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Biological units, repeated visits, and technical assay replicates

Example

Consider eight enrolled participants indexed by $i \in {1, \dots, 8}$ . Each participant is both the biological unit and the assignment unit; all eight participants are included, so this example does not posit probability sampling from a wider population. Exactly four participants are assigned to treatment ( $Z_{i} = 1$ ), uniformly over $Z = {z \in {0, 1}^{8} : \sum_{i = 1}^{8} z_{i} = 4},$ which has $(\binom{8}{4}) = 70$ assignments. The response is recorded at baseline $t = 0$ and four weeks after assignment $t = 1$ . At each participant-visit combination one specimen is read in three technical assay replicates $r \in {1, 2, 3}$ .

For each $i, t, a$ , let $u_{i t} (a) \in R$ be a fixed response value under arm $a \in {0, 1}$ . Let $d_{i t r} \in R$ be a fixed reading offset for technical replicate $r$ . The stipulated deterministic observation mechanism is $R_{i t r} = u_{i t} (Z_{i}) + d_{i t r} .$ The full data object has one row per participant and six response cells indexed by $(t, r)$ ; for this example the participant is the analysis unit as well as the biological and assignment unit. The assignment vector $Z$ is random under $π$ , so the response vector $R$ is random under $P_{θ}$ . Both $Z$ and the 48 response coordinates are recorded. All $u_{i t} (a)$ and $d_{i t r}$ are fixed latent model quantities; in particular, $u_{i t} (1 - Z_{i})$ is unobserved. The target population is these eight participants, and the estimand is the finite-schedule mean contrast at the four-week visit, $ψ (u, d) = \frac{1}{8} \sum_{i = 1}^{8} (u_{i 1} (1) - u_{i 1} (0)) .$ The experiment law averages over the stated assignment distribution. Its support is the finite set of records generated by the 70 assignments. The assignment law is stipulated, with no implementation evidence supplied; the deterministic measurement rule is an unvalidated modeling assumption. No independence between the 48 readings or generalization beyond the eight participants is asserted.

Facts & Assumptions

Given: The finite assignment mechanism and the deterministic record rule above.

[F1]

Scientific statistical model contracts: a contract specifies the observation space, distinct unit and index roles, target and estimand, observation mechanism, support and law, conditioning convention, and the epistemic status of adequacy assumptions; it proves no empirical or physical applicability.

[F2]

Statistical experiments: an experiment consists of a measurable space, a parameter set, and a family of probability measures on that space.

[F3]

Measurable spaces and measurable sets: a measurable space is a set together with a sigma-algebra.

[F4]

Measures on sigma-algebras: countable additivity on pairwise disjoint measurable sets is a defining condition for a measure.

[F5]

Probability measures and probability spaces: a probability measure is a measure of total mass one.

[F6]

A measurable function between measurable spaces: a map is measurable when the preimage of every measurable target set is measurable in the source.

[F7]

Random elements and real random variables: a random element is a measurable map from a probability space into a measurable space.

[F8]

Law or distribution of a random element: the law of a random element assigns to each measurable target set the probability of its preimage.

[F9]

The law of a random element is a probability measure: the law of a random element is a probability measure.

[F10]

Parametric statistical models: a parametric statistical model is the parameter map from the parameter set to the experiment's probability measures.

[F11]

Estimands on a statistical model: an estimand is a specified target-valued function on the parameter set of a parametric model.

Verification

1.1givenF4F5algebraconstruct

The set $Z$ has $(\binom{8}{4}) = 70$ members. Put $Ω = Z$ and equip it with $F = P (Ω)$ , which is a sigma-algebra because a full power set is closed under complements and countable unions. Define $π (B) = ∣ B ∣ / 70$ for $B \in F$ . Then $π (\emptyset) = 0$ and $π (Ω) = 1$ . Any pairwise disjoint sequence of subsets of the finite set $Ω$ has at most 70 nonempty terms, so cardinality is countably additive; hence $π$ is a probability measure by [F4] and [F5].

1.2givenF1F11algebra

Let the parameter set be $Θ = {(u, d) : u : {1, \dots, 8} \times {0, 1} \times {0, 1} \to R, d : {1, \dots, 8} \times {0, 1} \times {1, 2, 3} \to R},$ so $θ = (u, d)$ indexes all stated real outcome and offset arrays. Each participant contributes $2 \cdot 3 = 6$ readings, giving 48 response coordinates in eight participant rows. The participant is the biological, assignment, and analysis unit; $t$ and $r$ index repeated visits and technical readings within it, as required by [F1]. The displayed $ψ$ is a function of $θ$ into $R$ , so it is the stated estimand by [F11].

2.1givenF2F3F6F7F8F9F10step 1.1construct

Let $X = Z \times R^{48}, A = P (X),$ where the 48 reading coordinates are indexed by $(i, t, r)$ . The full power set $A$ is a sigma-algebra, so $(X, A)$ is a measurable space by [F3]. The assignment and response extractors $J : X \to Z$ , $J (z, r) = z$ , and $Q : X \to R^{48}$ , $Q (z, r) = r$ , are measurable to $(Z, P (Z))$ and $(R^{48}, P (R^{48}))$ : every preimage is a subset of $X$ and hence belongs to $A$ by [F6]. For $θ = (u, d) \in Θ$ , the record map $H_{θ} : Ω \to X$ sends $z$ to $(z, (u_{i t} (z_{i}) + d_{i t r})_{i, t, r})$ . Every preimage under this map is a subset of $Ω$ , hence belongs to $F$ , so $H_{θ}$ is measurable and is a random element by [F6] and [F7]. Define $P_{θ} (B) = π (H_{θ}^{- 1} (B))$ for $B \in A$ ; [F8] defines this law and [F9] makes it a probability measure. The extractors $J$ and $Q$ are therefore random elements under $P_{θ}$ by [F6] and [F7], and their assignment and response values are recorded in the data object. Thus $(X, A, Θ, (P_{θ}))$ is a statistical experiment by [F2] and a parametric model by [F10].

3.1givenF1F8step 1.1construct∎

Let $S_{θ} = {H_{θ} (z) : z \in Z}$ . Since $A = P (X)$ , $S_{θ} \in A$ ; its preimage under $H_{θ}$ is $Ω$ , so $P_{θ} (S_{θ}) = π (Ω) = 1$ by [F8] and step 1.1. This is the support field required by [F1]. The experiment averages over $π$ by construction; it makes no independence assertion about the readings.

Remarks

The construction is finite and choice-free. Here $π$ is a stipulated finite design law; no physical protocol or implementation is asserted. In an empirical contract, this assignment is design-controlled only if a protocol imposes it and implementation records confirm it. The deterministic measurement rule is empirically assessable with calibration or validation measurements, but none are supplied here, so the construction establishes no physical adequacy.

Depends on

Used by

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Dependency tree · two levels

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