Deriving Error Control From Independent Increments

Why Independent Increments Matter for Error Control

A group sequential design permits examination of accumulating data at several planned looks. Each look creates another opportunity to reject $H_0$, so a boundary that controls the rejection probability at any one look does not, by itself, control the probability of rejection across all looks. The central accounting question is: under $H_0$, what is the probability that the trial has crossed an efficacy boundary by look $k$?

For the derivation here, consider a one-sided upper efficacy boundary at each of $K$ planned looks. The result depends on calibrating those boundaries under the joint null distribution of the sequence of test statistics. In the canonical model, the underlying score has independent increments, although the standardized statistics at different looks are correlated. That distinction is what makes the probability accounting work.

Definition: Let $t_k$ be the information fraction at look $k$, with $0<t_1<\cdots<t_K=1$, and let $Z_k$ be the standardized test statistic at that look. An upper-boundary design rejects at look $k$ if $Z_k\geq b_k$, provided no earlier boundary has been crossed. The first-crossing event at look $k$ is $$ C_k=\{Z_1<b_1,\ldots,Z_{k-1}<b_{k-1},\ Z_k\geq b_k\}. $$ For $k=1$, this is simply $C_1=\{Z_1\geq b_1\}$.

The cumulative spending target at look $k$ is denoted $a_k$. It is the desired probability, under $H_0$, of having rejected by that look. The targets are nondecreasing, and $a_0=0$. Thus the intended probability of a first crossing specifically at look $k$ is $a_k-a_{k-1}$, not $a_k$. For example, if cumulative spending is .010 by look 1 and .025 by look 2, the additional probability allocated to first crossing at look 2 is .015.

The Canonical Joint Distribution

Under the canonical null model, the score accumulated by information $I_k$ has independent increments. In particular, write the score at look $k$ as $S_k$, with $S_k-S_{k-1}$ independent of the earlier increments and normally distributed with mean 0 and variance $I_k-I_{k-1}$. Here $I_k$ is the information at look $k$, and $I_0=0$. The standardized statistic is $Z_k=S_k/\sqrt{I_k}$. Under $H_0$, each $Z_k$ has a standard normal distribution, and for $i<j$,

\operatorname{Cov}(Z_i,Z_j)=\sqrt{\frac{I_i}{I_j}}.

When information fractions are defined by $t_k=I_k/I_K$, this covariance is $\sqrt{t_i/t_j}$. The statistics are therefore not independent: they share the score accumulated before the earlier look. Rather, the independent objects are the increments $S_k-S_{k-1}$. Under the Gaussian canonical model, these assumptions specify the joint distribution used to evaluate crossing probabilities.

The model is an approximation in many applications. Its use presumes that the planned information times are correctly specified and that the null distribution of the score process is adequately represented by the canonical Gaussian model. If the statistics have a different joint distribution, boundary probabilities must be evaluated under that appropriate distribution; independent increments alone do not justify using incorrect canonical correlations.

Deriving Cumulative Error Control

A rejection by look $k$ means that at least one boundary has been crossed at or before that look. Every such trial has a unique first crossing: it cannot first cross at two different looks. Consequently, the events $C_1,\ldots,C_k$ are mutually exclusive, and their union is exactly the event of rejection by look $k$.

Theorem: Under the specified joint null distribution, if the boundaries are calibrated so that $$ P_0(C_k)=a_k-a_{k-1},\qquad k=1,\ldots,K, $$ then the probability of rejecting $H_0$ by look $k$ is $a_k$. In particular, the overall Type I error probability by the final look is $a_K$.

The justification is a direct partition of the rejection event. Because first crossings are mutually exclusive,

P_0\left(\bigcup_{j=1}^{k} C_j\right) =\sum_{j=1}^{k}P_0(C_j) =\sum_{j=1}^{k}(a_j-a_{j-1}) =a_k-a_0 =a_k.

The telescoping sum is the error-control argument. Independent increments matter because they give a tractable canonical joint distribution for calculating each $P_0(C_k)$. They do not make the looks independent, and the theorem does not add marginal tail probabilities. It adds probabilities of disjoint first-crossing events.

