A semiparametric joint model for longitudinal and survival data with application to hemodialysis study

Backward joint model and dynamic prediction of survival with multivariate longitudinal data

An important approach to dynamic prediction of time-to-event outcomes using longitudinal data is based on modeling the joint distribution of longitudinal and time-to-event data. The widely used joint model for this purpose is the shared random effect model. Presumably, adding more longitudinal predictors improves the predictive accuracy. However, the shared random effect model can be computationally difficult or prohibitive when a large number of longitudinal variables are used. In this paper, we study an alternative way of modeling the joint distribution of longitudinal and time-to-event data. Under this formulation, the log-likelihood involves no more than one-dimensional integration, regardless of the number of longitudinal variables in the model. Therefore, this model is particularly suitable in dynamic prediction problems with large number of longitudinal predictors. The model fitting can be implemented with tractable and stable computation by using a combination of pseudo maximum likelihood estimation, Expectation-Maximization algorithm, and convex optimization. We evaluate the proposed methodology and its predictive accuracy with varying number of longitudinal variables using simulations and data from a primary biliary cirrhosis study.

Survival models and health sequences

Survival studies often generate not only a survival time for each patient but also a sequence of health measurements at annual or semi-annual check-ups while the patient remains alive. Such a sequence of random length accompanied by a survival time is called a survival process. Robust health is ordinarily associated with longer survival, so the two parts of a survival process cannot be assumed independent. This paper is concerned with a general technique-reverse alignment-for constructing statistical models for survival processes, here termed revival models. A revival model is a regression model in the sense that it incorporates covariate and treatment effects into both the distribution of survival times and the joint distribution of health outcomes. The revival model also determines a conditional survival distribution given the observed history, which describes how the subsequent survival distribution is determined by the observed progression of health outcomes.

Joint Modeling of Repeated Measures and Competing Failure Events In a Study of Chronic Kidney Disease

We are motivated by the Chronic Renal Insufficiency Cohort (CRIC) study to identify risk factors for renal progression in patients with chronic kidney diseases. The CRIC study collects two types of renal outcomes: glomerular filtration rate (GFR) estimated annually and end stage renal disease (ESRD). A related outcome of interest is death which is a competing event for ESRD. A joint modeling approach is proposed to model a longitudinal outcome and two competing survival outcomes. We assume multivariate normality on the joint distribution of the longitudinal and survival outcomes. Specifically, a mixed effects model is fit on the longitudinal outcome and a linear model is fit on each survival outcome. The three models are linked together by having the random terms of the mixed effects model as covariates in the survival models. EM algorithm is used to estimate the model parameters and the non-parametric bootstrap is used for variance estimation. A simulation study is designed to compare the proposed method with an approach that models the outcomes sequentially in two steps. We fit the proposed model to the CRIC data and show that the protein-to-creatinine ratio is strongly predictive of both estimated GFR and ESRD but not death.

Dynamic Prediction of Renal Failure Using Longitudinal Biomarkers in a Cohort Study of Chronic Kidney Disease

In longitudinal studies, prognostic biomarkers are often measured longitudinally. It is of both scientific and clinical interest to predict the risk of clinical events, such as disease progression or death, using these longitudinal biomarkers as well as other time-dependent and time-independent information about the patient. The prediction is dynamic in the sense that it can be made at any time during the follow-up, adapting to the changing at-risk population and incorporating the most recent longitudinal data. One approach is to build a joint model of longitudinal predictor variables and time to the clinical event, and draw predictions from the posterior distribution of the time to event conditional on longitudinal history. Another approach is to use the landmark model, which is a system of prediction models that evolve with the follow-up time. We review the pros and cons of the two approaches, and present a general analytical framework using the landmark approach. The proposed framework allows the measurement times of longitudinal data to be irregularly spaced and differ between subjects. We propose a unified kernel weighting approach for estimating the model parameters, calculating predicted probabilities, and evaluating prediction accuracy through double time-dependent Receiver Operating Characteristics (ROC) curves. We illustrate the proposed analytical framework using the African American Study of Kidney Disease and Hypertension (AASK) to develop a landmark model for dynamic prediction of end stage renal diseases or death among patients with chronic kidney disease.

