Why Dependence and Monotonicity Matter
In a multi-arm, multi-stage trial, evidence for different arms is often dependent. For example, treatment-versus-control statistics can share a control group, and successive looks can use overlapping data. This dependence affects the probability of at least one false rejection. When rejection events are positively associated, the chance of at least one rejection is no greater than it would be if the events were independent with the same marginal probabilities. This can support less conservative calibration, but only when the assumed dependence structure and rejection rules are justified.
The key structural condition is monotonicity: increasing any test statistic in the direction of benefit must not remove a rejection that was already possible. Ordinary upper-bound rules have this property. It allows results about positive dependence among statistics to be translated into results about joint boundary crossings.
For an arm \(i\) measured at stages \(1,\ldots,K\), write its upper efficacy boundary at stage \(k\) as \(b_{ik}\). Its rejection event is \(A_i=\{\text{there is a }k\text{ such that }Z_{ik}\ge b_{ik}\}\). This event is coordinatewise increasing: raising any stage statistic cannot turn a crossing into a non-crossing. The boundary values may differ across arms and stages. Monotonicity does not require the numerical boundaries to be ordered over time.
A multivariate normal vector with nonnegative pairwise covariances is associated. This gives a useful sufficient condition for Gaussian test statistics: check that the statistics have the intended joint normal distribution and that their covariance entries are nonnegative. Nonnegative correlation is not automatic in every design or after every data-dependent operation.
A Familywise-Error Bound From Positive Association
Let \(I_0\) be the set of true null hypotheses, and let \(A_i\) be the event of rejecting hypothesis \(i\). The familywise error rate (FWER) for this configuration is \(P(\bigcup_{i\in I_0}A_i)\). If the vector of true-null statistics is associated and each \(A_i\) is coordinatewise increasing, then the non-rejection events \(A_i^c\) are coordinatewise decreasing. Association also gives the product inequality for events that are all decreasing. Thus,
Taking complements yields the bound
The expression on the right is the FWER that independent arm-level rejection indicators would have if their marginal rejection probabilities were the same \(p_i\). The inequality is an upper bound, not an assertion that the actual FWER equals the independence calculation. In general, the FWER is at least \(\max_{i\in I_0}p_i\), since the union contains each individual rejection event.
Worked Example: Two Arms at One Look
Suppose two true-null, standard normal test statistics each use the threshold 1.96. Under independence, the probability of at least one false rejection is \(1-\Phi(1.96)^2\). The marginal upper-tail probability is \(1-\Phi(1.96)\approx0.024997895\), so
This is the independence FWER for the stated threshold. If the marginal probability is rounded to 0.0250 first, the resulting approximate calculation is \(1-(1-0.0250)^2=0.049375\); that rounded value is not the exact FWER for threshold 1.96.
If the two statistics are positively associated, the bound above shows that their actual FWER is no greater than 0.0493709. It is at least the single-arm probability, approximately 0.0249979. Without specifying and calculating their joint distribution, these bounds do not determine the exact FWER.
Worked Example: Three Multi-Stage Arm-Crossing Events
Consider a hypothetical three-arm design with multiple looks. Suppose each arm’s full stagewise rule has already been checked to give the following true-null marginal probabilities of crossing any efficacy boundary: \(p_1=0.0200\), \(p_2=0.0150\), and \(p_3=0.0100\). Assume the joint vector of all relevant stage statistics is associated, and each arm’s crossing event is coordinatewise increasing. These assumptions are essential; the marginal probabilities alone do not establish them.
The association bound is
Thus the FWER is at most 0.044353 under the stated assumptions. If the arm-level events were independent, the FWER would equal 0.044353. Positive association permits an upper-bound comparison with that independence value; it does not establish equality. The FWER is at least \(0.0200\), the largest of the three marginal probabilities.
Monotone Boundaries and Their Consequences
A boundary-crossing rule can also be described directly in terms of its rejection region. For a fixed arm, the region \(A_i\) contains every statistic vector that crosses at least one boundary. Raising one or more boundary values shrinks this region. Formally, if \(b'_{ik}\ge b_{ik}\) at every stage, then the crossing event under \(b'\) is contained in the crossing event under \(b\). Consequently, its probability cannot increase, for any distribution of the statistics.
