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In transplantation studies, often longitudinal measurements are collected for important markers prior to the actual transplantation. Using only the last available measurement as a baseline covariate in a survival model for the time to graft failure discards the whole longitudinal evolution. We propose a two-stage approach to handle this type of data sets using all available information. At the first stage, we summarize the longitudinal information with nonlinear mixed-effects model, and at the second stage, we include the Empirical Bayes estimates of the subject-specific parameters as predictors in the Cox model for the time to allograft failure. To take into account that the estimated subject-specific parameters are included in the model, we use a Monte Carlo approach and sample from the posterior distribution of the random effects given the observed data. Our proposal is exemplified on a study of the impact of renal resistance evolution on the graft survival.

Many medical studies involve analyzing responses together with event history data collected for each patient. A well-known and broadly studied example can be found in AIDS research, where CD4 cell counts taken at different time points are related to the time to death. These data need to be analyzed using a joint modeling approach in order to properly take into account the association between the longitudinal data and the event times. The requirement for a joint modeling approach is twofold. Namely, when focus is on the longitudinal outcome, events cause nonrandom dropout that needs to be accounted for in order to obtain valid inferences. When focus is on the event times, the longitudinal responses cannot be simply included in a relative risk model because they represent the output of an internal time-dependent covariate [

In this paper, we focus on a setting that shares some similarities with the standard joint modeling framework described above, but also has important differences. In particular, we are interested in the association between longitudinal responses taken before the actual followup for the time-to-event has been initiated. This setting is frequently encountered in transplantation studies, where patients in the waiting list provide a series of longitudinal outcomes that may be related to events occurring after transplantation. A standard analysis in transplantation studies is to ignore the longitudinal information and use only the last available measurement as a baseline covariate in a model for the allograft survival. It is, however, evident that such an approach discards valuable information. An alternative straightforward approach is to put all longitudinal measurements as covariates in the survival model. Nevertheless, there are several disadvantages with this approach. First, it would require spending many additional degrees of freedom, one for each of the longitudinal measurements. Second, patients with at least one missing longitudinal response need to be discarded, resulting in a great loss of efficiency. Finally, we may encounter multicollinearity problems due to the possibly high correlation between the longitudinal measurements at different time points.

Nowadays, when it comes to measuring the association between a longitudinal marker and an event-time outcome, a standard approach is to use the joint model postulated by Faucett and Thomas [

However, in contrast with the standard joint modeling setting, in our case (i.e., transplantation studies) the longitudinal responses do not constitute an endogenous time-dependent variable measured at the same period as the time to event. In particular, since the longitudinal measurements are collected prior to transplantation, occurrence of an event (i.e., graft failure after transplantation) does not cause nonrandom dropout in the longitudinal outcome. Nevertheless, the problem of measurement error still remains. Ignoring the measurement error affects not only the standard errors of the estimates of interest, but also it can cause attenuation of the coefficients towards zero [

Our research is motivated by data from an international prospective trial on kidney-transplant patients. The study has two arms, where in the first arm donors’ kidneys were administered to cold storage and in the second arm they were administered to machine perfusion (MP). The advantage of machine perfusion is the possibility of measuring different kidney’s parameters reflecting the state of the organ. One of the parameters of interest is renal resistance level (RR), which has been measured at 10 minutes, 30 minutes, 1 hour, 2 hours, 4 hours, and just before transplantation. Our aim here is to study the association of the renal resistance evolution profile with the risk of graft failure. The time of last measurement was different for different patients and often unknown exactly. However, based on the medical consult and visual inspection of the individual profiles, the last measurement was chosen to be taken at 6 hours for each patient.

The rest of the paper is organized as follows. Section

Let

For the event process, we postulate the standard relative risk model of the form:

In the particular transplantation setting that will be analyzed in the following study,

In the standard joint modeling framework, the estimation is typically based on maximum likelihood or Bayesian methods (MCMC). This proceeds under the following set of conditional independence assumptions:

Maximum likelihood methods use the joint likelihood and maximize the log-likelihood function

Here we will proceed under the Bayesian paradigm to estimate the model parameter. Under the conditional independence assumption (

The implementation of the Cox and piecewise constant hazard models is typically based on the counting process notation introduced by Andersen and Gill [

In order to assess convergence for the full Bayesian model, standard MCMC diagnostic plots were used. The burn-in size was set to 10000 iterations, which was chosen based on the visual inspection of the trace plots and confirmed by the the Raftery and Lewis diagnostics. The same number of iterations were used for constructing the summary statistics. Based on the autocorrelation plots, we have chosen every 30th iteration. Therefore, in total to obtain 10000 iterations for the final inferenc 300000 iterations were required after the burn-in part. Additionally, we run a second parallel chain and used Gelman and Rubin diagnostic plots to assess the convergence.