Multiphasic effects of blood pressure on survival in hemodialysis patients

Dialysis patients exhibit an inverse, L- or U-shaped association between blood pressure and mortality risk, in contrast to the linear association in the general population. We prospectively studied 9333 hemodialysis patients in France, aiming to analyze associations between predialysis systolic, diastolic, and pulse pressure with all-cause mortality, cardiovascular mortality, and nonfatal cardiovascular endpoints for a median follow-up of 548 days. Blood pressure components were tested against outcomes in time-varying covariate linear and fractional polynomial Cox models. Changes throughout follow-up were analyzed with a joint model including both the time-varying covariate of sequential blood pressure and its slope over time. A U-shaped association of systolic blood pressure was found with all-cause mortality and of both systolic and diastolic blood pressure with cardiovascular mortality. There was an L-shaped association of diastolic blood pressure with all-cause mortality. The lowest hazard ratio of all-cause mortality was observed for a systolic blood pressure of 165 mm Hg, and of cardiovascular mortality for systolic/diastolic pressures of 157/90 mm Hg, substantially higher than currently recommended values for the general population. The 95% lower confidence interval was approximately 135/70 mm Hg. We found no significant correlation for either systolic, diastolic, or pulse pressure with myocardial infarction or nontraumatic amputations, but there were significant positive associations between systolic and pulse pressure with stroke (per 10-mm Hg increase: hazard ratios 1.15, 95% confidence interval 1.07 and 1.23; and 1.20, 1.11 and 1.31, respectively). Thus, whereas high pre-dialysis blood pressure is associated with stroke risk, low pre-dialysis blood pressure may be both harmful and a proxy for comorbid conditions leading to premature death.

Quantifying the Race Stratified Impact of Socioeconomics on Graft Outcomes in Kidney Transplant Recipients

BACKGROUND

Socioeconomic status (SES) is a significant determinant of health outcomes and may be an important component of the causal chain surrounding racial disparities in kidney transplantation. The social adaptability index (SAI) is a validated and quantifiable measure of SES, with a lack of studies analyzing this measure longitudinally or between races.

METHODS

Longitudinal cohort study in adult kidney transplantation transplanted at a single-center between 2005 and 2012. The SAI score includes 5 domains (employment, education, marital status, substance abuse and income), each with a minimum of 0 and maximum of 3 for an aggregate of 0 to 15 (higher score → better SES).

RESULTS

One thousand one hundred seventy-one patients were included; 624 (53%) were African American (AA) and 547 were non-AA. African Americans had significantly lower mean baseline SAI scores (AAs 6.5 vs non-AAs 7.8; P < 0.001). Cox regression analysis demonstrated that there was no association between baseline SAI and acute rejection in non-AAs (hazard ratio [HR], 0.92; 95% confidence interval [95% CI], 0.81-1.05), whereas it was a significant predictor of acute rejection in AAs (HR, 0.89; 95% CI, 0.80-0.99). Similarly, a 2-stage approach to joint modelling of time to graft loss and longitudinal SAI did not predict graft loss in non-AAs (HR, 1.01; 95% CI, 0.28-3.62), whereas it was a significant predictor of graft loss in AAs (HR, 0.23; 95% CI, 0.06-0.93).

CONCLUSIONS

After controlling for confounders, SAI scores were associated with a lower risk of acute rejection and graft loss in AA kidney transplant recipients, whereas neither baseline nor follow-up SAI predicted outcomes in non-AA kidney transplant recipients.

Joint multiple imputation for longitudinal outcomes and clinical events that truncate longitudinal follow-up

Longitudinal cohort studies often collect both repeated measurements of longitudinal outcomes and times to clinical events whose occurrence precludes further longitudinal measurements. Although joint modeling of the clinical events and the longitudinal data can be used to provide valid statistical inference for target estimands in certain contexts, the application of joint models in medical literature is currently rather restricted because of the complexity of the joint models and the intensive computation involved. We propose a multiple imputation approach to jointly impute missing data of both the longitudinal and clinical event outcomes. With complete imputed datasets, analysts are then able to use simple and transparent statistical methods and standard statistical software to perform various analyses without dealing with the complications of missing data and joint modeling. We show that the proposed multiple imputation approach is flexible and easy to implement in practice. Numerical results are also provided to demonstrate its performance. Copyright © 2015 John Wiley & Sons, Ltd.

