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

Abstract

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 the predicted probabilities, and evaluating the prediction accuracy through double time-dependent receiver operating characteristic curves. We illustrate the proposed analytical framework using the African American study of kidney disease and hypertension to develop a landmark model for dynamic prediction of end-stage renal diseases or death among patients with chronic kidney disease.

Appendices

Appendix 1

We prove the consistency of the sensitivity estimator (8). The proof for the specificity estimator (9) is similar and omitted. The proof makes use of the following regularity conditions:

• (A1) The kernel function K(.) is a symmetric density function centered around zero with compact support.

• (A2) The variable U has bounded support; its density function $f_U(.)$ is Lipschitz continuous and $f_U(s) > 0$, a.s.

• (A3) $h_1 \rightarrow 0$, $nh_1^2 \rightarrow \infty$, and $h_1$ satisfies the conditions in Lemma 1 below.

According to [6], the kernel weighted Kaplan–Meier estimator is consistent under standard regularity conditions: $\hat{S}_{\tilde{T}}(t| U_i, Q_i ) = S_{\tilde{T}}(t|U_i, Q_i) + o_p(1)$. Therefore,

$$\begin{aligned} \hat{W}(U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i) =&1 - (1-\tilde{\delta }_i) \dfrac{ S_{\tilde{T}}( \tau _1 | U_i, Q_i ) + o_p(1) }{ S_{\tilde{T}}( \tilde{Y}_i | U_i, Q_i ) + o_p(1) } \\ =&W(U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i) + o_p(1) \end{aligned}$$

$$\begin{aligned} \hat{Se}(s, c) =&\dfrac{ \sum _{i \in \mathcal {R}(s)} K_{h_1}(U_i, s) W( U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i)1(Q_i> c) + o_p(1) }{ \sum _{i \in \mathcal {R}(s)} K_{h_1}(U_i, s) W( U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i) + o_p(1) } \\ =&\dfrac{ \sum _{i \in \mathcal {R}(s)} K_{h_1}(U_i, s) W( U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i)1(Q_i> c) }{ \sum _{i \in \mathcal {R}(s)} K_{h_1}(U_i, s) W( U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i) } + o_p(1) \\ =&\dfrac{ \sum _{i=1}^n K_{h_1}(U_i, s) W( U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i)1(Q_i> c)1(\tilde{Y}_i> s) }{ \sum _{i=1}^n K_{h_1}(U_i, s) W( U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i) 1(\tilde{Y}_i > s) } + o_p(1). \end{aligned}$$

Using Lemma 2 below, the equation above equals

$$\begin{aligned} \hat{Se}(s, c) =&~ \dfrac{ E\left\{ W(U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i) 1(Q_i> c) 1(\tilde{Y}_i> s) ~|~ s \right\} f_U(s) }{ E\left\{ W(U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i) 1(\tilde{Y}_i > s) ~|~ s \right\} f_U(s) } + o_p(1), \end{aligned}$$

where $f_U(.)$ is the density function of $U_i$. Replacing $W(U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i)$ by $E\left( 1(\tilde{T}_i \le \tau _1) ~|~ U_i, Q_i, \tilde{Y}_i, \tilde{\delta }_i \right)$ in the equation above, we have

$$\begin{aligned} \hat{Se}(s, c) =&\dfrac{ E\left\{ 1(\tilde{T}_i \le \tau _1) 1(Q_i> c) 1(\tilde{Y}_i> s) ~|~ s \right\} }{ E\left\{ 1(\tilde{T}_i \le \tau _1) 1(\tilde{Y}_i> s) ~|~ s \right\} } + o_p(1) \\ =&P( Q_i> c | \tilde{T}_i \in (s, s + \tau _1] , \tilde{Y}_i > s, s ) + o_p(1) \nonumber . \end{aligned}$$

This completes the proof of consistency.

