Web-based Supplementary Material for A Two-Part Joint. Model for the Analysis of Survival and Longitudinal Binary. Data with excess Zeros

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1 Web-based Supplementary Material for A Two-Part Joint Model for the Analysis of Survival and Longitudinal Binary Data with excess Zeros Dimitris Rizopoulos, 1 Geert Verbeke, 1 Emmanuel Lesaffre 1 and Yves Vanrenterghem 2 1 Biostatistical Centre, Catholic University of Leuven, Belgium 2 Department of Nephrology, University Hospital Gasthuisberg, Leuven, Belgium 1. Web Simulation Study Details 1.1 Web Simulation Study Design A simulation study has been performed to empirically investigate the performance of the maximum likelihood estimates under the proposed two-part joint model in finite samples as well as the effect of copula misspecification. In particular, a two group comparison is considered with two choices for the sample size, a moderate one with n = 200 and a small one with n = 50. The submodels specification for the two-part joint model is as follows. First, the degenerate group indicator d i is simulated according to the logistic model logit {P r(d i = 1; θ d )} = θ d0 + θ d1 T i, where T i denotes the treatment indicator, and (θ d0, θ d1 ) = (1, 0.8). Second, for the survival process a Weibull model with a frailty term is assumed log T i = γ 0 + γ 1 T i + γ d d i + b ti + σ t ε i, where (γ 0, γ 1, γ d ) = (0.5, 1.5, 0.5), σ t = 0.5, and ε i follows an extreme value distribution. The censoring mechanism follows an exponential distribution with mean 11, which results in 30% censoring on average. Third, the non-degenerate part of the longitudinal process has the form logit {P r(y ij = 1 b yi, d i = 1)} = β 0 + β 1 t ij + β 2 T i t ij + b y0i + t ij b y1i, 1

2 where t ij denotes the time points at which the y ij measurements are taken, T i t ij is the interaction term between T i and t ij, and (β 0, β 1, β 2 ) = (0.1, 1.5, 1). The maximum number of repeated measurements per individual is 20, with t ij = seq(0, 4, 20), where seq(a, b, c) denotes a regular sequence from a to b of length c (e.g., seq(0, 2, 5) = 0, 0.5, 1, 1.5, 2). Taking into account the censoring and the degenerate individuals (i.e., {i (1, 2,...) : d i = 0}), the average number of measurements per individual is 9.1 with standard deviation 7.6 measurements. Finally, for the random-effects model the following scenarios are considered: (i) the normal copula, with correlation α = sin(0.5π/2) = 0.707; (ii) the Student s-t copula, with 4 degrees of freedom and correlation α = sin(0.5π/2) = 0.707; (iii) the Clayton copula, with parameter α = 2; (iv) the Frank copula, with parameter α = The value of α for each copula has been chosen such that the association between b yi and b ti equals 0.5 in terms of Kendall s-τ. Moreover, both marginal distributions H y (b yi ; ω y ) and H t (b ti ; ω t ) are taken to be normal with zero mean, and with covariance matrix parameters ω y = {σby0 2 = var(b y0i) = 4; σby1 2 = var(b y1i) = 1; cor by = cor(b y0i, b y1i ) = 0.4} and variance ωt 2 = 4, respectively. For each scenario and each sample size setting 500 data-sets have been simulated. 1.2 Web Simulation Analysis Models Each simulated data-set has been analyzed under four two-part joint models. In particular, for the degenerate component, the event process and the longitudinal measurement process the correct model specifications, as described in the previous section, are assumed. However, for the random-effects the normal, the Student t-t (df = 4), the Clayton and the Frank copulas are fitted for each data-set. Thus, under each scenario (i) to (iv), the true and three misspecified random-effects models are fitted. The MLEs for the joint models parameters are obtained using the EM algorithm described in Section 3 of the paper, using 13 quadrature points for the Gauss-Hermite rule. 1.3 Web Simulation Results For the data-sets considered in the simulation study the average computer time for fitting each model was min. (standard deviation min.) for n = 200, and min. (standard deviation min.) for n = 50. Computations have been performed in 2

