Finite randomized treatment-assignment models

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.