Design of Experiments Intro: Explained with Examples

Design of Experiments (DOE) is the discipline of planning an experiment before running it, so you learn the maximum from the minimum of time, money, and materials. You lay out a structured plan — which factors to vary, at which levels, in which order — and let statistics do the heavy lifting.

This matters because experiments are expensive. A designed experiment extracts answers — including how factors interact — from a fraction of the runs that one-by-one tinkering would need. Fisher invented these ideas for agriculture, and industry has relied on them ever since.

Key Concepts

• Factor and level: a factor is a variable you control (fertiliser type); a level is one of its settings (A, B, C).
• Treatment: one specific combination of factor levels applied to an experimental unit.
• Experimental unit: the smallest thing receiving a treatment — one plot, one patient, one machine run.
• Replication: repeating each treatment on several units, so you can measure natural variation.
• Randomisation: assigning treatments to units by chance, which neutralises hidden biases (a shady corner of the field, a skilled operator).
• Blocking (local control): grouping similar units into blocks and comparing treatments within each block, filtering out known sources of variation like soil fertility gradients.
• Basic designs: CRD (completely randomised design), RBD (randomised block design), and LSD (Latin square design).

How It Works

1. State the objective precisely: "which fertiliser maximises wheat yield?" beats "let's try some fertilisers".
2. List factors and levels: decide what varies (fertiliser: A, B, C) and what stays fixed (seed variety, watering).
3. Choose the design: if plots vary in fertility, use a randomised block design with fertility as the block.
4. Replicate: assign each fertiliser to several plots — say 4 plots each, 12 total.
5. Randomise within blocks: draw lots for which plot gets which fertiliser, so no systematic bias creeps in.
6. Run and analyse: collect yields and run ANOVA — the design you chose dictates which ANOVA table to compute.

Worked Example

Compare 3 fertilisers (A, B, C) on wheat yield. The field slopes, so the top is drier than the bottom.

1. Block: divide the field into 4 horizontal strips (blocks); each strip has uniform moisture.
2. Units: split each strip into 3 plots → 12 plots total.
3. Randomise: within each strip, randomly assign A, B, C to the 3 plots. Every fertiliser appears once per strip — 4 replications each.
4. Collect: suppose mean yields are A: 3.2, B: 3.8, C: 3.5 tonnes/hectare.
5. Analyse: a randomised-block ANOVA separates fertiliser effects from strip (moisture) effects; without blocking, the moisture gradient would inflate the error and could hide real differences.
6. Conclude: if the F-test for fertiliser is significant, follow up with pairwise comparisons to see which wins.

Common Misconceptions

• More data does not beat better design. A thousand haphazard observations can be worth less than twelve well-randomised ones.
• Randomisation is not optional decoration. It is the mathematical basis that makes p-values and F-tests valid — skip it and your "significance" is fiction.
• One-factor-at-a-time misses interactions. Fertiliser A might win only when watering is high — factorial designs catch such interactions; sequential tinkering never does.

Real-World Applications

• Agriculture: crop variety and fertiliser trials on experimental farms worldwide still use Fisher's designs.
• Pharmaceuticals: clinical trials randomise patients to drug vs placebo within blocks (hospitals, age groups).
• Manufacturing: finding the best combination of temperature, pressure, and speed with a handful of designed runs.
• Marketing: factorial email tests varying subject line, image, and send time simultaneously.

Design of Experiments is thinking before doing: replicate to measure error, randomise to kill bias, block to filter noise. An hour of design saves weeks of confused data.