3 R version 2.4.1, on an AMD Opteron Cluster, consisting of 164 dual Opteron250 servers running Linux (kernel version ), with 2GB RAM, several nodes with 16GB RAM, and 4 to 8 CPUs with frequencies varying from 1.8 to 2.6 GHz. Thus, we feel that our proposed model requires reasonable computing time considering its complexity. The bias and root mean square error for each parameter of the two-part joint model, presented in Web Section 1.1, are given in Web Tables 1 to 4 for sample size n = 200, and Web Tables 5 to 8 for sample size n = 50. For all scenarios we observe that the true model has a relatively good performance compared to the joint models with misspecified copula function. Moreover, the performance of the two elliptical copulas (i.e., normal and Student s-t) seems to be better than the one of the Archimedean ones (i.e., Clayton and Frank). For example, in Web Table 3 we observe that the Clayton copula, even though is the correct one, shows greater bias for γ d, β 1 and β 2 than the elliptical copulas; however, for the variance components it performs better than the other copulas. In addition, some sensitivity of the results regarding the copula choice is apparent in the parameter estimates of all three submodels. For instance, in Web Table 1 the estimated association parameter τ has less bias and root mean square error for the misspecified Student s-t copula rather than for the normal copula, which is the true one. Another example can be found in Web Table 5 in which the Clayton and Frank copulas provide better estimates for γ 0 than the true normal copula. For sample size n = 50, we observed the same behaviour for the two-part joint models but, as expected, with larger values for the root mean square error, and in some cases with also more bias. Finally, the sensitivity regarding the choice of copula is also evident in the number of times each copula has been selected as the best fitting copula according to the AIC, for each scenario. Even though the AIC the majority of times selects the true model, the number of times it fails to do so is not negligible. These observations reinforce our statement, presented in Section 2.2 of the paper, that sensitivity analysis is necessary in the joint modelling framework in order to investigate the impact of the modelling assumptions. 3

4 2. Web Appendices 2.1 Web Appendix A The integrals involved in the specification of the E-step do not have a closed form solution and thus are approximated using the Gauss-Hermite quadrature rule. In particular, E {A(b yi, b ti ) y i, T i, δ i, d i } = A(b yi, b ti ) p(b yi, b ti y i, T i, δ i, d i ) db yi db ti 2 q/2 h t A(t 2) p(t 2 yi, T i, δ i, d i ) exp( t 2 ), t 1 t q where q denotes the integral dimension, is used as shorthand for, t = t 1 t q t 1 t q (t 1,..., t q ) are the abscissas with corresponding weights h t, and 2 denotes the square of the Euclidean distance. A known problem of the Gaussian-Hermite rule (Pinheiro and Bates, 1995), is that it assumes that the main mass of the integrand is around zero, which might not be the case for certain individuals. The adaptive Gauss-Hermite rule solves this problem by centering and rescaling the integrand in each iteration, increasing however dramatically the computational burden. In order to avoid both the poor approximation of the simple Gauss-Hermite rule and the computational complexity of the adaptive rule, we use the Empirical Bayes (EB) estimates and their standard error from the ignorable models, to center and scale the integrand. Even though this procedure is not a fully adaptive rule, we expect that the ignorable EB estimates provide a good approximation to the patients standing in the random-effects dimension, resulting in an acceptable integral approximation with a moderate number of quadrature points. 2.2 Web Appendix B Here we present the form of u/ ω y = H y (b yi ; ω y )/ ω y, used in the M-step of the EM algorithm, where H y (b yi ; ω y ) denotes the normal cumulative distribution function (cdf) with zero mean and variance components parameterized through ω y. We present two cases; univariate and bivariate random-effects. First, in the univariate case, with b yi representing a random-intercepts term, we get ω y H y (b yi ; ω y ) = b yi ω y p(b yi ; ω y ), 4

