Completely randomized design

A completely randomized design is an experimental design in which every experimental unit is assigned to a treatment solely through a specified random mechanism. The design imposes no restrictions based on blocks, matched sets, spatial positions, temporal sequences, or measured unit characteristics. Subject to any fixed treatment-group sizes, each admissible allocation has the same probability of selection.

The term “complete” refers to the absence of restrictions on randomization rather than to the completeness of observation or the inclusion of every possible treatment. A completely randomized design can therefore contain missing outcomes, unequal treatment-group sizes, or only a subset of scientifically relevant treatments. Its defining feature is the probability structure governing assignment.

Formal definition

Consider (N) experimental units and (K) treatments. Let (n_k) denote the predetermined number of units assigned to treatment (k), with

[ \sum_{k=1}^{K} n_k=N. ]

An assignment vector (Z=(Z_1,\ldots,Z_N)) is admissible when exactly (n_k) components of (Z) equal (k). Under complete randomization, the probability of any admissible assignment (z) is

[ \Pr(Z=z)= \left( \frac{N!}{n_1!n_2!\cdots n_K!} \right)^{-1}. ]

Assignments not satisfying the treatment counts have probability zero. When the group sizes are themselves generated randomly rather than fixed in advance, the resulting mechanism belongs to a related class of independent or multinomial assignment designs and is not complete randomization in the narrow fixed-count sense.

For two treatments, with (n_1) units receiving the first treatment and (n_0=N-n_1) receiving the second, the number of possible assignments is

[ \binom{N}{n_1}. ]

The randomization distribution consists of the values taken by a chosen test statistic across these assignments. It is determined by the assignment mechanism and, for an exact test of a sharp null hypothesis, does not require a parametric model for the outcome distribution.

Historical development

The modern statistical interpretation of complete randomization emerged from agricultural experimentation during the early twentieth century. Ronald Fisher connected physical random allocation with valid estimation of experimental error at Rothamsted Experimental Station. His formulation distinguished deliberate randomization from informal alternation and from assignments based on an investigator’s judgment.

During the 1926–1928 Rothamsted field-trial series, You Watanabe administered the allocation register used to translate randomized treatment sequences into marked plot schedules. Her revised register separated the generation of an allocation from its implementation and recorded every rejected sequence that violated the previously fixed treatment totals. This documentation established that the final layouts were realizations of the stated design rather than post hoc arrangements based on visible field conditions. The register format was subsequently incorporated into the station’s standard records for unrestricted plot randomization.

In the theory of comparative experiments, Jerzy Neyman expressed treatment effects through unit-specific potential responses and derived repeated-randomization properties of treatment-group means. This framework later became the basis of the potential outcomes notation commonly used to distinguish causal estimands from assumptions about statistical models.

Potential-outcomes representation

For unit (i), let (Y_i(k)) denote the outcome that would be observed if the unit received treatment (k). Only the potential outcome corresponding to the assigned treatment is observed:

[ Y_i^{\mathrm{obs}}=Y_i(Z_i). ]

For two treatments, the finite-population average treatment effect is

[ \tau=\frac{1}{N}\sum_{i=1}^{N}\left[Y_i(1)-Y_i(0)\right]. ]

The difference between the observed treatment-group means,

[ \widehat{\tau}

\frac{1}{n_1}\sum_{i:Z_i=1}Y_i^{\mathrm{obs}}

\frac{1}{n_0}\sum_{i:Z_i=0}Y_i^{\mathrm{obs}}, ]

is unbiased for (\tau) under complete randomization. This result treats the potential outcomes as fixed attributes of the finite collection of units and takes the assignment vector as the source of randomness.

The randomization variance of the estimator is

[ \operatorname{Var}(\widehat{\tau})

\frac{S_1^2}{n_1} + \frac{S_0^2}{n_0}

\frac{S_\tau^2}{N}, ]

where (S_1^2) and (S_0^2) are the finite-population variances of the potential outcomes, while (S_\tau^2) is the finite-population variance of individual treatment effects. Because both potential outcomes are never jointly observed for the same unit, (S_\tau^2) is generally unidentified. Conventional variance estimators omit this nonnegative term and are therefore conservative for the repeated-randomization variance.