Lemma 1

Let $\{(X_i, Y_i), i=1,\ldots ,n\}$ be independent and identically distributed random vectors, where the $Y_i$’s are scalar random variables. Assume that $E|Y|^s<\infty$ and $\text {sup}_x\int |Y|^s f(X, Y)dY < \infty$, where f denotes the joint density of (X, Y). K is a bounded positive function with a bounded support, satisfying a Lipschitz condition. Then

$$\begin{aligned} \underset{x}{\mathbf {sup}} \Bigl |\dfrac{1}{n}\sum _{i=1}^{n} \bigl [ K_h(X_i-x)Y_i - E\{ K_h(X_i-x)Y_i \} \bigr ] \Bigr | = O_p\Bigl ( \Bigl \{ \dfrac{\log (1/h)}{nh} \Bigr \}^{1/2} \Bigr ) \end{aligned}$$

provided that $n^{2\epsilon -1}h \rightarrow \infty$ for some $\epsilon < 1-s^{-1}$. (This lemma was used in [43]).

Lemma 2

Suppose $\{(Y_i, T_i), i=1,\ldots ,n\}$ satisfy the conditions in Lemma 1; $T_i$’s satisfy the regularity conditions (A1)–(A3); E(Y|T) is Lipschitz continuous. Then as $n\rightarrow \infty$:

$$\begin{aligned} n^{-1}\sum _{i=1}^{n}K_h(T_i - t_0)Y_i = E(Y|t_0)f(t_0) + O_p(h^2) + O_p\Bigl ( \Bigl \{ \dfrac{\log (1/h)}{nh} \Bigr \}^{1/2} \Bigr ) . \nonumber \end{aligned}$$

The proof of this result uses Lemma 1 and a Taylor series expansion of E(Y|T)f(T) around $t_0$. (This lemma was used in [19]).

Appendix 2

With irregularly spaced measurement times, some subjects may not have an adequate number of measurements within the history window $( s-\tau _2, s)$ for calculating the slope, or rate of change, of a biomarker. We outline a solution to this problem. Suppose we want to make a prediction on a subject at time s. The repeated biomarker measurements and the corresponding measurement times within the history window are denoted by vectors $\mathbf {Z}_{0}$ and $\mathbf {t}_0$, respectively. Here we focus on the case when s equals the last measurement time in $\mathbf {t}_{0}$. The case when s is not a measurement time has been discussed in Sect. 2.2. Let $\mathcal {R}(s)$ denote the set of subjects in the training dataset that are at risk at time s, i.e., $\mathcal {R}(s) = \{ i | i=1,2,\ldots ,n; Y_i > s \}$. For the i-th subject in $\mathcal {R}(s)$, let $t_{ij}$ be the measurement time that is the closest to s, i.e., $| t_{ij} - s | \le | t_{ij^{'}} - s |$ for any $j^{'} \in \{ 1,2,\ldots ,n_i \}$ (in case of ties, make a random choice). Let $\mathbf {Z}_{i}$ and $\mathbf {t}_{i}$ be the corresponding vectors of biomarker measurements and measurement times in the history window, respectively (by definition, the last element in $\mathbf {t}_{i}$ is $t_{ij}$). Fit a random intercept and slope model to the data $\{ \mathbf {Z}_i, \mathbf {t}_i, i \in \mathcal {R}(s) \}$, with the i-th subject being weighted by the kernel weight $W_i = h^{-1}K\left( (t_{ij} - s)/h \right)$. This is done by maximizing the following weighted multivariate Gaussian log-likelihood with respect to $\mathbf {\alpha }(s)$, $\mathbf {\Sigma }(s)$, and $\sigma (s)$:

$$\begin{aligned}&\underset{i \in \mathcal {R}(s) }{\sum } \left( -\dfrac{W_i}{2} \right) \log \left| \mathbf {D}_i \mathbf {\Sigma }(s) \mathbf {D}_i^T + \sigma (s)^2 \mathbf {I} \right| \nonumber \\&\quad + \underset{i \in \mathcal {R}(s) }{\sum } \left( -\dfrac{W_i}{2} \right) \left( \mathbf {Z}_i - \mathbf {D}_i\mathbf {\alpha }(s) \right) ^T \left( \mathbf {D}_i \mathbf {\Sigma }(s) \mathbf {D}_i^T + \sigma (s)^2 \mathbf {I} \right) ^{-1} \left( \mathbf {Z}_i - \mathbf {D}_i\mathbf {\alpha }(s) \right) ,\qquad \quad \end{aligned}$$

