Translating Beta Spending Into Lower Z Boundaries

From a Spending Target to a Boundary

A cumulative futility-spending function sets how much probability of stopping for futility may have accrued by each analysis, under a specified design alternative. A lower Z boundary makes that target operational: at look $i$, a sufficiently low standardized statistic signals that the trial is unlikely to achieve its objective and may stop. The translation is direct at the first look, but at later looks it must account for the statistics’ dependence and for participants who have not already crossed a futility boundary.

The calculation therefore has two parts. First define the event whose probability the spending target controls. Then find the boundary that makes this event’s probability equal the target. The calculation below concerns futility crossing under the design alternative; it is not, by itself, a calculation of the complete trial’s power or of crossing probabilities when efficacy stopping is also incorporated.

Definition: Let $K$ be the number of analyses, and let $t_i=I_i/I_K$ be the information fraction at look $i$, with $0<t_1<\cdots<t_K=1$. Let $\Delta$ be the mean of the final-look Z statistic under the design alternative. Under the canonical joint distribution, the interim statistics satisfy $$ (Z_1,\ldots,Z_K)\sim N_K(\boldsymbol{\mu},R),\qquad \mu_i=\Delta\sqrt{t_i},\qquad R_{ij}=\sqrt{\frac{t_i}{t_j}}\quad (i\leq j), $$ with $R_{ji}=R_{ij}$. Thus each $Z_i$ has variance 1, and the statistics are correlated across looks. A lower futility boundary $l_i$ means futility is crossed at look $i$ when $Z_i\leq l_i$.

Write $B_i=B(t_i)$ for the cumulative futility-spending target at look $i$, evaluated under this design alternative, and set $B_0=0$. In the futility-only calibration, $B_i$ is the probability that at least one lower boundary has been crossed by look $i$:

B_i=P_{\Delta}\!\left(\bigcup_{j=1}^{i}\{Z_j\leq l_j\}\right).

Consequently, $B_i-B_{i-1}$ is the probability of a first futility crossing at look $i$, not the probability of crossing $l_i$ without regard to earlier looks. A statistic already below an earlier lower boundary belongs to the earlier cumulative spending, not again to the increment at look $i$.

The Sequential Boundary Equation

A first futility crossing at look $i$ requires the statistic to have stayed above every earlier futility boundary and to be at or below the new boundary. If $f_i$ denotes the joint density of $(Z_1,\ldots,Z_i)$ under the design alternative, the later-look boundary therefore solves

\int_{z_1>l_1,\ldots,z_{i-1}>l_{i-1}} \int_{-\infty}^{l_i} f_i(z_1,\ldots,z_i)\,dz_i\,dz_{i-1}\cdots dz_1 = B_i-B_{i-1}.

The distribution in this integral is the canonical joint normal distribution, including its covariance among looks. Replacing it with independent marginal normals generally gives the wrong probability. At the first look, there are no earlier boundaries, so the equation reduces to a single normal quantile:

P_{\Delta}(Z_1\leq l_1)=B_1,\qquad l_1=\Delta\sqrt{t_1}+\Phi^{-1}(B_1),

where $\Phi$ is the standard normal cumulative distribution function. For $i>1$, the integral is usually evaluated numerically and a one-dimensional root-finding method is used for $l_i$. The left side is continuous and nondecreasing in $l_i$: raising the boundary can only add outcomes to the event $Z_i\leq l_i$. This gives a well-defined numerical target whenever the requested increment is attainable.

Feasibility check: Since the probability of having survived earlier futility boundaries is $1-B_{i-1}$, the required increment must satisfy $$ 0\leq B_i-B_{i-1}\leq 1-B_{i-1}. $$ Equivalently, cumulative targets must be nondecreasing and no greater than 1. A zero increment corresponds to $l_i=-\infty$ (no first crossing at that look); an increment equal to the entire surviving probability corresponds to $l_i=+\infty$. Strictly intermediate increments give finite boundaries.

Worked Calculations

Worked Example: A First-Look Boundary

Suppose a hypothetical design has $t_1=0.40$, final-look mean $\Delta=2.40$ under the design alternative, and first-look cumulative futility spending $B_1=0.025$. The canonical model gives $Z_1\sim N(2.40\sqrt{0.40},1)$. The assumptions needed here are simply the specified design alternative and the stated canonical marginal normal distribution.

The interim mean is $2.40\sqrt{0.40}=1.518$. Since $\Phi^{-1}(0.025)=-1.960$, the boundary is $$ l_1=1.518-1.960=-0.442. $$

Thus the futility-only calibration puts the boundary at approximately $-0.442$ on the Z scale: under the design alternative, $P_{\Delta}(Z_1\leq -0.442)$ is about 0.025. This is a probability under the specified alternative, not under the null hypothesis.

Worked Example: A Second-Look Boundary With Correlated Looks

Consider a hypothetical two-look design with $t_1=0.50$, $t_2=1$, and $\Delta=2.00$. Suppose the first boundary is $l_1=0.50$, and the cumulative targets are $B_1=0.180$ and $B_2=0.300$. The model has means $\mu_1=2\sqrt{0.50}=1.414$ and $\mu_2=2.000$, and correlation $R_{12}=\sqrt{0.50}=0.7071$. These values specify the canonical bivariate normal distribution used in the calculation.

