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An orthogonal block design has centered treatment-indicator vectors orthogonal to centered block-indicator vectors on the experimental units. With treatment replication $r_i$, block size $k_j$ and total $n$, this is equivalent to $n_{ij}=r_i{k}_j/n$. It permits additive block adjustment without changing treatment estimates.

An orthogonal block design has treatment contrasts orthogonal to block contrasts after removing the grand mean. This is a statement about vectors on the experimental units, with the usual inner product; the two full spaces are not orthogonal because both contain the constant vector.

For a counting criterion, let $n_{ij}$ count experimental units receiving treatment $i$ in block $j$, let $r_i={\sum }_j{n}_{ij}$, let $k_j={\sum }_i{n}_{ij}$, and let $n={\sum }_i{r}_i$. The inner product of the centered indicators for treatment $i$ and block $j$ is $n_{ij}-r_i{k}_j/n$. ConsequentlyIn particular, equal-sized blocks must contain each treatment in the same proportion. Under an additive block-and-treatment model, adjustment for blocks then does not change the fitted treatment contrasts.

The constant subspace has one statistical degree of freedom. Day contrasts have $5-1=4$, and the within-block ANOVA stratum has $50-5=45$. Because the orthogonal block design puts all four independent dose contrasts in that last ANOVA stratum, the residual has $45-4=41$ statistical degrees of freedom.

The corrected total has 49 statistical degrees of freedom; dose is tested against the within-day residual. The quantitative dose scale also permits the dose component to be split into linear, quadratic, cubic and quartic orthogonal polynomial contrasts, each with one statistical degree of freedom. This is optional and does not assume the response is linear in dose.

Use a randomized complete block design, with day as block and each biological cell as an experimental unit. Each day should contain two biological cells at each of the five dose levels. Each level then has ten replicates overall and occurs equally often in every day: the resulting block design is an orthogonal block design.

This spreads any day-to-day changes in preparation, temperature or measurement across all levels instead of confounding dose with day. Treat the zero level as a control with otherwise comparable handling. Keep the treatment-to-measurement delay standardized because the response variable is a diffusion rate measured immediately after preparation.

A randomized complete block design places every treatment in every block, with a common replication pattern, and uses independent within-block randomization. The usual basic version has one occurrence of each treatment per block; equal repeated occurrences give a replicated complete-block version. Such equal-proportion designs are orthogonal block designs.