Weighted log rank type statistics for comparing survival curves when there is a time lag in the effectiveness of treatment

“The sample size was determined by inverting the test statistic proposed by Zucker and Lakatos.(18) With a type I error as 0.”

“Examples include the linear combinations (Gastwirth, 1985; Zucker and Lakatos, 1990), maximum of several standardized tests (Tarone, 1981; Fleming and Harrington, 1984, 1991), maximum after orthogonalization (Breslow, Elder, and Berger, 1984), or χ2 test of the simultaneous null hypothesis for the…”

“trials (Gail 1985; Zucker and Lakatos 1990).”

“…the data is uncertain, but a family of scientifically plausible models and their normally distributed tests are available (Gastwirth, 1966, 1985; Davies, 1977; Zucker and Lakatos, 1990; Whittemore and Tu, 1998; Freidlin and others, 1999; Gastwirth and Freidlin, 2000; Shih and Whittemore, 2001).”

“This finding is consistent with Zucker and Lakatos’s results [14], where the study power can be severely over-estimated when the true changing point t∗ is above the specified.”

“Zucker and Lakatos [4] proposed a maximum efficiency robust statistic to test the treatment effect with a time lag.”

“Although a simple algorithm when C is the class of linear combination of {Zi : i ∈ I} was given in Gastwirth [9] (see also Zucker and Lakatos [27]), the computation of ZMERT is more complicated as it is related to quadratic programming algorithms.”

“For the two-sample problem, Zucker and Lakatos [2] investigated the log-rank test statistics for a treatment lag effect.”

“In addition to applications of the MERT in genetics, it has applications in other areas, including survival analysis [90, 116, 356].”

“A simple algorithm to obtain the MERT is given in Gastwirth (1985) and is used by Zucker & Lakatos (1990) to analyse survival data.”

“Although this condition may not be easily verified in many applications, the simpler MERT (3) has been discussed and applied in many settings [26, 27], including genetic studies.”

“For a given 𝜆2 and N, simulate the time-to-event data under the random lag duration scenario where t∗ind ∼ Uniform[1, 11] months and compute the empirical powers of the following testing methods: • GPW-Logrank test assuming a random lag time varying between 1 and 11 months, with weights w∗1(t) = 0, if t ≤ 1;w∗2(t) = (…”

“For example, the G tests, which are constructed assuming a class of survival distributions indexed by ρ and γ under the alternative hypothesis (Fleming and Harrington, 1991), a linear combination or the maximum of several test statistics (Tarone, 1981; Fleming and Harrington, 1984, 1991; Gastwirth, 1985; Zucker and La…”

“…and accelerated failure time (e.g., Cox and Oakes, 1984) models, the accelerated hazards (AH) model (Chen and Wang, 2000) was proposed for its ability to capture the gradual effects of a treatment; this lag period often exists before a treatment is fully effective (e.g., Zucker and Lakatos, 1990).”

“For randomized clinical trials with two treatment arms, Zucker and Lakatos [7] investigated the logrank test statistics for a delayed treatment effect.”

“For clinical trial data, Meinert (1986) and Zucker and Lakatos (1990) argue that the treatment effect may manifest only after a period of time.”

“In particular, a good approximation for (t) is n−1p(1 − p) × {number of failures in [0, t]} (Schoenfeld, 1983; Zucker and Lakatos, 1990).”

“Zucker and Lakatos (1990) proposed two weighted log-rank types of statistics designed to have good efficiency over a wide range of lag functions, which can be applied in situations where a delayed effect is expected but cannot be specified precisely in advance.”

“…(AFT) model (Cox and Oakes, 1984), the AH model is particularly useful to model the effect of treatments or other covariates on the failure time distribution when the treatment effect is gradually released and there is a lag period before the treatment is fully effective (Zucker and Lakatos, 1990).”

“For example, weighted log-rank test, restricted mean survival time (RMST), and other methods (Harrington and Fleming 1982; Zucker and Lakatos 1990; Yang and Prentice 2005; Fine 2007; Royston and Parmar 2011; Uno et al. 2014; Xu et al. 2017) are proposed to overcome the limitations of logrank test under nonproportional…”

“That is, there is a lag period before the treatment is fully effective (Zucker and Lakatos, 1990).”

“As another example, [2] describes analysis of the Physician’s Health Study for testing the effect of beta-carotene on cancer incidence.”

“The R codes to generate CDE based on Equations (10) and (11) is presented in Appendix 3.”

