Simultaneous confidence bands using model assisted Cox regression

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1 Simultaneous confidence bands using model assisted Cox regression Shoubhik Mondal and Sundarraman Subramanian Center for Applied Mathematics and Statistics Department of Mathematical Sciences New Jersey Institute of Technology USA Abstract In the first part, entitled Model assisted Cox regression and published in Journal of Multivariate Analysis (JMVA), it was shown that standard Cox regression, combined with Dikta s semiparametric random censorship models, provides an effective framework for obtaining improved parameter estimates. Here, this methodology is exploited to construct simultaneous confidence bands (SCBs) for subject-specific survival curves. Simulation results are presented to compare the performance of the proposed SCBs with the SCBs that are based only on standard Cox. The new SCBs provide correct empirical coverage and are more informative. The proposed SCBs are illustrated with two real examples. An extension to handle missing censoring indicators is also outlined. KEY WORDS: Counting process; Empirical coverage; Equal-precision; Martingale; Gaussian multiplier bootstrap; Strong consistency. 1 Introduction In biomedical applications it is common practice to report pointwise confidence intervals (PCIs) for survival curves to facilitate comparison of two treatments. PCIs over any desired region are easy to calculate and hence very attractive to practitioners. However, they can lead to incorrect judgment regarding treatment efficacy when treatments have time-varying effect, say, when a treatment may have high early survival but lower long term survival. Consider the example of comparing allogenic bone marrow transplant (BMT) versus conventional chemotherapy (CC) for chronic myelogenous leukemia (CML), discussed in Zhang and Klein (21). PCIs indicated that the threshold for BMT inefficacy was up to 4.5 years, after which period it appeared that the BMT group would have a lower mortality rate than the CC group. However, the simultaneous confidence bands (SCBs) that Zhang and Klein (21) presented indicated a more conservative threshold of about 6 years, implying that the BMT group had 1

2 1 INTRODUCTION a higher mortality rate for up to 6 years. Although PCIs produce smaller widths than SCBs, which, when extended to a region, produce the perception of increased discriminative ability through smaller enclosed areas than SCBs, this may hardly be the case in reality. Specifically, PCIs guarantee correct coverage for each isolated point separately but not for a multitude of points jointly. In the case of the CML example referred above, for example, the PCIs of the difference of the two survival curves computed at time points 4.5 years and 4.75 years would not cover the true difference at 4.5 and 4.75 years jointly with 95% confidence. The smaller enclosed area of a PCI over a region, therefore, is an artifact of this limitation. SCBs would give a better picture of global variability than PCIs. In their analysis of the Mayo data base, Dickson et al. (1989) make a strong case for computing subject-specific survival function estimates accompanied by SCBs. The Cox proportional hazards (PH) model, because of its simplicity, is widely used to investigate the effect of covariates on the survival time. For Cox PH regression, under the framework of the random censorship model (RCM), Burr and Doss (1992), Lin, Fleming, and Wei (1994), and Lai and Su (26) developed SCBs for subject-specific survival and quantile functions; Zhang and Klein (21) developed SCBs for the difference of two survival functions. Dabrowska and Ho (2) developed SCBs for the transition probabilities in a Markov chain model with intensities specified by the Cox PH model. Wei and Schaubel (28) constructed SCBs under nonproportional hazards to compare hemodialysis and peritoneal dialysis of endstage renal disease patients, and debated the conclusions of previous studies. Gilbert et al. (22) analyzed data from a cholera vaccine study and developed SCBs for the log-hazard ratios of the cholera vaccine and placebo, using which they concluded that the vaccine lost its efficacy after 3 years and suggested that a more durable immune response needed to be developed in reformed cholera vaccines. Shen and Cheng (1999), Yin and Hu (24), and Lee and Hyun (211) also developed SCBs, but under the additive risk model. In the case of homogeneous right censored data, SCBs for cumulative hazard, survival and quantile functions have been extensively investigated; see the recent paper of Subramanian and Zhang (213), who developed one sample model-based SCBs for survival curves, for a list of past work. In this paper, we develop new subject-specific SCBs for survival curves from standard Cox PH regression assisted by semiparametric random censorship models (SRCMs), introduced by Dikta (1998) for the homogeneous case. Two-sample SCBs when the group-specific hazards are proportional are a special case, where the single covariate is the group indicator. We exploit the methodology proposed by Mondal and Subramanian (214), who incorporated SRCMs into standard Cox PH regression, to produce more efficient estimators of the regression parameter vector and baseline cumulative hazard function. In fact, the reported gain in efficiency should be expected, given the recent finding that the SRCM-based survival function estimator for the homogeneous case is semiparametric efficient (Dikta, 214). In simulations, the model assisted estimators provided a relative improvement amounting to between 5% and 8% over the partial likelihood and Breslow (1972) estimators. Furthermore, an illustration with recidivism data available in Rossi, Berk and Lenihan (198) showed how the new method can improve upon 2 S. Mondal and S. Subramanian

