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TABLE 3 Frequency distribution between failure, withdrawal, and transplant healthier conditions. In these cases, it is unlikely that their expected survival times are the same as those for patients not chosen for transplantation, since the waiting times for surgery can be larger than the failure times associated with those experiencing the worse health conditions. Therefore, we investigate whether transplant and withdrawal are informative to failure in our analysis. Finally, since patients in the same facility share the same patterns and practices, their survival times are expected to be correlated and the frailty models, presented in Section 2, arise here as natural approaches for this data set.

Table 3 Frequency distribution between failure, withdrawal, and transplant healthier conditions. In these cases, it is unlikely that their expected survival times are the same as those for patients not chosen for transplantation, since the waiting times for surgery can be larger than the failure times associated with those experiencing the worse health conditions. Therefore, we investigate whether transplant and withdrawal are informative to failure in our analysis. Finally, since patients in the same facility share the same patterns and practices, their survival times are expected to be correlated and the frailty models, presented in Section 2, arise here as natural approaches for this data set.

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In the M step, the (r + 1)-th estimate of 0, denoted by 6+), is then obtained through the direct maximization of the likelihood function 0, (0|0) in (11). Since 62, 0 = (w™, B™) and © = Qw©, B© , b) appearing in (4) are orthogonal, they can be estimated separately. In order to estimate o7, consider the score function
In the M step, the (r + 1)-th estimate of 0, denoted by 6+), is then obtained through the direct maximization of the likelihood function 0, (0|0) in (11). Since 62, 0 = (w™, B™) and © = Qw©, B© , b) appearing in (4) are orthogonal, they can be estimated separately. In order to estimate o7, consider the score function
It is important to emphasize that the functions in (14), (16), and (17) are nonlinear, regardless of the choice of the baseline functions ACW) and AO Cfy). On the other hand, the linearity of (15) and (18) depends on which baseline function has been assumed for the problem.
It is important to emphasize that the functions in (14), (16), and (17) are nonlinear, regardless of the choice of the baseline functions ACW) and AO Cfy). On the other hand, the linearity of (15) and (18) depends on which baseline function has been assumed for the problem.
In terms of score functions, replacing (19) and (20) in (15) and (18) provides the following sets of nonlinear equations.
In terms of score functions, replacing (19) and (20) in (15) and (18) provides the following sets of nonlinear equations.
In this section, we present a simulation study to evaluate the models presented in the previous section. All models were fitted in R (R Core Team, 2017). The M step was implemented using the multiroot function of the package rootSolve (Soetaert, 2016), and the E step was carried out using the arms function available in the package dim (Petris, 2015). eee itera amaranth ttre inreieeiarnmans = Madama naman iain mamma ‘ities —~ nine, etaned: eins” iol With the purpose of assessing the impact of different correlation structures on the magnitude and the direction of bias, three scenarios were considered: positive correlation (¢ = 1), negative correlation (@ = —1), and null correlation (¢ = 0) between failure and dropout times. For each scenario, we have generated 500 data sets with a sample size of 200 subjects randomly dis- tributed in 40 clusters with 5 subjects per cluster. In the process to generate each data set, we have considered the Weibull baseline hazard function with parameters a7) = 2, y™ = 0.1 and a = 2, y© = 0.02 for the failure and dropout times, respectively. For the regression structure, we have assumed that x = xo = X, ~ Unif(—10, 10) and x a x = X, ~ Bern(0.5), and the following sets of regression coefficients: B'7) = (0.1, —1.4) and B© = (0.2, 1.2). Finally, the administrative censoring times were generated through the Unif(0, 10) distribution. The log-frailty terms were obtained from the Normal(0, 1) distribution. A short simulation study exploring the strength of the correlation as a function of ¢ is shown in the appendix. Bax weak, anawmemn aasoreahed akwoes hn wermnwened PVileavherll awn DEN wandalae eon Geren +A tis Aas core) CA ARAM RAE TR.
