8.4 One-Factor Experiments: Completely Randomized, Randomized Block, and Latin Square Designs
Key Takeaways
- A completely randomized design assigns treatments entirely at random and is correct when no nuisance variable is known.
- A randomized block design controls exactly one known nuisance variable by running a complete set of treatments within each block.
- A Latin square controls two nuisance variables simultaneously using k squared runs instead of k cubed.
- Blocking removes nuisance variation from the error term, which increases the sensitivity of the test for the treatment effect.
- A Latin square assumes no interaction among the treatment and the two blocking factors.
Choosing by the number of nuisance variables
| Nuisance variables to control | Design | Runs for $k$ treatments | Error degrees of freedom |
|---|---|---|---|
| 0 | Completely randomized (CRD) | $k \times n$ replicates | $N - k$ |
| 1 | Randomized complete block (RCBD) | $k \times b$ blocks | $(k-1)(b-1)$ |
| 2 | Latin square | $k^2$ | $(k-1)(k-2)$ |
| 3 | Graeco-Latin square | $k^2$ | $(k-1)(k-3)$ |
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:
| Source | df |
|---|---|
| Treatment | $k - 1$ |
| Error | $N - k$ |
| Total | $N - 1$ |
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: $y_{ij} = \mu + \tau_i + \beta_j + \varepsilon_{ij}$.
| Source | df |
|---|---|
| Treatment | $k - 1$ |
| Block | $b - 1$ |
| Error | $(k-1)(b-1)$ |
| Total | $kb - 1$ |
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.
| Lot 1 | Lot 2 | Lot 3 | Lot 4 | Lot 5 | |
|---|---|---|---|---|---|
| Order within block | Randomized | Randomized | Randomized | Randomized | Randomized |
| Coatings run | A, B, C, D | A, B, C, D | A, B, C, D | A, B, C, D | A, B, C, D |
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:
| Day 1 | Day 2 | Day 3 | Day 4 | |
|---|---|---|---|---|
| Operator 1 | A | B | C | D |
| Operator 2 | B | C | D | A |
| Operator 3 | C | D | A | B |
| Operator 4 | D | A | B | C |
Sixteen runs control both operator and day effects. A full factorial of treatment by operator by day would need 64.
| Source | df |
|---|---|
| Treatment | $k - 1$ |
| Rows | $k - 1$ |
| Columns | $k - 1$ |
| Error | $(k-1)(k-2)$ |
| Total | $k^2 - 1$ |
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
| Situation | Design |
|---|---|
| Comparing four solder pastes on a single stable line, one shift, one day | CRD |
| Comparing four solder pastes where board lot strongly affects the result | RCBD, blocking on lot |
| Comparing four solder pastes where both operator and day matter | Latin square |
| Comparing four pastes where paste behaves differently by operator | Factorial with operator as a factor -- the additive assumption fails |
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.
Four tool coatings are compared, and bar-stock lot is known to strongly affect tool life. Which design is appropriate, and what is the benefit?
In a 4 by 4 Latin square design, how many runs are required and how many error degrees of freedom result?
A randomized block design shows that treatment A is best in blocks 1 and 2 but worst in blocks 3 and 4. What does this indicate?