Design of experiments: CRD, RCBD and Latin square
Fisher's principles of field experiments, when to use a completely randomised, randomised block or Latin square design, and a worked analysis of variance.
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A field trial asks a simple question, such as which variety yields most or how much fertiliser pays, but the answer is clouded by variation: no two plots of land are identical, and no two plants grow alike. Experimental design is the art of arranging an experiment so that the effect of the treatments can be separated from that background variation, and measured honestly.
The basic designs, the completely randomised design, the randomised complete block design and the Latin square, and the analysis of variance used with them, are taught in every agricultural statistics course and are among the most reliably examined topics in agronomy, plant breeding and statistics papers.
Fisher's three principles
Modern experimental design was developed by R. A. Fisher at the Rothamsted Experimental Station in England in the 1920s and set out in The Design of Experiments (1935). Three principles underlie every design.
- Replication: each treatment is applied to more than one plot. Replication allows experimental error to be estimated and makes treatment means more precise; the standard error of a mean falls with the square root of the number of replicates.
- Randomisation: treatments are allocated to plots by a random process. This protects against bias from unknown patterns in the field and makes the error estimate valid.
- Local control (blocking): plots are grouped so that those within a group are as alike as possible, and each group receives every treatment. Variation between groups is then removed from the error.
Some terms are needed throughout:
- Treatment: what is being compared (varieties, doses, practices).
- Experimental unit or plot: the smallest unit to which a treatment is applied independently.
- Experimental error: the variation among plots treated alike.
- Block: a group of plots that are relatively homogeneous.
Completely randomised design (CRD)
Treatments are allocated to plots completely at random over the whole experimental area, with any number of replicates (equal or not).
- When to use it: when the experimental material is uniform: laboratory, greenhouse and pot experiments, or a genuinely uniform field.
- Advantages: simplest layout and analysis; maximum error degrees of freedom; missing plots cause no difficulty.
- Disadvantage: if the field is not uniform, all of its variation goes into the error, making it hard to detect real differences.
Randomised complete block design (RCBD)
The field is divided into blocks, each containing one plot of every treatment, and treatments are randomised separately within each block. Blocks are laid out to capture a known source of variation: perpendicular to a fertility or slope gradient, so that each block is as uniform as possible inside.
- When to use it: most field experiments, where there is one main direction of variation. It is the most widely used design in agricultural research.
- Advantages: removes block-to-block variation from the error; flexible, simple to analyse.
- Disadvantages: becomes less efficient with many treatments, because large blocks are less uniform; then incomplete block designs (lattices, alpha designs) are used.
Latin square design (LSD)
Plots are arranged in a square of t rows and t columns, and each treatment appears once in every row and once in every column.
- When to use it: when variation runs in two directions at right angles, such as a slope one way and an irrigation gradient the other, or in animal experiments with two blocking factors (for example animals and periods).
- Advantages: removes both row and column variation from the error.
- Disadvantages: the number of replicates must equal the number of treatments, so it is practical only for about 4 to 8 treatments; error degrees of freedom are few for small squares; analysis is harder with missing plots.
Analysis of variance
The analysis of variance (ANOVA), also Fisher's, splits the total variation among plots into parts due to each source, and tests whether treatment differences are larger than error would explain.
| Source | CRD (t treatments, r replicates) | RCBD | Latin square (t × t) |
|---|---|---|---|
| Blocks / rows | — | r − 1 | t − 1 (rows) |
| Columns | — | — | t − 1 |
| Treatments | t − 1 | t − 1 | t − 1 |
| Error | t(r − 1) | (r − 1)(t − 1) | (t − 1)(t − 2) |
| Total | tr − 1 | tr − 1 | t2 − 1 |
These degrees of freedom are among the most commonly asked exam items.
Analysing a randomised block experiment
Four varieties (A–D) are grown in three blocks. Yields, in t/ha:
| Block | A | B | C | D | Block total |
|---|---|---|---|---|---|
| I | 4.2 | 5.0 | 5.6 | 4.6 | 19.4 |
| II | 4.6 | 5.4 | 6.0 | 5.0 | 21.0 |
| III | 4.0 | 5.2 | 5.8 | 4.4 | 19.4 |
| Variety total | 12.8 | 15.6 | 17.4 | 14.0 | G = 59.8 |
- Correction factor: CF = G2 ⁄ (rt) = 59.82 ⁄ 12 = 298.003.