3 1 INTRODUCTION marginal decisions produced by a standard Cox analysis. Finally, they showed that model assisted Cox PH regression is more flexible than standard Cox in that it handles right censored data with missing censoring indicators (MCIs) without undue extra computational effort. Two real examples highlight the need to factor SRCMs into Cox PH regression. First, for the well-known Stanford heart transplant data, Yuan (25) noted that longer surviving patients are more likely to drop out from the study and thus are more probable to be censored. Secondly, for the recidivism data (Mondal and Subramanian, 214), it seems likely that a prisoner s susceptibility for re-arrest, within the follow-up period of one year since the last release, would be related to the number of prior convictions. Here, absence of re-arrest constitutes censoring and number of prior convictions is a covariate. Incorporation of SRCMs into Cox PH regression involves replacing the censoring indicator with a model-based estimate of its conditional expectation given the observed minimum and other covariates. For a parametric assist, the logistic or Cauchy link would usually suffice (Subramanian, 212; Subramanian and Zhang, 213; Mondal and Subramanian, 214), but alternatives such as the probit, complementary log-log, and generalized PH models (Dikta, 1998) may also be explored to arrive at a satisfactory specification. Plugging in the new estimators into a standard representation leads to ˆΛ(t, z), our proposed estimator of the subject-specific cumulative hazard function. For subject with covariate z, we derive an asymptotic representation of the process Ĥ( ) = n1/2 (ˆΛ(, z ) Λ(, z )), from which the weak convergence of Ĥ to H, a zero-mean Gaussian process, can be deduced. The method of SCB construction relies on the capability to obtain the upper-α quantile of the distribution of Ĥ t 2 t1, the supremum of Ĥ over [t 1, t 2 ] [, τ H ), where τ H is the right endpoint of the support of the distribution of X, see section 2. Unlike for the homogeneous case, a standard Brownian bridge approximation to its distribution, however, is not a viable option, since H does not have independent increments. Lin, Fleming, and Wei (1994) faced a similar problem and employed the Gaussian multiplier bootstrap (GMB), which we will do here as well. The GMB seeks to approximate the asymptotic distribution of a normalized process by computing random realizations of its asymptotic representation repeatedly. Note that an asymptotic representation is a sum of centered independent and identically distributed random variables/vectors scaled by n 1/2, see, in particular, Eqs. (2.5) and (2.6). Each summand is multiplied by an independently generated standard normal random deviate to obtain a perturbed asymptotic representation, constituting one random realization. The required threshold for computing the SCB is obtained as the 1(1 α) value of the ordered realized values, where α =.5 typically. The GMB was introduced by Lin, Wei and Ying (1993), and has been employed by Parzen, Wei, and Ying (1997), McKeague and Zhao (25), Cai, Zeng, and Pan (21), Zhou (21), Bücher and Dette (213), Yang (214), among others. We apply the GMB to the derived asymptotic representation of H to obtain thresholds needed for constructing our proposed equal-precision (EP) and Hall Wellner (HW) type SCBs. Simulation and sensitivity studies presented in section 3 show that our proposed SCBs provide approximately correct coverage. For censoring rates (CRs) between 2% and 5% the pro- 3 S. Mondal and S. Subramanian

4 2 PROPOSED SIMULTANEOUS CONFIDENCE BANDS posed SCBs gave a percent relative reduction in estimated average enclosed areas (EAEAs) and estimated average widths (EAWs) amounting to between 5% and 15% over their only competitor, namely, the standard Cox based SCBs developed by Lin, Fleming and Wei (1994). We also provide an extension that handles MCIs. This paper is organized as follows. In section 2 we present our proposed bands and the asymptotic results. In section 3, a set of simulation studies are presented to showcase the performance of our proposed bands. Next we provide two illustrations using real datasets. Technical details, such as the asymptotic validity of the GMB, are given in the appendix. 2 Proposed simultaneous confidence bands For the Cox PH model, the conditional hazard function of the failure time given the vector Z takes the form λ(t Z = z) = λ (t)e βt z, where β is a p 1 regression parameter and λ (t) is an unspecified baseline hazard function. Under the RCM setup, the observed data constitute n independent and identically distributed triplets (X i, δ i, Z i ), i = 1,..., n, where X = min(t, C) is the minimum of the failure and censoring times, δ is the censoring indicator (1 when uncensored and when censored), and Z is the p 1 vector of covariates. When there are MCIs, we introduce a missingness indicator ξ (1 when δ is observed and when missing), with the observed data then being the n independent and identically distributed quintuplets (X i, ξ i, σ i, Z i ), i = 1,..., n, where σ = ξδ. SRCMs attach a model for m(x, z). Specifically, m(x, z, θ ) = P (δ = 1 X = x, Z = z), where m(x, z, θ) has a known form with unknown θ IR k, the true value of which is θ. The likelihood for θ, given by, n {m(x i, Z i, θ)} σ i {1 m(x i, Z i, θ)} ξ i σ i, (2.1) reduces to the standard one when there are no MCIs, see Mondal and Subramanian (214). Let ˆθ denote the maximizer of (2.1) and write N i (t) = I(X i t), Y i (t) = I(X i t). Then ˆβ, the SRCM-adjusted estimator of β, solves the equation S n (β, ˆθ) =, where [ τh n j=1 S n (β, ˆθ) = m(t, Z i, ˆθ) Z i Y ] j(t)e βt Z j Z j n j=1 Y dn i (t). (2.2) j(t)e βt Z j Under appropriate regularity conditions, Mondal and Subramanian (214) proved the existence of ˆβ that converges to β in probability as n, and that the asymptotic variance of a T ˆβ is no greater than its RCM counterpart. They also proposed the alternate Breslow-type estimator of the baseline cumulative hazard function, given by ˆΛ (t) = t m(s, Z i, ˆθ) n j=1 Y j(s)eˆβ T Z j dn i (s), (2.3) and proved the weak convergence and asymptotic efficiency of its normalized version in D[, τ H ). 4 S. Mondal and S. Subramanian

5 2 PROPOSED SIMULTANEOUS CONFIDENCE BANDS Let Λ(t, z ) denote the conditional cumulative hazard of T given Z = z. The new SCBs for subject-specific survival functions will be based on ˆΛ(t, ˆβ, ˆθ, z ), the estimate of Λ(t, z ): ˆΛ(t, ˆβ, ˆθ, z ) ˆΛ(t, z ) = t m(u, Z i, ˆθ) exp(ˆβ T z ) n j=1 Y j(u)eˆβ T Z j dn i (u). (2.4) Let M denote the martingale associated with the counting process N, see Eq. (A.2). In Appendix A.2, we show that Ĥ(t) = n1/2 (ˆΛ(t, z ) Λ(t, z )) admits the asymptotic representation Ĥ(t) = L n,1 (t) + L n,2 (t) + o p (1), uniformly for t [, τ H ), L n,1 (t) = n 1/2 L n,2 (t) = n 1/2 τh m(u, Z i, θ )a i (t, u)dm i (u) (2.5) δ i m(w i, θ ) m(w i, θ ) m(w i, θ ) bt (t)i 1 Grad θ (m(w i, θ )), (2.6) W = (X, Z T ) T, m(, θ) = 1 m(, θ), and I 1, a(t, u), and b(t) are given by Eqs. (A.1), (A.12), and (A.13) respectively. The vector Grad θ (m(w, θ)) is defined in Appendix A.2. To apply the GMB, suppose that G 1,..., G n denote independent standard normal random variables, generated independent of the data. Using ˆθ and consistent estimates â(t, u) and ˆb(t), form Ĥ (t) = L n,1(t) + L n,2(t), where L n,1(t) = n 1/2 m(w i, ˆθ)â i (t, X i )I(X i < τ H )G i, (2.7) L n,2(t) = n 1/2 δ i m(w i, ˆθ) m(w i, ˆθ) m(w i, ˆθ) ˆb T (t)i 1 (ˆθ)Grad θ (m(w i, ˆθ))G i. (2.8) Let [t 1, t 2 ] [, τ H ). In Appendix A.3 we show that, for almost all sample sequences, Ĥ ( ) has the same distribution as Ĥ( ). By the continuous mapping theorem, for almost all sample sequences, Ĥ t 2 t1 has the same distribution as Ĥ t 2 t1. For < α < 1, let q α denote the upper-α quantile of the distribution of Ĥ t 2 t1. A linear 1(1 α)% SCB for Λ(t, z ) is given by ) (ˆΛ(t, z ) n 1/2 q α, ˆΛ(t, z ) + n 1/2 q α. To construct more elaborate SCBs let φ be a known function whose derivative φ is continuous and nonzero in [t 1, t 2 ] [, τ H ). Let ĝ(t, z ) denote a weight function that converges uniformly in [t 1, t 2 ] to the nonnegative bounded function g(t, z ). Defining ˆB(t) = n 1/2 ĝ(t, z )[φ(ˆλ(t, z )) φ(λ(t, z ))], by the functional delta method, the distribution of ˆB(t) can be approximated by ˆB (t) = ĝ(t, z )φ (ˆΛ(t, z ))Ĥ (t). 5 S. Mondal and S. Subramanian

