Comparison of a time-varying covariate model and a joint model of time-to-event outcomes in the presence of measurement error and interval censoring: application to kidney transplantation

Models M1-M4 were fitted to the kidney transplant data. The three joint models (M1, M2, M3) had a longitudinal sub-model with TAC as the outcome, a fixed intercept and slope for time (f(t_ij)=b₀+b₁t_ij)) and a random intercept and/or slope for time (depending on the model) for variations across individuals. Spline terms for time were tested, but DIC and WAIC indicated that a linear trend was sufficient. For the survival sub-models, various baseline hazard functions were tested (exponential, Weibull, gamma, piecewise constant) and the Weibull yielded the best model fit (h₀(t)=αt^α−1, where α is the Weibull shape parameter). The Weibull distribution makes intuitive sense for this data, as the hazard can flexibly be increasing or decreasing over time and the risk of dnDSA is thought to decrease as time from transplant progresses. The parameters from the Weibull model can easily be interpreted as the change in log-relative risk. All survival sub-models had time to dnDSA as the interval censored outcome, random intercept(s) associated with the longitudinal sub-model, and a vector of baseline covariates: age, ethnicity, and number of HLA mismatches. See Table 1 for the description of these covariates. Baseline covariates were tested in both the longitudinal and survival sub-models, but none were significant predictors in the longitudinal sub-model, so they were removed.

The interval censored survival model with a time-varying covariate (M4) had dnDSA as the outcome, TAC as the time-varying covariate, and the same baseline covariates as listed above. After considering other baseline hazard functions for M4, we chose the Weibull due to model fit and to compare to M1, so \(\phantom {\dot {i}\!}h_{0tvc}(t)=\alpha _{tvc}t^{\alpha _{tvc}-1}\) where α_tvc is the Weibull shape parameter.

Table 2 displays the results of model comparisons based on the DIC and the WAIC. The survival sub-model fits in M1-M3 can be directly compared with the model fit for M4. The best fitting survival model according to both DIC and WAIC is Model 1, which shares both the random intercept and slope from the longitudinal model with the random intercept from the survival model.

Table 3 presents the results from the final model, M1, and the time-varying covariate model, M4. The longitudinal portion of M1 shows that on average, individuals start at a TAC level of about 7 ng/ml. For every one month post-transplant, the average individual’s TAC levels decrease by 0.05 ng/ml. There is a significant negative correlation between the random intercept and slope, ρ=−0.38, indicating that individuals with higher starting TAC have a lower slope. The survival portion of the model has a decreasing hazard over time, parameterized by α=0.57. The degree of HLA mismatching is significantly associated with dnDSA; for every one unit increase in mismatching, the risk of dnDSA increases 1.26-fold (95% CrI: 1.13, 1.40). Race is also associated with risk of dnDSA; compared to Caucasian individuals, African Americans and Hispanics have a higher risk of dnDSA (2.09 and 1.60, respectively). Age is also a factor, as middle-aged (30-49 years) and older-aged (50+ years) individuals both have a reduced risk of dnDSA compared to individuals under the age of 30 (Hazard Ratios: 0.57 and 0.30, respectively). These covariate effects are all conditional on the same initial level and slope of TAC, so these results are not explained by different TAC levels across HLA mismatches, race or age groups. The first association parameter, λ₀, explains the risk of dnDSA associated with a difference in overall average level of TAC. For every one ng/ml higher average TAC level, the risk of dnDSA lowers 0.64-fold (95% CrI: 0.52, 0.75). The second association parameter, λ₁, explains the risk of dnDSA associated with a difference in the slope of TAC levels. For every 0.05 ng/ml/month higher TAC slope, the risk of dnDSA lowers 0.43-fold (95% CrI: 0.32, 0.58).

The interval censored survival model M4 does not have a linear sub-model, but instead uses a time-varying covariate for TAC. The scale parameter of the survival model is α=0.46, which also results in a decreasing hazard of dnDSA over time. The effects of baseline covariates, γ^′, are similar in magnitude to β^′ in M1. The hazard ratio for the time-varying TAC is η=0.80. Within an individual, following a one unit increase in TAC level, the risk of dnDSA decreases 0.80-fold (95% CrI: 0.75, 0.85). Though this is still a significant finding, the effect size of η in M4 is smaller than λ₀ in M1. We explore this further in the “Simulation Study” section.

To assess the adequacy of model fit for M1-M3, we simulated 1000 datasets from the posterior predictive distribution and compared them to the observed data. We performed these posterior predictive assessments separately for the longitudinal and survival sub-models. This is usually done through a discrepancy function represented by a scalar or a vector, often represented graphically [32, 33]. For comparing the observed longitudinal TAC data with the posterior distribution of longitudinal measures on given individuals, we have conditioned on the posterior mean of the random effects at the individual level; e.g. for M1: \(\hat {a}_{0i}, \hat {a}_{1i}\) in (1) are considered fixed [34]. We compared 1000 posterior predictive TAC trajectories against the observed trajectory for 3 random individuals using M1, M2 and M3. Due to the large variability in TAC levels within and between individuals, the plots are all nearly identical, which is indicative that no one model fits the longitudinal trajectory better than another model. The plots are shown in Additional file 1: Figure S2.

