Randomized Assignment

An assignment mechanism is the chance process that decides which units are treated, and it is a design decision recorded before outcomes are seen. Because it is generated independently of the potential outcomes, it makes treated and control groups comparable in expectation, which is what licenses reading a difference in means as a causal effect. Bernoulli and complete randomization differ in what they hold fixed, and the difference changes the inference that follows.

Definition

An assignment mechanism gives the probability of each treatment vector $W = (W_{1}, \dots, W_{N})$ . Under Bernoulli assignment each unit is treated independently with probability $p$ , so $P (W = w) = \prod_{i} p^{w_{i}} (1 - p)^{1 - w_{i}}$ and the treated count is random. Under complete randomization exactly $N_{1}$ of the $N$ units are treated, every one of the $(\binom{N}{N_{1}})$ such vectors is equally likely, and all other vectors have probability zero.

Formal statement

$\hat{τ} = {\bar{Y}}_{1} - {\bar{Y}}_{0}$ where ${\bar{Y}}_{1} = \frac{1}{N_{1}} \sum_{i : W_{i} = 1} Y_{i}^{obs}$ and ${\bar{Y}}_{0} = \frac{1}{N_{0}} \sum_{i : W_{i} = 0} Y_{i}^{obs}$ ; under complete randomization $E [\hat{τ}] = τ_{S}$ .

Assumptions and scope

The mechanism must be known and must not depend on the potential outcomes. A rule that consults the outcome, such as treating whoever seems likeliest to benefit, destroys the comparability randomization provides, however carefully the arithmetic is then performed.
Observed covariate imbalance does not imply the assignment was not random. Under complete randomization some allocations are unbalanced, and discarding them changes the mechanism and therefore the reference distribution for any later inference.
Bernoulli assignment permits the all-treated and all-control vectors unless the design excludes them. With small $N$ this is a practical risk, not a technicality, and it is one reason complete randomization is usually preferred.
Unbiasedness of the difference in means is a statement about averaging over repeated assignments, not about the single realised experiment. A particular estimate can sit far from the effect it estimates without anything having gone wrong.

Worked material

Example

Three procedures, one of which is not randomization

A clinic flips a fair coin for each of 40 patients. Bernoulli assignment with $p = 0.5$ . The treated count is random: it could be 17, or 23, and with probability $2 \times {0.5}^{40}$ it is 0 or 40.

A clinic writes 40 slips, 20 marked "treat", shuffles, and deals one per patient. Complete randomization with $N_{1} = 20$ . Exactly twenty are treated in every possible realisation; there are $(\binom{40}{20}) \approx 1.4 \times 10^{11}$ equally likely allocations.

A clinic offers the treatment and records who accepts. Not an assignment mechanism at all. The probability of each vector is unknown and depends on the patients, who plausibly decide on grounds related to how much they expect to benefit, exactly the dependence on potential outcomes that randomization exists to exclude.

The first two support a causal reading of $\hat{τ}$ . The third supports a description of who accepted.

Non-example

Five procedures described as randomized that are not

Alternation. Patients are assigned treatment, control, treatment, control by order of arrival. The rule is deterministic: knowing a patient's position determines their assignment, and whoever schedules arrivals controls the allocation. It is also predictable, so a clinician who prefers a patient to receive treatment can delay them by one slot.

Assignment by date. Everyone admitted on an even-numbered day is treated. Admission date is not a chance device. It correlates with staffing, referral patterns, and the severity mix of who presents when.

Haphazard is not random. "We just split them however" has no stated probabilities. Without a mechanism there is nothing to average over, so no standard error or randomization test has a defined reference distribution, even if the split happens to look balanced.

Assignment after seeing outcomes. A pilot treats the first ten patients, observes poor responses, and moves later patients into control. The rule now depends on the outcomes, which is precisely the dependence randomization excludes. No amount of subsequent adjustment restores it.

Volunteering. Participants choose their arm. The probability of each vector is unknown and is plausibly related to expected benefit.

Contrast

Bernoulli, complete, and not random at all

Bernoulli against complete randomization.

	Bernoulli	Complete randomization
Treated count	Random	Fixed at $N_{1}$
Unit assignments	Independent	Dependent, since the total is fixed
Allowed vectors	All $2^{N}$	Only those with exactly $N_{1}$ treated
Probability of a vector	$p^{n_{1}} (1 - p)^{N - n_{1}}$	${(\binom{N}{N_{1}})}^{- 1}$
Degenerate allocations	Possible	Excluded by construction

Both are valid mechanisms and both are independent of the potential outcomes. They differ in what they hold fixed, and that difference propagates: the randomization distribution of $\hat{τ}$ is not the same under the two, so a test or standard error computed for one does not apply to the other.

The distinction that matters more. Either of these against a rule that consults the units. Bernoulli and complete randomization are two ways of being random; alternation, self-selection and clinician choice are not near-misses but a different kind of thing, and no analysis converts one into the other.

Common errors

Common misconception

If the treated and control groups differ noticeably on a baseline covariate, the randomization failed, so the groups should be re-randomized or the imbalance corrected before proceeding.

Randomized Assignment

Assumptions and scope

Worked material

Three procedures, one of which is not randomization

Five procedures described as randomized that are not

Bernoulli, complete, and not random at all

Common errors

Related units

Requires

Connected

Learn this topic

Used in

Sources

Settings

Focus timer

Today

Last seven days

Where the time went

Recent sessions

Settings

Notifications

Your history