Why Randomize Within Strata?
Randomization can produce groups that differ by chance on important baseline characteristics. If a trial includes several clinical sites, for example, one treatment group might happen to receive more participants from a site where usual care differs. Stratification uses information available before assignment to organize participants into groups, then randomizes separately within those groups. Its purpose is to target balance on the characteristics used to form the groups, not to guarantee that all participant characteristics will be balanced.
The key design decision is made before any treatment is assigned: define the strata from baseline information, and specify how randomization will occur within each stratum. The assignment mechanism must remain genuinely random. Stratification is not a substitute for randomization, and it does not make a nonrandom assignment valid.
A stratification factor is the baseline characteristic used to form the groups. It may be categorical already, such as study site, or it may be a quantitative characteristic divided into categories using pre-specified cut points, such as age below versus at least 65 years. When more than one factor is used, their combinations define the strata. For example, two sites and two severity categories produce four possible strata: each site crossed with each severity category.
The word “baseline” matters: the factor must be known before assignment and must not be defined using a post-treatment outcome. Defining strata using information affected by treatment would make group membership depend on events occurring after randomization and would no longer implement the intended baseline design.
What Separate Randomization Targets
Let \(h\) index strata. Write \(n_h\) for the number of participants in stratum \(h\), and \(N_{1h}\) and \(N_{0h}\) for the numbers assigned to treatment 1 and treatment 0 in that stratum. If the design uses the same target allocation fraction \(\pi\) in every stratum, it aims for \(N_{1h}/n_h\) to be near \(\pi\) in each stratum. The exact extent of balance depends on the allocation mechanism: randomization within strata by itself does not necessarily fix the treatment counts.
The balance target can be seen by considering the distribution of the stratification factor in the two treatment groups. If the treated and control groups have the same fraction from every stratum, then they have the same distribution of the stratification factor. For two strata, for instance, equal fractions from stratum 1 imply equal fractions from stratum 2 as well, since the stratum fractions in each arm sum to one.
Here \(N_1=\sum_h N_{1h}\) and \(N_0=\sum_h N_{0h}\) are the total numbers assigned to the two treatments. If this equality holds, any characteristic used to define the strata has the same category distribution in both arms. If a quantitative characteristic is divided into categories, however, equal category distributions do not imply equal means or identical values within each category.
A short expectation calculation clarifies what randomization alone can say. Suppose every participant in stratum \(h\) has probability \(\pi\) of assignment to treatment 1, under a specified randomized mechanism. Define \(Z_i=1\) if participant \(i\) receives treatment 1 and \(Z_i=0\) otherwise. Then \(N_{1h}\) is the sum of the \(Z_i\)'s in that stratum, so linearity of expectation gives the expected treatment count:
This is an expectation, not a promise about the realized count. A mechanism that only assigns each participant with probability \(\pi\) may still produce unequal treatment counts in a small stratum. To guarantee a particular count or proportion, the mechanism must impose that constraint. The definition of strata specifies where randomization is conducted; the allocation procedure specifies what assignments are allowed and how they are selected.
Worked Examples
Worked Example: Defining Strata by Site
A hypothetical two-treatment trial recruits at three hospitals. Before assignment, each participant’s hospital is recorded. Identify the strata and explain how the design targets balance by hospital.
Step 1: Define the factor. The factor is hospital, with categories North, Central, and South. Each participant belongs to exactly one hospital, so these categories are mutually exclusive and exhaustive for the participants in this example.
Step 2: Define the strata. There are three strata: North, Central, and South. For example, if 18 participants are recruited at North, then \(n_{\text{North}}=18\).
Step 3: State the target. Randomization is conducted separately within each hospital. If the trial targets a one-to-one treatment allocation within each stratum, the intended within-hospital treatment fraction is \(\pi=0.5\). The mechanism must specify whether it constrains counts or merely gives each participant an assignment probability; the factor definition alone does not establish exact counts.
Interpretation. Separating randomization by hospital targets a similar distribution of hospitals across treatments. It does not ensure balance on age, baseline severity, or other characteristics that were not used to define the strata.
Worked Example: Cross-Classifying Site and Severity
In a hypothetical trial, participants come from either Site A or Site B and are classified before assignment as having lower or higher baseline severity. Define the strata and identify what must be recorded to apply the design.
Step 1: List the factor categories. Site has two categories and severity has two categories. Taking every possible combination gives \(2\times2=4\) strata: Site A/lower severity, Site A/higher severity, Site B/lower severity, and Site B/higher severity.
Step 2: Assign each participant to one cell. A participant from Site B with higher severity belongs to the Site B/higher-severity stratum only. This uses baseline data, so the membership can be determined before treatment assignment.
Step 3: Randomize within the cells. The randomization procedure is applied separately in each of the four strata. Thus the design targets balance on site and on the defined severity category, considered together. It does not necessarily balance finer distinctions within a severity category.
What to record. The protocol and assignment records need to identify the factors, category definitions, cut points if any, each participant’s stratum, and the randomization procedure used there. Without these details, an assignment cannot be checked against the intended design.