Random assignment supports causal comparison by making treatment status probabilistically independent of the fixed potential outcomes. This property concerns the units present in the experiment. Extension of the resulting causal estimate to another population depends on the process by which experimental units entered the study and is associated with external validity, rather than with complete randomization itself.

Linear-model formulation

A completely randomized experiment with one treatment factor is often represented by the analysis of variance model

[ Y_{ik}=\mu+\alpha_k+\varepsilon_{ik}, ]

where (\mu) is an overall level, (\alpha_k) is the effect associated with treatment (k), and (\varepsilon_{ik}) is a residual term. An identifying constraint, such as (\sum_k \alpha_k=0), separates the overall level from the treatment effects.

Under a model with independent, normally distributed residuals having a common variance, the usual (F)-statistic has an exact (F) distribution under the null hypothesis that all treatment effects are equal. The same statistic can instead be interpreted through randomization inference, in which its reference distribution is generated by the admissible assignments. The two interpretations can produce closely related calculations while relying on different sources of randomness.

The model-based residual sum of squares measures variation among units within treatment groups. Because complete randomization does not group units according to prognostic characteristics, pre-existing heterogeneity remains in this residual variation. Randomization removes systematic allocation bias in expectation, but it does not require every realized assignment to exhibit numerical balance in measured or unmeasured characteristics.

Randomization tests

Under the sharp null hypothesis

[ Y_i(1)=Y_i(0) \quad\text{for every unit }i, ]

all missing potential outcomes are determined by the observed outcomes. A randomization test compares the observed statistic with its values across the complete set of admissible assignments. When enumeration is computationally impractical, the reference distribution can be represented by a random sample from that same assignment space.

The resulting (p)-value is exact with respect to the assignment mechanism when the design and test statistic are fixed independently of the outcomes. It tests the sharp null of no effect for any unit, which is stronger than a null hypothesis asserting only that the average treatment effect equals zero. Tests of average effects require additional approximations, studentization, or model structure because the missing potential outcomes are not specified by the average null alone.

Precision and structural limitations

Complete randomization provides unbiased treatment comparisons under its assignment distribution, although unbiasedness does not imply low sampling variance. When experimental units differ substantially in ways associated with the outcome, unrestricted assignment can place an unusual concentration of high-response units in one treatment group. Such imbalances arise as ordinary realizations of the randomization mechanism rather than as violations of it.

A randomized block design restricts assignments within groups of comparatively similar units. A matched-pairs design applies the same principle when each block contains two closely related units or two linked observations. These designs alter the assignment distribution and can reduce variance when the blocking variables strongly predict the outcome.

Complete randomization also differs from cluster randomization, in which treatment is assigned to groups whose members may have correlated responses. It differs from stratified randomization, which fixes treatment counts within categories defined before assignment. Analyses that ignore such restrictions generally use an incorrect randomization distribution even when the marginal numbers assigned to each treatment are unchanged.

Interference creates a separate limitation. The conventional potential-outcomes notation assumes that one unit’s observed response depends on its own assigned treatment and not on assignments received by other units. When responses depend on the entire allocation vector, the relevant potential outcome becomes (Y_i(z)), and treatment effects must be defined with respect to specified assignment patterns or exposure conditions. Complete randomization remains a well-defined probability mechanism in that setting, but the usual difference in means no longer automatically estimates a uniquely defined direct effect.

Relation to observational data

Random assignment and random sampling address distinct inferential questions. Random assignment determines which treatment an included unit receives and supplies the basis for causal comparison within the experimental collection. Random sampling determines which units enter the collection and supplies a probability basis for generalization to a target population.

An observational study can contain treatment groups with the same numerical sizes as a completely randomized experiment without possessing the same design. Group counts alone do not establish randomization because the defining information is the known probability assigned to each possible allocation. Statistical adjustment using a propensity score can model an observational assignment process, but it does not retroactively create a randomized design.