6 3 NUMERICAL RESULTS Let φ(x) = log x. Let ˆσ 2 (t, z ) denote a consistent estimate of the asymptotic variance function of Ĥ(t), see Eq. (A.14). Taking ĝ(t, z ) = ˆΛ(t, z )/ˆσ(t, z ), let q 1,α denote the upperα quantile of the distribution of ˆB t 2 t1. A 1(1 α)% SCB for log ˆΛ over [t 1, t 2 ] is given by log ˆΛ(t, z ) n 1/2 q 1,αˆσ(t, z )/ˆΛ(t, z ). Since log Λ = log( log S), we obtain an EP-type band (Nair, 1984) as Ŝ(t, z ) exp (±n 1/2 q 1,αˆσ(t,z )/ˆΛ(t,z )). (2.9) Taking ĝ(t, z ) = ˆΛ(t, z )/(1 + ˆσ 2 (t, z )), let q 2,α be the upper-α quantile of the distribution of ˆB t 2 t1. Then, a 1(1 α)% HW type SCBs for S(t, z ) are given by (Transformed) Ŝ(t, z ) exp (±n 1/2 q 2,α (1+ˆσ 2 (t,z ))/ˆΛ(t,z )) (Untransformed) Ŝ(t, z ) n 1/2 1 + ˆσ 2 (t, z ) q 2,α. (2.1) ˆΛ(t, z ) Remark When there are MCIs (Subramanian, 24), replace L n,2 (t) given by Eq. (2.6) with L n,2 (t) = n 1/2 ξ i (δ i m(w i, θ )) m(w i, θ ) m(w i, θ ) bt 1 (t)ĩ Gradθ (m(w i, θ )). (2.11) Similar adjustment will be needed for L n,2(t) given by Eq. (2.8). 3 Numerical results In this section we report the results of simulation studies, based on sample size 1. Each method is first examined for performance efficacy, in terms of its empirical coverage probability (), which is the proportion of 1, SCBs that include S(t, z ) for all t [t 1, t 2 ]. When the s were found to be close to the nominal value of 95%, further comparisons between the methods were then based on the two measures estimated average enclosed area (EAEA) and estimated average width (EAW). Suppose that u j, j = 1,..., n are the ordered observed minimums. Let 1 m 1 < n denote the integer such that u m1 t 1 < u m1 +1, and let m 2 denote the integer such that m 1 < m 2 < n and u m1 +1 < u m2 < t 2 u m2 +1. Then, as in Subramanian and Zhang (213), the EAW and EAEA are defined over the interval [u m1, u m2 ] by EAEA = 1 k k { m2 j=m 1 l j uj } i, EAW = 1 k k { m2 j=m 1 l j Sj where l j denotes the width of the band computed at u j, k denotes the number of replications, uj = u j+1 u j and Sj = S j S j 1. The endpoints t 1 and t 2 were chosen as the.25 and.75 quantiles of the ordered values of X 1,..., X n. The critical values q 1,α and q 2,α in Eqs. (2.9) (2.1) were based on 1, bootstrap replications. 6 S. Mondal and S. Subramanian } i,

7 3.1 Censoring indicators always observed 3 NUMERICAL RESULTS To showcase specificity (turning in correct s when m is correctly specified), the case of SRCMs with no misspecification was considered first; here, the exact models that were used to generate the censoring indicators were fitted. Since, in practice, misspecification is often the norm than the exception, the sensitivity of the proposed SCBs to misspecified models was also considered; here the fitted model for m was different from that which generated the indicators. The same approach was also employed when there were MCIs. However, in the absence of a competitor for the MCIs scenario, only s are presented. We also performed a robustness study that used an ill-fitting nonparametric model for the missingness probability. 3.1 Censoring indicators always observed Absence of MCIs permits comparison of the proposed SCBs with the SCBs developed by Lin et. al (1994). We shall denote their EP-type SCBs as Cox EP, and their HW-type transformed and untransformed SCBs as Cox HW1 and Cox HW2 respectively Specificity study The one-dimensional covariate Z was taken to be uniformly distributed over (1, 2). The conditional event-time and censoring hazards given Z = z were exp(z) and exp(γz) respectively, where γ was selected to give censoring rates (CRs) between 2% and 5%. The baseline hazard was taken to be unity. With these specifications, the true model for the conditional probability m(x, z) = P (δ = 1 X = x, Z = z) turns out to be the logit model given by m(z, θ) = 1/(1 + exp( θ T w)), where θ = (θ, θ 1 ) T = (, 1 γ) T and w = (1, z) T. This true model was fitted and SCBs for S(t, z ), t [t 1, t 2 ], were computed at two covariate levels z = 1.2 and z = 1.8. The and the percentage relative reduction (PRR) in proposed EAEA and EAW values over the standard Cox counterparts are plotted as a function of the CR in figure 1. The PRR is given by the formula 1(Cox Proposed)/Cox. The upper panel of graphs are for z = 1.2 and the lower panel are for z = 1.8. Figure 1 here Both methods gave SCBs providing s close to the nominal 95%. The PRR in EAEA and EAW values provided by the proposed SCBs over their standard Cox counterparts, for both Z values, varied from 3.5% to 12%. Thus, the proposed SCBs are more informative than the standard Cox ones, especially for higher censoring rates Sensitivity study Here Z was taken as before, one dimensional and having the uniform distribution over (1, 2). The conditional event-time hazard given Z = z was taken as a Cox PH model with covariate log z, the scalar regression parameter β = 1, and the baseline hazard equal to 1. The conditional censoring time was taken as uniform over (, γz), where γ was chosen to give 7 S. Mondal and S. Subramanian

