Design Types and the Replication Structure

Choosing the right design structure is mostly about matching the unit of randomization and the unit of replication to your question, and respecting any nesting in the analysis. This file walks through the standard structures and then treats the single most common fatal error — pseudoreplication — in depth.

Table of contents

• Completely randomized design
• Randomized complete block design
• Latin square
• Repeated-measures and crossover
• Split-plot designs
• Cluster / group-randomized designs
• Nested designs and pseudoreplication

Completely randomized design

Units are assigned to treatments purely at random, no blocking. Simplest design; appropriate when units are homogeneous and there's no identifiable nuisance factor. Analyze with one-way ANOVA / regression. If units are not homogeneous, the nuisance variation inflates error — block instead.

Randomized complete block design

Group units into blocks of similar units (day, batch, litter), and randomize all treatments within each block. Every treatment appears once per block. The between-block variation is removed from the error term, sharply increasing precision when blocks differ. Analyze with treatment + block in the model. This is the default upgrade over a completely randomized design whenever a nuisance factor exists.

Latin square

Controls two nuisance factors simultaneously with a square layout: each treatment appears exactly once in every row and every column. Classic uses: row = day, column = position/order, cell = treatment. Requires #treatments = #rows = #columns, and assumes no interactions between the blocking factors and treatment. Efficient when both nuisance dimensions matter and runs are limited. (Graeco-Latin squares extend this to three nuisance factors.)

Repeated-measures and crossover

Each subject receives more than one condition, serving as its own control. This removes between-subject variation — usually the largest noise source — so these designs can improve precision when within-subject correlation and carry-over assumptions support them; the benefit is not automatic.

• Repeated measures: the same units measured under several conditions or over time.
• Crossover: each subject receives each treatment in sequence, with washout periods between to clear carry-over. Subjects are randomized to treatment orders (e.g. an AB/BA crossover; or a Williams square for ≥3 treatments to balance order).

Watch for:

• Carry-over / residual effects — an effect of the previous treatment persisting into the next period. Adequate washout is essential; otherwise the design is biased.
• Period effects — systematic change over time (learning, fatigue, disease progression). Balanced orders let you separate period from treatment.
• Correlation within subject — the repeated observations are not independent; the analysis must model it (mixed model / repeated-measures ANOVA). Sample-size/power for these depends on the within-subject correlation — use simulation in the statistical-power skill.

Split-plot designs

Arises when some factors are hard to change (applied to large units) and others are easy to change (applied to sub-units). The hard-to-change factor is randomized to whole plots; the easy factor is randomized to subplots within each whole plot. Example: oven temperature (whole plot — you can't re-set it per sample) × coating type (subplot — applied per sample). Crucially there are two different error terms — one for whole-plot factors, one for subplot factors — and the analysis must use both. Treating a split-plot as a completely randomized factorial gives wrong (usually anticonservative) tests for the whole-plot factor. Industrial DOE and agricultural trials are full of accidental split-plots; recognize when a factor can't be reset per run.

Cluster / group-randomized designs

When the intervention is delivered to a group (a clinic's protocol, a classroom curriculum, a village water supply), you can only randomize at the group level. The cluster is the unit of randomization, and because members of a cluster are correlated, it is effectively the unit of replication too.

• Power depends on the number of clusters far more than the number of individuals, and on the intraclass correlation (ICC). Adding people to existing clusters helps much less than adding clusters.
• The design effect DEFF = 1 + (m − 1)·ICC (m = common cluster size) approximates the variance inflation for an equal-size parallel design with exchangeable correlation; unequal sizes or other designs require a suitable calculation. Even a small ICC with large clusters costs dearly. Power these by simulation (see statistical-power).
• Analyze with a method that accounts for clustering (mixed model with a cluster random effect, or GEE). Analyzing individuals as independent is pseudoreplication.

Nested designs and pseudoreplication

Pseudoreplication is treating non-independent measurements as independent replicates. It is a serious design and analysis error in experimental biology. Statistical modeling cannot create missing independent replication. If enough independent units were collected, a mistaken analysis can be corrected by respecting the nesting; no mixed model rescues one independently treated unit per arm.

The principle: the replicate is whatever the treatment is independently applied and randomized to. Measurements below that level can be biological subsamples (cells within mice) or technical repeated measurements. They characterize that unit but do not become independent treatment replicates.

Worked examples:

• One dish per treatment, 50 cells imaged. Treatment applied to the dish ⇒ n = 1 per treatment. The 50 cells describe that one dish; they are not 50 independent tests of the treatment. You need multiple independently treated dishes.
• 3 mice per group, 100 cells each. n = 3 (mice) for a treatment given to the mouse, not 300 (cells). Average within mouse, or use a mixed model with mouse as a random effect.
• One tank of fish given a diet, every fish measured. The tank is the unit (the diet was randomized to the tank) ⇒ n = number of tanks, not number of fish. Shared tank water, temperature, and social effects make fish within a tank correlated.
• Repeated measurements over time on the same subject are nested within subject; the subject is the replicate.

How to avoid it:

1. Identify the experimental unit = the smallest physical entity to which a treatment level is independently and randomly assigned.
2. Replicate at that level — more independently treated units, not more measurements per unit (though technical replicates can reduce measurement noise).
3. Analyze with the nesting respected — average to the unit level, or fit a mixed model with random effects for the nesting (cells in mice, fish in tanks, time in subjects). Inference must still use a suitable covariance model and small-sample degrees of freedom; few clusters can make routine asymptotic mixed-model or GEE tests unreliable.

Technical replicates are still worth taking — they sharpen each unit's estimate — but report and analyze them as what they are, never as independent biological replicates. For sample size of nested/clustered designs, use simulation in statistical-power.

See the NC3Rs experimental-unit guidance for animal, litter, cage, and split-plot examples. Different treatment factors can have different experimental units within one study.