For exact calibration, each boundary is chosen to make its first-crossing probability equal to the planned increment. If a boundary is chosen conservatively so that $P_0(C_k)\leq a_k-a_{k-1}$ at every look, the same partition shows that the overall rejection probability is at most $a_k$ by look $k$, and at most $a_K$ at the end.

Worked Example: One Look and a One-Sided Error Target

Suppose a hypothetical design has one planned look and a one-sided final Type I error target of .025. Under $H_0$, assume the test statistic $Z_1$ has a standard normal distribution. There are no earlier looks, so the first-crossing probability is the marginal upper-tail probability.

Choose $b_1$ to satisfy $P_0(Z_1\geq b_1)=.025$. Thus $b_1$ is the .975 quantile of the standard normal distribution:

b_1=\Phi^{-1}(1-.025)=\Phi^{-1}(.975)\approx1.960.

The null rejection probability is $P_0(Z_1\geq1.960)\approx.025$. Since $C_1$ is the entire rejection event in this one-look design, cumulative spending and first-crossing probability coincide. With more than one look, that coincidence no longer holds after the first look.

Calibrating a Later Boundary

At a later look, the calibration target is conditional on the trial not having rejected earlier, expressed as an unconditional first-crossing probability. Under the canonical model, the correlation between two standardized statistics allows that probability to be evaluated from their bivariate normal distribution. For two looks with correlation $\rho=\sqrt{t_1/t_2}$, the conditional distribution is

Z_2\mid Z_1=z\ \sim\ N\left(\rho z,\ 1-\rho^2\right).

Therefore, for fixed $b_1$, the probability of first crossing at look 2 is

P_0(C_2) =\int_{-\infty}^{b_1}\phi(z)\, \left[1-\Phi\left(\frac{b_2-\rho z}{\sqrt{1-\rho^2}}\right)\right]\,dz,

where $\phi$ and $\Phi$ are the standard normal density and distribution functions. The upper limit $b_1$ enforces the condition that look 1 did not cross. The boundary $b_2$ is found by solving $P_0(C_2)=a_2-a_1$. This is a joint-probability equation; setting $P_0(Z_2\geq b_2)$ equal to the spending increment would generally be wrong.

Worked Example: A Two-Look Design With Correct First-Crossing Calibration

Consider a hypothetical one-sided design with $t_1=.5$, $t_2=1$, cumulative spending $a_1=.010$ and $a_2=.025$. Assume the canonical null model applies. At look 1, choose

b_1=\Phi^{-1}(1-.010)=\Phi^{-1}(.990)\approx2.326.

The additional spending assigned to look 2 is $a_2-a_1=.025-.010=.015$. The canonical correlation is $\rho=\sqrt{.5/1}\approx.7071$, so the second-look boundary must solve

\int_{-\infty}^{2.326}\phi(z) \left[1-\Phi\left(\frac{b_2-.7071z}{\sqrt{1-.7071^2}}\right)\right]\,dz =.015.

Numerical solution gives $b_2\approx2.076$ (small variation is possible from rounding $b_1$ or the numerical integration). Substituting this value into the joint-probability integral gives $P_0(C_2)\approx.0150$. For a check on why the marginal tail is not the target, $P_0(Z_2\geq2.076)$ is about .0189; roughly .0039 of that tail probability corresponds to trials that already crossed at look 1. The remaining first-crossing probability is about $.0189-.0039=.0150$.

The overall null rejection probability is therefore $$ P_0(C_1)+P_0(C_2)=.010+.015=.025. $$ The design meets its final one-sided Type I error target under the stated canonical model. Using a boundary near 2.106 instead would not meet the specified increment: its second-look first-crossing probability is about .0138, not .0150.

Worked Example: Independent Increments Do Not Mean Independent Tests

A simple hypothetical Gaussian process makes the distinction visible. Let $X_1$ and $X_2$ be independent standard normal increments. Define $Z_1=X_1$ and $Z_2=(X_1+X_2)/\sqrt{2}$. Both standardized statistics have variance 1, while their correlation is $1/\sqrt{2}$. Set both upper boundaries to zero for this illustration.

