Representing Interim Looks as Numerical States

Why Represent Each Look as a State?

A group-sequential design produces a sequence of statistics, one at each planned interim analysis. To calculate the chance of crossing a boundary, it is not enough to know the final statistic’s distribution: the sequence of values matters, because stopping at an early look changes which later looks are reached. A numerical state representation makes that sequence explicit. At each look it records the standardized statistic, the information accumulated, and the stopping limits that determine whether the process continues.

This tutorial uses the canonical large-sample model for a fixed sequence of planned analyses. It explains what the state means and how to express survival and crossing events in terms of it. It does not require the raw observations to be modeled separately at every look, but it does require a valid information measure and a suitable joint model for the statistics.

Definition: At planned look \(k\), let \(U_k\) be an efficient score statistic for the treatment effect and let \(I_k\) be its accumulated Fisher information. The canonical Z-statistic state is \(Z_k=U_k/\sqrt{I_k}\). For \(K\) planned looks, the sequential state is the ordered vector \((Z_1,\ldots,Z_K)\), together with the information values and the stopping limits at those looks.

Canonical Model and Information Time

Assume the planned information values satisfy \(0<I_1<I_2<\cdots<I_K\). In the canonical model, the score at look \(k\) has mean \(\theta I_k\) and variance \(I_k\), where \(\theta\) is the standardized effect parameter per square root of information. The score increments between successive looks are independent and normally distributed, with mean \(\theta(I_k-I_{k-1})\) and variance \(I_k-I_{k-1}\); set \(I_0=0\). These assumptions imply the joint normal model

E_\theta(Z_k)=\theta\sqrt{I_k},\qquad \operatorname{Var}_\theta(Z_k)=1,\qquad \operatorname{Cov}_\theta(Z_i,Z_j)=\sqrt{\frac{I_i}{I_j}}\quad (i<j).

The correlation therefore depends on the information accumulated at the two looks, not merely on the number of participants enrolled at each one. The information time is the relative information

t_k=\frac{I_k}{I_K},\qquad 0<t_1<\cdots<t_K=1.

Information time is a convenient scale for locating an analysis within the planned information schedule. It is not necessarily the fraction of calendar time elapsed or the fraction of the planned sample size enrolled. Those quantities coincide only under conditions that make information proportional to time or sample size. For a one-sided design, a boundary on the unused side can be represented by an infinite limit.

Worked Example: From Information to State Distribution

Suppose a hypothetical two-look design has \(I_1=25\) and \(I_2=100\), with canonical effect parameter \(\theta=0.08\). These information values are strictly increasing, as required. Under the canonical model, calculate the mean of each Z-statistic, their variances, their correlation, and the information times.

At the first look,

E_\theta(Z_1)=0.08\sqrt{25}=0.08(5)=0.400.

At the second look,

E_\theta(Z_2)=0.08\sqrt{100}=0.08(10)=0.800.

Each statistic has variance 1. Their covariance, and hence their correlation because both variances are 1, is

\operatorname{Corr}_\theta(Z_1,Z_2) =\sqrt{\frac{25}{100}}=0.500.

The relative information times are \(t_1=25/100=0.250\) and \(t_2=100/100=1.000\). Thus, under this model, the pair has means \(0.400\) and \(0.800\), unit marginal variances, and correlation \(0.500\). The effect parameter shifts the means; it does not change the stated canonical correlation.

Why the Canonical Dependence Formula Holds

The covariance formula follows directly from the independent-increment score model. For \(i<j\), write \(U_j=U_i+(U_j-U_i)\). The second term is independent of \(U_i\), so the covariance between \(U_i\) and \(U_j\) is the variance of \(U_i\), namely \(I_i\). Standardizing both scores gives the correlation structure.

