Power and sample size

Assume AC. Let $Z\sim N(0,1)$ and $\mathrm{\Phi }(x)=P(Z\le x)$. Then $\mathrm{\Phi }$ is continuous and strictly increasing, with limits $0$ and $1$ at $-\mathrm{\infty }$ and $+\mathrm{\infty }$; $P(Z=x)=0$ for every $x\in \mathbb{R}$; and $\mathrm{\Phi }(-x)=1-\mathrm{\Phi }(x)$. For each $q\in (0,1)$, there is a unique real $z_q$ with $\mathrm{\Phi }(z_q)=q$, and $z_{1-q}=-z_q$, in particular $z_{1\mathrm{/}2}=0$.

Assume AC. Let $n\ge 1$, and let $X_1,\dots ,X_n$ be mutually independent real random variables on one probability space, each with law $N(\mu ,{\sigma }^2)$ for a fixed $\mu \in \mathbb{R}$ and $\sigma \ge 0$. Then ${\bar{X}}_n=n^{-1}{\sum }_{i=1}^n{X}_i$ has law $N(\mu ,{\sigma }^2\mathrm{/}n)$. If $\sigma >0$, then for every ${\mu }_0\in \mathbb{R}$, $\frac{\sqrt{n}({\bar{X}}_n-{\mu }_0)}{\sigma }\sim N\text{\hspace{0.17em}⁣}(\frac{\sqrt{n}(\mu -{\mu }_0)}{\sigma },1)\mathrm{.}$

Let $S\subseteq {\mathbb{N}}_{\ge 1}$ be nonempty. Then there is a unique $n_{\ast }\in S$ such that $n_{\ast }\le n$ for every $n\in S$.

Identify integers and naturals with their canonical copies in the reals. For every real $x$, put $\lceil x\rceil =-\lfloor -x\rfloor$, using Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$. Then $\lceil x\rceil$ is an integer, $x\le \lceil x\rceil$, and $\lceil x\rceil \le m⟺x\le m(m\in \mathbb{Z})\mathrm{.}$ For $a\ge 0$, the least positive integer $n$ with $n\ge a$ is $\max \{1,\lceil a\rceil \}$. In particular $\lceil 0\rceil =0$, and $\lceil k\rceil =k$ for every integer $k$.

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.

Assume AC. Let $X_1,\dots ,X_n$ be mutually independent with common law $N(\mu ,{\sigma }^2)$, where $n\ge 1$ and $\sigma >0$. For the simple null $H_0:\mu ={\mu }_0$ against the two-sided alternative $H_1:\mu \mathrm{\ne }{\mu }_0$, let $Z_n=\sqrt{n}({\bar{X}}_n-{\mu }_0)\mathrm{/}\sigma$, let $c_{\alpha }=z_{1-\alpha \mathrm{/}2}$ for $\alpha \in (0,1)$, and reject when $\mathrm{\mid }{Z}_n\mathrm{\mid }>c_{\alpha }$. This test has size $\alpha$. At $\mu ={\mu }_0+\delta$, where $\delta \in \mathbb{R}$, its rejection probability (the power function evaluated at this parameter) is $1-\mathrm{\Phi }(c_{\alpha }-\lambda )+\mathrm{\Phi }(-c_{\alpha }-\lambda ),\lambda =\frac{\sqrt{n}\delta }{\sigma }\mathrm{.}$ For $\delta \mathrm{\ne }0$ this is power at the stated alternative; for $\delta =0$ it is the null rejection probability. The first term is the upper-tail rejection probability and the second is the lower-tail rejection probability.

Input a planning specification Power and sample-size planning specifications, a finite ordered list $d_1,\dots ,d_m$ of its candidate plans with $n(d_1)\le \cdots \le n(d_m)$, and an exact evaluator that terminates on each listed plan and returns true exactly when that plan meets all requested power, precision, allocation, and resource conditions. The list order among equal-size plans is part of the input.

Inspect $d_1,\dots ,d_m$ in order. Return the first plan for which the evaluator returns true, together with its total size $n(d_j)$. If no entry passes, return $\text{none}$. If $m=0$, return $\text{none}$ without an evaluation.

This procedure certifies $\text{none}$ only for the supplied finite list. The exact evaluator is an input subroutine; no numerical approximation or universal termination claim for arbitrary probability laws is part of this algorithm.

Input the full planning specification Power and sample-size planning specifications and an exact evaluator $E$ on every positive integer $n$. Each call $E(n)$ must terminate and return a Boolean with the following meaning: $E(n)$ is true exactly when the specification has a feasible plan of total size $n$, including its sampling law/test, power or precision target, allowed allocation and resource conditions. With a fixed allocation rule this is the ordinary per-size power calculation. For several candidate plans at one size, this assumption requires exact resolution of whether any is feasible; it is not supplied by the procedure. The evaluator, and any stopping certificate below, are inputs, not a universal ability to compute probabilities of arbitrary models.

