Why Stratify by Calendar Period?
In a platform trial, the mix of patients, clinical practice, and background risk can change as calendar time passes. A treatment group enrolled mostly in one period and a control group enrolled mostly in another may therefore differ for reasons beyond treatment. Comparing treatment and control participants within prespecified periods helps keep the primary comparison anchored to patients enrolled at similar times. The controls used for each comparison should be concurrent: they are enrolled in the same period as the treatment participants being compared.
Stratification also makes the target of an overall summary explicit. Period-specific treatment effects need not be identical. A weighted average answers a defined question about a particular mix of periods; it does not assert that every period has the same effect.
The risk difference is expressed in percentage points when risks are expressed as percentages. A negative value means fewer outcome events under treatment than under control. The weights \(q_j\) describe the period distribution in the target population. For example, \(q_j\) might be the prespecified proportion of the target population expected to come from period \(j\). The observed proportions enrolled in the trial are not automatically the right weights; they are appropriate only if the intended target is the trial’s own period mix.
This definition targets a mixture across periods. It does not, by itself, make the trial participants in each period representative of a wider population within that period. If that broader transport is intended, additional assumptions and methods are needed.
Estimation Within Each Period
Suppose \(n_{1j}\) treatment participants and \(n_{0j}\) concurrent control participants have observed binary outcomes in period \(j\). Let \(x_{1j}\) and \(x_{0j}\) be their respective event counts. The estimated risks are \(\hat p_{1j}=x_{1j}/n_{1j}\) and \(\hat p_{0j}=x_{0j}/n_{0j}\). Estimate the period-specific risk difference by subtracting the concurrent control risk from the treatment risk, then combine those estimates using the target weights:
The design and analysis should specify periods before examining outcome results. Within each period, the treatment and control groups should be assigned through randomization, or the analysis must justify another basis for exchangeability. For a randomized comparison, the causal interpretation relies on correct implementation of randomization, a well-defined treatment and outcome, no interference between participants, and adequate outcome ascertainment. Both arms must have participants at risk of contributing data in every period with \(q_j>0\); without a concurrent control in a target period, its treatment effect cannot be estimated by this method.
For the simple variance formula below, also assume independent participants, complete binary outcomes, and independent samples across treatment and control groups and periods. If there is clustering, substantial missingness, or a more complex outcome, the variance calculation must reflect that feature rather than treating all observations as independent Bernoulli outcomes.
Worked Example: Combining Two Period-Specific Risk Differences
Consider a hypothetical randomized comparison with an adverse event as the binary outcome. In period 1, 18 of 120 treatment participants and 24 of 120 concurrent controls have an event. In period 2, the corresponding counts are 30 of 100 and 40 of 100. Suppose the target population has 60% of its members in period 1 and 40% in period 2.
The estimates within the two periods are:
Thus, the estimated treatment risk is 5 percentage points lower than the control risk in period 1 and 10 percentage points lower in period 2. The target-weighted estimate is:
For this target mix of periods, the estimated adverse-event risk is 7 percentage points lower under treatment. This is an average for the specified period distribution, not a claim that the effect equals 7 percentage points in either individual period.
Uncertainty and a Short Derivation
For independent binomial samples within period \(j\), the estimated variance of a sample risk is \(\hat p(1-\hat p)/n\). The two arm-specific estimates are independent, so the estimated variance of their difference is the sum of those two estimated variances. If the period-specific estimates are also independent, the variance of their weighted sum is the sum of their variances multiplied by the squared weights:
The squared weights arise because multiplying a random estimate by \(q_j\) multiplies its variance by \(q_j^2\). Independence across periods makes the covariance terms zero. An approximate 95% confidence interval is the estimate plus or minus 1.96 times its estimated standard error, where \(SE\) is the square root of the estimated variance. This normal approximation is most dependable when event and non-event counts are not too small.
Worked Example: Standard Error and Confidence Interval
Use the hypothetical data from the preceding example, with weights \(q_1=0.60\) and \(q_2=0.40\). In period 1, the estimated variance is:
In period 2, it is:
The variance and standard error of the weighted estimate are therefore:
The approximate 95% confidence interval is:
The interval ranges from a 14.8-percentage-point reduction to a 0.8-percentage-point increase in adverse-event risk for the defined target mix. It includes zero, so these data do not establish a nonzero average risk difference at the 5% two-sided level. The interval describes uncertainty in the estimated target-weighted effect under the stated assumptions; it is not a probability statement about where the fixed effect lies.