\operatorname{Cov}(Z_i,Z_j) =\operatorname{Cov}\left(\frac{U_i}{\sqrt{I_i}},\frac{U_j}{\sqrt{I_j}}\right) =\frac{I_i}{\sqrt{I_iI_j}} =\sqrt{\frac{I_i}{I_j}}.

This derivation explains why separate standard normal distributions for each look are not a complete model. Each \(Z_k\) is marginally normal with variance 1, but the statistics are correlated. A calculation that treats the looks as independent would generally misrepresent the probability of reaching and crossing later boundaries.

Surviving Regions and First Crossings

Let \(a_k\) and \(b_k\) be the lower and upper stopping limits at look \(k\), with \(a_k<b_k\). Under the convention used here, a statistic at or below \(a_k\) crosses the lower limit, a statistic at or above \(b_k\) crosses the upper limit, and a statistic strictly between them continues. Because the canonical model is continuous, including or excluding equality does not alter probabilities.

Definition: The surviving state region through look \(k\) is the set of state vectors satisfying \(a_j<Z_j<b_j\) at every look \(j=1,\ldots,k\). The event of a first crossing at look \(k\) consists of survival through look \(k-1\) and a crossing of at least one limit at look \(k\).

In symbols, survival through look \(k\) is the joint event

\{a_1<Z_1<b_1,\ldots,a_k<Z_k<b_k\}.

For \(k>1\), a first crossing at look \(k\) is the union of the two events

\{a_j<Z_j<b_j\text{ for }j=1,\ldots,k-1,\ Z_k\leq a_k\} \;\cup\; \{a_j<Z_j<b_j\text{ for }j=1,\ldots,k-1,\ Z_k\geq b_k\}.

At the first look there are no previous survival constraints. At later looks, the constraints from all earlier looks remain part of the event. In particular, the interval \((a_k,b_k)\) describes continuation at look \(k\), but it does not by itself describe survival through look \(k\). A path may have crossed earlier and later returned inside the interval; that later value is not a continuation state for the original design.

Worked Example: Writing the Surviving Region

Consider a hypothetical three-look design with limits \((-2.4,2.3)\), \((-2.2,2.1)\), and \((-2.0,2.0)\), respectively. A possible observed sequence is \(Z_1=0.6\), \(Z_2=1.4\), and \(Z_3=2.2\). Determine where the sequence survives and where it first crosses.

At look 1, \(-2.4<0.6<2.3\), so the process survives. At look 2, \(-2.2<1.4<2.1\), so it survives again. At look 3, \(2.2\geq2.0\), so the upper limit is crossed. The first-crossing event for this realized path is therefore the upper-crossing part of the look-3 event, subject to survival at looks 1 and 2:

\{-2.4<Z_1<2.3,\ -2.2<Z_2<2.1,\ Z_3\geq2.0\}.

The observed value \(Z_3=2.2\) is beyond the look-3 upper limit, but a calculation of the probability of this event also needs the joint distribution across the three looks. The sequence of inequalities defines the event; the canonical model supplies its probability.

Numerical States from Hypothetical Interim Data

The state representation can be formed from interim score and information values without changing the event definitions. Consider a hypothetical two-look study with \(I_1=16\), \(I_2=36\), and observed efficient scores \(U_1=4.2\), \(U_2=8.1\). Suppose the limits are \((-2.5,2.5)\) at look 1 and \((-2.1,2.1)\) at look 2. The information schedule is increasing. Assuming the score and information values are valid under the canonical approximation, the standardized states are

Z_1=\frac{4.2}{\sqrt{16}}=\frac{4.2}{4}=1.05,\qquad Z_2=\frac{8.1}{\sqrt{36}}=\frac{8.1}{6}=1.35.

Both values lie inside their corresponding limits, so the observed path survives both looks. Its information times are \(16/36=0.444\) and \(36/36=1.000\). The relative information increment is \(20/36=0.556\). This calculation identifies the numerical state and the survival event; it does not by itself estimate a crossing probability.

