Signature
RR_naive = (d_R / (PT_R + I)) / (d_N / PT_N); RR_split = (d_R / PT_R) / (d_N / (PT_N + I)); F = RR_naive / RR_split
| Inputs | Definition | Unit |
|---|---|---|
d_R | Deaths among responders, all of them after the response | count |
PT_R | Person-time of responders counted from the response to death or the end of follow-up, above zero | person-years |
I | Person-time of responders between entry and response, during which none of them could die and still be classed a responder; zero or above | person-years |
d_N | Deaths among patients who had not responded when they died, early deaths included, above zero | count |
PT_N | Person-time from entry of the patients who never responded, above zero | person-years |
RR_naive | Death rate of responders over the death rate of patients who never responded, with I added to the responders' person-time | ratio |
|---|---|---|
RR_split | Death rate of responders after response over the death rate in all person-time without a response, I included | ratio |
F | RR_naive divided by RR_split; below 1 whenever I is above zero | ratio |
Function
Landmark survival comparison by a status reached during follow-up and the immortal-time bias of grouping from the start
Maps follow-up data for patients grouped by a status reached after follow-up begins, such as tumour response, to a comparison of the groups that does not credit the time before the status to one group, made either by restarting the clock at a landmark time s for the patients still event free at s or by splitting person-time at the change of status, and to the bias of the naive grouping from the start. HE-FM-LMA-002 gives that bias as a factor F in a constant-hazard rate ratio of responders when the person-time responders spent before responding is counted as responder time, each rate ratio having the form of HE-FM-HR-005. The article's conditional survival S(t | T > s) = S(t) / S(s) is the live HE-FM-LTR-001, S_cond = S_t / S_l with the entry time l in the place of the landmark s, and is not restated. The article's whole-cohort value 0.495 / 0.85, about 0.582, is that formula applied, and so are its landmark proportions, since in the naive non-responder group of 130 patients survival to the landmark is 100/130 and 0.385 divided by 100/130 is 0.50. The fall in the naive survival gap is HE-FM-LTR-001 again, because the responder proportion 49/70 is the same in both groupings and the gap falls by S_cond minus S_t of the naive non-responder group, 0.50 minus 0.385, the 11.5 points between the article's gaps of 31.5 and 20 points; each gap has the form of HE-FM-ARR-001. The same ratio is the residual stay R_ab of HE-FM-MKA-002 and S_res of HE-FM-MKA-003 for geometric and exponential stays, and one minus it over a single cycle is the cycle probability of HE-FN-CUMH-001 and HE-FM-CUMH-002, since 1 minus exp(minus (H(t) minus H(t minus Delta))) equals 1 minus S(t) / S(t minus Delta). Relative survival HE-FM-BGM-001 divides by expected background survival at the same time, not by the cohort's own survival at an earlier time, so it is a different ratio. Landmark survival from patient records is the left-truncated Kaplan-Meier curve of HE-CF-LTR-001 run in each status group with every entry time set to s, built on the product-limit step of HE-FM-ADMC-002. The landmark Cox model h(t | x(s), T > s) = h_0s(t) exp(beta_s x(s)) is an ordinary Cox model fitted to the patients at risk at s, with its hazard ratio from the coefficient as in HE-FM-HR-002; it has no closed-form calculator line and is given for reading only. Notation follows the Landmark Analysis article.
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Implementations
Excel
Naive and time-split responder rate ratios and the bias factor from named person-time
With the counts and person-time in cells named DeathsResp, PersonTimeResp, ImmortalTime, DeathsNonResp and PersonTimeNonResp, the three formulas return RR_naive in a cell named RateRatioNaive, RR_split in RateRatioSplit and F in BiasFactorRR.
=(DeathsResp/(PersonTimeResp+ImmortalTime))/(DeathsNonResp/PersonTimeNonResp); =(DeathsResp/PersonTimeResp)/(DeathsNonResp/(PersonTimeNonResp+ImmortalTime)); =RateRatioNaive/RateRatioSplit
Assumptions
Constant hazards within each response group for the bias factor
Each death rate is events over person-time and estimates a constant hazard, as in HE-FM-HR-005. With hazards that change over follow-up the extended Cox model is needed, and the closed form of F describes only the crude rate ratio.
No deaths in the pre-response person-time of responders
A responder cannot die before responding, so every death before a response counts in d_N and d_R holds only deaths after the response. This is what makes the event counts equal in the naive and time-split ratios.