For the survival sub-model predictive check, we created artificial intervals (that resembled the kidney study data intervals) around the survival times taken from the 1000 posterior predictive datasets. Next, using these intervals, we plotted the predicted Weibull cumulative density functions from the 1000 posterior datasets and then overlaid the estimated cumulative density curve from the raw data. We chose to plot the cumulative density function instead of the survival function because of the low event rate in these sub-groups [35]. We had to choose certain values of baseline covariates for these checks, so we presented three different types of individuals that represent the most prevalent combinations of covariates. The results of the survival predictive checks for three particular combinations of baseline covariates, for all three models M1-M3, are found in the Additional file 1: Figure S3. The coverage of the observed survival curve by the posterior predictive survival curves is best for M1. This is indicative that M1 fits the survival data better than the other two joint models.

To determine if the choice of priors affected the estimates of M1 parameters, an alternative prior distribution of Uniform (U(−100,100)) was compared against a Normal (N(0,1000)) prior for b₀, b₁, β, λ₀ and λ₁, and the half-Cauchy (HC(0,25)) distribution for σ_e was compared against the uniform prior (U(0,100)) for this variance component. The results, tabulated in Additional file 1: Table S1, indicate the choice of priors has very little effect on the parameter estimates.

We conducted a simulation study to compare how the joint M1 and the time varying covariate model perform under (1) varying degrees of association, (2) varying amounts of interval censoring (IC), and (3) varying amounts of measurement error (ME). We simulated data using the joint model with shared random intercepts (M2) under nine scenarios, each with a different combination of measurement error (none \(\left [\sigma _{e}^{2}=0\right ]\), low \(\left [\sigma _{e}^{2}=1\right ]\), and high \(\left [\sigma _{e}^{2}=8\right ]\)) and degree of interval censoring (lighter [0-1 month, 1–6 month, 6–12 month, 12–24 month, yearly after], visits missed at random, and heavier [3 year intervals]). For the random interval censoring, we assumed patients had a fixed follow-up schedule matching the lighter IC scenario, but missed 50% of visits at random (we also set the missing-ness to 25% and the results were almost identical). All other variables were set to be similar to the kidney transplant data, with Weibull parameters α=0.50 and β₀=−2. Two hundred datasets were simulated under each of these six conditions, and the JM and TVC models were fitted to each dataset using MCMC. Each dataset contained 300 individuals, and approximately half developed dnDSA. The results from all simulations were obtained using two parallel chains; the TVC model had a burn-in of 1000 samples and 1000 samples for inference, while the JM had a burn-in of 2000 samples and 2000 samples for inference. The average model estimates from the light and heavy IC scenarios are reported in Table 4. The light and random IC scenarios did not differ significantly, so the results from the random IC scenarios are in the Additional file 1: Table S2.

To compare power for detecting an association between the longitudinal and the time to event outcomes, we varied the association between TAC and dnDSA (λ₀ in the JM and η in the TVC) between -0.50 and 0 within each scenario. We calculated power based on each model at each true association value using the credible intervals of λ₀ or η. For example, for simulation run m, m=1,...,M (M=200 for our simulations), let \(\hat {\lambda }_{0L}\) and \(\hat {\lambda }_{0U}\) denote the 2.5% and 97.5% percentiles of the distribution of the MCMC samples of \(\hat {\lambda }_{0}\). The power is calculated as: Power \(= 1-1/M\sum \limits _{m=1}^{M} I(\hat {\lambda }_{0L}<0<\hat {\lambda }_{0U})\), where I(E) is the indicator function of event E. The power curves for all nine scenarios are presented in Fig. 2.

As expected, the association parameter was consistently attenuated toward zero in the TVC compared to the JM. As more measurement error was introduced, the TVC underestimated the association even more (true value: λ₀=−0.50, low ME: \(\hat {\eta }\)= -0.32 and high ME: \(\hat {\eta }\)= -0.18), while there was no evidence of substantial bias for the JM (low ME: \(\hat {\lambda }_{0}= -0.53\) and high ME: \(\hat {\lambda }_{0}= -0.54\)). As heavier interval censoring was introduced, the association estimates were further from the true value in both modeling approaches, but the standard deviations also increased. The parameter that was most affected by heavier interval censoring was the Weibull shape parameter, α. Although the bias of α was lower in the TVC context, this model should not be preferred when there is high measurement error and heavy censoring, due to the underestimation of the association between the two outcomes, which is typically of more interest.

The power of the TVC model was consistently lower than the power of the JM (Fig. 2). When more measurement error was introduced, the power of the TVC model significantly decreased while the power of the JM stayed approximately the same. When visits were missed at random, the decrease in power on both approaches was minimal. The wider interval censoring around dnDSA resulted in a high type I error in the case of the TVC model (type I error=0.23 in the high ME and heavy IC simulation from Fig. 2). The average credible intervals and coverage probabilities for all simulations are reported in the Additional file 1: Figure S3.