Canonical group-sequential alpha-spending design

Assume the Axiom of Choice (The Axiom of Choice). Fix a finite integer maximum $K\ge 1$, prescheduled information fractions $0<t_1<\cdots <t_K=1$ (where $t_j$ is the fraction of total planned information available at look $j$), and a nominal level $\alpha \in (0,1)$. An alpha-spending function is a nondecreasing map $A:[0,1]\to [0,\alpha ]$ with $A(0)=0$ and $A(1)=\alpha$.

Under the stipulated simple null law $P_0$, the standardized look vector is $Z=(Z_1,\dots ,Z_K)\sim N_K(0,\mathrm{\Sigma })$, where ${\mathrm{\Sigma }}_{jk}=\sqrt{t_j\mathrm{/}{t}_k}(j\le k),{\mathrm{\Sigma }}_{jk}={\mathrm{\Sigma }}_{kj}(j>k)\mathrm{.}$ Here $N_K$ is the finite-dimensional multivariate normal law of Multivariate normal law, including singular covariance, whose projection characterization determines the law by Characteristic function of a multivariate normal law. The construction of this covariance matrix as a valid Gaussian covariance is proved in Group-sequential boundaries exist uniquely and spend alpha exactly ↗. The standardized coordinates are correlated; independence of the $Z_j$ is not assumed.

For a one-sided upper-boundary design, look $j$ rejects when $Z_j\ge b_j$. For a symmetric two-sided design, look $j$ rejects when $\mathrm{\mid }{Z}_j\mathrm{\mid }\ge c_j$. In either case, the procedure stops at its first crossing. For each $k$, its cumulative type-I probability is the probability under the stipulated null law of crossing at one of looks $1,\dots ,k$. The design is exactly calibrated to $A$ when this cumulative probability equals $A(t_k)$ for every $k$. If it is embedded in an experiment with a nonempty alternative and singleton null hypothesis, this null rejection probability is the test size in the sense of Test size and power.

The fractions and joint null law are assumptions of this definition. It does not derive a Gaussian model from an asymptotic central limit theorem, justify data-dependent look timing, justify estimated variance, or establish error control over an unspecified composite null. A two-sided design calibrates its two-sided crossing events separately; one-sided boundaries are not doubled to obtain two-sided boundaries.