Why Plan the Strata Before Planning Power?
A stratified Gray-test design compares the cumulative incidence of a target event between groups while accounting for prespecified strata. Stratification is useful when a factor such as study site or baseline risk group is expected to be related to the event process, or when randomization should preserve treatment balance within levels of that factor. It does not, by itself, guarantee greater power. The value of stratification depends on how the strata are defined, how participants and events are distributed across them, and how the treatment-group differences are expected to behave within them.
Planning therefore requires more than naming a stratification factor. Before choosing a sample size, specify the target event, the follow-up period, the groups being compared, the strata, and the plausible cumulative-incidence patterns within each stratum. In particular, ask whether the anticipated differences point in a common direction. A single overall contrast can be weak when sizeable within-stratum differences oppose one another.
What Must Be Specified in the Design?
For a two-group comparison, let \(F_{1g,h}(t)\) denote the CIF by time \(t\) for target event type 1, in group \(g\) and stratum \(h\), where \(g=0,1\). The stratified Gray-test null is that the target-event CIFs are equal between groups within each stratum over the tested time range. A useful design alternative states, at minimum, the expected direction and approximate size of the differences \(F_{11,h}(t)-F_{10,h}(t)\) over clinically relevant times.
These assumptions should be grounded in the scientific question and available planning information, not selected after looking at the trial outcomes. For each stratum, specify expected group sizes, target-event incidence, competing-event incidence, follow-up, and censoring. Those features affect the score’s expected contribution and its variance. The CIF is not a cause-specific hazard: a difference in CIF can reflect differences in the target-event process, competing-event process, or both.
The usual large-sample planning approximation combines stratum-specific score contributions. Write \(U_h\) for the signed Gray score contribution in stratum \(h\), oriented so that positive values favor group 1 having greater target-event incidence, and \(V_h\) for its variance under the null. Under the null, score contributions from independent strata have mean zero; if participants are independent across strata, their variances add. For \(H\) strata, the test statistic is approximately:
For power planning, replace \(U_h\) with its anticipated mean under the alternative, denoted \(\mu_h\). The resulting approximate noncentrality parameter is:
This is a planning approximation, not a recipe for deriving \(\mu_h\) from one CIF value alone. The score depends on the event-time pattern, competing events, censoring, and group sizes over follow-up. In practice, estimates of \(\mu_h\) and \(V_h\) should come from a justified approximation or suitable data-based planning method.
Worked Examples for Specifying Strata and Effects
Worked Example: Choosing a Risk-Stratification Factor
Suppose a hypothetical trial compares a new treatment with usual care for time to a target complication, with death before the complication treated as a competing event. Investigators believe baseline disease severity predicts both the complication and death. They propose two strata: lower severity and higher severity. At two years, they anticipate the following target-event CIFs:
| Baseline severity stratum | Usual-care CIF | Treatment CIF | Anticipated difference (treatment minus usual care) |
|---|---|---|---|
| Lower severity | 0.10 | 0.08 | -0.02 |
| Higher severity | 0.30 | 0.20 | -0.10 |
The proposed factor is a plausible design stratum because severity is measured at baseline and is expected to predict outcomes. The table gives more than an overall effect: it says that the treatment is expected to reduce target-event incidence in both strata, with a larger absolute difference in the higher-severity stratum. The differences are 0.08 − 0.10 = −0.02 and 0.20 − 0.30 = −0.10, respectively.
Before using this pattern for power calculations, investigators should also specify the expected number assigned to each group within each severity stratum, the competing-death pattern, censoring, and how the differences evolve over the full follow-up period. Two-year CIF values alone do not determine the Gray score’s mean or variance. If randomization is stratified by severity, the allocation plan should preserve meaningful representation of both groups within each stratum.
Worked Example: Opposing Stratum-Specific Differences
Consider a hypothetical two-stratum design. To illustrate the combination of score contributions, suppose planning calculations give anticipated score means \(\mu_1=9\) and \(\mu_2=-7\), with null variances \(V_1=16\) and \(V_2=9\). These are illustrative score-scale quantities, not CIF differences.
The combined anticipated score is \(9+(-7)=2\), and the combined variance is \(16+9=25\). Thus:
The modest combined signal arises because the stratum-specific expected scores point in opposite directions. The two-sided normal-approximation power at \(\alpha=0.05\) is approximately \(P(|Z|>1.96)\) for \(Z\sim N(0.400,1)\):
So the approximate power is 6.85%, only modestly above the 5% type I error rate. This example shows why a design should state the anticipated pattern within every stratum. A large effect in one stratum does not ensure a strong combined test if another stratum is expected to contribute in the opposite direction. The appropriate response is not to hide the conflict in an overall average: revisit the scientific alternative, target population, and proposed analysis.
