A Two-Look Spending Design

Why a Two-Look Design Needs Two Calibrated Boundaries

A two-look group sequential design allows a one-sided efficacy conclusion at an interim analysis and, if that boundary is not crossed, at the final analysis. The boundaries cannot generally be chosen by applying the same nominal significance level independently at each look: the chance of crossing at the second look depends on the first-look result. Calibration must account for both the information schedule and the dependence between the two test statistics.

The earlier material on alpha-spending functions distinguishes cumulative spending by a given information time from the probability of crossing at that particular look. Here we use those ideas to specify and check a complete two-look design. The example uses a one-sided total significance level of \(0.025\), an interim look at half the final information, and a linear spending curve.

Definition: Let \(I_1<I_2\) be the information at the interim and final looks, and let \(t_k=I_k/I_2\), so \(t_2=1\). A cumulative alpha-spending function \(A(t)\) is nondecreasing, has \(A(0)=0\), and satisfies \(A(1)=\alpha\). For upper efficacy boundaries \(b_1,b_2\), the design stops for efficacy at look 1 if \(Z_1\geq b_1\); otherwise it stops for efficacy at look 2 if \(Z_2\geq b_2\). Calibration requires \(P_{H_0}(Z_1\geq b_1)=A(t_1)\) and \(P_{H_0}(Z_1\geq b_1\text{ or }Z_2\geq b_2)=\alpha\).

Model and Calibration Equations

Assume the canonical group sequential model under \(H_0\): the standardized test statistics have marginal distributions \(Z_k\sim N(0,1)\), and their joint distribution is bivariate normal with correlation \(\rho=\sqrt{I_1/I_2}=\sqrt{t_1}\). This model follows when information accumulates in independent increments and the statistic is standardized by the square root of information. The information values must be correctly specified, the one-sided test direction must be chosen in advance, and the spending curve must be specified before calibration. No futility stopping is included here.

At the first look, the spending requirement directly determines the boundary. If \(\Phi\) is the standard normal cumulative distribution function, then \(P_{H_0}(Z_1\geq b_1)=1-\Phi(b_1)\). Thus \(b_1=\Phi^{-1}(1-A(t_1))\). At the final look, the marginal tail probability \(P(Z_2\geq b_2)\) is not the required cumulative error: some paths reaching the final boundary would already have crossed at look 1.

P_{H_0}(\text{cross by look 2}) =1-P_{H_0}(Z_1<b_1,\ Z_2<b_2)=\alpha.

Under the canonical model, conditional on \(Z_1=z\), \(Z_2\) has a normal distribution with mean \(\rho z\) and variance \(1-\rho^2\). Therefore the no-crossing probability used to solve for \(b_2\) is

\int_{-\infty}^{b_1} \phi(z)\, \Phi\!\left(\frac{b_2-\rho z}{\sqrt{1-\rho^2}}\right)\,dz =1-\alpha.

Here \(\phi\) and \(\Phi\) are the standard normal density and distribution functions. This equation is usually solved numerically. Equivalently, the probability of a new crossing at look 2 can be written by conditioning on \(Z_2\):

P_{H_0}(Z_1<b_1,\ Z_2\geq b_2) =\int_{b_2}^{\infty} \phi(z)\, \Phi\!\left(\frac{b_1-\rho z}{\sqrt{1-\rho^2}}\right)\,dz.

The first-look crossing probability plus this later, previously uncrossed probability must equal \(\alpha\). This decomposition is also a direct check of the calibrated design.

Worked Example: Calibrating a Linear-Spending Design

Consider a hypothetical one-sided trial with total \(\alpha=0.025\), interim information \(I_1=50\) information units, and final information \(I_2=100\) information units. Choose the linear spending curve \(A(t)=\alpha t\). The assumptions for the canonical model are that the two looks use the same prespecified one-sided test, the information increments are independent under \(H_0\), and the stated information values correctly describe the looks.

1. Information time and dependence. The interim information time and null correlation are

t_1=\frac{50}{100}=0.5, \qquad \rho=\sqrt{0.5}=0.7071.