Worked Example: Raising a Later-Look Boundary
For a simple hypothetical calculation, suppose one arm has two independent standard normal stage statistics. Independence is an explicit simplifying assumption here, not a general model for repeated looks. With boundaries \(b_1=2.4\) and \(b_2=2.0\), the crossing probability is
Now raise the second boundary to 2.1 while leaving the first unchanged. The new crossing probability is
The probability falls, as monotonicity requires. With dependent looks, the product formulas in this example would not apply; the event-inclusion conclusion would still hold. In an actual stagewise design, the joint distribution must be used to calculate the crossing probability.
The association argument and the boundary-monotonicity argument have different jobs. Monotonicity identifies the events to which an association result can be applied. Association then provides a comparison of their joint probability with the product of their marginal probabilities. Neither result supplies marginal crossing probabilities: those must come from the specified stagewise model and boundaries.
When Dependence-Aware Calibration Is Justified
A dependence-aware calibration uses a specified joint distribution to select boundaries so that the probability of one or more true-null rejections is controlled. For example, under a justified multivariate normal model, one can evaluate the probability of the union of the boundary-crossing regions and choose critical values to meet a target level. The positive-association bound can also justify a conservative comparison with the independence benchmark, provided its assumptions hold.
For strong FWER control, the guarantee must hold for every configuration of true and false null hypotheses, not just when all null hypotheses are true. Therefore, it is not enough to show positive association under the global null. For each configuration, the true-null marginal rejection probabilities must be controlled and the true-null rejection events must have the required association property under that configuration. If treatment effects, adaptive choices, or changes to the analysis alter the distribution of the true-null statistics, a global-null calculation alone may not establish strong control.
This distinction is particularly important with shared controls and repeated looks. Shared controls often induce positive correlation across arm statistics, and nested observations often induce positive correlation across looks, but neither observation replaces checking the full joint model. A covariance matrix that has negative entries, a data-dependent rule that makes the rejection region non-monotone, or a misspecified joint distribution can invalidate the argument. An exact joint calibration is only as trustworthy as its statistical model.
Compared with these approaches, Bonferroni control uses the inequality \(P(\bigcup_i A_i)\le\sum_i P(A_i)\), which does not require positive dependence. It can be more conservative, but it remains a useful fallback when the dependence structure is uncertain. Dependence-aware calibration can use a credible joint distribution to improve efficiency; it should not be chosen merely because the observed correlations happen to be positive.
List the true-null hypotheses under consideration and define each arm’s crossing event over all its stages.
Verify that increasing any statistic in the rejection direction cannot remove a rejection. If decisions involve additional data-dependent rules, check the full rule rather than only its numerical boundaries.
Justify the joint distribution and association property for the relevant true-null statistics under each configuration for which control is claimed.
Use the product bound only when its assumptions hold; otherwise use a valid alternative such as Bonferroni control or a justified joint-probability calculation.
Common Pitfall and Diagnostics
A common mistake is to observe positive correlations between arm statistics under the global null and conclude that independence-based calibration must control the FWER strongly. Positive association gives the stated union bound only when the rejection events are monotone and the association and marginal-probability conditions apply to the true-null events in the configuration being considered. A global-null check by itself is not a strong-control argument.
Before using the comparison, check that the statistics are oriented in the same rejection direction, that the rejection regions are increasing, and that the joint model represents shared controls and repeated looks. Check covariance signs in the model rather than inferring them from design intuition alone. If dependence is estimated, consider whether its uncertainty could materially affect the claimed calibration. A disagreement between model-based crossing probabilities and simulation under the planned design is a warning to investigate the model or implementation, not a reason to ignore the discrepancy.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Two positively associated true-null rejection events have marginal probabilities 0.03 and 0.02. What is the association-based upper bound on their union probability?
- Why does raising one efficacy boundary not increase that arm’s crossing probability, even if the stage statistics are dependent?
- Does positive association under the global null alone establish strong FWER control? Explain.
- When might Bonferroni control be preferable to dependence-aware calibration?
Solutions. (1) The bound is \(1-(1-0.03)(1-0.02)=1-(0.97)(0.98)=0.0494\). It is an upper bound; equality holds for independent events, not necessarily for positively associated events. (2) Raising a boundary makes the crossing event a subset of its former event. The probability of a subset cannot be greater, regardless of dependence. (3) No. Strong control requires the relevant marginal error probabilities and dependence conditions for every configuration of true and false null hypotheses; the global-null result alone covers only that configuration. (4) Bonferroni may be preferable when the joint dependence model is uncertain or the monotonicity or association conditions cannot be justified, because its union bound does not require positive dependence.