Classical Boundaries With Unequal Information Fractions

Why Information Fractions Matter

A group sequential design may schedule interim analyses at unequal amounts of information. For example, the first analysis might occur after 30% of the maximum planned information, the second after 65%, and the final analysis at 100%. A boundary is applied at each analysis using its information fraction, not simply its position in the sequence of looks. This distinction matters because classical boundaries are shaped in information time: two analyses with the same ordinal position can have different thresholds if their information fractions differ.

The formulas for Pocock, O’Brien–Fleming, and Wang–Tsiatis boundaries describe how thresholds change with information. They do not, by themselves, determine the constant needed to control the design’s overall type I error. That constant depends on the complete planned schedule and the error criterion.

Model, Notation, and Boundary Forms

Consider a design with \(K\) planned analyses at information amounts \(I_1,\ldots,I_K\), where \(0<I_1<\cdots<I_K=I_{\max}\). The information fraction at look \(k\) is \(t_k=I_k/I_{\max}\), so \(0<t_1<\cdots<t_K=1\). It is the planned information amount that defines \(t_k\); the fraction of participants recruited is only a substitute when it correctly represents the fraction of information.

For the standard canonical group sequential model under \(H_0\), the cumulative Z statistics have mean zero and, for \(i\leq j\), covariance \(\operatorname{Cov}(Z_i,Z_j)=\sqrt{t_i/t_j}\). This model is justified when information accumulates through independent increments and the standardized test statistic is approximately normal at each look. The schedule and analysis plan are fixed in advance, and the specified error criterion (one-sided or two-sided) matches the boundary being used.

Definition: An upper efficacy boundary is a sequence of critical values \(b_1,\ldots,b_K\). At look \(k\), stop for efficacy if \(Z_k\geq b_k\). Under the canonical null model, the familywise type I error is \(P_{H_0}(\text{at least one planned boundary is crossed})\). For a two-sided design, the crossing rule is typically \(|Z_k|\geq b_k\), and calibration must use that two-sided event.

The common upper-boundary shapes can be written as follows. The constants shown are design-specific: they must be calibrated for the planned information fractions and error criterion.

Boundary forms: Pocock uses \(b_k=c\), a constant threshold. O’Brien–Fleming uses \(b_k=c/\sqrt{t_k}\). The Wang–Tsiatis family uses \(b_k=c/t_k^{1/2-\Delta}\), where \(\Delta\) is the shape parameter. In this parameterization, \(\Delta=0\) gives the O’Brien–Fleming shape and \(\Delta=0.5\) gives the Pocock shape.

These expressions specify the relative shape across looks. A value of \(c\) that is suitable for one schedule is not automatically valid for another. Under the canonical null model, calibration chooses \(c\) so that the probability of crossing at least one boundary equals the target \(\alpha\):

P_{H_0}\left(\bigcup_{k=1}^{K}\{Z_k\geq b_k\}\right)=\alpha.

For a two-sided design, replace each event in the union by \(\{|Z_k|\geq b_k\}\). Because the Z statistics are correlated across looks, calibration is not generally achieved by assigning \(\alpha/K\) to each look. It uses their joint null distribution, or a validated numerical method based on that distribution.

Applying the Shapes at Unequal Looks

Worked Example: Comparing Pocock and O’Brien–Fleming Thresholds

Suppose a hypothetical three-look design plans analyses at \(t_1=0.30\), \(t_2=0.65\), and \(t_3=1.00\). Assume the design specification supplies schedule-calibrated constants \(c=2.35\) for a Pocock boundary and \(c=2.04\) for an O’Brien–Fleming boundary. The design uses an upper-boundary efficacy rule and the canonical information model.

For Pocock, \(b_k=c\) at every look, so the thresholds are \(2.350,\ 2.350,\ 2.350\).

For O’Brien–Fleming, calculate \(b_k=2.04/\sqrt{t_k}\) separately at each planned information fraction:

b_1=\frac{2.04}{\sqrt{0.30}}=3.725,\qquad b_2=\frac{2.04}{\sqrt{0.65}}=2.530,\qquad b_3=\frac{2.04}{\sqrt{1.00}}=2.040.

The early O’Brien–Fleming threshold is high, then falls as information accumulates. For example, if the observed Z statistic at the first look were \(3.60\), it would exceed the Pocock threshold \(2.350\), but not the O’Brien–Fleming threshold \(3.725\). These numbers illustrate applying supplied, schedule-specific calibrations; they do not establish that the two constants target the same \(\alpha\) unless the calibration specifications say so.

Worked Example: Applying a Wang–Tsiatis Boundary

A hypothetical design has planned information fractions \(0.36,\ 0.70,\ 1.00\). Its design specification gives a Wang–Tsiatis shape parameter \(\Delta=0.25\) and a calibrated constant \(c=2.10\). Assume the statistics follow the canonical null model and efficacy is defined by crossing an upper boundary.

Here \(1/2-\Delta=0.25\), so \(b_k=2.10/t_k^{0.25}\). The three thresholds are:

b_1=\frac{2.10}{0.36^{0.25}}=2.711,\qquad b_2=\frac{2.10}{0.70^{0.25}}=2.296,\qquad b_3=\frac{2.10}{1.00^{0.25}}=2.100.

Suppose the observed Z statistics are \(2.70,\ 2.40,\ 2.05\), respectively. At look 1, \(2.70<2.711\), so the efficacy boundary is not crossed. At look 2, \(2.40\geq2.296\), so the trial stops for efficacy at that look; the third statistic would not be used to reverse that decision. The example assumes that no unplanned analyses are conducted and that the stated information fractions and calibration apply to the actual design.