Nonparametric multistate representations of survival and longitudinal data with measurement error

This paper proposes a nonparametric procedure to describe the progression of longitudinal cohorts over time from a population averaged perspective, leading to multistate probability curves with the states defined jointly by survival and longitudinal outcomes measured with error. To account for the challenges of informative dropout and nonlinear shapes of the longitudinal trajectories, we apply a bias corrected penalized spline regression to estimate the unobserved longitudinal trajectory for each subject. We then estimate the multistate probability curves on the basis of the survival data and the estimated longitudinal trajectories. We further use simulation-extrapolation method to reduce the estimation bias caused by the randomness of the estimated trajectories. We develop a bootstrap test to compare multistate probability curves between groups. We present theoretical justification of the estimation procedure along with a simulation study to demonstrate finite sample performance. We illustrate the procedure by data from the African American Study of Kidney Disease and Hypertension, and it can be widely applied in longitudinal studies.

Joint modeling of multivariate longitudinal data and the dropout process in a competing risk setting: application to ICU data

BACKGROUND

Joint modeling of longitudinal and survival data has been increasingly considered in clinical trials, notably in cancer and AIDS. In critically ill patients admitted to an intensive care unit (ICU), such models also appear to be of interest in the investigation of the effect of treatment on severity scores due to the likely association between the longitudinal score and the dropout process, either caused by deaths or live discharges from the ICU. However, in this competing risk setting, only cause-specific hazard sub-models for the multiple failure types data have been used.

METHODS

We propose a joint model that consists of a linear mixed effects submodel for the longitudinal outcome, and a proportional subdistribution hazards submodel for the competing risks survival data, linked together by latent random effects. We use Markov chain Monte Carlo technique of Gibbs sampling to estimate the joint posterior distribution of the unknown parameters of the model. The proposed method is studied and compared to joint model with cause-specific hazards submodel in simulations and applied to a data set that consisted of repeated measurements of severity score and time of discharge and death for 1,401 ICU patients.

RESULTS

Time by treatment interaction was observed on the evolution of the mean SOFA score when ignoring potentially informative dropouts due to ICU deaths and live discharges from the ICU. In contrast, this was no longer significant when modeling the cause-specific hazards of informative dropouts. Such a time by treatment interaction persisted together with an evidence of treatment effect on the hazard of death when modeling dropout processes through the use of the Fine and Gray model for sub-distribution hazards.

CONCLUSIONS

In the joint modeling of competing risks with longitudinal response, differences in the handling of competing risk outcomes appear to translate into the estimated difference in treatment effect on the longitudinal outcome. Such a modeling strategy should be carefully defined prior to analysis.

Effect of trajectories of glycemic control on mortality in type 2 diabetes: a semiparametric joint modeling approach

Data on the effect of trajectories in long-term glycemia and all-cause mortality are lacking. The authors studied the effect of trajectories in long-term glycemic control on all-cause mortality in patients with type 2 diabetes. A cohort of 8,812 veterans with type 2 diabetes was assembled retrospectively using Veterans Affairs registry data. For each veteran in the cohort, a 3-month person-period data set was created from April 1997 to May 2006. The average duration of follow-up was 4.5 years. The overall mortality rate was 15.3%. Using a novel approach for joint modeling of time to death and longitudinal measurements of hemoglobin A1c (HbA1c) level, after adjustment for all significant baseline covariates, baseline HbA1c was found to be significantly associated with mortality (hazard ratio = 2.1, 95% confidence interval: 1.3, 3.6) (i.e., a 1% increase in baseline HbA1c level was associated with an average 2-fold increase in mortality risk). Similarly, the slope of the HbA1c trajectory was marginally significantly associated with mortality (hazard ratio = 7.3, 95% confidence interval: 0.9, 57.1) after adjustment for baseline covariates (i.e., a 1% increase in HbA1c level over 3 months was associated with a 22% increase in mortality risk). The authors conclude that a positive trajectory of long-term hyperglycemia is associated with increased mortality.