Lan-DeMets Boundary Construction

From Alpha Spending to Efficacy Boundaries

A spending function specifies how much of the overall type I error budget may have been used by a given information fraction. It does not, by itself, give the efficacy cutoff at an interim look. A boundary must also account for the fact that a trial reaching a later look has not crossed an efficacy boundary earlier. Lan-DeMets boundary construction connects these two pieces: it evaluates the spending function at each observed information fraction and calibrates the boundary so that the cumulative probability of crossing matches the cumulative spending target.

The distinction matters in a group sequential design because the test statistics at successive looks are dependent. A later cutoff based only on its marginal null distribution would count some paths that already crossed an earlier boundary. The appropriate calculation uses the probability of crossing at the current look among paths that stayed below all previous boundaries.

Definition: Let \(t_i=I_i/I_K\) be the information fraction at look \(i\), where \(I_i\) is the information available at that look and \(I_K\) is the planned maximum information. A cumulative alpha spending function \(A(t)\) gives the planned cumulative probability of rejecting \(H_0\) by information fraction \(t\), under \(H_0\). For a one-sided upper efficacy design, boundaries \(c_1,\ldots,c_K\) are calibrated to satisfy \(P_{H_0}(Z_1<c_1,\ldots,Z_{i-1}<c_{i-1}, Z_i\ge c_i)=A(t_i)-A(t_{i-1})\), with \(A(t_0)=0\). Equivalently, the cumulative probability of crossing by look \(i\) is \(A(t_i)\).

The construction assumes a specified null distribution for the sequence of test statistics and the information fractions at the looks. In the canonical group sequential model used here, under \(H_0\), each \(Z_i\) has a standard normal marginal distribution and, for \(j<i\), \(\operatorname{Corr}(Z_j,Z_i)=\sqrt{t_j/t_i}\). This model is appropriate when the information accumulates in the usual way and the standardized statistics have the stated joint null distribution. The spending function is evaluated at increasing information fractions, with \(A(0)=0\), \(A(1)=\alpha\), and cumulative spending that does not decrease. These conditions make the targets interpretable as cumulative error spending; they do not replace checking that the statistical model matches the design.

The tutorial uses one-sided upper boundaries: reject \(H_0\) at look \(i\) if \(Z_i\ge c_i\), provided no earlier boundary was crossed. A two-sided design requires a corresponding two-sided crossing calculation; its allocation is not obtained merely by reusing an upper-tail cutoff unchanged.

How Each Boundary Is Calibrated

At the first look there is no previous crossing to exclude, so the boundary is obtained directly from the spending target:

P_{H_0}(Z_1\ge c_1)=A(t_1), \qquad c_1=\Phi^{-1}\{1-A(t_1)\}.

At a later look, the new crossing probability must equal the increase in cumulative spending, \(\Delta A_i=A(t_i)-A(t_{i-1})\). The null probability is evaluated only over paths that survived earlier looks. At the second look, write \(\rho=\sqrt{t_1/t_2}\). The conditional distribution of \(Z_2\) given \(Z_1=z\) is normal with mean \(\rho z\) and variance \(1-\rho^2\), so the equation for \(c_2\) is

\int_{-\infty}^{c_1} \phi(z)\left[ 1-\Phi\left(\frac{c_2-\rho z}{\sqrt{1-\rho^2}}\right) \right]\,dz = A(t_2)-A(t_1).

The integral is the probability that \(Z_1\) stays below its boundary and \(Z_2\) crosses its boundary. For a fixed \(c_1\), the left side decreases as \(c_2\) increases, so numerical root-finding can locate the cutoff for a specified target. At later looks, the same principle applies: integrate the probability of crossing the current boundary over the joint null distribution restricted to paths that survived every earlier boundary. Then choose \(c_i\) so this new-crossing probability equals \(A(t_i)-A(t_{i-1})\).

Set the look targets.
Record the information fractions \(t_i\), evaluate \(A(t_i)\), and calculate \(\Delta A_i=A(t_i)-A(t_{i-1})\).

