Assignment randomization and sharp-null tests

Let $(\mathrm{\Omega },w)$ be a finite probability space, let $T:\mathrm{\Omega }\to \mathbb{R}$, and let $Z$ have law $w$. Define $q(z):={\sum }_{\omega \in \mathrm{\Omega }}w(\omega )1\{T(\omega )\ge T(z)\}\mathrm{.}$ Then $\mathrm{Pr}(q(Z)\le u)\le u$ for every $u\in [0,1]$. If an auxiliary random variable $U$ is independent of $Z$ and uniform on $[0,1]$, define $q^{+}(Z,U):={\sum }_{\omega \in \mathrm{\Omega }}w(\omega )1\{T(\omega )>T(Z)\}+U{\sum }_{\omega \in \mathrm{\Omega }}w(\omega )1\{T(\omega )=T(Z)\}\mathrm{.}$ Then $\mathrm{Pr}(q^{+}(Z,U)\le u)=u$ for every $u\in [0,1]$.

Let $N\ge 1$, and let $\mathcal{I}=\{1,\dots ,N\}$ index the fixed experimental units at the actual treatment-assignment level. Let $\mathcal{Z}$ be a nonempty finite set of allowed assignments. If treatment is assigned to clusters, each cluster is one indexed unit; no separate probability-sampling unit is present. The index $i$ identifies that unit, and no visit or repeated-measurement index is modeled here.

A known assignment law is a function $p:\mathcal{Z}\to [0,1]$ satisfying ${\sum }_{z\in \mathcal{Z}}p(z)=1$, so $(\mathcal{Z},p)$ is a finite probability space (Finite probability spaces, outcome weights, events, and event probabilities); assignments of probability zero may remain in $\mathcal{Z}$. A potential-outcome schedule is a function $\theta :\mathcal{I}\times \mathcal{Z}\to \mathbb{R}$, with $Y_i(z):=\theta (i,z)$. The schedule and the units are fixed; the assignment $Z$ is the only randomness supplied by the design, and the observed response vector is $Y_{\theta }(Z)=(Y_i(Z){)}_{i\in \mathcal{I}}$. The schedule is defined for every allowed assignment, including zero-probability assignments, and may depend on the full assignment vector, so no no-interference assumption is implicit. The scientific statistical model contract Scientific statistical model contracts requires the observation space, units, quantity status, observation mechanism, support, conditioning convention, and adequacy assumptions to be recorded; this model specifies those fields as follows.

To express this as a statistical experiment, take the data space $\mathcal{X}=\mathcal{Z}\times {\mathbb{R}}^{\mathcal{I}}$ with its full power-set sigma-algebra, which is a sigma-algebra by Sigma-algebras and makes $\mathcal{X}$ a measurable space (Measurable spaces and measurable sets), and parameter set $\mathrm{\Theta }={\mathbb{R}}^{\mathcal{I}\times \mathcal{Z}}$. For each $\theta \in \mathrm{\Theta }$ let $P_{\theta }$ be the law (Law or distribution of a random element) of the deterministic observation map $O_{\theta }(z)=(z,Y_{\theta }(z))$ from $(\mathcal{Z},p)$. The map is measurable because its domain has the full power set, in the preimage sense of A measurable function between measurable spaces, and its law is a probability measure by The law of a random element is a probability measure. Thus $((\mathcal{X},2^{\mathcal{X}}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ is an experiment in the sense of Statistical experiments. The assignment law asserts no response-error model, Gaussian law, superpopulation randomization, or sampling representativeness beyond the stated finite assignment mechanism. Its finite full-measure support set is $S_{\theta }:=\{(z,Y_{\theta }(z)):z\in \mathcal{Z},p(z)>0\},$ since $P_{\theta }(S_{\theta })={\sum }_{z:p(z)>0}p(z)=1$. The record contains the assigned vector and every unit's response; missing or censored outcomes and repeated visits require a separately declared observation mechanism. The inferential target is the assignment-invariance hypothesis for these $N$ units, not an estimand for a wider population.

Inference in this page holds $\mathcal{I}$ and $\theta$ fixed and averages over the declared law $p$; it does not condition on the realized assignment. The status of the quantities is explicit: $\theta$ and all $Y_i(z)$ are fixed, $Z$ is random, $(Z,Y_{\theta }(Z))$ is observed, and potential responses under unassigned vectors are fixed but latent. The assignment set, weights, and implemented randomization protocol are design-controlled and should be checked against protocol records. Assignment logs and response records are empirically assessable, but those checks do not prove that the declared law generated the data. Potential outcomes under unassigned vectors are fixed and latent in the realized record. This does not imply nonidentifiability from the observed-data law: for each $z$ with $p(z)>0$, the law assigns mass $p(z)$ to $(z,Y_{\theta }(z))$, and therefore determines $Y_{\theta }(z)$. Assignment invariance on this positive-probability set is empirically assessable, and the theorem tests the stronger invariance hypothesis on all allowed assignments. Values at assignments with $p(z)=0$, and restrictions on those values, are substantively untestable from this law, since changing them leaves $P_{\theta }$ unchanged. One realized assignment does not reveal responses at other assignments. These classifications do not make the design or scientific model empirically true.