2. Interim boundary. The cumulative spending by the interim look is \(A(0.5)=0.025(0.5)=0.0125\). Therefore

b_1=\Phi^{-1}(1-0.0125) =\Phi^{-1}(0.9875) \approx 2.2414.

This gives \(P_{H_0}(Z_1\geq b_1)=0.01250\). The interim crossing probability equals cumulative spending at that look because there has been no earlier look.

3. Final boundary. The incremental error available for a new crossing at look 2 is \(0.025-0.0125=0.0125\). Solve

\int_{b_2}^{\infty}\phi(z)\, \Phi\!\left( \frac{2.2414-0.7071z}{\sqrt{1-0.7071^2}} \right)\,dz =0.0125.

Numerical root-finding, with the unrounded interim quantile and correlation used in the calculation, gives \(b_2\approx 2.1257\). The integral is the probability of crossing at the final look while having remained below the interim boundary; it is not simply \(1-\Phi(b_2)\).

4. Verify total null error. The first-look crossing contributes \(0.01250\), and the newly crossing paths at the final look contribute \(0.01250\). Thus

P_{H_0}(\text{cross by look 2}) =0.01250+0.01250=0.02500.

Subject to numerical rounding, the design controls the one-sided Type I error at \(0.025\). The final boundary is lower than the interim boundary in this example; that is not an error. The joint dependence and the chosen allocation of cumulative error determine both values.

What the Spending Curve Changes

A spending curve determines how much cumulative Type I error is assigned by each information time. It does not directly prescribe the probability of crossing at each look after the first, because a later-look crossing is counted only on paths that have not crossed earlier. With two looks, the first boundary follows directly from \(A(t_1)\), while the final boundary must be recalibrated to make the total crossing probability equal \(\alpha\).

For the same \(\alpha=0.025\) and \(t_1=0.5\), a quadratic curve \(A(t)=0.025t^2\) is more back-loaded than linear spending: \(A(0.5)=0.00625\), which is less than the linear curve's \(0.0125\). The interim boundary is consequently higher, leaving more of the total error for possible final-look crossings. This comparison is about spending by the interim time; it does not imply a particular final boundary without solving the joint calibration equation.

Worked Example: A More Back-Loaded Spending Curve

Keep the hypothetical information schedule \(t_1=0.5\) and total \(\alpha=0.025\), but use \(A(t)=0.025t^2\). At the interim look,

A(0.5)=0.025(0.5)^2=0.00625, \qquad b_1=\Phi^{-1}(1-0.00625) =\Phi^{-1}(0.99375)\approx 2.498.

The interim null crossing probability is \(0.00625\), smaller than the \(0.01250\) under linear spending. Accordingly, the final boundary must be found by solving the same joint-null equation, now with \(b_1\approx2.498\), \(\rho=0.7071\), and total error \(0.025\). The remaining \(0.01875\) is the required probability of a new final-look crossing conditional on no earlier crossing, expressed as an unconditional probability over those paths. It is not the marginal tail probability at look 2.

Numerical Checks and Failure Modes

Calibration is a probability calculation, so a reliable implementation checks the result in the same model used to obtain the boundary. Evaluate the cumulative crossing probability at the candidate boundaries, and separately check the first-look and later-look contributions. If root-finding is used, confirm that the residual from the target probability is small and that nearby boundary values move the crossing probability in the expected direction. As \(b_2\) increases, the probability of a new final crossing decreases.

The calculation can be wrong even when each marginal normal tail looks plausible. A common error is to set the final boundary so that \(P(Z_2\geq b_2)=\alpha-A(t_1)\). This ignores paths that cross at both looks; those paths count only once in the trial-level Type I error. Another failure is to use the wrong correlation after changing an information time. Under the stated canonical model it is \(\sqrt{I_1/I_2}\), not an arbitrary correlation or the ratio \(I_1/I_2\).

