Scientific model contracts and applicability

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment (Statistical experiments). When the model is parametric, write $\mathsf{P}(\theta )=P_{\theta }$ (Parametric statistical models). A scientific statistical model contract is an explicit record of the following fields for a specified analysis.

1. Observation space and record. Give $(\mathcal{X},\mathcal{A})$ and say which values form one recorded data object. A response and each predictor used as a function of the data must be a specified measurable map from this space to a named measurable target. This is the map convention for a statistic in Statistics and estimators; it does not make a predictor causal or the response law correct.

2. Scientific, sampling, assignment, and analysis units. Name the biological or physical unit about which the scientific question is asked, the sample unit selected for observation when sampling occurs, the assignment unit receiving a randomized treatment when randomization occurs, and the analysis unit represented in each data record. These roles may differ. A technical replicate is a repeated reading or assay of the same specimen; a biological replicate is a distinct biological unit. Neither term asserts independence. State whether enrollment is a probability sample or a specified convenience/design cohort.

3. Indices and time. Give every unit, treatment, visit, and repeated measurement index, the object each index identifies, and the time origin. In particular, do not identify a participant index with a visit or assay index merely because all are written as rows in a data array.

4. Quantity status. Mark each model quantity as fixed (held nonrandom under the declared law), random (a measurable random element with its law specified, using Random elements and real random variables and Law or distribution of a random element), observed (recorded in the data object), or latent (part of a fuller model state but not recorded). A latent quantity must separately be declared fixed or random. If inference conditions on a design value or event, name it and state the conditional law being used.

5. Target and estimand. Name the target population of units, settings, and time period, and distinguish it from the enrolled or observed sample. Specify the target as a function $\psi :\mathrm{\Theta }\to \mathcal{T}$ when it is an estimand in the parametric model, as in Estimands on a statistical model. The contract alone does not show that the target is identifiable. Identifiability means that equality of sampling laws forces equality of target values (Identifiability of an estimand).

6. Observation mechanism. State how a fuller scientific state becomes the recorded data. A deterministic mechanism is a specified measurable map $O:(\mathcal{W},\mathcal{G})\to (\mathcal{X},\mathcal{A})$. A stochastic mechanism is specified by $K:\mathcal{W}\times \mathcal{A}\to [0,1]$ such that, for each $w$, $A\mapsto K(w,A)$ is a probability measure and, for each $A\in \mathcal{A}$, $w\mapsto K(w,A)$ is measurable. These clauses define the mechanism supplied by the contract; they do not assert that a conditional version exists for an arbitrary joint law.

7. Support and law. State the sampling law $P_{\theta }$ on $(\mathcal{X},\mathcal{A})$. For an asserted support set, give a measurable $S_{\theta }\subseteq \mathcal{X}$ with $P_{\theta }(S_{\theta })=1$. If “support” instead means topological support, name the topology and the conditions under which that convention is being used. If densities are used, name the dominating measure and state whether their positive sets agree across parameter values. For a randomized design, also state its assignment law; if inference averages over assignments, give the resulting joint law.

8. Conditioning convention. Say whether inference uses the joint law, averaging over the declared randomization, or a specified conditional law given the realized design. A conditional law at an event of probability zero must be supplied as a specified version with its existence hypotheses; it is not defined by an ordinary probability ratio.

9. Adequacy assumptions. List each scientific adequacy assumption and classify it as design-controlled (imposed by protocol and checked against implementation records), empirically assessable (data or validation measurements can challenge it, without proving it true), or substantively untestable from the declared observed-data law (external domain evidence or additional assumptions are needed). Split an assumption that has different parts into separate entries and justify each label.

These are bookkeeping and interpretation requirements. They do not assert that an empirical law equals a proposed model, that a design was implemented as specified, that calibration is valid, or that any scientific or physical application is adequate. No independence, exchangeability, missingness, causal-identification, or common-support assumption follows from the indexing or the classification labels. This definition uses no axiom of choice.