5 where p(b yi ; ω y ) denotes the normal probability density function with zero mean and standard deviation ω y. Second, in the bivariate case, where b yi = (b y1i, b y2i ), we use the parameterization of the bivariate normal cdf considered in Drezner and Wesolowsky (1989): H y (b y1i, b y2i ; ω y1, ω y2, ρ) = (ω y1ω y2 ) 1 2π 1 ρ 2 = H y (b y1i ; ω y1 )H y (b y2i ; ω y2 ) + 1 2π b y1i b y2i ρ 0 exp { h2 1 /ω2 y1 + h2 2 /ω2 y2 2ρh 1h 2 /ω y1 ω y2 2(1 ρ 2 ) } dh 1 dh 2 exp{ (b 2 y1i /ω2 y1 + b2 y2i /ω2 y2 2rb y1ib y2i /ω y1 ω y2 )/2(1 r 2 )} dr, 1 r 2 which leads to the following expressions for the partial derivatives with respect to the correlation ρ, and the standard deviations ω y1, and ω y2 H y (b y1i, b y2i ; ω y1, ω y2, ρ) ρ = exp{ (b2 y1i /ω2 y1 + b2 y2i /ω2 y2 2ρb y1ib y2i /ω y1 ω y2 )/2(1 ρ 2 )} 2π 1 ρ 2, H y (b y1i, b y2i ; ω y1, ω y2, ρ) + ρ 0 ω y1 = b y1i ω y1 p(b y1i ; ω y1 )H y (b y2i ; ω y2 ) B(r) exp{ (b 2 y1i /ω2 y1 + b2 y2i /ω2 y2 2rb y1ib y2i /ω y1 ω y2 )/2(1 r 2 )} 2π dr, 1 r 2 where B(r) = ( 2b y1i by1i ωy1 rb ) y2i, 2 1 r 2 ω y1 ω y2 and H y (b y1i, b y2i ; ω y1, ω y2, ρ)/ ω y2 is derived analogously. The integral over r can be easily approximated using an adaptive Gauss-Kronrod rule (Piessens et al., 1983). 2.3 Web Appendix C The form of l{(γ, γ d )} and l(σ t ) under the Weibull model is l{(γ, γ d )} = σ 1 t l(σ t ) = σ 1 t n {exp( ζ i ) δ i }ẅ i i=1 n à i (1 + ζ)δ i, i=1 where ζ i = (log T i w i γ d i γ d b ti )/σ t, with b ti = b ti p(b yi y i, T i, d i ) db ti, and A i = ζ i exp(ζ i ). 5

6 References Drezner, Z. and Wesolowsky, G. (1989). On the computation of the bivariate normal integral. Journal of Statistical Computation and Simulation 35, Piessens, R., dedoncker Kapenga, E., Uberhuber, C. and Kahaner, D. (1983). Quadpack: a Subroutine Package for Automatic Integration. Springer, New York. Pinheiro, J. and Bates, D. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics 4,

7 Web Table 1 Simulation results based on 500 data-sets under the normal scenario (i) with sample size n = 200. The bias and root mean square error for each parameter are presented. The line θ d θ d γ γ γ d σ t β β β σ by σ by cor by ω t τ AIC 236 (47.2%) 87 (17.4%) 103 (20.6%) 74 (14.8%) 7

8 Web Table 2 Simulation results based on 500 data-sets under the Student s-t scenario (ii) with sample size n = 200. The bias and root mean square error for each parameter are presented. The line θ d θ d γ γ γ d σ t β β β σ by σ by cor by ω t τ AIC 128 (25.6%) 223 (44.6%) 84 (16.8%) 65 (13.0%) 8

9 Web Table 3 Simulation results based on 500 data-sets under the Clayton scenario (iii) with sample size n = 200. The bias and root mean square error for each parameter are presented. The line θ d θ d γ γ γ d σ t β β β σ by σ by cor by ω t τ AIC 54 (10.8%) 31 (6.2%) 368 (73.6%) 47 (9.4%) 9

10 Web Table 4 Simulation results based on 500 data-sets under the Frank scenario (iv) with sample size n = 200. The bias and root mean square error for each parameter are presented. The line θ d θ d γ γ γ d σ t β β β σ by σ by cor by ω t τ AIC 59 (11.8%) 58 (11.6%) 76 (15.2%) 307 (61.4%) 10

11 Web Table 5 Simulation results based on 500 data-sets under the normal scenario (i) with sample size n = 50. The bias and root mean square error for each parameter are presented. The line θ d θ d γ γ γ d σ t β β β σ by σ by cor by ω t τ AIC 159 (31.8%) 89 (17.8%) 143 (28.6%) 109 (21.8%) 11

12 Web Table 6 Simulation results based on 500 data-sets under the Student s-t scenario (ii) with sample size n = 50. The bias and root mean square error for each parameter are presented. The line θ d θ d γ γ γ d σ t β β β σ by σ by cor by ω t τ AIC 120 (24.0%) 115 (23.0%) 141 (28.2%) 124 (24.8%) 12

13 Web Table 7 Simulation results based on 500 data-sets under the Clayton scenario (iii) with sample size n = 50. The bias and root mean square error for each parameter are presented. The line θ d θ d γ γ γ d σ t β β β σ by σ by cor by ω t τ AIC 59 (11.8%) 39 (7.8%) 293 (58.6%) 109 (21.8%) 13

14 Web Table 8 Simulation results based on 500 data-sets under the Frank scenario (iv) with sample size n = 50. The bias and root mean square error for each parameter are presented. The line θ d θ d γ γ γ d σ t β β β σ by σ by cor by ω t τ AIC 55 (11.0%) 53 (10.6%) 169 (33.8%) 223 (44.6%) 14