.01, with covariance 0.01. What is the estimated variance of their difference?
For four prespecified contrasts and familywise $\alpha=0.05$, what Bonferroni quantile is used in each two-sided interval?
Why does Bonferroni not require independent coefficient estimates?
What family is protected by a two-coefficient Wald ellipsoid?
If a simultaneous interval for a log hazard-ratio contrast is $[0.10,0.50]$, what is the corresponding interval for the ratio of hazard ratios?

Solutions. (1) The variance is $0.04+0.01-2(0.01)=0.03$. (2) Use $z_{1-0.05/(2\cdot4)}=z_{0.99375}$, approximately 2.498. (3) The probability of any failure is bounded by the sum of the individual failure probabilities, without an independence assumption. (4) It protects all linear contrasts of the two coefficients, under the Wald approximation and covariance assumptions. (5) Exponentiate the endpoints: $[\exp(0.10),\exp(0.50)]$, approximately $[1.105,1.649]$.