Why the Timing of Loss Matters
A planned Gray test needs enough information to detect the specified difference in cumulative incidence of the event of interest. Loss to follow-up can reduce that information, but the reduction depends on when participants become unobserved. Someone lost shortly after enrollment contributes little observed follow-up; someone lost near the end may contribute substantial information first. Thus, a single percentage lost can be a useful planning summary, but it does not always describe the information available to the test.
Loss to follow-up is not itself a competing event. A competing event is an observed event that prevents the event of interest from occurring in the defined way; loss to follow-up means that subsequent event status is unknown. For a Gray-test design, this distinction matters: a competing event belongs in the event-process model, whereas loss to follow-up is a censoring mechanism that can affect observation of that process.
Let \(N_0\) denote a total sample size obtained from the planned Gray-test calculation before allowing for loss to follow-up. It corresponds to specified group allocation, significance level, power, alternative cumulative-incidence pattern, and assumptions about event and censoring times. The adjustment below is an additional planning approximation, not a replacement for those design inputs.
Assumptions Behind an Adjustment
A simple or time-weighted adjustment is most defensible when the following conditions are reasonable:
- Loss to follow-up is independent of the event process, at least conditional on the group and measured design factors used in planning. In particular, participants at higher unobserved risk should not systematically be lost sooner after conditioning on those factors.
- The censoring model used for planning describes the anticipated study population and applies to the event follow-up period relevant to the Gray test.
- The specified event-of-interest and competing-event processes, group allocation, and follow-up schedule match the assumptions used to obtain \(N_0\).
- The approximation that information is reduced in proportion to the retained contribution is adequate. It is not an exact formula for the finite-sample variance of Gray’s statistic.
If loss differs by group, is associated with prognosis, or is much earlier than anticipated, a single retention multiplier can be misleading. The test’s variance and the observed subdistribution risk sets may change in ways not captured by a simple proportional adjustment. The calculations here are planning approximations; their assumptions should be checked against the intended design.
Simple Inflation by the Retained Fraction
The simplest calculation assumes that each enrolled participant has the same chance \(q\) of providing usable follow-up and that the expected information is approximately proportional to the number retained. If the expected fraction lost is \(L\), then \(q=1-L\). The corresponding enrollment target is
The ceiling is applied to the unrounded ratio, since a fractional participant cannot be enrolled. With equal allocation, it is often clearer to apply the calculation within each group and round each group target upward, then report their sum. This simple method treats loss as if it reduced each person’s contribution by the same average amount; it does not model when loss occurs.
Worked Example: Inflating for a Fixed Expected Loss
Suppose a hypothetical two-group design requires \(N_0=300\) participants under the planned no-loss approximation, with equal allocation. The investigators expect 10% of participants to be lost to follow-up and use the simple proportional-retention assumption.
The expected retained fraction is \(1-L=1-0.10=0.90\). The per-group target before rounding is \(150/0.90=166.667\). Rounding each group upward gives 167 participants per group, or 334 total.
Equivalently, the total calculation is \(N_{\text{plan}}=\lceil 300/0.90\rceil=\lceil333.333\ldots\rceil=334\). This target is appropriate only as a proportional-loss approximation. It does not imply that every participant lost removes exactly the same amount of information.
Allowing Retention to Vary Over Time
When loss accrues during follow-up, the key quantity is not merely the final retained percentage. A useful first-order approximation weights retention by the timing of information in the no-loss design. Let \(w(t)\) be a nonnegative information density under the planned alternative, scaled so that its integral over the study period represents the no-loss information contribution. Then define
Here \(\tau\) is the end of the relevant follow-up period. The approximation treats a contribution arising at time \(t\) as available only for participants still observed at that time, whose probability of remaining is \(G(t)\). It follows by multiplying each no-loss information contribution \(w(t)\,dt\) by its expected observed fraction \(G(t)\), then integrating. Thus expected retained information is approximated by \(\int_0^\tau w(t)G(t)\,dt\); dividing by the no-loss information \(\int_0^\tau w(t)\,dt\) gives \(R\).
This argument justifies the weighting idea, not an exact identity for Gray-test variance under every censoring pattern. The relevant information density depends on the specified alternative, event-of-interest and competing-event processes, and the test’s variance. In practice it should be obtained from the no-loss design calculation or a suitable approximation to it. If information is concentrated at later times, early loss can be especially costly. If information is concentrated earlier, the same loss rate may have a smaller impact.
Worked Example: Exponential Loss With Uniform Information Weight
Consider a hypothetical design with two years of follow-up. Suppose loss to follow-up has a constant hazard of \(0.08\) per year, so \(G(t)=e^{-0.08t}\), and suppose, for this illustration only, that information is uniformly distributed over the two years. Then \(w(t)\) is constant and cancels from the ratio.