As mentioned in Section

simulate

simulate

calculate

Steps

Step

We apply the proposed two-stage procedure and a fully Bayesian approach to the transplantation study introduced in Section

The individual profiles of 50 randomly selected kidney donors are presented in Figure

Individual profiles of renal resistance level for 50 sampled donors.

In the first step of our analysis, we aim to describe the evolution of the renal resistance level in time. Motivated by both biological expectation and Figure

To accommodate for the shapes of RR evolutions observed in Figure

The analysis was performed using

The results for the nonlinear-mixed model are presented in Table

Parameter estimates, standard errors, and 95% confidence intervals from the nonlinear mixed model for RR.

Effect | Parameter | Estimate | SE | (95% CI) |
---|---|---|---|---|

Fixed effects | ||||

| ||||

Constant | 2.838 | 0.094 | (2.654; 3.022) | |

Donor Age | 0.005 | 0.002 | (0.001; 0.009) | |

Donor Type (HB versus NHB) | −0.102 | 0.068 | (−0.235; 0.031) | |

Donor Region 1 versus 3 | −0.078 | 0.065 | (−0.205; 0.049) | |

Donor Region 2 versus 3 | −0.072 | 0.072 | (−0.213; 0.069) | |

| ||||

Constant | 3.510 | 0.211 | (3.096; 3.924) | |

Donor Age | 0.004 | 0.004 | (−0.004; 0.012) | |

Donor Type (HB versus NHB) | −0.064 | 0.154 | (−0.365; 0.238) | |

Donor Region 1 versus 3 | −0.107 | 0.147 | (−0.395; 0.181) | |

Donor Region 2 versus 3 | 0.033 | 0.163 | (−0.286; 0.352) | |

| ||||

Constant | 1.010 | 0.186 | (0.645; 1.375) | |

Donor Age | 0.003 | 0.003 | (−0.003; 0.009) | |

Donor Type (HB versus NHB) | 0.402 | 0.130 | (0.147; 0.657) | |

Donor Region 1 versus 3 | −0.284 | 0.125 | (−0.529; −0.039) | |

Donor Region 2 versus 3 | −0.032 | 0.138 | (−0.302; 0.238) | |

Random effects | ||||

| 0.396 | |||

| 0.955 | |||

| 0.572 | |||

| 0.257 | |||

| −0.053 | |||

| 0.023 | |||

| 7.507 |

In the second step of the analysis, we applied at first the naive approach and used the estimates

Parameter estimates, SE, and 95% confidence/credibility intervals from proportional hazards Cox model for graft survival for plug-in method (a), sampled covariates (b), and fully Bayesian approach (c, d, e).

