Appendix I: Joint Statistical Model

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Appendix I: Joint Statistical Model

The relationship between change in eGFR and risk of graft failure was assessed using the joint modeling framework.¹ Within the joint modeling framework, 2 models are fit to the data simultaneously: 1 for the longitudinal process (change in eGFR) and 1 for the event process (time-to-death-censored graft failure and time to all-cause graft failure).¹

The joint model consists of 2 submodels: 1. A longitudinal linear mixed effects (random intercept-slope) model for modeling patterns of change in eGFR (i.e., the “true” eGFR process); and 2. Cox’s hazards model for modeling time to death-censored graft failure and time to all-cause graft failure. Joint estimation of the submodels is achieved by assuming they are correlated via individual-level random effects.¹ The random effects in linear mixed effects model (the intercept and slope) are assumed to be multivariately normally distributed with mean 0 and unstructured covariance matrix.

The default specification of the joint model assumes the risk of an event at a moment in time depends on the current “true” value of the biomarker at that moment. This assumption may be reasonable; however, it is more likely the risks of death-censored graft failure and all-cause graft failure are related to dynamic characteristics of the longitudinal trajectory of eGFR, such as individual slopes. To accomplish this, the survival submodel was modified to include a slope parameterization of the eGFR process. This parameterization is of interest as it postulates that patients who show a steeper decease in eGFR (i.e., individual slope) are more likely to experience the event.

With Cox’s hazards model, the baseline hazard function is left completely unspecified to avoid the impact of miss-specifying the distribution of survival times. However, within the joint modeling framework, leaving the baseline hazard function unspecified may lead to underestimation of the standard errors of the parameter estimates.¹ To overcome this issue, the piece-wise constant baseline hazard function was used.¹

For modeling change in eGFR over time (the longitudinal process), prognostic factors included donor type (deceased versus living donor), donor age (per year increase), length of time from kidney transplantation to diagnosis of active AMR (per month increase), C4d positive stain at time of diagnosis of active AMR (yes versus no/unknown), and anti-HLA DSA positive (Class I only and Class II only versus both Class I and Class II). These factors were chosen for inclusion in the Linear Mixed Effects model their potential prognostic significance. For modeling time to death-censored graft failure and all-cause graft failure (the event process), prognostic factors included Linear Mixed Effects model-based prediction of eGFR at time of diagnosis of active AMR (i.e., baseline eGFR) and individual change in eGFR per month (slope parameterization).

Residual diagnostics were used to assess model assumptions. For the linear mixed effects submodel, subject-specific residuals were used to check the homoscedasticity and normality assumptions while marginal residuals were used to assess misspecification of the mean response structure.

Joint models were used to provide conditional survival predictions (death-censored free survival and graft survival) beyond 1 year following diagnosis of active AMR for a set of hypothetical eGFR dynamic characteristics (slopes). Predictions were used to determine change in eGFR over a 1-year time horizon (e.g., 1-year postdiagnosis of AMR) associated with a clinically meaningful difference in graft survival beyond 1 year. The 1-year time frame was chosen for the eGFR profile because this is the length of follow up that is typically used for pivotal trials. The set of hypothetical individual scenarios was based on the distribution of the individual slopes estimated from the longitudinal submodel. For each individual profile, the conditional survival beyond 1 year was estimated (e.g., at 5 years postdiagnosis of active AMR).

Survival rates for individuals with a stable eGFR profile were provided as the basis for quantifying a

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clinically meaningful difference in death censored and overall graft survival due to a progressive decrease in eGFR from baseline at 1 year. The hypothetical individual scenarios were used to address the following question: how do changes in the rate of eGFR affect graft survival prognosis?

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Appendix II: Definition of study visit windows for summarizing Estimate GFR in the First Year Postdiagnosis

Nominal Study Visit Laboratory Testing Window

Study Visit Window (Lab Date – AMR Diagnosis Date)

Day 0

(Diagnosis of active AMR)

-30 to +3 days Day -30 to Day 3

Month 1 (Day 30) ± 7 days Day 23 to Day 37

Above pattern is repeated for measurements beyond Month 36

± 30 days Above pattern is repeated for measurements beyond Month 36 Abbreviations: AMR, antibody-mediated rejection.

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Supplementary Tables

Table S1: Joint Modeling Results of the Estimated GFR Longitudinal Process and Risk of All-cause Graft Failure Composite Endpoint (Supplementary analysis set)

A. Linear Mixed Effects Submodel for the Longitudinal eGFR Process^a

B. Cox’s Hazards Submodel for Time to Death-censored Graft Failure Composite Endpoint

Abbreviations: eGFR = estimated glomerular filtration rate; DSA = donor-specific antibody; AMR = antibody mediated rejection. ^aEstimated GFR restricted to the first 12 months postdiagnosis of active AMR

Linear Mixed Effects Model for Longitudinal eGFR Process

Comparison Estimate Standard Error

P-value

Variable

Intercept 50.717 7.831 <0.0001

Time Per month -0.753 0.106 <0.0001

Prognostic factors

Site Barcelona vs

Wisconsin cohort

-0.414 5.880 0.944 Manitoba vs

Wisconsin cohort

11.555 3.830 0.003

Donor type Deceased vs Living -0.886 3.052 0.772

Donor age Per year increase -0.312 0.108 0.004

Length of time from kidney transplantation to AMR diagnosis

Per month increase -0.028 0.033 0.400 C4d positive stain at time of AMR

diagnosis

Yes vs no/unknown -0.376 4.389 0.932 Anti-HLA DSA Class at time of AMR

diagnosis

Class I only vs Class I and Class II

5.727 4.254 0.178

Class II only vs Class I and Class II

7.249 3.550 0.041

Event Process Comparison Hazard

ratio

95% Confidence Interval

P-value

Variable

Estimated GFR (predicted value) at time of diagnosis

Per unit increase

0.910 0.875, 0.947 <0.0001

Slope Per month increase

in eGFR 0.170 0.067, 0.429 <0.0001

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References

1. Rizopoulos D. Joint Models for Longitudinal and Time-to-Event Data: With Applications

Appendix I: Joint Statistical Model

in R. Boca Raton, FL: Chapman and Hall/CRC; 2012.