8 3.2 Censoring indicators missing at random 3 NUMERICAL RESULTS CRs between 1% and 4%. The true model for m is m(t, z) = z(γz t)/[z(γz t) + 1]. Misspecification was introduced by fitting the Cauchy link m(x, z, θ) = π arctan(θ + θ 1 x + θ 2 z) (3.12) to the generated censoring indicators. The results are shown in figure 2. Figure 2 here All the proposed SCBs still provided s close to the nominal 95%. The PRR in EAEA and EAW values provided by the proposed SCBs over their standard Cox counterparts varied between 2% and 7%, with increased reduction seen for higher CRs. Thus, even when the model was misspecified, the proposed SCBs are more informative than the standard Cox ones. 3.2 Censoring indicators missing at random Let π(x, z) denote the conditional expectation of the missingness indicator ξ given X = x and Z = z. We investigate four scenarios resulting from different combinations of specificity and sensitivity study for m and π. For example, we indicate specificity specificity to denote the case that there is no misspecification of m as well as π. In the absence of other methods to compare with the proposed, only s are reported for each scenario Specificity Specificity study Data were generated according to the scheme described in section and the true logit model was fitted for m as in section To impute missingness, the logit model π(x, z, α) = 1/(1 + exp( α(x + z))) was used, where α was chosen to give the two missingness rates (MRs) 1% and 4%. We fitted the true logit model π(x, z, θ) = 1/(1 + exp( θ T w)), where θ = (θ, θ 1, θ 2 ) T = (, α, α) T and w = (1, x, z) T. The graphs in the upper half of figure 3 give s as a function of the CR, for the two MRs 1% and 4% and for z = 1.2. The graphs in the lower half are for z = 1.8. The s are close to the nominal 95% Sensitivity Specificity study Figure 3 here Data were generated according to the scheme described in section and the Cauchy link, Eq. (3.12), was fitted for m. Thus, the model for m was misspecified. The model for π was not misspecified and fitting was as in section The plots of versus CR, shown in figure 4, indicate that the proposed SCBs provide s close to the nominal 95%. Figure 4 here 8 S. Mondal and S. Subramanian

9 3.3 A real example (Primary biliary cirrhosis data) 3 NUMERICAL RESULTS Specificity Sensitivity study Data were generated according to the scheme described in section Since m was correctly specified, its fitting was as in section We estimated π using the average of the ξ i s. This global nonparametric estimator was a misspecified model for π, which depends on both X and Z, see section Figure 5 shows the plot of versus CR for the two values z = 1.2 and z = 1.8, and the two MRs as described in section The plot indicates that even with the crude estimator of π the proposed SCBs provided s close to 95% Sensitivity Sensitivity study Figure 5 here Data were again generated according to the scenario described in section and the model for m is misspecified by fitting Cauchy link, Eq. (3.12), as before. Also, π was misspecified with nonparametric estimator given in section Figure 6 shows that s for the proposed SCBs are close to 95% except for untransformed bands, with z = 1.2, 4% MR and high CRs between 35% and 4%, when the s fell slightly below 93%. It must be noted, however, that such misspecification is unrealistic in practice. Parametric specifications of π (e.g., Cauchy or logit) will frequently root out such severe misspecification. Figure 6 here 3.3 A real example (Primary biliary cirrhosis data) We illustrate the proposed SCBs using the primary biliary cirrhosis (PBC) data from the Mayo clinic database. Lin et al. (1994) reported that the Cox PH model with five covariates, namely age, log(albumin), log(bilirubin), Oedema, and log(prothrombin time), denoted by Z 1,..., Z 5 respectively, provided a good fit for the PBC data. Our analysis was based on the 416 patients with complete data on the covariates Z 1,..., Z 5. We fitted several available binary response regression models for m, with link functions like Cauchy, logit, probit, and complementary log-log, and used the Akaike Information Criterion and the Bayesian Information Criterion to determine the most adequate one, which turned out to be the Cauchy model with covariates X, Z 1, Z 3, Z 4, Z 5. Note that Z 2 was found to be insignificant for the purposes of fitting m but was otherwise used in the analysis. We therefore fitted the model m(w, θ) =.5 + arctan(θ + θ 1 X + θ 2 Z 1 + θ 3 Z 3 + θ 4 Z 4 + θ 5 Z 5 ). (3.13) We have plotted in figure 7 the estimated subject-specific survival curves accompanied by 95% transformed HW- and EP-type SCBs for subjects with the covariate measurement z = (51, 3.4, 1.8,, 1.74) note that age was 51, log(albumin) was 3.4 and so on. We have plotted the standard Cox-based transformed HW- and EP-type bands for comparison. As noted by 9 S. Mondal and S. Subramanian