Choose one of these admissible modes.

1. Unbounded ascending search. Set $n=1$. Evaluate $E(n)$. If true, return $n$; otherwise replace $n$ by $n+1$ and repeat. This branch never returns $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ merely because a finite prefix failed.

2. Certified bounded search. Supply a nonnegative integer $B$ and a valid completeness certificate that, if any positive size is feasible, some feasible size is at most $B$. Evaluate $E(1),\dots ,E(B)$ in order, returning the first true index. If all are false, return $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$; for $B=0$, return $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ immediately. The certificate is a mathematical premise covering all positive sizes, not a claim inferred from those finite evaluations. A proved feasible witness of size $B$ is one sufficient certificate.

3. Certified infeasibility. Supply a valid proof that $E(n)$ is false for every positive integer $n$. Return $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ without an evaluation.

The output is a sample size, not a selected allocation. Selecting a plan at that size requires an explicit witness-producing evaluator or an independently specified selection rule. The extensional target remains the least feasible positive integer, or $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ if no such integer exists. Termination properties are separate proved results; no global infeasibility decision is asserted for unrestricted black-box input.

Assume AC. Let $X_1,\dots ,X_n$ be mutually independent with common law $N(\mu ,{\sigma }^2)$, where $n\in {\mathbb{N}}_{\ge 1}$, $\sigma >0$, and $\mu \in \mathbb{R}$. Fix ${\mu }_0\in \mathbb{R}$, $\alpha ,{\beta }_{\mathrm{I}\mathrm{I}}\in (0,1)$, and $\delta >0$, and put ${\mu }_1={\mu }_0+\delta$. Observe ${\bar{X}}_n=n^{-1}{\sum }_i{X}_i$ in the one-dimensional experiment with law $N(\mu ,{\sigma }^2\mathrm{/}n)$. For the upper-tail test of $H_0:\mu \le {\mu }_0$ against $H_1:\mu >{\mu }_0$ with rejection rule ${\phi }_n(x)=\mathbf{1}\text{\hspace{0.17em}⁣}\{\frac{\sqrt{n}(x-{\mu }_0)}{\sigma }>z_{1-\alpha }\},$ the test size is $\alpha$, and its power at any $\mu$ is ${\pi }_n(\mu )=\mathrm{\Phi }\text{\hspace{0.17em}⁣}(\frac{\sqrt{n}(\mu -{\mu }_0)}{\sigma }-z_{1-\alpha })\mathrm{.}$ The least positive integer sample size attaining target power $1-{\beta }_{\mathrm{I}\mathrm{I}}$ at ${\mu }_1$, with candidate sizes ${\mathbb{N}}_{\ge 1}$, is $n_{\ast }=\max \text{\hspace{0.17em}⁣}\{1,\lceil {\left(\frac{\sigma }{\delta }\right)}^2\max \text{\hspace{0.17em}⁣}{\{0,z_{1-\alpha }+z_{1-{\beta }_{\mathrm{I}\mathrm{I}}}\}}^2\rceil \}\mathrm{.}$ If $z_{1-\alpha }+z_{1-{\beta }_{\mathrm{I}\mathrm{I}}}\le 0$, every positive integer size meets the target and the least is $1$. This is an exact result for the specified known-$\sigma$ normal model; it makes no approximation or claim for an unknown-$\sigma$ $t$-test or another sampling law.

For the finite-candidate algorithm Finite-candidate power and sample-size search, suppose the input list is ordered by nondecreasing total size and its evaluator returns the exact feasibility truth value for every candidate. The algorithm terminates after at most $m$ evaluations. If it returns a plan, that plan is feasible and its total size is least among feasible plans in the supplied list; if it returns $\text{none}$, no plan in that list is feasible. When several feasible plans have the same least size, the algorithm returns the first such plan in the supplied tie order.

Use the exact terminating evaluator and admissible modes of Search over all positive sample sizes with explicit stopping certificates, and let $S=\{n\in {\mathbb{N}}_{\ge 1}:E(n)\text{ is true}\}$. The unbounded ascending mode terminates if and only if $S\mathrm{\ne }\mathrm{\varnothing }$, and then returns $\min S$ after exactly $\min S$ evaluations. If $S=\mathrm{\varnothing }$, it continues forever and returns no value, including no $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$.

With a valid finite completeness bound $B$, the bounded mode terminates after at most $B$ evaluations and returns the full-domain extensional answer: $\min S$ if $S\mathrm{\ne }\mathrm{\varnothing }$ and $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}$ if $S=\mathrm{\varnothing }$. The certified-infeasibility mode returns that answer immediately. No monotonicity is needed for these conclusions. If feasibility is nondecreasing and $E(B)$ is true, that witness is a valid bound, but monotonicity without a crossing witness or other stopping certificate does not guarantee termination in the empty case.