The first-look spending is $P_{\Delta}(Z_1\leq0.50)=\Phi(0.50-1.414)$, approximately 0.180, consistent with $B_1$. The additional first-crossing probability required at look 2 is $$ B_2-B_1=0.300-0.180=0.120. $$ For a proposed second boundary $c$, the conditional distribution is $Z_2\mid Z_1=z\sim N(1+0.7071z,\,1-0.7071^2)$, so its conditional standard deviation is $0.7071$. The probability equation to solve is $$ \int_{0.50}^{\infty} \phi(z-1.414)\, \Phi\!\left(\frac{c-(1+0.7071z)}{0.7071}\right)\,dz =0.120. $$

Numerical integration and root finding give $c$ approximately $1.29$ (rounding the boundary to two decimals). Substituting this boundary in the integral gives a first-crossing probability of about 0.120; adding the previous 0.180 gives cumulative spending of about 0.300. The integral starts at $Z_1=0.50$ because only paths that did not cross the first boundary are eligible for a first crossing at look 2.

The requested increment is feasible: $0.120$ is between zero and the surviving probability $1-0.180=0.820$. Ignoring the correlation or using $P_{\Delta}(Z_2\leq c)=0.120$ would answer a different question and produce a different boundary.

Worked Example: The Design Alternative Changes the Boundary

Suppose a hypothetical single-look calculation has $t_1=0.25$ and target $B_1=0.10$. First use a design alternative with $\Delta=1.20$. Its interim mean is $1.20\sqrt{0.25}=0.600$, and $\Phi^{-1}(0.10)=-1.282$. Therefore $$ l_1=0.600-1.282=-0.682. $$

Now keep the information fraction and spending target fixed but use $\Delta=2.00$. The interim mean becomes $2.00\sqrt{0.25}=1.000$, so $$ l_1=1.000-1.282=-0.282. $$

The boundary is higher under the more favorable alternative because a higher boundary is needed to retain the same 0.10 lower-tail probability when the statistic’s mean is higher. The target alone does not determine the boundary; the design alternative is an essential input.

Numerical Checks and Interpretation

A boundary calculation should be checked against both its probability definition and its inputs. For each look, verify that the information fractions increase, the cumulative targets do not decrease, and the means and correlations match the specified canonical model. After obtaining a numerical root, substitute it back into the probability equation and compare the attained cumulative probability with $B_i$, allowing for numerical integration and rounding error.

The root equation also provides a useful diagnostic. If the requested increment exceeds the probability of surviving earlier futility boundaries, no boundary can satisfy it. If the computed left side barely changes across trial values of $l_i$, check integration accuracy, the ordering of the integration limits, and whether the earlier-survival event has been coded correctly. A boundary that moves in the wrong direction when the spending target increases is another warning: for fixed prior boundaries, a larger increment should not yield a lower boundary.

This calibration uses the probability of futility crossing under the design alternative and does not include upper efficacy boundaries in the event. If a trial stops for efficacy before a possible futility crossing, the actual probability of stopping for futility under the complete stopping rule can differ. The calibrated $l_i$ should therefore be interpreted as a lower-boundary translation of the stated spending targets under the specified convention, not as a guarantee that every stopping rule has those same realized probabilities.

Choosing the Calculation That Matches the Question

Approach	What it calculates	When it is appropriate
First-look normal quantile	A lower-tail probability from one marginal normal distribution	At the first look, before any earlier futility boundary exists
Sequential joint-probability equation	First-crossing probability after surviving previous lower boundaries	For translating later cumulative spending targets under the canonical model
Marginal quantile at every look	$P_{\Delta}(Z_i\leq l_i)$, without conditioning on prior survival	Not a substitute for sequential cumulative spending when earlier boundaries exist
Simulation of the complete stopping rule	Operating characteristics under all implemented stopping conditions	As a check of a full design, with the complete rule represented correctly

The joint-probability equation is the appropriate translation when the target is cumulative futility crossing under the canonical design alternative. Simulation can check a complete implementation, but it does not remove the need to define which event the target refers to. A marginal quantile at every look is simpler, but it generally fails to account for earlier crossings and dependence.

Common Mistakes and Limitations

Using cumulative spending as the new-look probability. The look-$i$ increment is $B_i-B_{i-1}$; it is not $B_i$. The boundary equation must also exclude paths that crossed earlier.
Treating interim statistics as independent. Under the canonical model their correlation is determined by the information fractions. Ignoring it changes the joint probability and hence the boundary.
Using a null distribution for an alternative-based target. Beta spending for futility is calibrated under the stated design alternative. Changing the assumed alternative changes the mean vector and can change the boundaries.
Calling the cumulative futility-crossing probability the overall type II error. The former is the probability of a specified futility event under the design alternative. Overall failure to reject can include other pathways, such as reaching the final analysis without an efficacy crossing.
Reporting a boundary without its convention. State the alternative and whether the probability calculation includes efficacy stopping. Otherwise, the meaning of the number is unclear.

The method depends on the canonical joint distribution being an adequate representation of the standardized statistics at the planned information times. It also depends on a specified alternative and spending schedule. If the model, information times, or stopping event differs, the probability equation must be modified rather than reusing a boundary without checking its calibration.

Check Your Understanding

Answer each question, then compare with the solutions below.

At a first look, $t_1=0.36$, $\Delta=2.00$, and $B_1=0.05$. What is the lower boundary?
If $B_1=0.08$ and $B_2=0.13$, what first-crossing probability is assigned to look 2?
Why is $P_{\Delta}(Z_2\leq l_2)$ generally not the correct equation for the second boundary?
With $B_{i-1}=0.40$, can a proposed increment of $0.65$ be attained by any finite lower boundary? Explain.

Solutions. (1) The interim mean is $2\sqrt{0.36}=1.20$, and $\Phi^{-1}(0.05)=-1.645$, so $l_1=1.20-1.645=-0.445$. (2) The increment is $0.13-0.08=0.05$. (3) That marginal probability includes paths that may already have crossed the first boundary; the sequential equation restricts the probability to paths with $Z_1>l_1$ and uses the joint distribution. (4) No. The surviving probability is $1-0.40=0.60$, and an increment of $0.65$ exceeds it, so the target is infeasible under this futility-only definition.