10 3.4 A second real example (Kidney transplant data) 3 NUMERICAL RESULTS Lin et al. (1994), the estimated subject-specific cumulative hazard is very small for t < 2, due to which the lower bounds of the HW-type SCBs were very low over that region. We therefore set the lower bound for all t < 2 equal to the lower bound calculated at t = 2. The proposed HW-type SCBs are more informative (in the sense of being more tight) over the entire time span. Comparison of the EP-type bands show that the new SCBs are narrower for most of the time span except for the years between 6 and 8, where they are similar to their standard Cox counterparts. The PRR values, given in table 1, also indicate that the proposed HWand EP-type SCBs performed better than their standard Cox counterparts. Figure 7 here 3.4 A second real example (Kidney transplant data) For our second illustration we chose the data on the death time of 863 patients who underwent kidney transplant at the Ohio State University Transplant center between 1982 and Patients were censored if they were lost to follow-up or alive till June 3, The data are described in section 1.7 of Klein and Moeschberger (23). Three covariates, namely, age, race and gender of patients were listed. A model checking in terms of Cox-Snell residuals ensured that Cox proportional hazard model fitted well for this data set and only age came out as the significant covariate whereas gender and race were not important. The final model we fit for m is the logit model, m(x, Z, θ) = 1/(1 + exp( θ θ 1 time θ 2 age)). (3.14) We have plotted the estimated subject specific survival curve of the patient with age 43 (mean age of all patients) with two kinds of SCBs, transformed EP and HW type, in figure 8. The estimated subject specific cumulative hazards were very small for t < 162 days and this made lower bounds of HW-type SCBs were very low over that region. We set the lower bound for t < 162 is equal to the lower bound calculated at t = 162. Similar adjustment was done at t = 14 for Cox counterpart. Figure 8 shows that both EP and HW bands are narrower than the Cox counterparts. Also table 1 gives the PRR values for two types of bands which indicates that proposed bands performed significantly better than their Cox counterparts. EP type HW type Data EAEA EAW EAEA EAW Primary biliary cirrhosis Kidney transplant Table 1: Percentage relative reduction (PRR) in estimated average enclosed area (EAEA) and estimated average width (EAW) of proposed SCBs over standard Cox. Figure 8 here 1 S. Mondal and S. Subramanian

11 4 CONCLUSION 4 Conclusion Model assisted Cox PH regression offers significant strengthening of standard Cox PH analysis. In the first part published recently in JMVA, it was shown that with a correct parametric assist, the new estimators provided improved inference for the Cox PH model parameters. This work, constituting the second part, offers further evidence of the superiority of model assisted Cox PH regression over standard Cox. We have developed several SCBs for subject-specific survival, the construction of which requires the application of the GMB. The proposed SCBs are more informative than existing ones, even when there may be some parametric misspecification. Indeed, they have the potential to strengthen marginal conclusions obtained by standard Cox based SCBs. Furthermore, they are easy to compute in the presence of MCIs, a facility not shared by the existing SCBs; and can be extended to produce SCBs for the difference of two survival functions (Zhang and Klein, 21). 11 S. Mondal and S. Subramanian

12 A.1 Preliminaries APPENDIX Appendix We assume all the regularity conditions given in part I and recall some notation and asymptotic representations; see Mondal and Subramanian (214) for the details. In addition, we will assume that ˆθ is strongly consistent, see theorem 2.1 and corollary 2.2 of Dikta (1998). We will also need a strengthening of condition A.2 in Mondal and Subramanian (214): AA.2. There exists a neighborhood B of β such that, for j =, 1, 2, sup S j (β, t) s j (β, t) = o(1) a.s. β B, t [,τ H ) In subsection A.1 we obtain an alternate representation for n 1/2 (ˆβ β ) needed for the proofs. A.1 Preliminaries Let W = (X, Z T ) T and m(w, θ) = 1 m(w, θ). Let D r (m(w, θ)) denote the partial derivative of m(w, θ) with respect to θ r. Write Grad(m(w, θ)) = [D 1 (m(w, θ)),..., D k (m(w, θ))] T and let J θ (t, z) = [(Grad(m(t, z, θ))) 2 ], where a 2 = aa T. Vectors and matrices will be in bold. The information in the absence of MCIs [see Eq. (4) on page 257 of Dikta (1998)] and the presence of MCIs [see Eq. (3.11) on page 134 of Subramanian, 24] are given by ( I(θ ) I = E J θ (W ) m(w, θ ) m(w, θ ) ) ; Ĩ(θ ) Ĩ = E ( π(w )J θ (W ) m(w, θ ) m(w, θ ) ). (A.1) Write N(t) = I(X t) and Y (t) = I(X t). Furthermore, define S (m) (β, t) = 1 [Y i (t)e βt Z i Z m i ], s (m) (β, t) = E[Y (t)e βt Z Z m ]; m =, 1, 2; n ( ) Z(β, t) = S(1) (β, t) S () (β, t) ; z(β, t) = s(1) (β, t) s () (β, t) ; v(β, t) = s(2) (β, t) s (1) 2 s () (β, t) (β, t). s () (β, t) Recall that Λ X Z (t), the conditional cumulative hazard function of X given Z, satisfies the relation Λ X Z (t) = t eβ Z λ (u)du/m(u, z, θ ). For each i = 1,..., n, M i (t) = N i (t) t Y i (u)e βt Z i λ (u) du m(u, Z i, θ ) (A.2) is a martingale with respect to the sigma-field F t = σ{z i, I(X i s), i = 1,..., n : s t}, see page 81 of Bickel et al. (1993). The corresponding predictable covariation process is given by M, M (t) = t Y (u)e βt Z λ (u) du. m(u, Z, θ ) With fully observed censoring indicators, ˆθ admits the asymptotic representation n 1/2 (ˆθ θ ) = n 1/2 I 1 δ i m(w i, θ )) m(w i, θ ) m(w i, θ ) Grad(m(W i, θ )) + o p (1). (A.3) (A.4) 12 S. Mondal and S. Subramanian

