Stratified Complete Randomization With Unequal Stratum Sizes

Why Stratum Sizes Matter

When randomization is conducted separately within strata, each stratum receives a specified number of treatment assignments. If stratum sizes differ, the treatment count and the chance of treatment for an individual can differ from one stratum to another. The total number assigned to treatment may be fixed even though the overall treatment proportion is not exactly the same in every stratum.

The key is to distinguish the stratum's size from its treatment count. The randomization rule chooses a fixed-size subset of each stratum for treatment; it does not choose one fixed-size subset from the trial as a whole. We will calculate the resulting allocation probabilities and show how unequal stratum sizes affect them.

The design is conditional on the stratum rosters and the counts \(m_h\). It assumes that each participant belongs to exactly one stratum, the specified count is feasible in every stratum, and every subset of the required size within a stratum has the same chance of selection. Independence across strata means that a choice in one stratum does not alter the randomization probabilities in another. These are assumptions about the assignment mechanism; no outcome model is needed to calculate its probabilities.

Counting Allocations and Their Probabilities

There are \(\binom{n_h}{m_h}\) possible treatment subsets in stratum \(h\). Since the choices are independent across strata, the total number of complete allocations consistent with the stratum counts is the product of these numbers. Each such allocation has the same probability.

An allocation that violates even one required stratum count has probability zero. The total number assigned to treatment, \(M\), is fixed, but the stratum-specific counts are also fixed, so treatment counts cannot shift from one stratum to another under this design.

Individual and Joint Assignment Probabilities

Let \(A_{hi}=1\) indicate that participant \(i\) in stratum \(h\) is assigned to treatment, and let \(A_{hi}=0\) indicate control. Within a stratum, uniform selection of exactly \(m_h\) participants gives each participant the same marginal treatment probability, \(m_h/n_h\). To see this by counting, when \(1\leq m_h\leq n_h\), the number of allowed treatment subsets containing a specified participant is \(\binom{n_h-1}{m_h-1}\). Dividing by the total number of subsets gives the probability. The boundary case \(m_h=0\) must be handled separately: treatment is impossible, so the probability is \(0=m_h/n_h\).

Thus, the probability is determined by the treatment fraction in that participant's own stratum. If two strata have unequal treatment fractions, participants in those strata have unequal treatment probabilities. The same expression also covers \(m_h=n_h\), where every participant is assigned to treatment and the probability is 1.

For two distinct participants in the same stratum, both can be treated only if \(m_h\geq2\). When \(2\leq m_h\leq n_h\), counting subsets that contain both participants gives the joint probability below. If \(m_h=0\) or \(m_h=1\), the event is impossible and has probability zero; the expression on the right is also zero. This separate treatment of the boundary cases avoids using a binomial coefficient with a negative lower argument.

For \(n_h\geq2\) and \(m_h\in\{0,1\}\), the joint probability is \(0\), which equals \(m_h(m_h-1)/[n_h(n_h-1)]\). The formula on the right can therefore be used for any \(m_h\) when \(n_h\geq2\), with the counting derivation applying only when \(m_h\geq2\). There are no two distinct participants in a stratum of size one, so a same-stratum pair probability is not defined there.

For participants in different strata, independence of the stratum selections gives a product of their individual probabilities. This differs from the same-stratum case, where the fixed treatment count makes assignments dependent.

Justification and Dependence Within a Stratum

The individual probability follows from a simple counting identity. For \(1\leq m_h\leq n_h\), the number of subsets containing a specified participant is the number of ways to choose the other \(m_h-1\) treated participants from the remaining \(n_h-1\). The ratio of this count to the total number of subsets simplifies as follows:

The boundary \(m_h=0\) is not covered by that counting ratio, because it would involve \(\binom{n_h-1}{-1}\). Instead, no participant can be treated when the fixed treatment count is zero, so the probability is zero, as stated above. The same care is needed for the pair calculation: its counting ratio is valid for \(2\leq m_h\leq n_h\), while the cases \(m_h=0\) and \(m_h=1\) have probability zero directly.

Within a stratum, assignments are generally not independent. If one participant is treated, fewer treatment assignments remain for the others. For example, in the six-person stratum above, the probability that a second specified participant is treated given that the first is treated is \(1/5\), less than the unconditional probability \(1/3\). Across strata, by contrast, the design specifies independent selections, so the corresponding probabilities multiply.

Comparing the Design With Complete Randomization Overall

Under complete randomization over the entire trial, a fixed total of \(M\) participants would be selected uniformly from all \(\binom{N}{M}\) possible subsets. Stratified complete randomization restricts the allowed subsets further by fixing each stratum's count at \(m_h\). The number of allowed subsets is \(\prod_h\binom{n_h}{m_h}\), which is no larger than \(\binom{N}{M}\). The restricted design fixes the treatment count within every stratum; complete randomization overall fixes only the total.

In the two-stratum example, stratified randomization fixes counts at 2 of 4 and 1 of 3, with 18 allowed allocations. Complete randomization over all 7 participants with 3 assigned to treatment would allow \(\binom{7}{3}=35\) allocations. Under that alternative, the treatment count in either stratum could vary, and each participant's treatment probability would be \(3/7\), rather than the stratum-specific values \(1/2\) and \(1/3\).

The choice between these designs depends on the intended allocation rule. Stratification can ensure specified counts in important groups, while complete randomization overall allows those group counts to vary by chance. Neither comparison changes the arithmetic: the probabilities must be calculated from the actual set of allocations the design permits.

Diagnostics, Limitations, and Common Mistakes

Before interpreting assignment probabilities, check that the design is internally consistent. Each stratum count must satisfy \(0\leq m_h\leq n_h\); the stratum sizes must sum to the stated total \(N\); and the treatment counts must sum to the stated total \(M\). If a proposed allocation has a different count in any stratum, it is not an allowed allocation under this design. If a schedule is presented, count its assignments separately within each stratum rather than checking only the overall total.

A common mistake is to use \(M/N\) as the treatment probability for every participant. That is generally wrong: the probability for someone in stratum \(h\) is \(m_h/n_h\). The overall fraction \(M/N\) is a weighted average of those stratum-specific fractions. For example, the first worked example has \(M/N=7/16\), while the stratum-specific probabilities are \(1/2\), \(2/5\), and \(1/3\).

Another mistake is to multiply probabilities for two participants in the same stratum as if their assignments were independent. The fixed-count constraint creates dependence; use the pair formula or condition on the first assignment. Multiplication is justified for assignments in different strata because independence across strata is part of the design definition.

This design fixes stratum counts, but it does not guarantee that all participant characteristics are balanced within a stratum, nor does it make treatment and control groups identical on every feature. Its probabilities also rely on the stated strata and treatment counts: changing either changes the randomization distribution. In practice, unequal stratum sizes can make the stratum-specific treatment fractions differ when counts must be whole numbers, so the planned counts should be checked against the intended allocation proportions.