In this section, we present a simulation study to evaluate the models presented in the previous section. All models were fitted in R (R Core Team, 2017). The M step was implemented using the multiroot function of the package rootSolve (Soetaert, 2016), and the E step was carried out using the arms function available in the package dim (Petris, 2015). eee itera amaranth ttre inreieeiarnmans = Madama naman iain mamma ‘ities —~ nine, etaned: eins” iol With the purpose of assessing the impact of different correlation structures on the magnitude and the direction of bias, three scenarios were considered: positive correlation (¢ = 1), negative correlation (@ = —1), and null correlation (¢ = 0) between failure and dropout times. For each scenario, we have generated 500 data sets with a sample size of 200 subjects randomly dis- tributed in 40 clusters with 5 subjects per cluster. In the process to generate each data set, we have considered the Weibull baseline hazard function with parameters a7) = 2, y™ = 0.1 and a = 2, y© = 0.02 for the failure and dropout times, respectively. For the regression structure, we have assumed that x = xo = X, ~ Unif(—10, 10) and x a x = X, ~ Bern(0.5), and the following sets of regression coefficients: B'7) = (0.1, —1.4) and B© = (0.2, 1.2). Finally, the administrative censoring times were generated through the Unif(0, 10) distribution. The log-frailty terms were obtained from the Normal(0, 1) distribution. A short simulation study exploring the strength of the correlation as a function of ¢ is shown in the appendix. Bax weak, anawmemn aasoreahed akwoes hn wermnwened PVileavherll awn DEN wandalae eon Geren +A tis Aas core) CA ARAM RAE TR.
Positive correlation (¢ = 1); failure (37%), dropout (31%) administrative censoring (32%) TABLE 1 Results of the MC study for survival data with dependent censoring
Positive correlation (¢ = 1); failure (37%), dropout (31%) administrative censoring (32%) TABLE 1 Results of the MC study for survival data with dependent censoring
‘SE is the empirical standard error calculated from 500 parameter estimates. SEM is the mean of the 500 estimated standard errors. CP is the empirical coverage probability of the estimated 95% confidence intervals. RD is the relative difference calculated using the results in the “proposed model” column as the gold standard.
‘SE is the empirical standard error calculated from 500 parameter estimates. SEM is the mean of the 500 estimated standard errors. CP is the empirical coverage probability of the estimated 95% confidence intervals. RD is the relative difference calculated using the results in the “proposed model” column as the gold standard.
?SE is the empirical standard error calculated from 500 parameter estimates; SEM is the mean of the 500 estimated standard errors; CP is the empirical coverage probability of the estimated 95% confidence intervals; RD is the relative difference calculated using the results in the “Proposed model” column as the gold standard. TABLE 2 MC study for data generated with null correlation (¢ = 0); failure (39%), dropout (32%), administrative censoring (29%) A The analysis of both relative bias and relative difference in Tables | and 2 suggests that the empirical biases associated with the proposed models are considerably smaller than those obtained by the standard models under dependent censoring, resulting in large relative differences. On the other hand, when the censoring mechanism is independent, the empirical bias associated with the proposed and standard models are quite similar and, consequently, the empirical relative difference is close to zero. Based on this behavior, one can consider the strategy of using the relative difference to choose between the standard and the proposed models in real applications , where the censoring mechanism is often unknown.
?SE is the empirical standard error calculated from 500 parameter estimates; SEM is the mean of the 500 estimated standard errors; CP is the empirical coverage probability of the estimated 95% confidence intervals; RD is the relative difference calculated using the results in the “Proposed model” column as the gold standard. TABLE 2 MC study for data generated with null correlation (¢ = 0); failure (39%), dropout (32%), administrative censoring (29%) A The analysis of both relative bias and relative difference in Tables | and 2 suggests that the empirical biases associated with the proposed models are considerably smaller than those obtained by the standard models under dependent censoring, resulting in large relative differences. On the other hand, when the censoring mechanism is independent, the empirical bias associated with the proposed and standard models are quite similar and, consequently, the empirical relative difference is close to zero. Based on this behavior, one can consider the strategy of using the relative difference to choose between the standard and the proposed models in real applications , where the censoring mechanism is often unknown.
TABLE 4 Parameter, standard error and confidence interval estimates for the DOPPS data 'CI denotes the 95% confidence interval, RD(%) is calculated using the results in the “Proposed model” column as the gold standard.
TABLE 4 Parameter, standard error and confidence interval estimates for the DOPPS data 'CI denotes the 95% confidence interval, RD(%) is calculated using the results in the “Proposed model” column as the gold standard.
Considering the same configuration of true values used in Section 3, we simulated synthetic data taking into account different values of @, and evaluated three types of correlations (Pearson, Spearman, and Kendall). The resulting correlations as displayed in Figure Al. As it can be seen, the correlations increase approximately from —0.18 to 0.25 as ¢@ increases from —1 to 1.
Considering the same configuration of true values used in Section 3, we simulated synthetic data taking into account different values of @, and evaluated three types of correlations (Pearson, Spearman, and Kendall). The resulting correlations as displayed in Figure Al. As it can be seen, the correlations increase approximately from —0.18 to 0.25 as ¢@ increases from —1 to 1.
The other elements of the matrix are null.
The other elements of the matrix are null.
The second-order partial derivatives are given by The other elements of the matrix are null.
The second-order partial derivatives are given by The other elements of the matrix are null.
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