Why Factor Choice Matters
Stratification creates separate groups, or strata, defined by baseline characteristics; randomization is then carried out within those groups. This can help prevent important characteristics from becoming unevenly distributed between treatment arms by chance. But every additional factor can divide the participants into more strata. If too many combinations are unlikely to recruit many participants, some strata will contain very few participants or none at all.
The design question is therefore not simply whether a characteristic is clinically interesting. It is whether stratifying on it is likely to improve balance enough to justify the additional strata it creates. Clinical relevance, the number of categories, anticipated enrollment, and the joint distribution of the factors all matter.
This tutorial concerns choosing factors before assignment. The introductory material on stratified randomization establishes how randomization can be conducted within strata; the focus here is whether the proposed strata are sensible and recruitable.
Begin With Clinical Relevance and Feasibility
A candidate factor should have a clear design rationale. For example, a baseline disease-severity category may be important if severity is strongly related to prognosis or if clinicians regard severity as essential to balance between treatments. A site factor may be relevant when participating centers differ meaningfully in patient mix or care. By contrast, a characteristic that is only weakly relevant, difficult to classify consistently, or available only after randomization may not justify dividing the sample.
Useful screening questions include:
- Is the characteristic measured before randomization and defined consistently?
- Is there a clinical or design reason to expect balance on it to matter?
- How many categories does it have, and are those categories likely to be used often enough?
- Will it be known for essentially every participant at the time of assignment?
- What combinations will be created when this factor is crossed with the factors already selected?
The number of strata grows multiplicatively. If factor \(i\) has \(K_i\) categories, the maximum number of possible strata is the product of the category counts. This is a count of possible combinations, not a claim that every combination will occur or be equally common.
Here \(m\) is the number of selected factors, \(K_i\) is the number of categories for factor \(i\), and \(J_{\max}\) is the maximum number of combinations. The actual number of populated strata can be lower if some combinations are impossible, but the design still needs to consider whether the possible combinations are clinically and operationally sensible.
Estimate Cell Sizes From the Joint Distribution
For planning, let \(N\) be the anticipated total enrollment and let \(p_j\) be the anticipated proportion of participants in stratum \(j\). The expected number of participants in that stratum is \(Np_j\). This calculation is only as good as the assumed joint distribution: knowing the marginal proportions of separate factors does not, by itself, determine the proportions in their crossed cells.
If a stratum is defined by site, severity, and a biomarker category, its \(p_j\) is the proportion expected to be in that particular site-severity-biomarker combination. Multiplying the three marginal proportions gives that joint proportion only under an additional assumption, such as mutual independence of the three characteristics. Without such an assumption, use anticipated joint proportions from a relevant source or state that the cell-size estimate is uncertain.
Expected counts are planning summaries, not promises about realized enrollment. Under a simple model in which each of \(N\) participants independently has probability \(p_j\) of belonging to stratum \(j\), the probability that the stratum is empty is \((1-p_j)^N\). When \(p_j\) is small, a nonzero expected count can coexist with a substantial chance of observing no participants in that stratum.
Worked Example: Crossing Three Factors
Suppose a hypothetical trial expects \(N=120\) participants. The proposed factors are site (2 categories), disease class (2 categories), and biomarker category (3 categories). Assume, explicitly for this planning calculation, that the three factors are mutually independent; each site and each disease class has probability \(0.50\), and the biomarker categories have probabilities \(0.75\), \(0.20\), and \(0.05\).
The maximum number of strata is \(2\times2\times3=12\). Under the stated mutual-independence assumption, the joint proportion for a particular site, disease class, and biomarker category is the product of their probabilities. For a cell in the rare biomarker category, this is \(0.50\times0.50\times0.05=0.0125\). Its expected enrollment is \(120\times0.0125=1.50\) participants.
For a cell in the most common biomarker category, the expected enrollment is \(120\times(0.50\times0.50\times0.75)=22.5\). Thus, the same 12-cell design has expected counts ranging from 1.50 to 22.5 under these assumptions. The four rare-category cells have 1.50 expected participants apiece, for 6 expected participants total across the rare biomarker category. These are expectations, not guaranteed counts.
The site-by-disease-class combinations each have expected count \(120\times0.50\times0.50=30.0\), after summing over biomarker categories. That larger count does not remove the sparse-cell concern if all three factors are crossed: each rare biomarker combination remains small. The design team should therefore ask whether biomarker category adds enough value as a stratification factor to justify those rare cells.
A Short Justification for the Expected-Count Calculation
For a fixed stratum \(j\), define \(I_r=1\) if participant \(r\) belongs to that stratum and \(I_r=0\) otherwise. The stratum count is \(N_j=\sum_{r=1}^{N} I_r\). If each participant has probability \(p_j\) of belonging to stratum \(j\), then \(E(I_r)=p_j\). By additivity of expectation,
This expectation calculation does not require independence among the indicators. Independence is needed for the simple empty-cell probability \((1-p_j)^N\), because that probability treats participants’ stratum memberships as independent with the same probability \(p_j\). In practice, enrollment patterns may depart from that model, so the empty-cell calculation is a diagnostic approximation rather than a guarantee.
