Why Compare Follow-Up Before Interpreting a Gray Test?
A Gray test compares cumulative incidence functions across groups for a specified event type, accounting for competing events. Its validity depends on censoring conditions, as discussed in the earlier material on censoring conditions for Gray’s test. Follow-up diagnostics help assess whether those conditions are plausible and whether the groups have comparable information over the time range of interest.
The key distinction is between different amounts of follow-up and informative censoring. One group can have shorter follow-up on average without violating the test’s censoring assumption. For example, participants may enter that group later and therefore reach an administrative study close sooner. The concern is that censoring within a group may be related to the event process, so that people who remain observed no longer represent the people who were still event-free and under observation.
The diagnostics in this tutorial are descriptive checks, not formal tests that can prove independent censoring. Read them together: risk-set sizes show how much data remain, censoring summaries describe how observation ends, and the timing and reasons for censoring can reveal patterns worth investigating.
What to Inspect by Group
Begin with a table that gives, at relevant times, the number at risk and the number of each type of event and censoring. A count at risk is not a count of people who have not had the event of interest: people who have had a competing event are no longer at risk for a first event. Likewise, a decline in the risk set can reflect events as well as censoring. Inspecting risk counts alone cannot identify why the counts differ.
Useful summaries include the following:
- Numbers at risk at common time points. Large or rapidly emerging differences can show that the later portions of a comparison are supported by very different amounts of observed data.
- Censoring counts and timing. Report how many participants are censored, when censoring occurs, and, where available, the recorded reason. A single overall percentage hides whether censoring is concentrated early or late.
- Follow-up distributions. Summaries such as the median potential follow-up can help compare observation time across groups. Do not substitute the mean observed time \(Y_i\) without considering that events end observation too.
- Study design and calendar time. Staggered entry, different recruitment periods, site closures, or changes in data collection may explain differences in follow-up. They may also be related to prognosis and therefore merit investigation.
A reverse Kaplan–Meier estimate is a common way to summarize potential follow-up. To calculate it, treat censoring as the event of interest and treat observed events of any cause as censored observations. At each censoring time, the estimated probability of remaining uncensored is multiplied by one minus the number censored at that time divided by the number at risk just before it. The estimated median potential follow-up is the first time this product is at or below one half.
This summary has a specific purpose: it describes the distribution of censoring times while accounting for observed events that otherwise shorten measured follow-up. It is not evidence that censoring is independent of the event process. In particular, interpreting it as a distribution of potential censoring times relies on assumptions that cannot be verified from the observed follow-up summaries alone.
Worked Example: Similar Censoring Counts, Different Risk-Set Patterns
Consider a hypothetical study with 100 participants in each of two groups. Counts are summarized in three intervals. In each interval, the next risk-set count equals the previous count minus all observed events and censorings in that interval.
| Group | At risk at start | Events in interval | Censored in interval | At risk next time |
|---|---|---|---|---|
| A, 0–6 months | 100 | 12 | 6 | 82 |
| A, 6–12 months | 82 | 15 | 9 | 58 |
| A, 12–18 months | 58 | 10 | 17 | 31 |
| B, 0–6 months | 100 | 10 | 8 | 82 |
| B, 6–12 months | 82 | 12 | 10 | 60 |
| B, 12–18 months | 60 | 8 | 20 | 32 |
For group A, the risk-set arithmetic is \(100-12-6=82\), then \(82-15-9=58\), then \(58-10-17=31\). For group B it is \(100-10-8=82\), then \(82-12-10=60\), then \(60-8-20=32\). By 18 months, 32 of 100 participants in A and 38 of 100 in B have been censored. These overall proportions are \(32/100=0.32\) and \(38/100=0.38\).
The totals are not very different, and the risk-set counts are close through 18 months. This does not show that censoring is independent in either group; the timing and reasons for censoring still matter. It does show why comparing only overall percentages can be misleading: group B has more censoring in the last interval, while event counts also differ. A careful report would show these components separately and consider whether the later part of the comparison has adequate support.
Reverse Kaplan–Meier Follow-Up Summary
Here is a small hypothetical calculation. At each listed time, the table gives the number at risk just beforehand and the number censored. Observed events of either cause are treated as censored for this reverse Kaplan–Meier calculation.
Worked Example: Comparing Reverse Kaplan–Meier Medians
In group A, 10 participants begin follow-up. There are 2 censorings at month 6, 2 observed events at month 8, 3 censorings at month 10, and 3 observed events at month 12. In group B, there are 2 observed events at month 8, 1 censoring at month 10, 2 observed events at month 12, 3 censorings at month 16, and 2 observed events at month 20.
For group A, the reverse Kaplan–Meier estimate of remaining uncensored follow-up is 1.00 before month 6. At month 6, the risk set is 10 and 2 are censored, so the estimate becomes \(1.00(1-2/10)=0.80\). The 2 events at month 8 are treated as censored observations for this calculation; they reduce the risk set to 6 but do not change the estimate. At month 10, 3 of those 6 are censored, giving \(0.80(1-3/6)=0.40\). This is below 0.50, so the estimated median potential follow-up is 10 months.