Why the Schedule Enters Calibration

The schedule affects the joint distribution of the statistics used to calculate the probability of crossing. For \(i\leq j\), the canonical correlation is \(\sqrt{t_i/t_j}\). Changing an information fraction changes those correlations, as well as the threshold in any boundary shape that depends on \(t_k\). Thus, retaining the same constant after changing the schedule can change the overall type I error.

The calibration principle can be justified directly. Under \(H_0\), the vector \((Z_1,\ldots,Z_K)\) has the canonical joint normal distribution specified by the information fractions. For a proposed constant \(c\), the boundary formulas determine \(b_1(c),\ldots,b_K(c)\). The probability of no efficacy crossing is the joint probability that \(Z_k<b_k(c)\) at every look. Therefore:

P_{H_0}(\text{at least one crossing}) =1-P_{H_0}(Z_1<b_1(c),\ldots,Z_K<b_K(c)).

A numerical calibration selects \(c\) so that this probability equals the target \(\alpha\). Since both the joint distribution and the thresholds involve the planned \(t_k\), the calibrated constant belongs to that schedule and boundary shape. For two-sided designs, the probability calculation instead uses \(-b_k(c)<Z_k<b_k(c)\) for every look.

Worked Example: Using Actual Information Rather Than Look Number

Suppose a hypothetical four-look O’Brien–Fleming design plans information fractions \(0.20,\ 0.55,\ 0.80,\ 1.00\), and the supplied schedule-calibrated constant is \(c=2.00\). The canonical model and an upper efficacy rule are assumed. At the second look, the correct threshold uses \(t_2=0.55\):

b_2=\frac{2.00}{\sqrt{0.55}}=2.696.

If the observed statistic there is \(Z_2=2.75\), the boundary is crossed because \(2.75\geq2.696\). A common but incorrect shortcut is to treat the second of four looks as information fraction \(2/4=0.50\). That would give \(2.00/\sqrt{0.50}=2.828\), leading to the opposite decision. The correct fraction is the planned information fraction \(0.55\), not the look index divided by the number of looks.

For comparison, the same supplied O’Brien–Fleming constant gives \(b_1=2.00/\sqrt{0.20}=4.472\), \(b_3=2.00/\sqrt{0.80}=2.236\), and \(b_4=2.000\). The sequence decreases as information accumulates, but that shape does not remove the need for schedule-specific calibration.

Diagnostics, Alternatives, and Limitations

Before applying a boundary, verify the planned look schedule, the information fraction at each look, the boundary family and shape parameter, the calibrated constant, and whether the rule is one-sided or two-sided. A threshold table should use increasing \(t_k\), end at \(t_K=1\), and have the expected shape: a Pocock boundary is flat, while O’Brien–Fleming and the usual Wang–Tsiatis shapes with \(\Delta<0.5\) decrease toward the final analysis.

A key diagnostic is whether the analysis actually has the information assumed by the schedule. Participant counts need not be proportional to information: unequal outcome variability, unequal allocation, missing data, or changes in the analysis model can disrupt that correspondence. If actual information differs materially from the planned information, simply substituting a participant fraction or reusing the old table may not preserve the intended error rate. The design’s monitoring and recalibration procedures must address such deviations.

The principal alternatives here are different boundary shapes, not different ways to ignore the schedule. Pocock spends more of its efficacy opportunity early through a relatively flat threshold; O’Brien–Fleming is more conservative early and approaches its final threshold gradually; Wang–Tsiatis allows intermediate shapes through \(\Delta\). Each requires calibration for the chosen schedule. A nominal single-look p-value threshold at each analysis is not a substitute for an overall group sequential boundary: repeated testing changes the probability of at least one false crossing.

These classical boundaries also have limitations. Their operating characteristics rely on the assumed statistic, information pattern, and planned analysis schedule. They are not automatically valid for unplanned looks, a changed endpoint or model, or a materially different information process. The boundary determines an efficacy stopping rule; it does not by itself specify futility stopping, estimation after stopping, or a response to operational changes.

Common Mistakes

Using \(k/K\) as the information fraction. The look number does not define information. Use \(t_k=I_k/I_{\max}\) from the design.
Reusing a calibrated constant after changing the schedule. The crossing probability depends on the joint distribution across the planned looks. Recalibrate for the new schedule.
Treating the shape formula as the full calibration. A boundary’s functional form does not establish its familywise type I error; the constant must be calibrated for the design.
Continuing after a first efficacy crossing. Under the stated stopping rule, efficacy is declared at the first crossing; later planned statistics do not undo that decision.

The practical principle is simple: compute each threshold using the information fraction attached to that analysis, and use only a constant calibrated for the same boundary family, schedule, and error criterion.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A three-look design has information fractions \(0.25,\ 0.60,\ 1.00\). What information fraction is used for its second-look threshold?
For an O’Brien–Fleming boundary with supplied \(c=2.10\), calculate the threshold at \(t=0.49\).
A Wang–Tsiatis boundary has \(\Delta=0.50\). What shape does its formula produce, and what additional design quantity is still needed?
Why can a calibrated constant not generally be carried over unchanged when the planned information fractions change?

Solutions. (1) Use \(t_2=0.60\), as specified by the planned information schedule. (2) \(b=2.10/\sqrt{0.49}=2.10/0.70=3.000\). (3) Since \(1/2-\Delta=0\), \(b_k=c\), the flat Pocock shape; \(c\) still needs calibration for the schedule and error criterion. (4) The fractions affect both the canonical correlations among the sequential statistics and, for non-flat shapes, the thresholds themselves, so the probability of any crossing can change.