Worked Example: Marginal Proportions Are Not Enough
A second hypothetical trial expects 200 participants. A candidate factor has two categories, A and B, each with marginal probability \(0.50\); another factor has categories X and Y, also each with marginal probability \(0.50\). It would be incorrect to conclude from these margins alone that all four crossed cells have probability \(0.25\).
For example, one possible joint distribution is: \(P(A,X)=0.50\), \(P(B,Y)=0.50\), and \(P(A,Y)=P(B,X)=0\). Both factors still have the stated \(0.50/0.50\) marginal proportions, but two of the four possible cells are structurally empty.
A different possible joint distribution is \(0.25\) in each of the four cells. Under that joint distribution, each cell has expected count \(200\times0.25=50\). The two sets of expected counts are entirely different despite identical marginal proportions. Therefore a cell-size calculation based on multiplying margins must state and justify independence (or another joint-distribution assumption); otherwise, use anticipated joint proportions.
If the design team has no reliable joint data, it should report that limitation rather than presenting the product of marginal proportions as known. Clinical consultation and screening data may help identify plausible combinations, but uncertainty should remain visible in the planning decision.
Compare Candidate Sets, Not Just Individual Factors
A factor can look feasible on its own but become problematic when crossed with other factors. Consider a hypothetical trial expecting 240 participants at four equally enrolling sites. Severity is expected to be high for 25% and low for 75% of participants. Assume site, severity, and a three-category marker are mutually independent for this example; marker proportions are 80%, 17%, and 3%.
Worked Example: What One Rare Category Does to the Cells
With site and severity alone, there are \(4\times2=8\) strata. A high-severity stratum at any one site has expected count \(240\times0.25\times0.25=15.0\). A low-severity stratum at any one site has expected count \(240\times0.25\times0.75=45.0\). Under the assumptions, these counts do not suggest the same degree of sparsity.
Adding the marker creates \(4\times2\times3=24\) possible strata. A cell at one site, in high severity, and in the 3% marker category has expected count \(240\times0.25\times0.25\times0.03=0.450\). There are four such high-severity cells, one per site, so their combined expected count is 1.80; the value 0.450 is the expected count in each individual cell.
The calculation indicates that the rare marker category would be especially sparse when crossed with site and severity. It does not prove that the marker should never be used. Rather, the team should compare its clinical importance with the likely consequences of fragmenting the design into these small cells. If the marker is not essential for randomization balance, a simpler factor set may be more defensible.
The comparison should include the whole proposed set of factors. A useful table lists each candidate set, the number of possible strata, the assumptions or data used for joint proportions, and the smallest anticipated cell counts. It can also flag combinations believed to be rare or impossible. This makes clear whether a proposed factor improves the design or mostly multiplies the number of cells.
Diagnostics, Alternatives, and Limits
Before adopting a factor set, inspect recruitment or screening data by the proposed joint categories, not only by each factor separately. Look for cells with small counts, cells never observed, categories that are inconsistently recorded, and sites or subgroups whose enrollment is substantially below plan. Recalculate anticipated counts under plausible departures from the initial assumptions. If recruitment changes after the design is set, revisit the forecast rather than treating the original cell sizes as fixed facts.
The main alternative is to use fewer stratification factors. Complete randomization without stratification avoids dividing the sample into many cells, but it also does not enforce balance within the selected categories. Stratification on a smaller, clinically important set preserves within-stratum randomization for those factors while avoiding some of the fragmentation. Neither choice eliminates chance variation or guarantees that every realized subgroup is large.
There is no universal expected-count cutoff that makes a factor automatically acceptable or unacceptable. The consequences depend on the trial’s enrollment, allocation design, recruitment pattern, and the purpose of the factor. In particular, expected counts are not realized counts, and a large total sample does not ensure adequate enrollment in every cross-classified cell. This tutorial does not replace a full design review; it provides a disciplined way to detect when a proposed set deserves further scrutiny.
Common mistake: “Each factor has common categories, so their crossed strata will also be common.” This is not necessarily true. Marginal proportions do not specify joint proportions, and even under independence a rare category can produce tiny cells when crossed with other factors. The correction is to count the possible combinations, use joint enrollment assumptions explicitly, and report cell-specific expected counts rather than relying on factor margins alone.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A trial expects 160 participants. Two factors have 2 and 3 categories. How many possible strata are there?
- Two binary factors each have marginal proportions 0.50 and 0.50. Can you infer that each crossed cell has proportion 0.25? Explain.
- Under a stated independence assumption, a particular stratum has joint probability 0.02 and anticipated enrollment is 300. What is its expected count?
- Why can an expected count of 1.5 not be interpreted as a guarantee that at least one participant will enter the stratum?
- Name two reasons, besides category count, to question a candidate stratification factor.
Solutions. (1) \(2\times3=6\) possible strata. (2) No. The marginal proportions do not determine the joint distribution; the value 0.25 per cell follows only with an appropriate additional assumption such as independence. (3) \(300\times0.02=6.00\) expected participants. (4) An expected count is an average under the planning model, not a minimum or a realized count; a stratum with positive probability can still be empty. (5) For example, the factor may have little clinical rationale, may be measured inconsistently, may not be known at randomization, or may create rare combinations when crossed with other selected factors.