For group B, the 2 events at month 8 reduce the risk set from 10 to 8 without changing the reverse estimate. At month 10, 1 of 8 is censored, so the estimate is \(1.00(1-1/8)=0.875\). The 2 events at month 12 reduce the risk set to 5. At month 16, 3 of those 5 are censored, giving \(0.875(1-3/5)=0.350\). Thus the estimated median potential follow-up is 16 months.
The six-month difference indicates that the distribution of observed censoring times differs between these groups. It does not, by itself, mean Gray’s test is invalid: independent censoring can hold within each group even when censoring distributions differ between groups. The next step is to investigate why follow-up differs and whether censoring is plausibly independent of the event process within each group.
Why Unequal Follow-Up Is Not Automatically a Violation
A useful justification comes from the independent-censoring condition. Within group \(g\), suppose \(C\) is independent of the pair \((T,J)\). Among participants who are event-free at \(t\) and still observed at \(t\), the distribution of their future event time and cause is then the same as among participants who are event-free at \(t\) in that group:
To see why, conditional independence means that, within group \(g\), the probability of an event outcome in a set \(A\) and continued observation to \(t\) factors into separate terms once we condition on \(T\ge t\). Dividing by the probability of continued observation cancels the censoring term. Therefore, although censoring reduces the number observed, it does not selectively remove participants with particular future event outcomes under this condition.
The assumption is needed within groups; it does not require the censoring-time distributions to be identical across groups. However, if censoring depends on relevant measured factors, an unconditional comparison may not meet the condition. Whether a conditional analysis is appropriate depends on the design and analysis plan; descriptive follow-up checks do not fix a violation.
Worked Example: A Warning From Censoring Reasons
Suppose a hypothetical study follows 100 participants in each of two groups to 12 months. In group A, 10 are censored before 12 months; 2 of those withdrawals are recorded as following clinical deterioration. In group B, 30 are censored; 24 of those withdrawals follow documented deterioration. By 12 months, 25 participants in A and 18 in B have had an observed event, and 65 in A and 52 in B remain event-free and under observation. The accounting checks: \(25+10+65=100\) and \(18+30+52=100\).
The censoring proportions are \(10/100=0.10\) in A and \(30/100=0.30\) in B. More importantly, \(24/30=0.80\) of B’s censorings follow deterioration, compared with \(2/10=0.20\) of A’s. If deterioration predicts a subsequent event, this pattern raises concern that censoring may be related to prognosis, particularly in B. The much smaller risk set in B at 12 months also means that later portions of the comparison rely on fewer observed participants.
These observations do not prove informative censoring: deterioration might have been recorded after a decision unrelated to future event risk, and the summaries do not describe unmeasured factors. They do justify examining the study’s withdrawal process and considering whether the censoring condition is plausible. Reporting a reverse Kaplan–Meier median alone would not resolve this concern.
Diagnostics, Alternatives, and Limitations
A practical review should ask whether differences are concentrated in a particular group, time period, site, or reason for withdrawal. Plotting or tabulating numbers at risk beside cumulative-incidence estimates helps show where the data become sparse. A rapid drop in risk-set size late in follow-up is a warning about precision and support at those times; it is not, on its own, evidence of bias. Also check for data problems such as impossible follow-up times, inconsistent event and censoring records, or a group with no participants at risk over a time range used in interpretation.
The main summaries answer different questions. The raw median observed time summarizes \(Y_i\), so it can be shortened by either an event or censoring. The reverse Kaplan–Meier median focuses on censoring times while accounting for observed events as censored observations. The numbers-at-risk table shows how many people remain available at particular times, but not why others left. Censoring counts by reason and interval provide context but may be incomplete or subject to recording error. No single summary substitutes for the others.
Common limitations are important. Follow-up summaries cannot establish independent censoring, especially when the reason for dropout is unrecorded or related to unmeasured prognosis. Unequal risk sets can limit the range over which a comparison is informative, even if the test’s assumptions hold. Conversely, similar risk sets do not guarantee valid censoring. When the diagnostics raise concern, the appropriate response is to describe the concern, investigate the design and missingness process, and consider suitable sensitivity analyses or methods for the specific situation—not to treat a diagnostic statistic as proof or as a correction.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Two groups have different reverse Kaplan–Meier medians. Does this alone show that Gray’s test is invalid? Why or why not?
- At 12 months, a group has 40 people at risk. Does that mean 60 people were censored if the group began with 100? Explain.
- In one group, 15 of 20 censored participants withdrew after a worsening health measure. What does this pattern suggest, and what does it not establish?
- Why can the raw median observed time be a poor summary of potential follow-up?
Solutions. (1) No. It shows different distributions of censoring times, but censoring can be independent of the event process within each group even when those distributions differ. Investigate timing and reasons for censoring. (2) No. The other 60 may include participants who had the event of interest, experienced a competing event, or were censored; the risk-set count alone does not separate these outcomes. (3) It raises concern that censoring may be related to prognosis and merits investigation. It does not prove informative censoring or establish how much a Gray-test comparison is affected. (4) Observed time ends at either an event or censoring, so events can shorten it even when potential follow-up is long; the raw median combines both processes.