Graft survival, plug-in

Effect | Parameter | log(HR) | SE | (95% CI) |
---|---|---|---|---|

0.052 | 0.022 | (0.009; 0.095) | ||

−0.005 | 0.005 | (−0.015; 0.005) | ||

0.053 | 0.158 | (−0.257; 0.363) |

Graft survival, sampling two-stage

Effect | Parameter | log(HR) | SE | (95% CI) |
---|---|---|---|---|

0.053 | 0.024 | (0.006; 0.100) | ||

−0.006 | 0.008 | (−0.022; 0.010) | ||

0.055 | 0.185 | (−0.308; 0.418) |

Graft survival, fully Bayesian, Weibull

Effect | Parameter | log(HR) | SE | (95% HPD) |
---|---|---|---|---|

0.058 | 0.023 | (0.013; 0.103) | ||

−0.005 | 0.008 | (−0.020; 0.011) | ||

0.056 | 0.180 | (−0.299; 0.411) |

Graft survival, fully Bayesian, Cox

Effect | Parameter | log(HR) | SE | (95% HPD) |
---|---|---|---|---|

0.056 | 0.023 | (0.010; 0.101) | ||

−0.006 | 0.008 | (−0.022; 0.010) | ||

0.055 | 0.171 | (−0.280; 0.390) |

Graft survival, fully Bayesian, piecewise constant hazard

Effect | Parameter | log(HR) | SE | (95% HPD) |
---|---|---|---|---|

0.054 | 0.024 | (0.007; 0.102) | ||

−0.005 | 0.009 | (−0.022; 0.012) | ||

0.054 | 0.179 | (−0.297; 0.405) |

We observe that the estimates are close or almost identical as in plug-in model. SE of the Cox regression coefficients for the model with sampling are greater than SE from the plug-in model in Table

Even though it is hard to compare exactly the computational time for the two approaches, the rough estimation of the total computational time needed to estimate and assess the convergence (2 chains) of the full Bayesian model was about 21.6 hours and depended on the implemented survival model. A similar computational time was needed for the full Bayesian model with the Cox survival model and piecewise constant hazard model with a slightly more time needed for the parametric Weibull model. For the two-stage approach, the total computational time was about 10 hours using the Intel(R) Core(TM)2 Duo T9300 2.5 GHz and 3.5 GB RAM.

We have conducted a number of simulations to investigate the performance of our proposed two-stage method. In particular, we compared the plug-in method that uses the Empirical Bayes estimates

For the longitudinal part, the data were simulated for 500 patients from model (

In Table

Bias and residual mean squared error (RMSE) for the method with true

7 time points | ||||||

20 | 40 | |||||

0.5 | ||||||

GS | 0.00 (0.04) | −0.02 (0.03) | 0.01 (0.03) | −0.01 (0.04) | 0.02 (0.04) | −0.02 (0.04) |

plug-in | −0.05 (0.06) | −0.04 (0.05) | 0.06 (0.07) | −0.08 (0.09) | −0.04 (0.05) | 0.12 (0.12) |

sampling | −0.04 (0.05) | 0.03 (0.08) | 0.02 (0.07) | −0.05 (0.11) | −0.02 (0.06) | 0.03 (0.10) |

Bayesian | −0.03 (0.04) | −0.02 (0.04) | 0.01 (0.02) | −0.01 (0.04) | −0.02 (0.04) | 0.02 (0.07) |

1 | ||||||

GS | 0.04 (0.05) | 0.04 (0.07) | −0.03 (0.07) | −0.05 (0.09) | −0.04 (0.06) | −0.03 (0.05) |

plug-in | −0.07 (0.08) | −0.08 (0.09) | 0.07 (0.09) | −0.10 (0.12) | −0.08 (0.09) | 0.08 (0.11) |

sampling | −0.07 (0.09) | −0.06 (0.10) | −0.02 (0.11) | −0.05 (0.12) | 0.05 (0.11) | −0.03 (0.12) |

Bayesian | 0.01 (0.03) | 0.05 (0.06) | −0.03 (0.07) | 0.05 (0.06) | 0.04 (0.06) | −0.04 (0.07) |

5 | ||||||

GS | 0.04 (0.06) | 0.05 (0.06) | 0.04 (0.08) | 0.05 (0.10) | 0.01 (0.05) | −0.02 (0.06) |

plug-in | −0.09 (0.10) | −0.10 (0.11) | 0.08 (0.11) | −0.20 (0.22) | −0.21 (0.22) | 0.14 (0.18) |

sampling | 0.08 (0.13) | 0.06 (0.12) | −0.05 (0.12) | 0.07 (0.14) | −0.05 (0.13) | −0.11 (0.18) |

Bayesian | 0.09 (0.10) | 0.05 (0.09) | −0.09 (0.10) | −0.09 (0.10) | 0.08 (0.12) | −0.12 (0.18) |

14 time points | ||||||

20 | 40 | |||||

0.5 | ||||||

GS | −0.03 (0.03) | 0.00 (0.02) | −0.02 (0.03) | 0.02 (0.03) | −0.03 (0.04) | 0.02 (0.04) |

plug-in | −0.02 (0.03) | −0.03 (0.04) | 0.05 (0.07) | −0.02 (0.04) | −0.03 (0.04) | 0.05 (0.06) |

sampling | 0.03 (0.04) | 0.02 (0.06) | 0.02 (0.07) | 0.02 (0.04) | 0.04 (0.05) | 0.02 (0.08) |