13 A.1 Preliminaries APPENDIX When there are MCIs, however, ˆθ admits the modified asymptotic representation given by n 1/2 (ˆθ θ ) = n 1/2 Ĩ 1 ξ i (δ i m(w i, θ )) m(w i, θ ) m(w i, θ ) Grad(m(W i, θ )) + o p (1). (A.5) Note that Eq. (A.25) of Mondal and Subramanian (214) gave an asymptotic representation for n 1/2 (ˆβ β ) that was helpful for proving efficiency in their proposition 1. Here, however, we derive an alternate, simpler, representation that we will need in our proofs, namely n 1/2 (ˆβ β ) = Σ 1 C [U n,1(β, θ ) + U n,2 (β, θ )] + o p (1), where Σ C is the asymptotic covariance matrix of n 1/2 (ˆβ C β ) and U n,1 (β, θ ) = n 1/2 δ i m(w i, θ ) m(w i, θ ) m(w i, θ ) B I 1 Grad θ (m(w i, θ )), U n,2 (β, θ ) = n 1/2 τh m(u, Z i, θ )(Z i z(β, u))dm i (u). (A.6) (A.7) (A.8) Note that ˆβ C is the Cox partial likelihood estimator of β and B is given by Eq. (A.9) below. To derive representation (A.6), it suffices to show from Eq. (A.26) of Mondal and Subramanian (214) that S n (β, θ ) = U n,2 (β, θ ) + o p (n 1/2 ). From Eqs. (2.2) and (A.2), we have S n (β, θ ) = + τh τh m(t, Z i, θ )(Z i Z(β, t))dm i (t) (Z i Z(β, t))y i (t)e βt Z i λ (t)dt. The first term equals U n,2 (β, θ ) + o p (1), the negligibility of the remainder term following from an application of Lemma 2 of Gilbert et al. (28); see the top of page 295 of Mondal and Subramanian (214) for a similar calculation. The second term is algebraically. For facilitating our derivations, we will also need the following quantities: [ τh ] B = E [Z z(β, t)][grad(m(w, θ ))] T dn(t), (A.9) [ t exp (β T ] z ) E (t, β ) = E s () (β, u) [z z(β, u)]m(u, Z, θ )dn(u), (A.1) [ t F (t, β ) = E a(t, u) = I(X t)eβt z s () (β, u) exp (β T z ) s () (β, u) Grad θm(u, Z, θ )dn(u) + E T (t, β )Σ 1 C (Z z(β, u)), ], (A.11) (A.12) b T (t) = F T (t, β ) + E T (t, β )Σ 1 C B, (A.13) σ 2 (t, z ) = E [ m 2 (W, θ )a 2 (t, X)I(X < τ H ) ] + b T (t)i 1 b(t). (A.14) 13 S. Mondal and S. Subramanian

14 A.2 Derivation of the limiting distribution APPENDIX A.2 Derivation of the limiting distribution Henceforth, where convenient, we shall suppress the appearance of z. For example, we write ˆΛ(t, ˆβ, ˆθ, z ) simply as ˆΛ(t, ˆβ, ˆθ). Note, therefore, that Λ(t, z ) Λ(t) = exp(β T z )Λ (t), where Λ (t) is the baseline cumulative hazard function. Let θ (β ) denote a value on the line segment joining ˆθ ) ) and θ (ˆβ and β ). Also, let Grad β (ˆΛ(t, β, θ) and Grad θ (ˆΛ(t, β, θ) denote the vector of partial derivatives of ˆΛ(t, β, θ) with respect to β and θ respectively. Taylor s expansion of ˆΛ(t, ˆβ, ˆθ) about θ and β yields the three terms, denoted for easy reference by T 1, T 2 and T 3, on the right hand side (RHS) of the following equation: ) ) (ˆΛ(t, β, θ ) Λ(t)) (ˆβ β ˆΛ(t, ˆβ, ˆθ) Λ(t) = + Grad θ (ˆΛ(t, ˆβ, θ ) + Grad β (ˆΛ(t, β, θ ), ) ), (ˆθ θ. (A.15) Let Λ (t) = t I{ n Y i(x) > }λ (x)dx. From page 3 of Fleming and Harrington (25), Λ (t) Λ (t) = o p (n 1/2 ), (A.16) uniformly for t [, τ H ). The first term on the RHS of Eq. (A.15) can be expressed as T 1 = t m(u, Z i, θ )e βt z dn ns () i (u) e βt z Λ (β, u) (t) + e βt z (Λ (t) Λ(t)), which, on applying Eq. (A.2) and then Eq. (A.16) twice, equals T 1 = t m(u, Z i, θ )e βt z dm ns () i (u) + o p (n 1/2 ). (β, u) Applying Lenglart s inequality to the main term on the RHS of the above equation, we obtain T 1 = ˆΛ(t, t m(u, Z i, θ )e βt z β, θ ) Λ(t) = dm ns () i (u) + o p (n 1/2 ). (A.17) (β, u) By the consistency of ˆθ and ˆβ, and the strong law of large numbers, it follows that Grad β (ˆΛ(t, β, θ )) = 1 t e βt z (z Z(β, u)) m(u, Z n S () i, θ )dn i (u) + o p (1), (A.18) (β, u) Grad θ (ˆΛ(t, ˆβ, θ )) = 1 t e βt z Grad θ (m(u, Z i, θ )) dn n S () i (u) + o p (1). (A.19) (β, u) Now apply conditions A2 and A3 of Mondal and Subramanian (214), to conclude from Eqs. (A.18) and (A.19) that Grad β (ˆΛ(t, β, θ )) = E (t, β ) + o p (1), (A.2) Grad θ (ˆΛ(t, ˆβ, θ )) = F (t, β ) + o p (1). (A.21) 14 S. Mondal and S. Subramanian

15 A.3 Large-sample justification of the GMB APPENDIX From Eqs. (A.6) and (A.2), it follows that the second term on the RHS of Eq. (A.15) equals T 2 = E T (t, β )Σ 1 C [U n,1(β, θ ) + U n,2 (β, θ )] + o p (n 1/2 ). (A.22) From Eqs. (A.4) and (A.21), it follows that the third term on the RHS of Eq. (A.15) equals T 3 = δ i m(w i, θ ) m(w i, θ ) m(w i, θ ) F T (t, β )I 1 Grad θ (m(w i, θ )) + o p (n 1/2 ). (A.23) Combining the dominant term on the RHS of Eq. (A.17) and the second dominant term on the RHS of Eq. (A.22), and using Eqs. (A.8) and (A.12), we get L n,1 (t) given by Eq. (2.5). Likewise, combining the dominant term of Eq. (A.23) and the first dominant term of Eq. (A.22), and using Eqs. (A.7) and (A.13), we get L n,2 (t) given by Eq. (2.6). We now compute the covariance function of the limiting Gaussian process. We have E (L n,1 (t 1 )L n,1 (t 2 )) = E = E = E τh τh τh It is straightforward to show that m 2 (u, Z, θ )a(t 1, u)a(t 2, u) d M(u), M(u) m 2 (u, Z, θ )a(t 1, u)a(t 2, u) Y (u)λ (u)e βt Z du m(u, Z, θ ) m 2 (u, Z, θ )a(t 1, u)a(t 2, u) d[n(u) M(u)] [by Eq. (A.3)] = E [ m 2 (W, θ )a(t 1, X)a(t 2, X)I(X < τ H ) ]. (A.24) E(L n,2 (t 1 )L n,2 (t 2 )) = b T (t 1 )I 1 b(t 2 ) (A.25) Using iterated conditional expectation with conditioning by W, the cross product term is : E(L n,1 (t 1 )L n,2 (t 2 )) = E(L n,1 (t 2 )L n,2 (t 1 )) = The final form of the covariance function of the limiting Gaussian process is given by C(t 1, t 2 ) = E [ m 2 (W, θ )a(t 1, X)a(t 2, X)I(X < τ H ) ] + b T (t 1 )I 1 b(t 2 ). (A.26) When t 1 = t 2 = t the above reduces to the variance function V (t), given by V (t) = C(t, t) = E [ m 2 (W, θ )a 2 (t, X)I(X < τ H ] ) + b T (t)i 1 b(t). (A.27) A.3 Large-sample justification of the GMB Note that ˆβ C, the partial likelihood estimator of β, is strongly consistent, see Tsiatis (1981). Asymptotic validity of the GMB requires strong consistency of ˆβ, not proved in Mondal and Subramanian (214). Here, in subsection A.3.1, we first show the strong consistency of ˆβ. We then show, in subsection A.3.2, that the GMB is asymptotically valid. 15 S. Mondal and S. Subramanian