- Total SS = Σy2 − CF = 302.520 − 298.003 = 4.517.
- Block SS = (19.42 + 21.02 + 19.42) ⁄ 4 − CF = 0.427.
- Variety SS = (12.82 + 15.62 + 17.42 + 14.02) ⁄ 3 − CF = 3.983.
- Error SS = 4.517 − 0.427 − 3.983 = 0.107.
| Source | df | SS | Mean square | F |
|---|---|---|---|---|
| Blocks | 2 | 0.427 | 0.213 | 12.0 |
| Varieties | 3 | 3.983 | 1.328 | 74.7 |
| Error | 6 | 0.107 | 0.0178 | |
| Total | 11 | 4.517 |
The table value of F at 5 per cent for 3 and 6 degrees of freedom is 4.76, so the varieties differ significantly. Blocking was worthwhile: the block F of 12.0 exceeds F(2, 6) = 5.14, so a CRD would have left that variation in the error.
Comparing means. The critical difference (least significant difference) at 5 per cent = t0.05,6 × √(2 × MSE ⁄ r) = 2.447 × √(2 × 0.0178 ⁄ 3) = 0.27 t/ha. The means are C 5.80, B 5.20, D 4.67 and A 4.27 t/ha; every pair differs by more than 0.27, so all four varieties are significantly different from each other.
Coefficient of variation = √MSE ⁄ grand mean × 100 = 0.133 ⁄ 4.98 × 100 ≈ 2.7 per cent, a very precise trial (field trials often have CVs of 5 to 15 per cent).
Choosing a design
- Uniform material, few constraints: CRD.
- One known gradient in the field: RCBD, blocks across the gradient.
- Two gradients at right angles, few treatments: Latin square.
- Many treatments (such as breeding lines): incomplete block designs, augmented designs.
- Two or more factors (for example variety × nitrogen): factorial treatments, laid out in RCBD, or as a split-plot when one factor needs large plots (irrigation, tillage).
Whatever the design, good practice is the same: decide the question and the treatments first, choose plot size and replication to give enough precision (more replicates usually help more than bigger plots), randomise properly, record carefully, and check the assumptions of the analysis (normal errors, equal variances, additivity).
Key points
- Fisher's principles: replication (estimates error, improves precision), randomisation (avoids bias, validates error) and local control (blocking reduces error).
- CRD: random allocation over the whole area; for uniform conditions; error df = t(r − 1).
- RCBD: every treatment once per block, randomised within blocks; for one gradient; error df = (r − 1)(t − 1).
- Latin square: each treatment once per row and column; two gradients; error df = (t − 1)(t − 2).
- ANOVA partitions total variation; F = treatment mean square ÷ error mean square.
- After a significant F, compare means with a critical difference; report the CV.
What the exams ask
- School-leaving and entrance papers: the principles of experimentation and the definitions of treatment, replication and randomisation.
- Degree and research-fellowship entrances: choosing among CRD, RCBD and Latin square, their advantages and limits, degrees of freedom, and completing an ANOVA table.
- Statistics, plant breeding and agronomy postgraduate examinations: full analyses with critical differences and CVs, efficiency of blocking, missing plots, and extensions to factorial and split-plot designs.
Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.
Questions people ask
Why randomise if I can arrange plots evenly myself?
A deliberate arrangement can line up treatments with hidden patterns in the field, biasing results, and it invalidates the statistical test. Randomisation makes bias a matter of chance, which the analysis can account for.
How many replicates are enough?
Enough to give the error degrees of freedom and precision needed to detect a difference that matters. Many field trials use three to six replicates; a power calculation with an estimate of the CV gives a better answer.
Is a high F value always meaningful?
It shows that differences are unlikely to be due to chance, not that they are large or useful. Always look at the size of the differences, their critical difference and their practical value.
Sources
- Fisher, R. A. The Design of Experiments. Oliver and Boyd, 1935.
- Gomez, K. A. and Gomez, A. A. Statistical Procedures for Agricultural Research, 2nd edition. Wiley, 1984.
- Cochran, W. G. and Cox, G. M. Experimental Designs, 2nd edition. Wiley, 1957.
- Montgomery, D. C. Design and Analysis of Experiments, 10th edition. Wiley, 2019.
- Mead, R., Gilmour, S. G. and Mead, A. Statistical Principles for the Design of Experiments. Cambridge University Press, 2012.
Discussion
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