Bayesian | −0.03 (0.04) | −0.02 (0.04) | −0.02 (0.04) | 0.02 (0.04) | 0.03 (0.04) | −0.05 (0.06) |

1 | ||||||

GS | −0.03 (0.04) | −0.03 (0.04) | −0.01 (0.03) | 0.00 (0.03) | −0.02 (0.04) | 0.05 (0.06) |

plug-in | −0.09 (0.06) | −0.05 (0.06) | 0.06 (0.07) | −0.02 (0.04) | −0.04 (0.05) | 0.11 (0.11) |

sampling | 0.04 (0.08) | 0.02 (0.08) | −0.02 (0.07) | −0.02 (0.04) | −0.02 (0.08) | 0.04 (0.09) |

Bayesian | −0.03 (0.04) | 0.04 (0.05) | −0.03 (0.05) | 0.02 (0.04) | 0.03 (0.05) | 0.06 (0.07) |

5 | ||||||

GS | −0.03 (0.04) | −0.03 (0.04) | 0.01 (0.04) | −0.01 (0.04) | −0.02 (0.04) | 0.05 (0.06) |

plug-in | −0.05 (0.06) | −0.10 (0.11) | 0.07 (0.09) | −0.10 (0.11) | −0.09 (0.10) | 0.11 (0.12) |

sampling | 0.04 (0.09) | 0.04 (0.11) | −0.05 (0.11) | 0.07 (0.12) | 0.05 (0.11) | −0.06 (0.16) |

Bayesian | 0.03 (0.05) | 0.03 (0.08) | −0.05 (0.10) | 0.02 (0.04) | 0.06 (0.10) | −0.09 (0.14) |

Since the calculations for the simulation study were highly computationally intensive, we have used the cluster with about 20 nodes with AMD Quad-Core Opteron 835X, 4 × 2 GHz, and 16 GB RAM per node. The analysis for the the 200 simulated data sets for a single scenario took about 65.5 hours using the Bayesian approach and 31.2 hours using the two-stage approach.

We have proposed a two-stage method that can be used in a joint analysis of longitudinal and time to event data when the longitudinal data are collected before the start of followup for survival, and the interest is in estimation of the impact of longitudinal profiles on survival. The modeling strategy is based on specification of two separate submodels for the longitudinal and time to event data. First, the longitudinal outcome is modeled using a random effects model. Then the survival outcome is modeled using the Empirical Bayes estimates of the subject-specific effects from the first stage. The variability of the estimates from the first stage is properly taken into account using a Monte Carlo approach by sampling from the posterior distribution of the random effects given the data.

As it was demonstrated, ignoring the additional variability of the subject-specific estimates when modeling survival leads to some bias, and in particular, attenuates the regression coefficients towards zero [

Since the sampling in the proposed method relies strongly on the results of the first part, the accurate estimation of all parameters of nonlinear mixed model is a key feature and should be performed carefully. This can be problematic when the deviation from normality of the random effects, is suspected. However, it was shown that even for the nonnormal random effects one can still use a standard software such as

In comparison with the two-stage approach proposed by Tsiatis et al. [

Another advantage of the proposed two-stage method is that it can be easily generalized from survival to other types of models as it was applied for the binary Delayed Graft Failure (DGF) indicator (results not shown) in the analysis of the renal data. For that purpose in the second step of the two-stage procedure, the survival model was replaced by the logistic regression model for the DGF indicator. The first stage of the proposed approach could be also modified allowing for other types of longitudinal response and other types of mixed models. Therefore, instead of using a nonlinear mixed model a linear mixed model or generalized linear mixed model (GLMMs) can be considered depending on the type and the shape of the longitudinal response. In the presented real data example, we have chosen the three parameters that described the evolution of the longitudinal response. However, for the particular question of interest, one can easily choose the most convenient parametrization for the longitudinal model and use the selected parameters to analyze the nonlongitudinal response in the second stage.

The authors thank J. M. Smits from Eurotransplant International Foundation in Leiden for providing the data set analyzed in the paper and for the medical consult regarding the application of the proposed method.