The Wang-Tsiatis Boundary Family

Why Use a Power-Shaped Boundary?

A group sequential design needs a critical value at each planned analysis. Those values cannot be calibrated as if each look were a separate test: the test statistics are correlated, and the design’s Type I error is the probability of crossing at least one boundary under $H_0$. The Wang-Tsiatis family specifies how the critical values change with information accrued, then uses a common scale parameter to calibrate that joint crossing probability.

This tutorial concerns a one-sided efficacy boundary for a design with $K$ planned looks. It assumes that the information fractions are specified in advance, increase strictly, and end at one. The power-shaped rule gives a compact way to describe the boundary’s shape across those looks; its calibrated scale sets the overall height needed for the chosen significance level.

Definition: Let $t_i$ be the information fraction at look $i$, with $0<t_1<\cdots<t_K=1$. The Wang-Tsiatis upper efficacy boundary is $$ c_i = a\,t_i^{\Delta-1/2},\qquad i=1,\ldots,K, $$ where $a>0$ is the scale parameter and $\Delta$, conventionally between 0 and $1/2$, is the shape parameter. At look $i$, efficacy is declared if the one-sided group sequential statistic $Z_i$ is at least $c_i$.

The information fraction is the information available at a look divided by the planned maximum information. It is not automatically the fraction of participants enrolled; the earlier material on information fractions explains why that distinction matters. The final look has $t_K=1$, so its cutoff is $c_K=a$, whatever the value of $\Delta$. At earlier looks, both $t_i$ and $\Delta$ determine the multiplier of $a$.

Model and Calibration

To calibrate the scale, the design must specify the joint null distribution of the statistics. Under the standard canonical group sequential model, the statistics have mean zero and variance one under $H_0$, and for $i\leq j$,

(Z_1,\ldots,Z_K)\text{ is jointly normal},\qquad \operatorname{Corr}_0(Z_i,Z_j)=\sqrt{\frac{t_i}{t_j}}.

This model corresponds to independent increments of the underlying score process, standardized by information. In particular, the correlation structure follows the information fractions, not merely the calendar spacing between analyses. If the design’s actual analysis statistics do not support this canonical model, its error probability must instead be calibrated under an appropriate joint model.

For a one-sided design with target familywise significance level $\alpha$, the scale is chosen to satisfy

P_0\!\left(\bigcup_{i=1}^{K}\{Z_i\geq a\,t_i^{\Delta-1/2}\}\right)=\alpha.

The probability is a joint probability over the planned looks. The event is “cross at least one efficacy boundary,” not “cross the boundary at a particular look.” For fixed information fractions and shape, increasing $a$ raises every cutoff, so the crossing probability decreases. A numerical root-finding procedure can therefore select the scale that achieves the target. The calibrated $a$ depends on the shape, the planned looks and their information fractions, and the target $\alpha$; it is not a universal constant.

Specify the design.
List $t_1,\ldots,t_K$, the shape $\Delta$, and the target one-sided familywise level $\alpha$.

Compute candidate cutoffs.
For a trial value of $a$, calculate $c_i=a\,t_i^{\Delta-1/2}$ at every look.

Evaluate joint crossing probability.
Use the canonical joint normal distribution, or an appropriate design-specific joint model, to calculate the probability of crossing at least once.

Calibrate and verify.
Adjust $a$ until the crossing probability is $\alpha$, then recompute and report the resulting critical values.

Why the Calibration Uses the Joint Distribution

The event of no efficacy crossing is the event that all statistics remain below their corresponding cutoffs. Thus, under the canonical null model, the crossing probability is one minus a multivariate normal probability:

P_0(\text{at least one crossing}) =1-P_0(Z_1<c_1,\ldots,Z_K<c_K).

This identity justifies calibrating one common scale against the full design rather than assigning the same nominal test level independently to each look. For two looks, it simplifies to a bivariate normal probability. If $\Phi_2(x,y;\rho)$ denotes the probability that two standard normal variables with correlation $\rho$ are respectively below $x$ and $y$, then

P_0(\text{at least one crossing}) =1-\Phi_2(c_1,c_2;\rho), \qquad \rho=\sqrt{\frac{t_1}{t_2}}.

For more looks, the same complement principle applies, but the probability is multivariate. It can be evaluated using multivariate normal integration or the independent-increments representation of the canonical model. The calibration should use the actual planned information fractions, including unequal fractions when the design has unequal spacing in information.

Worked Example: Calculating Critical Values From a Specified Scale

Consider a hypothetical three-look design with information fractions $t_1=0.25$, $t_2=0.60$, and $t_3=1$. Suppose the selected shape is $\Delta=0.30$ and a calibration has supplied $a=2.04$. Calculate the three Wang-Tsiatis cutoffs.

The exponent is $\Delta-1/2=0.30-0.50=-0.20$, so $c_i=2.04t_i^{-0.20}$. At the first look,

c_1=2.04(0.25)^{-0.20} =2.04(1.320) \approx 2.69.

At the second look,

c_2=2.04(0.60)^{-0.20} =2.04(1.108) \approx 2.26.

At the final look, $t_3=1$, so $c_3=2.04(1)^{-0.20}=2.04$. The cutoffs are therefore approximately 2.69, 2.26, and 2.04. This calculation determines critical values for the specified scale; it does not by itself show that $a=2.04$ controls the familywise error at any particular target. That requires a joint-probability calibration.