The information-retention fraction is \(R=(1/2)\int_0^2 e^{-0.08t}\,dt =(1-e^{-0.16})/0.16\). Since \(e^{-0.16}\approx0.8521438\), this gives \(R\approx(1-0.8521438)/0.16=0.9241013\), or about 92.41%.
If the no-loss target is \(N_0=240\), with 120 per group, then the adjusted target per group is \(\lceil120/0.9241013\rceil=\lceil129.856\ldots\rceil=130\). The total target is therefore 260. This result uses uniform information over time; it should not be carried over unchanged to a design whose information is concentrated at different times.
For comparison, using only the two-year retention \(G(2)=0.8521438\) would give \(\lceil240/0.8521438\rceil=282\) participants. That calculation treats all information as if it arose at the end of follow-up, so it is more conservative in this particular uniform-information example. It is not a general rule that the end-of-study fraction is always conservative for every design.
Worked Example: Weighting Several Follow-Up Periods
Suppose a hypothetical no-loss calculation assigns 20%, 50%, and 30% of its information to three successive periods. For a simple grouped approximation, use representative follow-up times of 0.25, 1, and 2 years, respectively. Assume an exponential loss hazard of \(0.12\) per year, giving retention values \(e^{-0.03}\), \(e^{-0.12}\), and \(e^{-0.24}\) at those times.
The weighted retention fraction is \(R=0.20e^{-0.03}+0.50e^{-0.12}+0.30e^{-0.24}\). Using full precision in the sum, \(R\approx0.20(0.9704455)+0.50(0.8869204)+0.30(0.7866279)\) \(=0.1940891+0.4434602+0.2359884=0.8735377\), or about 87.35%.
For a no-loss target of 300 participants with equal allocation, the approximate adjusted total is \(N_{\text{plan}}=\lceil300/0.8735377\rceil=\lceil343.431\ldots\rceil=344\). Planning for 172 participants per group gives that total. The period weights and representative times are hypothetical design inputs; in an actual plan they should reflect the anticipated information pattern rather than being chosen merely for convenience.
Diagnostics, Alternatives, and Limits
Before accepting an adjusted target, inspect whether the censoring model and information weights fit the proposed study. Useful checks include plotting or tabulating expected retention over time by group, checking whether the loss hazard is plausibly constant if an exponential model is used, and comparing the event-information pattern with the assumed weights. If retention differs materially between groups, calculate group-specific retention and assess whether a single weighted fraction remains defensible. If loss is informative, independent-censoring calculations do not resolve the bias; larger enrollment alone cannot guarantee a valid comparison.
The main alternatives answer different planning needs:
| Approach | What it represents | Main limitation |
|---|---|---|
| Simple inflation by \(1/(1-L)\) | A single expected retained fraction applied uniformly to the no-loss target | Ignores the timing of loss and assumes proportional information reduction |
| Time-weighted retention | Retention evaluated across follow-up and weighted by the design’s information pattern | Depends on credible retention and information models; remains an approximation |
| Recalculate or evaluate the full design under the censoring model | Allows censoring to affect risk sets and variance more directly | Requires a specified event and censoring model and more computation |
A frequent mistake is to divide by the percentage still observed at the end of follow-up and call the result the required sample size. This silently assumes that all information is contributed at the end. The correction is to use simple inflation only when proportional retention is a reasonable approximation, or to weight retention over time when the information pattern is known. Another mistake is to count a competing event as loss to follow-up; it is an observed event and must remain part of the competing-risks design.
Finally, a scalar adjustment cannot repair misspecified alternatives, group-dependent informative loss, or an implausible event-time distribution. When those features could materially affect power, the design should be recalculated under assumptions that represent them. Report the assumed loss model, retention fraction, information weights, rounding rule, and resulting group targets so that the enrollment calculation is reproducible.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A design requires 180 participants per group without loss. If the expected loss fraction is 10% and simple proportional inflation is justified, what is the enrollment target per group?
- Why can two designs with the same overall percentage lost have different information-retention fractions?
- For uniform information over two years and exponential loss hazard \(0.08\) per year, what is the approximate information-retention fraction?
- Is an observed competing event a loss to follow-up? Explain the consequence for planning.
Solutions. (1) The target is \(\lceil180/(1-0.10)\rceil=\lceil200\rceil=200\) per group. (2) The timing of loss relative to the information pattern differs: loss before information-contributing times removes more contribution than loss after those times. (3) \(R=(1-e^{-0.16})/0.16\approx0.9241\), about 92.41%. (4) No. A competing event is observed and belongs to the competing-risks event process; loss to follow-up censors subsequent observation. Treating the competing event as ordinary loss would misrepresent the design.