In the finite randomized treatment-assignment model Finite randomized treatment-assignment models, a schedule satisfies the Fisher sharp null when for each unit $i$ there is a value $y_i$ such that $Y_i(z)=y_i$ for every allowed assignment $z$. The full vector $y=(y_i{)}_{i\in \mathcal{I}}$ is then imputable under every assignment.

A predeclared ordering score is a real-valued statistic $T:\mathcal{Z}\times {\mathbb{R}}^{\mathcal{I}}\to \mathbb{R}$ (Statistics and estimators) fixed before the assignment and outcomes are inspected. Its upper-tail ordering includes ties: assignment $z^{\mathrm{\prime }}$ is at least as extreme as $z$ when $T(z^{\mathrm{\prime }},y)\ge T(z,y)$. For a signed score $S$ and fixed null center $s_0$, the corresponding two-sided ordering is explicitly $T(z,y)=\mathrm{\mid }S(z,y)-s_0\mathrm{\mid }$. The center is part of the predeclared ordering; the sharp null alone does not force $s_0=0$ for an arbitrary signed score. A lower-tail ordering is represented by using $-S$ as the score. A statistic in this setting is a measurable map in the sense of Statistics and estimators; the full power-set data sigma-algebra makes every such real-valued map measurable.

Consider the no-interference binary-treatment submodel of Finite randomized treatment-assignment models in which the allowed assignment set is $\{0,1{\}}^{\mathcal{I}}$ and unit $i$'s response under assignment vector $z$ is $Y_i(z)=Y_i(z_i)$ for fixed real potential responses $Y_i(0)$ and $Y_i(1)$. For the fixed units, define the finite-population average treatment effect by ${\tau }_F:=\frac{1}{N}{\sum }_{i\in \mathcal{I}}(Y_i(1)-Y_i(0))\mathrm{.}$ The weak zero-average-effect null is the hypothesis ${\tau }_F=0$; it constrains the average of the unit-level effects, not each effect separately. The Fisher sharp null in this submodel, defined in Fisher sharp null and predeclared test orderings, requires $Y_i(1)=Y_i(0)$ for every unit. No probability distribution on the fixed potential responses is part of either definition.

In a finite randomized treatment-assignment model, fix any real-valued predeclared ordering score $T(z,y)$. For observed data $(z,y)$ define $p_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}(z,y):={\sum }_{z^{\mathrm{\prime }}\in \mathcal{Z}}p(z^{\mathrm{\prime }})1\{T(z^{\mathrm{\prime }},y)\ge T(z,y)\}\mathrm{.}$ Let $\lambda$ be the uniform probability measure on the Borel subsets of $[0,1]$. For each $\theta$, equip $\mathcal{X}\times [0,1]$ with the product sigma-algebra $2^{\mathcal{X}}\otimes \mathcal{B}([0,1])$ and define ${\tilde{P}}_{\theta }(E):={\sum }_{z\in \mathcal{Z}}p(z)\lambda (\{u\in [0,1]:((z,Y_{\theta }(z)),u)\in E\})\mathrm{.}$ Let $U$ be the $[0,1]$ coordinate on this extension. Define $p^{+}(z,y,U):={\sum }_{z^{\mathrm{\prime }}\in \mathcal{Z}}p(z^{\mathrm{\prime }})1\{T(z^{\mathrm{\prime }},y)>T(z,y)\}+U{\sum }_{z^{\mathrm{\prime }}\in \mathcal{Z}}p(z^{\mathrm{\prime }})1\{T(z^{\mathrm{\prime }},y)=T(z,y)\}\mathrm{.}$ For every schedule satisfying the Fisher sharp null and every $u\in [0,1]$, $P_{\theta }\text{\hspace{0.17em}⁣}(p_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}(Z,Y_{\theta }(Z))\le u)\le u,$ whereas the added randomization gives ${\tilde{P}}_{\theta }\text{\hspace{0.17em}⁣}(p^{+}(Z,Y_{\theta }(Z),U)\le u)=u\mathrm{.}$ Thus $p_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}$ is a valid finite-sample p-value for the sharp null, and randomized ties give an exactly uniform p-value when the additional independent uniform randomizer is used. For a two-sided test, take $T(z,y)=\mathrm{\mid }S(z,y)-s_0\mathrm{\mid }$ with its center $s_0$ fixed in advance. The result uses the actual, possibly unequal assignment probabilities; random assignment supplies no Gaussian error law, and this theorem makes no validity claim under the weak zero-average-effect null from Finite-population average treatment effect and weak null unless it is also Fisher-sharp.