16 A.3 Large-sample justification of the GMB APPENDIX A.3.1 Strong consistency of ˆβ Note that ˆβ maximizes the adjusted partial likelihood function ln (β, ˆθ) = 1 { ( τh )} m(t, Z i, n ˆθ) β T Z i log Y j (t)e βt Z j dn i (t). Since l n (β, ˆθ) is free of β, it follows that ˆβ maximizes l n (β, ˆθ) = l n (β, ˆθ) l n (β, ˆθ) 1 τh { ( )} m(t, Z i, n ˆθ) S (β β ) T () (β, t) Z i log dn S () i (t). (β, t) We first show that l n (β, ˆθ) l(β, θ ) = o(1) almost surely (a.s.), where τh [ [ ]] s l(β, θ ) = E m(t, Z, θ ) (β β ) T () (β, t) Z log dn(t). s () (β, t) By the triangle inequality, it suffices to introduce the intermediate function ln (β, θ ) = 1 τh [ [ ]] s m(t, Z i, θ ) (β β n ) T () (β, t) Z i log dn s () i (t), (β, t) and instead show that the following two equations hold: j=1 l n (β, ˆθ) l n (β, θ ) = o(1) a.s., (A.28) l n (β, θ ) l(β, θ ) = o(1) a.s. (A.29) First, Eq. (A.29) follows from strong law of large numbers. To prove Eq. (A.28), Taylor s expansion of l n (β, ˆθ) about θ yields ( ) l n (β, ˆθ) l n (β, θ ) = [l n (β, θ ) l n (β, θ )] + θ l n(β, θ) (ˆθ θ ), (A.3) θ=θ We apply conditions AA.2 and A.3 to deduce that the first term of Eq. (A.3) is o(1) a.s. To show that the second term is also o(1) a.s. for each β B, we shall assume that p = 1. Note that condition D.1 implies that Grad θ (m(x, z, θ )), (ˆθ θ ) km(x, z) ˆθ θ. Then, n Grad θ(l n (β, θ )), (ˆθ θ ) = Grad θ (m(x i, Z i, θ ) { k ˆθ θ constant + { (β β ) T Z i log sup t [,τ H ),β B ( )} S () (β, X i ) I(X S () i < τ H ), (ˆθ θ ) (β, X i ) ( ) } s () log (β, t) M(X s () i, Z i ) + o(n) = o(n) a.s., (β, t) 16 S. Mondal and S. Subramanian

17 A.3 Large-sample justification of the GMB APPENDIX by A.1 and D.1, together with the strong law of large numbers and strong consistency of ˆθ. Since the random concave function l n (β, ˆθ) converges pointwise to the concave function l(β, θ ) a.s., theorem 1.8 of Rockafellar (197) guarantees that the convergence a.s. is uniform in β [ M, M]. Also, the limit function l(β, θ ) has unique (global) maximum at β, see Mondal and Subramanian (214). We now follow the concluding part of the strong consistency proof given by Tsiatis (1981). Fix a δ > and note that for a δ-neighborhood around β we have l(β, θ ) l(β, θ ). From the uniform convergence proved above, it follows that l n (β, ˆθ) l n (β, ˆθ) converges a.s. to l(β, θ ) l(β, θ ). This implies that for almost all realizations there exists n, depending on the realization, such that for all n n any β on the boundary of the δ-neighborhood cannot be a local maximum. In turn, since l n (β, ˆθ) is continuous and differentiable over the interval β β δ, there must be a local maximum in the interior. That is, d l n (β, ˆθ)/dβ =, which is satisfied by ˆβ. This argument can be repeated for each shrinking δ to obtain a consistent sequence ˆβ converging a.s. to β. A.3.2 Asymptotic justification Recall from section 2 that Ĥ (t) = ˆL n,1(t) + ˆL n,2(t), where the RHS quantities are defined by Eqs. (2.7) and (2.8). Also, note that Ê(t), a strongly consistent estimate of E (t, β ), can be obtained by replacing β and θ in Eq. (A.18) with ˆβ and ˆθ respectively. Likewise, ˆF (t, ˆβ) can be obtained from Eq. (A.19). Furthermore, strong consistent estimates of B and b T (t) can be obtained from Eqs. (A.9) and (A.13) respectively. Finally, â i (t, u) will be obtained by replacing β, s () (β, u), E (t, β ), Σ C and z(β, u) in Eq. (A.12) with ˆβ, S () (ˆβ, u), Ê (t), ˆΣ C and Z(ˆβ, u) respectively. Let P G, E G, Cov G, Var G be the probability measure, expectation, covariance, and variance with respect to G, that is, conditioned on the sample (X i, δ i, Z i ) 1 i n. We have where Cov G (Ĥ (t 1 ), Ĥ (t 2 )) = E G (ˆL n,1(t 1 )ˆL n,1(t 2 )) + E G (ˆL n,2(t 1 )ˆL n,2(t 2 )) E G (ˆL n,1(t 1 )ˆL n,1(t 2 )) = 1 n E G (ˆL n,2(t 1 )ˆL n,2(t 2 )) = 1 n + E G (ˆL n,1(t 1 )ˆL n,2(t 2 )) + E G (ˆL n,1(t 2 )ˆL n,2(t 1 )), (A.31) m 2 (X i, Z i, ˆθ)â i (t 1, X i )â i (t 2, X i )I(X i < τ H ), ( δ i m(w i, ˆθ) m(w i, ˆθ) m(w i, ˆθ) ) 2 (ˆb T (t 1 )Î 1 (ˆθ)Grad θ (m(w i, ˆθ))) (ˆb T (t 2 )Î 1 (ˆθ)Grad θ (m(w i, ˆθ))), E G (ˆL n,1(t 1 )ˆL n,2(t 2 )) = 1 (δ i m(w i, ˆθ))m(W i, ˆθ) n m(w i, ˆθ) m(w i, ˆθ) ˆb T (t 2 )Î 1 (ˆθ)Grad θ (m(w i, ˆθ)) â i (t 1, X i )I(X i < τ H ). 17 S. Mondal and S. Subramanian