Worked Example: Calibrating a Two-Look Scale

Suppose a hypothetical one-sided design plans analyses at $t_1=0.50$ and $t_2=1$, and targets familywise $\alpha=0.025$. Use the shape $\Delta=0$. The cutoffs for a candidate scale $a$ are

c_1=a(0.50)^{-1/2}=a\sqrt{2},\qquad c_2=a.

Under the canonical null model, the correlation is $\rho=\sqrt{0.50/1}=\sqrt{0.50}\approx0.7071$. Therefore, choose $a$ to solve

1-\Phi_2(a\sqrt{2},a;0.7071)=0.025.

Numerical solution gives $a\approx1.98$, with small differences possible from rounding. The resulting cutoffs are $c_1\approx1.98\sqrt{2}\approx2.80$ and $c_2\approx1.98$. Substituting the unrounded root into the bivariate normal expression gives a crossing probability of 0.025. These values are joint-design critical values: the early cutoff is not calibrated to have a 0.025 marginal tail probability on its own.

The assumptions used here are the planned fractions $0.50$ and $1$, the stated one-sided familywise target, and the canonical joint normal null model. If the statistic’s null correlation were not approximately 0.7071, this calibration would not be justified without an alternative joint model.

Worked Example: Applying a Calibrated Boundary

Continue with the hypothetical three-look boundary from the first example: $t=(0.25,0.60,1)$, $\Delta=0.30$, and the already calibrated $a=2.04$. Suppose the observed statistics are $Z_1=1.40$ and $Z_2=2.31$.

At look 1, $1.40<2.69$, so the efficacy boundary is not crossed. At look 2, $2.31\geq2.26$, so the boundary is crossed. The design’s prespecified efficacy rule therefore declares efficacy at look 2 and stops for efficacy; a final-look statistic is not needed for that decision.

This conclusion is conditional on the cutoffs having been correctly calibrated for the planned information fractions, shape, and target error rate. A statistic exceeding a cutoff is a boundary-based decision under the design; it does not mean that the null hypothesis has a particular probability of being true.

Diagnostics, Alternatives, and Limitations

Several checks help identify an incorrectly specified or incorrectly used boundary:

Confirm that the information fractions are strictly increasing and that the final fraction is exactly one. A fraction based on participant count is not a substitute when information does not accrue proportionally to enrollment.
Check that the calibrated cutoffs use the same $\Delta$, $t_i$, number of looks, and one-sided error target as the design being implemented.
Verify that the joint null model reflects the analysis statistics. The canonical correlations depend on the information fractions; using independence or marginal tail probabilities generally gives the wrong crossing probability.
Recalculate the joint crossing probability from the reported cutoffs. A mismatch with $\alpha$ can reveal rounding, a coding error, or a mismatch between planned and implemented information.
Check that the efficacy direction and inequality are consistent. For a lower-tail test, the boundary and crossing rule must be formulated in the lower tail instead.

The principal alternative boundaries in the classical setting include Pocock’s flat critical-value rule and the O’Brien-Fleming information-scaled rule. Those are specified by different boundary shapes; the Wang-Tsiatis family instead uses a power of information fraction and calibrates its common scale to the design’s joint null crossing probability. Comparing or selecting among these geometries requires attention to the design’s objectives, not just their formulas.

The family is not a substitute for a sound information model. Its critical values are only as appropriate as the assumed information fractions, joint distribution and analysis plan. It also does not, by its formula alone, determine power under an alternative, expected sample size, or the consequences of changing the number or timing of analyses. Those quantities require additional calculations under specified alternatives and stopping rules.

Common Pitfall

Common mistake: “The critical values are given by the power formula, so each look has its own correctly controlled Type I error.”

Correction: The formula specifies candidate cutoffs once $a$, $\Delta$, and the information fractions are known. The scale still must be calibrated using the joint null distribution so that the probability of crossing at least one cutoff is the target $\alpha$. Treating the looks as separate tests or calibrating each marginal tail alone does not, in general, control the design’s familywise Type I error.

Key takeaway: The Wang-Tsiatis rule assigns $c_i=a\,t_i^{\Delta-1/2}$. The shape and information fractions determine the relative cutoffs, while the scale is calibrated against the probability of crossing anywhere in the planned design.

Check Your Understanding

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

A design has $t=(0.40,1)$, $\Delta=0.25$, and $a=2.10$. Write the formula for each cutoff and calculate both to three significant figures.
For a two-look design with $t_1=0.64$ and $t_2=1$, what is the canonical null correlation between $Z_1$ and $Z_2$?
Why is calibrating each look to have marginal upper-tail probability $\alpha$ not the same as calibrating the familywise Type I error to $\alpha$?
A statistic at an interim look is below its Wang-Tsiatis cutoff. What can and cannot be concluded from that comparison alone?
List two design details that must match the calibration when applying its cutoffs.

Solutions. (1) The exponent is $0.25-0.50=-0.25$. Thus $c_1=2.10(0.40)^{-0.25}=2.10(1.257)\approx2.64$, and $c_2=2.10(1)^{-0.25}=2.10$. (2) The correlation is $\sqrt{0.64/1}=0.800$. (3) The familywise event is crossing at least one boundary, and the statistics are correlated. Its probability is a joint probability, not generally the marginal probability at a single look or the sum of identical marginal probabilities. (4) It means the efficacy boundary was not crossed at that look; it does not establish that the null hypothesis is true or guarantee that a later boundary will not be crossed. (5) Examples include the information fractions, shape parameter, number of looks, one-sided target $\alpha$, and assumed joint null distribution.