Worked Example: Translating Planned Score Contributions Into Power
In a separate hypothetical design, suppose two strata have anticipated score means \(\mu_1=7\) and \(\mu_2=5\), and null variances \(V_1=9\) and \(V_2=16\). The expected contributions point in the same direction. Then:
For a two-sided test with \(\alpha=0.05\), the approximate power is:
The design’s approximate power is therefore 67.0% under these planning inputs. If the same allocation pattern and event-time assumptions held as sample size increased, and the score means and variances scaled approximately in proportion to sample size, then \(\lambda\) would scale approximately with the square root of sample size. To target 80% power, the required multiplier would be approximately:
Under those assumptions, the total sample size would need to increase by about 36.3%. This is a local scaling calculation, not a substitute for recalculating the stratum-specific event and censoring information at the larger sample size.
How to Make Stratification Useful
A design’s strata should be few enough to support reliable within-stratum comparisons and meaningful enough to reflect the scientific or randomization problem. Common candidates include baseline risk categories and study sites, but a factor should not be included automatically merely because it is available. If many sites are expected to enroll only a few participants or events, site-specific strata can produce unstable contributions and a weak large-sample approximation. Combining categories may be reasonable when justified before outcomes are examined and when the combined categories remain scientifically interpretable.
For each proposed stratum, planning should record the expected group allocation, target-event frequency, competing-event frequency, censoring, and anticipated CIF curves or differences across time. A factor that predicts event incidence may help describe heterogeneity, but it does not automatically increase power. Stratification changes how the comparison is formed; it cannot create events or repair a poorly specified alternative.
The design should also anticipate whether effects are directionally consistent. A common-direction alternative is often easier for a single combined score to detect than an alternative with large but opposing stratum-specific effects. If the scientific question genuinely concerns different effects across strata, a single overall stratified test may not be the only relevant analysis. The interaction or stratum-specific questions need to be planned explicitly; they should not be inferred from a global test alone.
Name the target event, competing events, treatment groups, and relevant follow-up period.
Use baseline factors with a clear design rationale, and check that both groups can be represented in each stratum.
Describe the expected CIF patterns, including the direction and timing of group differences in every stratum.
Project group sizes, target events, competing events, follow-up, and censoring by stratum; assess whether sparse cells or informative censoring are plausible.
Combine anticipated score means and variances, and examine whether opposing contributions or imprecise planning inputs weaken the result.
Diagnostics, Alternatives, and Limitations
A design review should flag a stratum with very few participants, few target events, or little overlap in group representation. Such strata can make estimated score contributions unstable and can undermine the normal approximation. Check also that anticipated censoring is plausible within each group and stratum. If loss to follow-up differs in ways related to unobserved event risk, the noninformative-censoring assumption may fail; simply stratifying on a baseline factor does not solve that problem.
The main alternative is an unstratified Gray test. It avoids dividing the comparison into strata but does not use the same within-stratum structure. Stratification is most defensible when the factor and the expected within-stratum patterns are specified in advance and the data can support the planned divisions. If the factor is weakly prognostic or the strata would be sparse, an unstratified comparison may be preferable. A log-rank test is not an interchangeable version of Gray’s test: it targets a different survival comparison, whereas Gray’s test compares cumulative incidence in the presence of competing events.
A stratified Gray test is a test, not a complete summary of treatment effects. Its p-value does not give a pooled absolute CIF difference, and a nonsignificant result does not establish that all stratum-specific CIFs are equal. Report or inspect CIF estimates by group and stratum alongside the test, with appropriate uncertainty summaries. Avoid treating an overall score as proof that effects are identical across strata.
Check Your Understanding
Answer each question using the design principles above. Brief solutions follow.
- Why should the strata be specified before outcomes are observed?
- Two strata have anticipated score means of 6 and −4 and variances of 9 and 16. What is the approximate noncentrality parameter?
- What additional planning information is needed if investigators have specified only the two-year CIFs?
- Give one reason a design with many site strata may have an unreliable power approximation.
Solutions. (1) Prespecification avoids choosing strata to fit observed outcome patterns and makes the planned comparison interpretable. (2) The combined mean is \(6-4=2\), and the combined variance is \(9+16=25\), so \(\lambda=2/\sqrt{25}=0.400\). (3) Specify the pattern over follow-up, group allocation, competing-event incidence, censoring, and expected information in each stratum; the two-year CIF values alone do not determine score means and variances. (4) Many sites may have very few participants or events, making stratum-specific contributions unstable and the large-sample approximation doubtful.