Functions & Formulae

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.