8.4 One-Factor Experiments: Completely Randomized, Randomized Block, and Latin Square Designs

Choosing by the number of nuisance variables

The progression is the point: each additional blocking factor removes another known source of variation from the error term, at the cost of degrees of freedom and additional assumptions.

Completely randomized design

Every experimental unit has an equal chance of receiving any treatment, and run order is fully randomized. This is the correct design when experimental units are homogeneous and no nuisance variable is known.

Model: $y_{ij} = \mu + \tau_i + \varepsilon_{ij}$, where $\tau_i$ is the effect of treatment $i$.

Analysis is a one-way ANOVA:

Strengths: simplest to run and analyze, tolerates unequal group sizes, and maximizes error degrees of freedom. Weakness: any nuisance variation stays in the error term, so a real treatment effect can be masked.

Randomized complete block design

A block is a group of units expected to be more homogeneous within than between: one raw material lot, one day, one operator, one machine. Every treatment appears exactly once in every block, and run order is randomized within each block.

Model: $y_{ij} = \mu + \tau_i + \beta_j + \varepsilon_{ij}$.

Worked example. Four cutting-tool coatings ($k = 4$) are compared, and tool life is known to depend strongly on the bar-stock lot. Using five lots as blocks ($b = 5$) requires 20 runs, with each coating run once per lot.

Because lot-to-lot variation is estimated as its own term, it leaves the error, and a coating difference that a CRD would have missed becomes detectable. Error degrees of freedom drop from 16 in a CRD with the same 20 runs to $(4-1)(5-1) = 12$; the trade is almost always worth it when the blocking factor genuinely matters.

Assumption: no treatment-by-block interaction. If one coating is best in some lots and worst in others, the additive model is wrong and a factorial treating lot as a factor is required.

Complete means every treatment appears in every block. When blocks are too small to hold all treatments -- a furnace that holds three parts but four treatments are being compared -- a balanced incomplete block design is used instead.

Latin square

Controls two nuisance variables at once. Treatments are arranged in a $k \times k$ grid so each treatment appears exactly once in every row and exactly once in every column.

A $4 \times 4$ Latin square with operators as rows and days as columns:

Sixteen runs control both operator and day effects. A full factorial of treatment by operator by day would need 64.

Constraints and assumptions:

• The number of treatments must equal the number of rows and columns.
• No interaction is permitted among treatment, row, and column factors; the model is strictly additive.
• Error degrees of freedom are small for small $k$. A $3 \times 3$ square leaves only 2, which is rarely enough; $k \ge 4$ is preferred, or replicate the square.
• The particular square used should be selected at random from the available squares of that size.

A Graeco-Latin square extends the idea to three nuisance variables by superimposing a second set of treatment labels, still in $k^2$ runs, but the error degrees of freedom shrink further to $(k-1)(k-3)$.

Choosing in practice

The last row is the decision that matters. Blocking assumes the nuisance factor shifts the response by a constant amount regardless of treatment. If the nuisance factor genuinely interacts with the treatment, it is not a nuisance factor -- it is a factor, and the design must treat it as one.