Biological and technical replication

For a specified scientific question, a biological unit is one unit at the level whose biological variation is relevant to the target population. A biological replicate is a distinct biological unit at that same level under the sampling law being modeled. The unit receiving randomized treatment is the randomization unit; when treatment is assigned to a cage, litter, plot, or batch, that cluster is the randomized replicate even if members or subsamples within it are measured separately. A technical replicate is another reading or assay of the same specimen or material within one biological unit. These roles follow the scientific and assignment design recorded by Scientific statistical model contracts; their names alone do not assert statistical independence.

Let $n,m\ge 1$ be integers, let $i\in \{1,\dots ,n\}$ index biological units, and let $j\in \{1,\dots ,m\}$ index technical measurements of unit $i$. On one probability space, take square-integrable real random variables $B_i$ and $E_{ij}$, in the sense of Random elements and real random variables, with $Y_{ij}=\mu +{\tau }_i+B_i+E_{ij}\mathrm{.}$ Here $\mu$ and the treatment contributions ${\tau }_i$ are fixed real numbers when the analysis conditions on the assigned treatment schedule. The random effect $B_i$ represents biological variation shared by all measurements of unit $i$; $E_{ij}$ is technical measurement error. The complete finite family $(B_i{)}_{i=1}^n,(E_{ij}{)}_{i\le n,j\le m}$ is mutually independent in the sense of Independent random elements, and $\mathbb{E}[B_i]=\mathbb{E}[E_{ij}]=0,\mathrm{Var}(B_i)={\sigma }_b^2,\mathrm{Var}(E_{ij})={\sigma }_e^2$ for all indices, where ${\sigma }_b^2,{\sigma }_e^2\ge 0$. Mean, variance, and square-integrability use Expectation of a nonnegative or integrable random variable and Moments, variance, and covariance on a probability space. No Gaussian assumption is made.

With the rows indexed by $(i,j)$, this is a special case of the finite- dimensional linear mixed model Finite-dimensional linear mixed model with explicit longitudinal roles: the random-intercept design row selects $B_i$, and the fixed mean at that row is $\mu +{\tau }_i$. When inference averages over randomized assignments, the vector $\mathit{\tau }=({\tau }_1,\dots ,{\tau }_n)$ is instead a random vector under the declared assignment law; its covariance and any dependence on $B$ or $E$ must then be retained as specified by the result that uses it.

The equal-variance, mutual-independence, and balanced-measurement assumptions belong to this particular model. Role labels do not imply them. Shared plate, visit, cluster, batch, or serial-assay effects require additional terms or a different covariance structure. This definition and its variance consequences do not establish that the model fits an empirical study, that its sampling law is representative, or that a treatment contrast is causally identified.

Assume the balanced model of Biological and technical replication model, with $n,m\ge 1$, and set $\bar{Y}=\frac{1}{nm}{\sum }_{i=1}^n{\sum }_{j=1}^m{Y}_{ij},\bar{B}=\frac{1}{n}{\sum }_{i=1}^n{B}_i,\bar{E}=\frac{1}{nm}{\sum }_{i=1}^n{\sum }_{j=1}^m{E}_{ij},\bar{\tau }=\frac{1}{n}{\sum }_{i=1}^n{\tau }_i\mathrm{.}$ For a fixed treatment schedule $({\tau }_i{)}_{i=1}^n$, $\mathbb{E}[\bar{Y}]=\mu +\bar{\tau },\mathrm{Var}(\bar{Y})=\frac{{\sigma }_b^2}{n}+\frac{{\sigma }_e^2}{nm}\mathrm{.}$ If instead $\mathit{\tau }=({\tau }_1,\dots ,{\tau }_n)$ is square-integrable and random under the assignment-averaged law, then $\mathbb{E}[\bar{Y}]=\mu +\mathbb{E}[\bar{\tau }],\mathrm{Var}(\bar{Y})=\frac{{\sigma }_b^2}{n}+\frac{{\sigma }_e^2}{nm}+\mathrm{Var}(\bar{\tau })+2\mathrm{Cov}(\bar{\tau },\bar{B})+2\mathrm{Cov}(\bar{\tau },\bar{E})\mathrm{.}$ Thus, when $\bar{\tau }$ is independent of $\bar{B}$ and $\bar{E}$, the assignment contribution is $\mathrm{Var}(\bar{\tau })=n^{-2}{\sum }_{i,k=1}^n\mathrm{Cov}({\tau }_i,{\tau }_k)$. If, in addition, the ${\tau }_i$ are mutually independent with common variance ${\sigma }_{\tau }^2$, the full variance is ${\sigma }_b^2\mathrm{/}n+{\sigma }_e^2\mathrm{/}(nm)+{\sigma }_{\tau }^2\mathrm{/}n$. Without independence of the treatment average from $\bar{B},\bar{E}$, the two covariance terms remain. No Gaussian assumption is required.

The proof uses only finite sums and moment identities, so it is choice-free. The assignment contribution is not generally ${\sigma }_{\tau }^2\mathrm{/}n$: under a constrained or otherwise dependent assignment law, its covariance sum must be evaluated under that law. The identity is conditional on the stated model assumptions; it does not validate them or provide an exact normal inference procedure.