Rounding Lower Boundaries In Practice

Why Boundary Rounding Needs a Design Check

A computed lower futility boundary is often stored or reported to fewer decimal places than the software used to calculate it. The numerical change may look negligible, but a lower boundary determines which interim results trigger stopping: the statistic stops for futility when it is at or below the boundary. Moving the boundary upward therefore includes additional outcomes in the stopping region. Whether this change matters depends on the probability mass near the boundary and on the design’s joint monitoring rule.

The relevant check is not merely whether the rounded number is close to the computed number. It is whether using the rounded boundary changes the operating characteristics enough to affect the design’s intended error rates, power, or futility-spending targets. This tutorial focuses on that check. Earlier material on binding and nonbinding futility, attained beta, and simulation provides the decision-rule and operating-characteristic context.

For clarity, suppose \(l_i^{(r)}> \tilde l_i\), as happens when a negative boundary is rounded upward toward zero. Then the rounded rule stops for futility on the additional interval \(\tilde l_i<Z_i\leq l_i^{(r)}\). Rounding downward has the reverse effect: it removes outcomes from the futility-stopping region. These statements concern the boundary rule itself; overall effects can also depend on whether earlier efficacy or futility decisions prevent the trial from reaching look \(i\).

Model and Conditional Probability Change

For the calculations below, use the standard canonical group sequential model at increasing information fractions \(0<t_1<\cdots<t_K=1\). Under a fixed standardized effect \(\theta\), assume the vector of standardized statistics is jointly normal, each statistic has variance one, and

This model is appropriate when the information-based normal approximation and the planned analysis schedule are adequate. It is not a substitute for checking the actual analysis model when, for example, the statistic has a materially different distribution or the information fractions do not represent the planned information. At each look, assume the stopping rule is applied to the statistic and boundary actually specified in the monitoring plan.

In this model, the conditional distribution at a later look, given the immediately preceding statistic, is also normal. If \(t_{i-1}<t_i\), then

Write the conditional mean as \(m\) and the conditional standard deviation as \(s_c\), the square root of the variance shown above. For an upward-rounded boundary \(l^{(r)}>l\), the conditional increase in the chance of crossing the futility boundary, given that history, is exactly

This expression gives a direct local sensitivity calculation. It is not, by itself, the change in the overall probability of stopping for futility: the overall calculation must also account for the probability of reaching this look and the distribution of earlier statistics among trials that continue.

Worked Boundary Changes

From a Local Change to Operating Characteristics

An operating characteristic is a probability defined by the complete stopping rule under a specified parameter value. Examples include the probability of rejecting \(H_0\) under the null, the probability of rejecting under an alternative (power), and the probability of stopping for futility. A rounded boundary can affect these probabilities differently. In particular, futility crossing probability is not automatically the same as overall type II error: the decision rule and whether futility is binding determine how the events relate.

For a binding futility rule, an exact calculation partitions the joint normal distribution into the regions that correspond to stopping for efficacy, stopping for futility, and continuing. For example, with lower boundary \(l_1\) and upper efficacy boundary \(u_1\) at the first look, continuation requires \(l_1<Z_1<u_1\). If the lower boundary is rounded upward, the first-look futility region expands, and the continuation region contracts. At later looks, only histories that have not already stopped contribute to that look’s unconditional stopping probability.

A direct verification compares the original and rounded designs under the same model, schedule, parameter values, and efficacy boundaries. Evaluate the full stopping rule once with the computed \(l_i\) values and once with the stored \(l_i^{(r)}\) values. Check the planned quantities under \(H_0\) and under the design alternative, as well as any futility-spending targets used to construct the boundaries. A check only at the alternative can miss a change in type I error; a check only under the null may not reveal a material change in power or futility behavior.

Choosing a Rounding Direction

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to check the probability mass between the full-precision and implemented boundaries. If that mass is tiny under the relevant conditional distributions, the local effect is likely to be small; if substantial density lies near a boundary, even a small numerical change may matter. The overall impact also depends on how often the design reaches the look and on the prior history of trials that continue. Consequently, a tiny absolute change in a boundary is not, on its own, evidence that the operating characteristics are unchanged.

Exact joint-normal integration is the direct verification approach when the canonical model is appropriate and the calculation is available. It avoids Monte Carlo sampling error and permits close comparisons of the original and rounded designs. Simulation is a useful independent check and can accommodate more complex implementation details, but finite simulation produces random error. When comparing two designs by simulation, use the same generated paths for both where possible; this paired comparison reduces noise in their difference. Report the number of simulated trials and the uncertainty of estimated probabilities rather than treating simulation estimates as exact.

Common warning signs include a rounded lower boundary that is numerically below the computed value when the documented rule says “round upward,” an unexpected decrease in futility stopping after upward rounding, or an unexplained discrepancy between code and the monitoring plan. Also investigate a rounded boundary that meets or exceeds the efficacy boundary: the intended continuation region may then be empty or incoherent. These checks can catch sign, precision, and inequality errors before they affect decisions.

Rounding is not the only source of discrepancy. The normal model may be an approximation; information fractions or effect assumptions may be misspecified; and numerical integration or simulation may be inaccurate. Rounding checks establish the consequences of using particular thresholds under the model being evaluated. They do not establish that the model itself is appropriate, nor do they replace a prespecified tolerance for acceptable changes in operating characteristics.

Common Mistake