Calibrate the first cutoff.
Set the upper-tail probability at look 1 equal to \(A(t_1)\), giving \(c_1=\Phi^{-1}\{1-A(t_1)\}\).

Calibrate each later cutoff.
Under the joint null model, calculate the probability of crossing at that look after surviving all earlier looks. Solve for the cutoff that makes this probability \(\Delta A_i\).

Check cumulative calibration.
Verify that the sum of the disjoint new-crossing probabilities through look \(i\) equals \(A(t_i)\), and that the final cumulative target is \(\alpha\).

Worked Constructions

Worked Example: Calibrating a Two-Look Boundary

Suppose a hypothetical one-sided design has maximum type I error \(\alpha=0.05\), with looks at \(t_1=0.25\) and \(t_2=1\). Use the linear spending function \(A(t)=0.05t\). Under \(H_0\), assume the canonical joint normal model described above.

At the first look, the cumulative target is \(A(0.25)=0.05(0.25)=0.0125\). Therefore

c_1=\Phi^{-1}(1-0.0125)=\Phi^{-1}(0.9875)\approx 2.2414.

At the final look, cumulative spending is \(A(1)=0.05\). The additional crossing probability required at look 2 is \(0.05-0.0125=0.0375\). Since \(\rho=\sqrt{0.25/1}=0.5\), the boundary solves

\int_{-\infty}^{2.2414} \phi(z)\left[ 1-\Phi\left(\frac{c_2-0.5z}{\sqrt{0.75}}\right) \right]\,dz = 0.0375.

Numerical integration and root-finding give \(c_2\approx1.735\). Thus the probability of crossing at look 1 is \(0.0125\), and the probability of crossing at look 2 after surviving look 1 is \(0.0375\). Their sum is \(0.0500\), the planned overall type I error. The marginal upper-tail probability \(P(Z_2\ge1.735)\) is about \(0.0414\), not \(0.0375\); the difference reflects paths that already crossed at look 1 and are excluded from the new-crossing probability.

Worked Example: Turning a Spending Curve Into Look Targets

Consider a hypothetical design with \(\alpha=0.05\), linear spending \(A(t)=0.05t\), and looks at \(t_1=0.40\), \(t_2=0.70\), and \(t_3=1\). First evaluate the cumulative spending:

A(t_1)=0.020,\qquad A(t_2)=0.035,\qquad A(t_3)=0.050.

The respective new-crossing targets are \(0.020\), \(0.035-0.020=0.015\), and \(0.050-0.035=0.015\). The first boundary is

c_1=\Phi^{-1}(1-0.020)=\Phi^{-1}(0.980)\approx2.0537.

For look 2, the correlation under the null is \(\sqrt{0.40/0.70}\approx0.7559\). The second boundary must solve the two-look survival-region integral for a new-crossing probability of \(0.015\). For look 3, calibration must restrict the calculation to paths satisfying both \(Z_1<c_1\) and \(Z_2<c_2\), then set the new-crossing probability to \(0.015\). This example shows why the spending curve supplies targets, while the joint distribution and earlier boundaries determine the actual later cutoffs.

Worked Example: Recalculating the First Boundary at an Observed Look

Suppose a hypothetical study’s first analysis occurs at information fraction \(t=0.30\), and its prespecified spending function is \(A(t)=0.05t\). Under the one-sided standard normal null model, the cumulative target at this look is \(0.05(0.30)=0.015\). Because it is the first look, there are no earlier crossings to subtract:

c_1=\Phi^{-1}(1-0.015)=\Phi^{-1}(0.985)\approx2.1701.

The decision rule at this analysis is to reject \(H_0\) if \(Z_1\ge2.1701\). The cutoff corresponds to a 1.5% upper-tail probability under \(H_0\), not to the full 5% maximum error budget. If the analysis were instead at \(t=0.25\), the same spending function would give a target of \(0.0125\) and a first-look cutoff of approximately \(2.2414\). The cutoff changes because the spending target changes with the information fraction.