18 A.3 Large-sample justification of the GMB APPENDIX Strong consistency of ˆθ, ˆβ, Î(ˆθ), Ê(t), ˆF (t), ˆB and ˆb(t), coupled with condition AA.2 and the strong law of large numbers ensures that, for almost all samples, the first two terms on the RHS of Eq. (A.31) converge to the terms on the RHS of Eqs. (A.24) and (A.25) respectively. The same argument, followed by an application of iterated conditional expectation with conditioning by W, also ensures that the two cross-moment terms in Eq. (A.31) are each zero. Therefore, the process Ĥ ( ) has the same limiting covariance structure given by Eq. (A.26). It remains to show that the process Ĥ ( ) converges weakly to a zero-mean Gaussian process. We shall verify Lindeberg s condition and tightness. Recall that Ĥ (t) = L n,1(t) + L n,2(t), where L n,1(t) and L n,2(t) are given by Eqs. (2.7) and (2.8) respectively. Combining the terms on the RHS of Eqs. (2.7) and (2.8), we can write Ĥ (t) = n 1/2 n k i(t)g i. To verify Lindeberg s condition we need to show that, for almost all sample sequences, 1 n [ E G k 2 i (t)g 2 i I ( k i (t)g i n 1/2 ϵ )] as n. (A.32) Applying Cesaro means it suffices to show that each summand on the LHS of Eq. (A.32) tends to as n. By problem 2 on page 46 of Chung (21), however, it is enough to show that for almost all sample sequences P G ( ki (t)g i n 1/2 ϵ ) as n. This follows because P G ( ki (t)g i n 1/2 ϵ ) = 2Φ ( n 1/2 ϵ/ k i (t) ) as n. Recall that µ 4 = 3µ 2 2, where µ i is the ith central moment of the normal distribution. To verify tightness, apply formula (3) on p. 52 of Shorack and Wellner (1986): For t 1 < t 2, [ ] [Ĥ ] 4 4 lim E G (t 2 ) n Ĥ (t 1 ) = lim E G n 1/2 (k i (t 2 ) k i (t 1 ))G i n = lim n 3 lim n [ K n [ 1 n ] 2 (k i (t 2 ) k i (t 1 )) 2 ] 2 (k i (t 2 ) k i (t 1 )) 2. (A.33) where K 3. From Eq. (A.1), it will be convenient to define, for t 1 < t 2, Ê ((t 1, t 2 ]) = 1 exp (ˆβ T z )(z Z(ˆβ, X i )) m(x i, Z i, n ˆθ)I(t 1 < X i t 2 ). S () (ˆβ, X i ) The quantities ˆF ) ((t 1, t 2 ], ˆβ, ˆb ((t1, t 2 ]), â ((t 1, t 2 ], u) are likewise defined as above, based on Eqs. (A.11) (A.13). It can now be shown that the RHS of inequality (A.33) equals K (k i (t 2 ) k i (t 1 )) 2 = K [ m(x i, Z i, n n ˆθ)â i ((t 1, t 2 ], X i ) I(X i < τ H ) + δ i m(w i, ˆθ) ˆθ))] 2 m(w i, ˆθ) m(w i, ˆθ) ˆb T ([t 1, t 2 )) Î 1 (ˆθ)Grad θ (m(w i,, 18 S. Mondal and S. Subramanian

19 A.3 Large-sample justification of the GMB APPENDIX which equals the RHS of Eq. (A.27), after replacing the first argument of a and b there with (t 1, t 2 ]. Therefore, the LHS of Eq. (A.33) is finite. Tightness is verified. 19 S. Mondal and S. Subramanian

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23 GRAPHS Empirical Coverage EAEA EAW Cox EP Cox HW1 Cox HW2 PRR PRR Empirical Coverage EAEA EAW Cox EP Cox HW1 Cox HW2 PRR PRR Figure 1: MCIs absent and the model for m is correctly specified. The and relative reduction in EAEA and EAW values of proposed over standard Cox SCBs are plotted against censoring rate. 23 S. Mondal and S. Subramanian

24 GRAPHS Empirical Coverage EAEA EAW Cox EP Cox HW1 Cox HW2 PRR PRR Empirical Coverage EAEA EAW Cox EP Cox HW1 Cox HW2 PRR PRR Figure 2: MCIs absent and the model for m is misspecified. The and relative reduction in EAEA and EAW values of proposed over standard Cox SCBs are plotted against censoring rate. 24 S. Mondal and S. Subramanian

25 GRAPHS Z = 1.2, MR= 1% Z = 1.2, MR = 4% Z = 1.8, MR = 1% Z = 1.8, MR = 4% Figure 3: Specificity Specificity study. Correctly specified models for both m and missingness probability. The s of the proposed SCBs are plotted against censoring rate. 25 S. Mondal and S. Subramanian

26 GRAPHS Z = 1.2, MR= 1% Z = 1.2, MR= 4% Z = 1.8, MR= 1% Z = 1.8, MR= 4% Figure 4: Sensitivity Specificity study. Correctly specified model for the missingness probability but model for m is misspecified. The s of the proposed SCBs are plotted against censoring rate. 26 S. Mondal and S. Subramanian

27 GRAPHS Z = 1.2, MR = 1% Z = 1.2, MR = 4% Z = 1.8, MR = 1% Z = 1.8, MR = 4% Figure 5: Specificity Sensitivity study. Correctly specified model for m but the model for the missingness probability is misspecified. The s of the proposed SCBs are plotted against censoring rate. 27 S. Mondal and S. Subramanian