Worked Example: Why Stratum Composition Can Matter
Consider a hypothetical trial with 40 higher-severity participants and 60 lower-severity participants. In one possible realization under overall complete randomization, 28 of the 50 participants assigned to treatment 1 are higher-severity and 12 of the 50 assigned to treatment 0 are higher-severity. Calculate the higher-severity proportions and compare them.
Treatment 1: The higher-severity proportion is \(28/50=0.56\), or 56%.
Treatment 0: The higher-severity proportion is \(12/50=0.24\), or 24%.
The difference is \(0.56-0.24=0.32\), or 32 percentage points. This is a possible chance imbalance under overall complete randomization, not evidence that complete randomization systematically favors either treatment. If the design instead constrains the treatment fractions within each severity stratum to be equal, the higher-severity proportions will be equal across arms whenever both arms receive participants from that stratum.
This comparison illustrates the target of stratification. It does not establish that a particular stratified allocation mechanism will always produce equal counts: that conclusion requires a count constraint, not merely the creation of strata.
Design Assumptions and a Practical Check
Before using stratified randomization, check the conditions that make the design coherent and implementable:
- The factors and their categories are specified before treatment assignment, using information available at baseline.
- The strata form a mutually exclusive and exhaustive partition of eligible participants.
- The assignment procedure is specified for every stratum and uses random selection rather than investigator choice.
- Stratum membership is recorded accurately, so each participant is randomized under the intended rule.
- The design’s claimed balance is no stronger than what its allocation mechanism actually enforces.
A useful audit is to compare the recorded baseline category and stratum with the assignment record, then tabulate treatment counts within every stratum. A participant assigned under the wrong stratum’s rule is a protocol deviation even if the overall treatment totals look acceptable. Conversely, unequal counts in a stratum are not automatically a deviation if the specified randomization mechanism permits them. The protocol, not an informal expectation of “balance,” determines whether the assignment followed the design.
If a continuous baseline factor is categorized, verify that the recorded value falls on the correct side of the pre-specified cut point. A boundary rule must be unambiguous; for example, “below 65” and “65 or older” assign a value of exactly 65 to one stated category. Changing cut points after looking at assignments or outcomes undermines the planned design.
How Stratification Compares With Other Randomization Approaches
| Approach | How assignment is organized | What balance it targets |
|---|---|---|
| Complete randomization | Randomize across the full set of participants without conditioning on strata. | No particular baseline factor is specifically targeted; chance imbalances may occur. |
| Stratified randomization | Define baseline strata and randomize separately within each one. | Targets balance in the factors defining the strata; exact balance depends on the within-stratum mechanism. |
| Permuted-block randomization | Use blocks to organize assignments in a sequence and control treatment counts within blocks. | Targets allocation balance over the sequence; it does not by itself define baseline strata. |
Stratification and permuted blocks address different design features and can be combined: strata organize participants by baseline characteristics, while a within-stratum procedure may control assignment counts over time. The details of such combined procedures depend on the specified randomization mechanism. Neither strategy balances every unmeasured or unused covariate.
Common Mistakes and Limitations
- Claiming exact balance from stratification alone. Forming strata does not fix the treatment counts. Check whether the actual procedure constrains those counts.
- Using post-assignment information to define strata. Stratum membership should be determined from baseline factors, before treatment assignment.
- Confusing balance targets with causal conclusions. Stratification helps organize randomized assignment; it does not, by itself, show that one treatment is more effective.
- Assuming balance on one factor means balance on all factors. Only the chosen categories are directly targeted, and random differences in other characteristics can remain.
- Ignoring the level of categorization. If age is grouped into broad categories, equal group proportions do not ensure equal mean ages.
Stratification is most useful when the chosen baseline factors are important to the design and can be measured and categorized reliably. Its limitations follow from that same choice: it targets only the defined factors, and each additional factor can create more combinations to manage. The main design principle is therefore precise rather than expansive: specify strata that have a clear baseline interpretation, then describe accurately what the chosen randomization rule does within them.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A trial uses two hospitals and two baseline severity categories. How many strata are defined, and what are they?
- In a stratum of 20 participants, each participant has probability \(0.5\) of assignment to treatment 1. What is the expected number assigned to treatment 1? Does that expectation guarantee an observed count of 10?
- Explain why equal proportions from every stratum in the two treatment groups give the groups the same stratum composition.
- A researcher defines a stratum using a symptom score measured after treatment starts. What is wrong with this design choice?
Solutions. (1) There are \(2\times2=4\) strata: each hospital crossed with each severity category. (2) The expected count is \(20(0.5)=10\). An expectation does not guarantee the realized count is 10; that depends on whether the allocation mechanism constrains the count. (3) If each treatment group has the same fraction from every stratum, its distribution over strata is identical by definition; the fractions across all strata sum to one in each group. (4) The score is measured after treatment starts, so it is not a baseline factor available for defining strata before assignment. Membership based on it does not implement baseline stratified randomization.