Response time known for every responder in the person-time split
Splitting person-time needs each responder's response time. Where response is assessed only at set visits, such as the article's six-month assessment, the assessment time is used, and I is the person-time responders spent before it.
Worked examples
Article's cohort with illustrative timing of deaths and responses
Using the article's counts with timing assumptions added here for illustration (response at the six-month assessment, early deaths at three months on average and later deaths one year after the landmark on average), responders contribute 35 person-years before response and 119 after it, with 21 deaths, and patients without a response contribute 207.5 person-years with 80 deaths. The naive rate ratio is 0.354 and the time-split ratio 0.535, so F is 0.661 (computed here for illustration).
d_R = 21; PT_R = 119; I = 35; d_N = 80; PT_N = 207.5; RR_naive = 0.3537; RR_split = 0.5349; F = 0.6612
No pre-response person-time leaves no bias in the responder rate ratio
With the same deaths and person-time but no immortal time, as when status is known at entry, both ratios are 0.4577 and F is 1, a limiting case that checks an implementation (computed here for illustration).
d_R = 21; PT_R = 119; I = 0; d_N = 80; PT_N = 207.5; RR_naive = 0.4577; RR_split = 0.4577; F = 1
Rate-ratio bias factor with 50 person-years of immortal time
With illustrative inputs of 20 deaths in 200 person-years after response, 50 person-years before it, and 60 deaths in 300 person-years among patients who never responded, the naive ratio is 0.4 and the time-split ratio 0.5833, so F is 0.6857 (computed here for illustration).
d_R = 20; PT_R = 200; I = 50; d_N = 60; PT_N = 300; RR_naive = 0.4; RR_split = 0.5833; F = 0.6857
Common errors
Pre-response time counted as responder time in a rate ratio
In the article-based illustration the naive rate ratio of 0.354 suggests a death rate 65% lower for responders, against 47% lower with person-time split at the response (computed here for illustration). Suissa reports that the bias in the rate ratio from misclassified immortal time increases with the duration of immortal time.
Immortal time dropped instead of moved to the side without a response
Removing I from the responders without adding it to the person-time without a response gives a rate ratio of 0.458 in the illustration, still below the time-split 0.535 (computed here). Suissa names excluded as well as misclassified immortal time as a source of bias in the rate ratio.
Sources
Giobbie-Hurder and colleagues on the extended Cox model with a time-varying covariate
Giobbie-Hurder A, Gelber RD, Regan MM. Challenges of guarantee-time bias. Journal of Clinical Oncology. 2013;31(23):2963-2969. doi:10.1200/JCO.2013.49.5283 (full text read in PubMed Central, PMC3732313). Analytic Methods, Extended Cox Model With Time-Varying Covariates, which states that all patient data are used, that patients who develop the classifying event are switched into its group at the time it occurs and remain there, and that the resulting hazard ratio compares patients after the event with those who had not yet or never had it. The person-time split and the closed form of F are derived here for constant hazards.
Giobbie-Hurder and colleagues on guarantee time in the transplantation example
Giobbie-Hurder A, Gelber RD, Regan MM. Challenges of guarantee-time bias. Journal of Clinical Oncology. 2013;31(23):2963-2969. doi:10.1200/JCO.2013.49.5283 (full text read in PubMed Central, PMC3732313). Background and Figure 1, which state that patients in the transplantation group were guaranteed to survive at least until a donor was available, that their survival reflects the total time before and after transplantation, and that the extra waiting time is guarantee time.
Suissa on misclassified and excluded immortal time in the rate ratio
Suissa S. Immortal time bias in pharmaco-epidemiology. American Journal of Epidemiology. 2008;167(4):492-499. doi:10.1093/aje/kwm324. Abstract, which defines immortal time as a span of cohort follow-up during which, because of exposure definition, the outcome could not occur, and states that the bias in the rate ratio from misclassified or excluded immortal time increases with the duration of immortal time. The closed form of F is derived here and is not taken from the paper.
Anderson, Cain and Gelber on comparing survival by tumour response
Anderson JR, Cain KC, Gelber RD. Analysis of survival by tumor response. Journal of Clinical Oncology. 1983;1(11):710-719. doi:10.1200/JCO.1983.1.11.710. Abstract, which states that the usual method of comparing responders and nonresponders is biased in favour of responders, that two valid methods are discussed, and that a comparison by response category should not be used for inference about treatment effectiveness. The person-time form of the bias in HE-FM-LMA-002 is derived here and is not taken from the paper.
Canonical Identity
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