“These include the maximin efficiency tests (Gastwirth, 1985; Zucker and Lakatos, 1990), the supremum version tests (Gill, 1980; Fleming, Harrington and O’Sullivan, 1987), the maximum of a finite cluster of statistics (Tarone, 1981; Fleming and Harrington, 1984; Lee, 1996) and the maximum of a large class of statistics…”

“We considerMantel’s log-rank test and several others that account for delayed treatment effects: (i) a lagged log-rank test that ignores the first 10% of observed followup times (Zucker and Lakatos, 1990); (ii) the Fleming– Harrington G0,1 test, which gives more weight to late events (Fine, 2007); (iii) the adaptive l…”

“Finally, for RCTs with delayed treatment effect, some research works focused on various weighted log-rank tests (He and Su, 2015; Xu et al., 2017; Zucker and Lakatos, 1990).”

“The sample size formula (7) for the MERT derived by Ye and Yu(5) underestimated the sample size even when δ>0.”

“2013–2:40pm] [1–8] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/MSCJ/Vol00000/130005/APPFile/SG-MSCJ130005.”

“However, as Gastwirth (1985), Zucker and Lakatos (1990), and Zucker (1992) have pointed out, there are special infinite-dimensional families of (contiguous) alternatives with respect to which the minimum ARE of a MERT test will still be reasonably good (say, 60-70%).”

“The incubation and immune ramp-up periods could be explicitly included in the likelihood; if this lag is known, it can be used to construct optimal weights [Zucker and Lakatos, 1990].”

“For this reason some authors have suggested to introduce weights changing sign before and after the crossing point [10, 11, 24, 25].”

“Zucker and Lakatos (1990), Freedman et al. (1993) and Luo et al. (1997) have discussed several animal experiments and clinical trials for evaluating the clinical benefits of several compounds for the prevention of cancer.”

“This model has been considered previously by several authors (Murhpy and Sen, 1991; Hastie and Tibshirani, 1990; Zucker and Lakatos, 1990; Cai and Sun, 2003; Winnett and Sasieni, 2003; Scheike and Martinussen, 2004).”

“Efficiency and Robustness For the continuous case, the efficiency of a test with weight W(t) relative to the optimal test with weight W*(t) is [Zucker and Lakatos, 1990, equation (2)] f (u) W*(u)W(u)h(u) du '0 .”

“where τ is the total time of the study, w (t) is the weight function at time t, Ri(t) is the number of items/individuals at risk at time t in the i group, R(t) is the overall number of items/individuals at risk at time t, and Ni(t) is the number of items/individuals which underwent the event of interest by time t in t…”

“Studies discussing issues related to non-PH go back to the 1980’s and 1990’s; thus, even before the use of IO therapies [42, 52, 62, 69, 78].”

“For hypothesis testing, weighted log-rank tests have been considered that have a larger power than the unweighted test, if more weight is allocated to time points where the effect is larger [13, 14, 15, 16].”

“Under the alternative hypothesis defined by each of the lag models, we calculate μ w and σ 2 w using λ 1 t ð Þ in Equation (5) and is the optimal weight in terms of detecting treatment effect over control (Zucker and Lakatos 1990).”

“For example, Zucker and Lakatos (1990) proposed to use a linear combination of weighted log-rank tests or a combination of maximum efficiency to account for a broad range of lags time functions.”

“Kaplan-Meier(KM)analysis[47]withLog-rank-typetest[48]is thenperformed.”

“The American Statistician, May 2015, Vol. 69, No. 2 115 Zucker and Lakatos (1990) and Zucker (1992) proposed using weighted log-rank tests to deal with nonproportional hazards.”

“We consider Mantel’s log-rank test and several others which account for delayed treatment effects: i) a lagged log-rank test that ignores the first 10% of observed follow-up times (Zucker and Lakatos, 1990); ii) the Fleming-Harrington G test, which gives more weight to late events (Fine, 2007); iii) the adaptive log-r…”

“Later on, Zucker and Lakatos (1990) used the procedure for the comparison of treatments with time lag effects in survival analysis.”

“To compare the hazard rate functions of the two groups, we conduct the hypothesis testing of (5) using both the Wald test (6), the likelihood ratio test (7), the 6 existing methods mentioned above related to (9), and the method by Zucker and Lakatos (1990). For the 6 existing methods considered, besides the regular co…”

“In cases when the time-lag period can be pre-specified, Zucker and Lakatos 18 proposed two weighted Log-rank tests for comparing the related hazard curves.”