Why the Period Weights Matter
A simple pooled comparison calculates one event risk in each arm across all periods and subtracts them. That comparison implicitly uses each arm’s own observed period composition. If those compositions differ, the resulting contrast mixes treatment differences with changing background risk. Period stratification avoids that particular mismatch by comparing arms within periods first. The chosen \(q_j\) then states how those comparisons are combined for the target.
The same logic explains why precision weights are not automatically target-population weights. Giving greater weight to a period because its estimate is more precise may improve precision for a particular statistical summary, but it changes the period mix unless those weights happen to match the target. To estimate the stated target effect, use its prespecified \(q_j\), not weights selected solely from sample sizes or standard errors.
Worked Example: A Crude Comparison Can Reverse the Conclusion
In a hypothetical trial, both periods have a 10-percentage-point lower event risk under treatment than under concurrent control. Period 1 has 100 events among 1,000 treatment participants and 180 among 900 controls. Period 2 has 540 events among 900 treatment participants and 70 among 100 controls.
The period-specific estimates are:
For an equally weighted target mix, the estimate is \(0.5(-0.10)+0.5(-0.10)=-0.10\), a 10-percentage-point reduction. But pooling participants separately by arm gives:
The crude pooled comparison suggests an 8.68-percentage-point increase, despite the lower treatment risk in each period. The arms have very different period compositions, and the background risks differ greatly between periods. The within-period comparisons and the target-weighted estimate preserve the concurrent comparisons and make the intended period mix explicit.
Diagnostics, Alternatives, and Limitations
Before interpreting a period-stratified estimate, check that the periods were prespecified, that both arms have data in each target period, and that the controls are genuinely concurrent. Tabulate participant numbers and event counts by arm and period. Very sparse cells make risk differences and normal-approximation intervals unstable; a nominally present control group with only a few participants may provide little useful information. Review whether outcome definitions, follow-up, eligibility, and treatment delivery remained comparable enough for the period-specific contrasts to answer the intended question.
Period stratification is not a repair for every source of bias. Randomization supports exchangeability between treatment and control within a period, but it does not ensure that trial participants represent a different target population within that period. Nor does period stratification correct a systematic change in outcome measurement or differential missingness. If such problems are plausible, the assumptions and analysis must address them directly.
| Approach | What it summarizes | Main caution |
|---|---|---|
| Period-stratified estimates with target weights | Within-period concurrent comparisons combined for a specified period mix | Requires adequate data in each target period and defensible weights |
| Crude pooling by arm | Difference between the observed arm-specific overall risks | Can confound treatment with unequal period composition |
| Regression adjustment for calendar time | A model-based adjusted comparison using calendar time | Depends on model specification; the resulting estimand may not equal a prespecified weighted period average |
A final limitation is interpretive: one weighted average can conceal different period-specific effects. Always report the stratum-specific estimates and their data support alongside the weighted summary. The summary answers the target-mixture question; it should not be used to imply that the effect is constant across periods.
Common Mistake
Wrong reasoning: “The treatment and control data can be pooled across the whole platform because both arms have large total sample sizes.” Large totals do not remove bias from unequal period composition, and they do not create a concurrent control for a period in which none exists.
Correction: Compare treatment with concurrent control within each prespecified period, then combine those estimates using weights that define the target population. If a target period has no concurrent comparison, do not assign it a period-specific effect from this analysis without additional assumptions and methods.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Two periods have estimated risk differences of \(-0.02\) and \(0.05\). If the target weights are \(0.30\) and \(0.70\), what is the target-weighted estimate?
- Why is it important that the control group used for a period-specific comparison be concurrent?
- A target period has treatment participants but no control participants. Can its period-specific risk difference be estimated by this method?
- Why should inverse-variance weights not automatically replace target-period weights?
Solutions. (1) \(0.30(-0.02)+0.70(0.05)=-0.006+0.035=0.029\), a 2.9-percentage-point increase in risk for that target mix. (2) A concurrent comparison aligns treatment and control participants in calendar time, helping prevent time-related changes in background risk or care from being mistaken for a treatment effect. (3) No. Without a control observation in that period, the within-period contrast is unavailable; estimating it would require additional assumptions or a different method. (4) Inverse-variance weights prioritize precision, whereas target weights define the population mix. Substituting one for the other generally changes the estimand.