Worked Example: A First-Look Crossing Probability

Suppose a hypothetical design has a single-sided upper stopping limit of \(2.00\) at its first look, and the null model is \(Z_1\sim N(0,1)\). The information and null distribution meet the canonical assumptions for this first-look calculation. Find the probability of crossing at that look under \(H_0\).

The upper-crossing event is \(Z_1\geq2.00\). Standardizing under the stated null distribution gives

P_{H_0}(Z_1\geq2.00)=1-\Phi(2.00)\approx1-0.9772=0.0228.

Thus, the probability of crossing this first-look limit under the null is approximately \(0.0228\), or \(2.28\%\). This is the probability for the first look alone. In a multi-look design, the overall probability of an upper crossing includes first crossings at later looks and must account for survival at every preceding look.

Assumptions, Diagnostics, and Alternatives

The canonical representation is useful when the statistic at each planned look is approximately normal, the information values are meaningful and increasing, and score increments behave approximately independently under the specified model. It is most direct for a fixed analysis schedule and a fixed definition of the statistic. In an applied analysis, check that the information measure matches the score being standardized, that the planned look order is correctly recorded, and that the assumptions supporting the normal approximation are credible for the sample sizes and data structure.

The canonical model may be inadequate if information is estimated poorly, the statistic has a strongly nonnormal distribution, or dependence between increments is not represented by the model. It also should not be transferred mechanically to a design in which an unplanned adaptation changes the statistic or its sampling distribution. Such settings may require a different or explicitly adapted joint model. A useful diagnostic is to verify that each look has unit marginal variance after standardization and that the implied correlation \(\sqrt{I_i/I_j}\) is plausible for the design. A non-increasing information schedule, for example, signals a bookkeeping or definition problem that must be resolved before interpreting the state sequence.

The state-vector description is more informative than a terminal-only analysis because it retains the earlier values and their stopping constraints. It is also more disciplined than treating each look as an unrelated Z-test, which discards the dependence and the pathwise stopping rule. Raw score and information values remain useful for checking how a state was obtained, while information time is a compact way to describe its location in the planned information schedule. None of these representations removes the need to specify the actual boundaries and the joint model.

Common Mistake

Common mistake: Treating \(a_k<Z_k<b_k\) as the full definition of survival through look \(k\). That only says the current statistic is inside the current limits. Survival through look \(k\) requires \(a_j<Z_j<b_j\) at every look \(j\leq k\). The correction matters because a path that crossed previously cannot be counted as continuing merely because a later statistic returns inside its limits.

Key takeaway: A sequential state consists of the standardized statistic at each planned look, its information position, and the boundary constraints applied along the path. The canonical model specifies the joint dependence; the sequence of inequalities specifies survival and first crossing.

Check Your Understanding

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

If \(I_1=9\) and \(I_2=36\), what are the information times and the canonical correlation between \(Z_1\) and \(Z_2\)?
With limits \((-2,2)\) at both of two looks, a path has \(Z_1=2.1\) and \(Z_2=0.3\). Did it survive through look 2? Explain.
Under \(H_0\), \(Z_1\sim N(0,1)\) and the first-look upper limit is \(1.96\). What is the approximate upper-crossing probability at that look?
Why does a state vector with marginally standard normal statistics not imply that the looks are independent?

Solutions. (1) \(t_1=9/36=0.250\), \(t_2=1.000\), and \(\operatorname{Corr}(Z_1,Z_2)=\sqrt{9/36}=0.500\). (2) No. Since \(Z_1=2.1\geq2\), the path crossed at look 1; the later value does not undo that crossing. (3) \(P(Z_1\geq1.96)=1-\Phi(1.96)\approx0.0250\), or \(2.50\%\). (4) Marginal distributions describe each look separately. Under the canonical model the looks have correlation \(\sqrt{I_i/I_j}\) for \(i<j\), which is generally nonzero, so their joint behavior is not independent.