At look 1, the first-crossing probability is $P(X_1\geq0)=.5$. At look 2, first crossing requires $X_1<0$ and $X_1+X_2\geq0$. In the $(X_1,X_2)$ plane, the joint density is circularly symmetric. The stated inequalities describe a wedge of angle $\pi/4$, so its probability is its angle divided by $2\pi$:

P_0(C_2)=\frac{\pi/4}{2\pi}=\frac18=.125.

Thus the cumulative probability of crossing by look 2 is $.5+.125=.625$. In this deliberately high-error illustration, the increments $X_1$ and $X_2$ are independent, but $Z_1$ and $Z_2$ are correlated. It is the disjoint first-crossing probabilities—not an assumption that the tests at the looks are independent—that produce the cumulative total.

Practical Checks and Common Pitfalls

The derivation is simple, but its use depends on checking the event and model being calibrated. At each look, the boundary calculation must include all earlier noncrossing conditions. The information times used to specify the canonical covariance must correspond to the design’s actual analysis plan. If a numerical integration or root-finding calculation is used, the resulting boundary should be checked by evaluating the first-crossing probability again, then summing those probabilities through each look.

Replacing first-crossing probability with a marginal tail. The event $Z_k\geq b_k$ includes some paths that crossed earlier. The correction is to impose $Z_j<b_j$ for every earlier look when calculating the probability assigned to look $k$.
Adding overlapping events as if they were disjoint. The events $\{Z_k\geq b_k\}$ can overlap across looks. Partition rejection by the unique first-crossing look instead.
Treating standardized statistics as independent. Independent score increments do not imply independent cumulative $Z$-statistics. Use the canonical joint distribution, including its correlations.
Using an incorrect information schedule. The covariance depends on information fractions. Incorrect information times can make a numerically accurate boundary solve target the wrong joint distribution.
Checking only the final total. A design may have the intended final error but spend it differently across looks. Check cumulative rejection probabilities at each planned look against $a_k$, as well as the final value.

Compared with a Bonferroni allocation, which controls error by assigning marginal tail probabilities whose sum is no larger than the desired total, spending-function calibration uses the joint distribution and allocates probability to first crossings. It can use the dependence between looks directly, but it relies on specifying that dependence correctly. A single final-look test is simpler but does not provide the planned interim opportunities to stop for efficacy. Neither comparison changes the basic requirement: the interpretation of the calibrated error rate is conditional on the null model and the planned monitoring rule being appropriate.

Key takeaway: Under the canonical null model, independent score increments determine the joint distribution of the sequential statistics. Calibrating each boundary to its cumulative-spending increment makes the mutually exclusive first-crossing probabilities sum to the planned cumulative Type I error.

Check Your Understanding

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

If cumulative spending is .008 at look 1 and .020 at look 2, what first-crossing probability is assigned to look 2?
Why is $P_0(Z_2\geq b_2)$ generally not the correct target for the second-look boundary?
In the canonical model, are $Z_1$ and $Z_2$ independent when the score increments are independent? Explain.
If $P_0(C_1)=.006$, $P_0(C_2)=.009$, and $P_0(C_3)=.010$, what is the null probability of rejection by look 3?
Name two assumptions or design inputs that should be checked before interpreting the calibrated total as the intended Type I error.

Solutions. (1) The increment is $.020-.008=.012$. (2) The marginal event $Z_2\geq b_2$ includes paths that may already have crossed at look 1; the target is $P_0(Z_1<b_1,Z_2\geq b_2)$. (3) No. The standardized statistics share accumulated score, with canonical correlation $\sqrt{t_1/t_2}$; it is the underlying score increments that are independent. (4) First-crossing events are disjoint, so the probability is $.006+.009+.010=.025$. (5) Appropriate checks include whether the canonical Gaussian null model is adequate and whether planned information times correctly represent the monitoring schedule. The earlier noncrossing conditions and the stated stopping rule must also be included in calibration.