Worked Example: Diagnosing an Incorrect Final Boundary

Return to the linear-spending design with \(b_1\approx2.2414\) and \(\rho\approx0.7071\). Suppose a calculation mistakenly uses \(b_2=2.100\). The interim error remains \(0.01250\). The probability of crossing for the first time at the final look is

\int_{2.100}^{\infty}\phi(z)\, \Phi\!\left( \frac{2.2414-0.7071z}{\sqrt{1-0.7071^2}} \right)\,dz \approx 0.0134.

The total null crossing probability is therefore approximately \(0.01250+0.0134=0.0259\), which exceeds the target \(0.025\). The boundary \(2.100\) is too low: it admits too many new final-look crossings. Raising it to the calibrated value, approximately \(2.1257\), reduces the later contribution to \(0.01250\) and restores the target total error. This check also shows why rounding a computed boundary should be followed by recalculating the resulting error.

How to Use and Interpret the Design

The calibrated boundaries describe a stopping rule: at each planned look, compare the observed one-sided standardized statistic with that look's boundary, and declare efficacy only if it reaches or exceeds the boundary. Under the null model and assumptions stated above, the probability of at least one efficacy crossing across both looks is \(\alpha\). This is a repeated-testing error rate for the whole design, not an \(\alpha\) level that can be applied afresh at each look.

A fixed-sample one-look test is an alternative when no interim efficacy opportunity is needed; its single critical value is calibrated from a single null distribution. A two-look spending design instead allocates error across planned information times and accounts for their dependence. Different spending curves can yield different interim boundaries and power behavior while preserving the same total null error, provided each design is correctly calibrated. The choice of curve is a design decision, not a repair for incorrect information or an incorrectly implemented stopping rule.

The two-look calculation is limited to the specified one-sided efficacy test under its canonical null model. If actual information times differ from the schedule, the correlation and boundary calibration may no longer match the design. Likewise, adding or removing a look changes the joint crossing calculation; the existing two-look boundaries should not simply be reused. Before interpreting results, check the look schedule, information values, test direction, spending curve, boundary comparisons, and numerical cumulative error.

Specify the design.
Record \(\alpha\), the one-sided direction, \(I_1,I_2\), and the cumulative spending curve \(A(t)\). Confirm that \(0<t_1<1\) and \(A(1)=\alpha\).

Calibrate look 1.
Set \(t_1=I_1/I_2\), calculate \(\rho=\sqrt{t_1}\), and obtain \(b_1=\Phi^{-1}(1-A(t_1))\).

Calibrate and verify look 2.
Solve the joint crossing equation for \(b_2\), then check that the first-look and new final-look crossing probabilities sum to \(\alpha\).

Key takeaway: In a two-look one-sided design, the first boundary is determined by cumulative spending at the interim information time. The final boundary is determined jointly with the first boundary, because total Type I error counts paths crossing at either look only once.

Check Your Understanding

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

For a one-sided design with \(\alpha=0.02\) and \(A(t)=0.02t\), what is the cumulative spending by an interim look at \(t_1=0.4\), and how is \(b_1\) defined?
Under the canonical model, what is the correlation between the two statistics if \(I_1/I_2=0.64\)?
Why is setting the final marginal upper-tail probability equal to \(\alpha-A(t_1)\) generally not a valid calibration?
If a candidate final boundary produces cumulative null crossing probability \(0.027\) for a design with \(\alpha=0.025\), what direction should the boundary be changed, assuming the model and first boundary are correct?

Solutions. (1) \(A(0.4)=0.02(0.4)=0.008\), so \(b_1=\Phi^{-1}(1-0.008)=\Phi^{-1}(0.992)\), approximately \(2.409\). (2) \(\rho=\sqrt{0.64}=0.8\). (3) The marginal final tail includes paths that already crossed at the interim look; the final contribution must count only paths with \(Z_1<b_1\). (4) Raise the final boundary. A higher upper boundary reduces the probability of a new final crossing and thus lowers the cumulative error toward \(0.025\).

What You'll Learn