(10)

where $\mathbf {D}_i$ is a two-column matrix consisting of a vector of 1’s and $\mathbf {t}_i$; $\mathbf {\alpha }(s)$ is a $2\times 1$ vector of the fixed effect coefficients for intercept and slope; and $\mathbf {\Sigma }(s)$ is the $2\times 2$ variance matrix of the random effects. Given the estimated parameters, the rate of change may be defined to be the second element of the following $2 \times 1$ vector:

$$\begin{aligned} \hat{\mathbf {\alpha }}(s) + \hat{\mathbf {\Sigma }}(s) \hat{D}_0^T \left( \mathbf {D}_0 \hat{\mathbf {\Sigma }}(s) \mathbf {D}_0^T + \sigma (s)^2 \mathbf {I} \right) ^{-1} \left( \mathbf {Z}_0 - \mathbf {D}_0 \hat{\mathbf {\alpha }}(s) \right) , \end{aligned}$$

(11)

where $\mathbf {D}_0$ is a two-column matrix consisting of a vector of 1’s and $\mathbf {t}_0$. Equation (11) is the best linear unbiased prediction of the random intercept and slope. Since $\mathbf {\alpha }(s)$, $\mathbf {\Sigma }(s)$, and $\sigma (s)$ can be estimated at any landmark time s, their trajectories characterize the temporal change of the at-risk population.

We emphasize that the procedure above is not intended to provide a consistent or unbiased estimate of the slope of the biomarker trajectory at time s for each subject, or the mean slope of the at-risk subjects at time s, because the trajectory may be non-linear and $\tau _2$ does not go to 0 as n increases. The procedure should be viewed as a principled way of defining biomarker slope other than using the least squares estimate. It causes the individual least squares slope to shrink toward the mean slope of the at-risk subjects and thus increases the stability of individual slopes. When the individual biomarker trajectory is non-linear and smooth, estimating the slope as a smooth time-varying function is difficult in both the landmark modeling and joint modeling frameworks, unless strong assumptions are made for the trajectory shape or large numbers of repeated measures per subject are available. In clinical practice, the concept of a time-varying first-order derivative of a non-linear smooth “true” trajectory is rarely used in prognostic evaluation. Physicians often use the least square slope in the history window to describe, in broad strokes, the overall trend of the biomarker during that time, even when in reality most of the biomarker trajectories are non-linear to certain extent and their first-order derivatives are never constant within the history window. Examples include the prostate-specific antigen (PSA) velocity [41] and GFR slope [31]. The procedure above is an algorithmic formulation of the concept of rate of change used in clinical practice. Therefore, this linear mixed model with time-varying parameters should be viewed as a working assumption.

Even when the “true” individual trajectory is indeed linear within the history window with a constant underlying “true” slope, the BLUP is not the true slope and their difference is often viewed as a manifestation of measurement error in the biomarker [21]. The magnitude of the error is of order $\sigma _i/\sqrt{m_i}$, where $\sigma _i$ is the residual variance of subject i and $m_i$ is the number of eGFR measurements within the history window. The measurement error model literature showed that covariate measurement error does not cause bias in prediction if the distribution of the observed data is the same in the training and testing datasets [4]. Therefore, despite the measurement error, we can still use the BLUP for prediction purpose. This is the justification for the proposed biomarker slope definition. However, the assumption that the training and testing datasets have the same distribution implies that the distribution of the measurement times $t_{ij}$ must also be the same, as the measurement times are treated as random in this paper (Sect. 2.1 ). If, for example, we develop a landmark model out of a training dataset where every subject follows a six-month visit schedule with some random deviation, and apply it to a new dataset where every patient follows a three-month visit schedule, then it would not be appropriate to use the BLUP with a time-varying linear mixed model developed from the training data.