Power and sample-size planning specifications

Let ${\mathsf{E}}_d=(({\mathcal{X}}_d,{\mathcal{A}}_d),\mathrm{\Theta },(P_{\theta ,d}{)}_{\theta \in \mathrm{\Theta }})$ be the specified statistical experiment for each candidate plan $d$ in a nonempty or empty candidate set $\mathcal{D}$. A plan records its sampling or assignment law, a permitted allocation vector $\mathbf{n}(d)=(n_1,\dots ,n_g)$ for an integer $g\ge 1$, with nonnegative integer counts $n_j$, and its total sample size $n(d)={\sum }_{j=1}^g{n}_j\in {\mathbb{N}}_{\ge 1}$. Which allocations are permitted, including any positive lower bounds, capacity limits, or costs, is part of $\mathcal{D}$.

For a test plan, fix disjoint nonempty hypotheses ${\mathrm{\Theta }}_0,{\mathrm{\Theta }}_1\subseteq \mathrm{\Theta }$, a measurable randomized test function ${\phi }_d:{\mathcal{X}}_d\to [0,1]$ (Randomized tests), and a nonempty target set ${\mathrm{\Theta }}_{\star }\subseteq {\mathrm{\Theta }}_1$. Its power at $\theta$ is ${\pi }_d(\theta )=E_{\theta ,d}[{\phi }_d(X)]$, its size is $s_d={\sup }_{\theta \in {\mathrm{\Theta }}_0}{\pi }_d(\theta )$, and its type-II error at a specified alternative $\theta$ is $1-{\pi }_d(\theta )$ (Test size and power). A target effect is the declared alternative parameter or contrast, for example $\delta ={\mu }_1-{\mu }_0$ in a scalar location model. The sidedness is fixed by the test's predeclared rejection region: an upper or lower tail, or a stated two-sided region such as $\{\mathrm{\mid }{T}_d-c_0\mathrm{\mid }>c\}$; no conversion from a one-sided cutoff to a two-sided cutoff is implicit.

A test-based request specifies nominal size $\alpha \in [0,1]$, target type-II error $\beta \in [0,1]$, and the target set ${\mathrm{\Theta }}_{\star }$. A plan meets it when $s_d\le \alpha$ and ${\pi }_d(\theta )\ge 1-\beta$ for every $\theta \in {\mathrm{\Theta }}_{\star }$, together with every other declared design constraint.

An optional precision request specifies a scalar estimand $\psi :\mathrm{\Theta }\to \mathbb{R}$, measurable endpoint functions $L_d,U_d:({\mathcal{X}}_d,{\mathcal{A}}_d)\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ with $L_d\le U_d$ and interval $C_d(X)=[L_d(X),U_d(X)]$, a target set ${\mathrm{\Theta }}_{\star }$, coverage target $1-\gamma$ with $\gamma \in [0,1]$, maximum width $w>0$, and width assurance $1-\eta$ with $\eta \in [0,1]$. It is met when, for every $\theta \in {\mathrm{\Theta }}_{\star }$, $P_{\theta ,d}\{\psi (\theta )\in C_d(X)\}\ge 1-\gamma \text{and}{P}_{\theta ,d}\{U_d(X)-L_d(X)\le w\}\ge 1-\eta \mathrm{.}$ Thus precision is a property of the stated interval procedure and sampling law, not just a target effect label.

Let $\mathcal{F}\subseteq \mathcal{D}$ be the plans meeting all selected test, precision, allocation, and resource conditions. The planned sample size is $\text{none}$ if $\mathcal{F}=\mathrm{\varnothing }$, and otherwise is the least integer in $\{n(d):d\in \mathcal{F}\}$. Among plans tied at that size, the specification may return any one or a declared tie-breaking choice. The definition is an extensional selection rule over its declared candidates; a finite search algorithm is a separate procedure.

The condition that a plan meets a target is always tied to its declared model, observation and assignment laws, test or interval, effect, allocation, sidedness, and constraints. No model-adequacy or universal power guarantee follows from the planning record.

The least-size or none characterization is extensional on the full candidate set. Operational evaluation over all positive sizes is supplied by Search over all positive sample sizes with explicit stopping certificates and its separate correctness theorem. Its ascending branch requires a feasible size to terminate; reporting none over the whole domain requires a sound global certificate. The finite-list algorithm retains its distinct supplied-list guarantee.