Why the Conditional Crossing Calculation Matters

A Lan-DeMets design is not a list of marginal significance thresholds. Its defining calibration is cumulative: by each look, the probability of having crossed at least one efficacy boundary under \(H_0\) matches the spending function at that look. Because crossing events at different looks are disjoint when defined as first crossings, their probabilities add. In particular,

P_{H_0}(\text{cross by look }i) = \sum_{j=1}^{i}P_{H_0}(\text{first cross at look }j) = \sum_{j=1}^{i}\{A(t_j)-A(t_{j-1})\} = A(t_i).

This identity justifies the incremental targets, but it does not say that each target is the marginal tail probability at that look. At the first look the two probabilities coincide. At later looks, calibration must account for the previous survival region.

Compared with classical boundaries planned for a fixed schedule, spending-based construction evaluates the cumulative target at the information fractions supplied to the calculation. A fixed classical boundary instead uses cutoffs calibrated to a particular planned schedule. Direct incremental alpha allocation may state how much error probability is assigned to each look, but a boundary still requires the joint null crossing calculation. Lan-DeMets construction makes the cumulative spending curve the design target and uses that calculation to obtain efficacy cutoffs.

Diagnostics, Limitations, and a Common Mistake

Before using computed boundaries, check that the information fractions are ordered, the spending targets are cumulative and nondecreasing, and the increments sum to the intended final error budget. Verify that the first boundary reproduces its stated one-look tail probability. At later looks, recompute the first-crossing probabilities under the joint null model and confirm that their cumulative sums match \(A(t_i)\). Numerical integration tolerance, root-finding tolerance, and rounding can cause small discrepancies, so comparisons should allow for stated numerical precision.

A zero spending increment implies that no additional crossing probability is allocated at that look; a finite cutoff should not be reported as though it spends a positive amount. If a computed crossing probability does not decrease as a proposed cutoff increases, or the cumulative probability exceeds its target, check the integration region, dependence structure, and whether earlier crossings have been excluded. Results also depend on the assumed null distribution and information fractions; a calibrated boundary is not automatically appropriate if those inputs do not describe the design.

Wrong reasoning: “The incremental target at look 2 is 0.0375, so choose \(c_2\) with \(P(Z_2\ge c_2)=0.0375\).”
Correction: That marginal calculation ignores paths that crossed at look 1. Choose \(c_2\) so that the probability of \(Z_2\ge c_2\) among paths that survived look 1 is 0.0375.

The principal limitation is that spending-function calibration controls the stated null crossing probability only under the model and information quantities used in the calculation. The spending curve alone does not determine the joint distribution, nor does it guarantee that the design is sensible for every scientific objective. The boundary computation should therefore be reported alongside its spending function, information fractions, null model, and numerical calibration checks.

Key takeaway: Evaluate cumulative spending at each information fraction, take differences to obtain first-crossing targets, and calibrate each later cutoff over the null paths that survived previous looks. The sum of those calibrated first-crossing probabilities is the cumulative type I error spent.

Check Your Understanding

Answer each question, then compare with the solutions below.

For \(\alpha=0.04\) and linear spending \(A(t)=0.04t\), what is the cumulative target at \(t=0.5\), and what is the first-look upper boundary if this is the first look?
Why is a later look’s incremental spending generally not equal to its marginal upper-tail probability?
A design has cumulative spending \(A(t_1)=0.01\) and \(A(t_2)=0.03\). What is the new-crossing target at look 2?
State one numerical check that can detect a failure in boundary calibration.

Solutions. (1) The target is \(0.04(0.5)=0.020\), so \(c_1=\Phi^{-1}(0.980)\approx2.0537\). (2) The marginal tail includes paths that may already have crossed an earlier boundary; the new-crossing probability excludes those paths. (3) The increment is \(0.03-0.01=0.020\). (4) Recompute the null probability of each first crossing and check that their cumulative sum through look \(i\) equals \(A(t_i)\), allowing for numerical tolerance.