eGFR Decline and Kidney Failure

Why Model eGFR and Kidney Failure Together?

Estimated glomerular filtration rate (eGFR) is repeatedly measured to describe kidney function over time. In a kidney-failure study, those measurements are not equally precise reflections of a participant’s underlying kidney function: biological variation and laboratory measurement error can make the observed values fluctuate around the underlying trajectory. Meanwhile, a participant’s risk of kidney failure may be related to that evolving trajectory. A joint model represents both processes and links the event hazard to the estimated underlying eGFR, rather than treating each observed measurement as exact.

The distinction matters for interpretation. A model relating event risk to the latest observed eGFR can be sensitive to measurement noise and to when that measurement happened to be collected. A joint model instead uses the longitudinal data to estimate a smooth, participant-specific trajectory and relates the event process to that trajectory. Baseline kidney function can also be included, so the association with the evolving eGFR is interpreted conditional on initial kidney status and other specified covariates.

A Joint Model for eGFR and Kidney Failure

For illustration, suppose time is measured in years since study entry, \(B_i\) denotes participant \(i\)’s latent true baseline eGFR, and \(B_*\) is a reference baseline value. If a baseline eGFR is measured with error, \(B_i\) should be understood as the underlying baseline level inferred by the measurement model, not automatically as an error-free recorded value. A linear mixed longitudinal model can be written as

Here \(\epsilon_{ij}\) represents measurement and short-term variation around the trajectory; a common assumption is \(\epsilon_{ij}\sim N(0,\sigma^2)\), independently across measurements conditional on the participant’s random effects and covariates. The participant-specific random slope, \(b_{1i}\), is commonly modeled as a mean-zero normal variable. Thus participants may differ in their rate of change. The fixed effects describe the average trajectory and how it varies with baseline kidney function.

One possible event submodel is a proportional hazards model with a current-value association:

In this example, \(h_i(t\mid\cdot)\) is the instantaneous kidney-failure hazard at time \(t\), conditional on being event-free just before \(t\). The baseline hazard \(h_0(t)\) describes the hazard at the reference values \(B_i=B_*\), \(m_i(t)=90\), and \(Z_i=0\). The covariates \(Z_i\) may include measured factors such as age or treatment group. The coefficient \(\gamma\) is the log hazard ratio for a 10 mL/min/1.73 m\(^2\) difference in baseline eGFR, conditional on the current latent eGFR and other covariates. The coefficient \(\alpha\) is the log hazard ratio per 1 mL/min/1.73 m\(^2\) difference in current latent eGFR, conditional on baseline kidney function and other covariates.

Assumptions and What the Joint Link Means

The model’s conclusions depend on assumptions about both the repeated measurements and the event process. The longitudinal trajectory must be adequately represented by the selected time function and random-effects structure; for example, a straight-line trajectory may be inadequate if eGFR decline accelerates or changes after treatment. Conditional on the trajectory and covariates, the measurement errors must have an appropriate distribution and dependence structure. The event submodel must also describe the hazard adequately, including its baseline hazard and covariate effects.

The shared-random-effects formulation assumes that the dependence between the repeated eGFR measurements and kidney failure is captured by the shared latent trajectory or its random effects, as specified by the association structure. It also requires an appropriate model for the random effects and a defensible assumption about censoring: censoring should be independent of the event process conditional on the modeled information. If loss to follow-up is related to worsening kidney function in a way not captured by the model, the event association can be biased.

A joint model does not make the observed eGFR values error-free. Instead, it estimates the latent trajectory using the repeated measurements and accounts for their residual variability. Its uncertainty calculations should reflect uncertainty in the longitudinal and event components together. The introductory material on shared-random-effects models provides the general framework; here the substantive question is how to interpret that framework for eGFR and kidney failure.

Interpreting an Association as a Hazard Ratio

The current-value association coefficient is on the log-hazard scale. If two participants have the same baseline kidney function and other covariates, but their latent current eGFR values differ by \(d\), the model-implied hazard ratio for the participant with the lower eGFR by \(d\) units is \(\exp(-\alpha d)\). This follows by subtracting their log hazards: the baseline-hazard and shared covariate terms cancel, leaving the difference in the current-value terms.

Why Adjust for Baseline Kidney Function?

Baseline adjustment distinguishes participants who reach the same current eGFR by different histories. For instance, a current latent eGFR of 75 may represent a modest decline from a high starting level or a smaller change from a lower starting level. In the specified model, \(\alpha\) describes the current-value association conditional on baseline kidney function; \(\gamma\) describes the additional modeled association with baseline level conditional on current latent eGFR. These are model-dependent conditional associations, not necessarily independent biological mechanisms.

Baseline and current eGFR are likely to be correlated, so their coefficients can be imprecise or sensitive to model specification. If baseline eGFR is measured with error but inserted as if exact, that error can distort the baseline adjustment and affect the estimated current-value association. A measurement model that accounts for uncertainty in baseline function, such as using replicate measures or modeling baseline observations as part of the longitudinal process, can be preferable. The baseline definition and measurement approach should be stated clearly.

Diagnostics, Alternatives, and Limitations

A joint model can still be poorly specified. Diagnostics should address the longitudinal and event components as well as their link. Compare observed eGFR profiles with fitted population and participant-specific trajectories; inspect residuals for trends over time, changing variance, outliers, or serial dependence. Check whether the selected time function captures curvature or changes in decline, and whether the random-effects distribution and covariance structure are plausible. For the event component, examine model fit and whether the chosen baseline hazard and covariate effects adequately represent observed event patterns. Numerical warnings, unstable estimates, or large changes under reasonable alternative specifications also deserve investigation.

The endpoint definition is especially important. For this example, kidney failure could be defined as initiation of chronic dialysis or kidney transplantation, with death before kidney failure treated according to the study’s prespecified analysis plan. If the event is instead defined partly by an eGFR threshold, using eGFR as the longitudinal predictor can create a close mathematical link between predictor and endpoint. That design may be appropriate for some questions, but interpretation must acknowledge the endpoint construction and the possibility that the association partly reflects it.

Joint modeling does not eliminate confounding, correct a poor trajectory specification, or automatically solve informative dropout. It also does not guarantee that an estimated association is causal. The results depend on how time, baseline function, censoring, and the kidney-failure endpoint are defined. If death precludes kidney failure and is scientifically relevant, treating it as ordinary independent censoring may be inappropriate; the analysis must reflect the estimand and event process of interest.

Common Pitfall

Wrong reasoning: “The latest observed eGFR is 10 units lower, so the participant’s kidney-failure hazard is exactly the model’s 10-unit hazard ratio higher.” This treats one noisy observation as the latent current value and ignores the model’s conditional structure.

Correction: The association parameter applies to the modeled latent trajectory, and its hazard-ratio interpretation holds conditional on baseline kidney function and the other included covariates. A noisy observation informs the estimate of that trajectory but is not automatically equal to it. Likewise, the coefficient does not establish that changing eGFR would cause the corresponding change in hazard.