{title2=$\operatorname{Var}(\overline Y_g)=\tau^2/n+\sigma^2/(nm)$}

For $n$ pots per <treatment> and $m$ <leaves> per pot, the group <sample mean> decomposes into two averages:
$$
\overline Y_g=\frac1{nm}\sum_{i=1}^n\sum_{j=1}^mY_{gij}
=\mu+\alpha_g+\frac1n\sum_{i=1}^nU_{gi}
+\frac1{nm}\sum_{i=1}^n\sum_{j=1}^m\varepsilon_{gij}.
$$
The <variance of a sum> adds the <variances> of the <independent> terms. Every pot <random effect> has weight $1/n$, whereas each remaining error has weight $1/(nm)$, giving the short proof
$$
\operatorname{Var}(\overline Y_g)
=n\frac{\tau^2}{n^2}+nm\frac{\sigma^2}{n^2m^2}
=\frac{\tau^2}{n}+\frac{\sigma^2}{nm}.
$$
This is also the <variance> of the <arithmetic mean> of the $n$ pot <sample means>, each with <variance> $\tau^2+\sigma^2/m$. Equal numbers of <leaves> per pot make the two averaging procedures identical.

If all $nm$ <leaf> measurements were falsely treated as <independent>, the predicted <variance> would instead be $(\tau^2+\sigma^2)/(nm)$. The ratio of correct to false <variance> is
$$
\frac{m\tau^2+\sigma^2}{\tau^2+\sigma^2}
=1+(m-1)\rho.
$$
Thus <pseudoreplication> can underestimate a <standard error> by the square root of this ratio. Equality holds when $m=1$ or $\tau^2=0$; sharing a pot creates an error here precisely when the shared influence has positive <variance>.

Choose $\tau^2=4$, $\sigma^2=1$, $n=4$ and $m=3$ as constructed model parameters in squared <length> units. These are assumptions rather than fitted experimental values. For each group,
$$
\operatorname{Var}(\overline Y_g)=\frac44+\frac1{12}=\frac{13}{12},
\qquad
\operatorname{Var}_{\mathrm{false}}(\overline Y_g)=\frac5{12},
\qquad \rho=\frac45.
$$
The <treatment contrast> $\widehat\Delta=\overline Y_B-\overline Y_A$ has <expected value> $\alpha_B-\alpha_A$. Since the group errors are <independent> conditional on the allocation, its <variance> is twice the group <variance>, namely $13/6$, whereas the false calculation gives $5/6$. Its correct <standard error> is $\sqrt{13/6}\approx1.472$ <length> units; the false value is $\sqrt{5/6}\approx0.913$. The <variance> ratio is $13/5=2.6$.

\Image[/experimental-replication-variance-budget.png]
{title=Constructed <variance> predictions: adding pots reduces uncertainty faster than adding <leaf> measurements within four pots}
{description=Each curve uses $\tau^2=4$ and $\sigma^2=1$; the horizontal axis counts measurements per <treatment>. The solid designs agree at four pots with three <leaves> each. The dashed curve assumes all measurements are <independent> and therefore illustrates <pseudoreplication>; none of the curves are observed experimental data.}

With four pots fixed, increasing $m$ drives the error contribution $\sigma^2/(4m)$ towards zero, but the <limit> of the group <variance> remains $\tau^2/4=1$. Measuring twelve <leaves> on each of four pots gives $49/48$. Using the same forty-eight measurements on sixteen pots, with three <leaves> per pot, gives $13/48$. More independent <experimental units> reduce both contributions. Extra <nested measurements> are useful for characterizing each <experimental unit>, but cannot average away its shared <random effect>. This is why sample collection effort and the number of independent <treatment> <replicates> answer different questions.<Pseudoreplication and the Design of Ecological Field Experiments by Stuart H. Hurlbert>[Pseudoreplication and the Design of Ecological Field Experiments, p. 201, “Multiple samples per experimental unit”]{ref}


Variance of nested measurements ( $Var (\overline{Y}_{g}) = τ^{2} / n + σ^{2} / (nm)$ )

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