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

Abstract

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 nonparametric 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.

Appendices

Appendix 1: Covariance Matrix for the Joint Distribution

According to models (1) and (2), we have

$$\begin{aligned}&Cov\left( {\begin{array}{*{20}{c}} b\\ \epsilon \\ {{r_1}}\\ {{r_2}}\\ y\\ \begin{array}{l} {x_1}\\ {x_2} \end{array} \end{array}} \right) \\&\quad = \left( {\begin{array}{*{20}{c}} D&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {D{Z^T}}&{}\quad {D{\lambda _1}}&{}\quad {D{\lambda _2}}\\ 0&{}\quad {{\sigma ^2}I}&{}\quad 0&{}\quad 0&{}\quad {{\sigma ^2}I}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad {s_1^2}&{}\quad 0&{}\quad 0&{}\quad {s_1^2}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad {s_2^2}&{}\quad 0&{}\quad 0&{}\quad {s_2^2}\\ {ZD}&{}\quad {{\sigma ^2}I}&{}\quad 0&{}\quad 0&{}\quad {ZD{Z^T} + {\sigma ^2}I}&{}\quad {ZD{\lambda _1}}&{}\quad {ZD{\lambda _2}}\\ {\lambda _1^TD}&{}\quad 0&{}\quad {s_1^2}&{}\quad 0&{}\quad {\lambda _1^TD{Z^T}}&{}\quad {\lambda _1^TD{\lambda _1} + s_1^2}&{}\quad {\lambda _1^TD{\lambda _2}}\\ {\lambda _2^TD}&{}\quad 0&{}\quad 0&{}\quad {s_2^2}&{}\quad {\lambda _2^TD{Z^T}}&{}\quad {\lambda _2^TD{\lambda _1}}&{}\quad {\lambda _2^TD{\lambda _2} + s_2^2} \end{array}} \right) \end{aligned}$$

We suppress the subscript i for clarity.

Appendix 2: Expectation of the Sufficient Statistics

We first derive the expectations assuming both $x_1$ and $x_2$ are observed. We will then revise the formula accounting for the scenario when either or both $x_1$ or $x_2$ are censored. We use $Y_{comp}$ to denote the full data including $\mathbf {y}, x_1$ and $x_2$.

According to models (1) and (2) and the covariance structure in the appendix, we have

$$\begin{aligned} \begin{array}{l} E\left( {b|Y_{comp},\hat{\theta }} \right) = {C_{12}}C_{22}^{ - 1}\left( {\begin{array}{*{20}{c}} {y - T\hat{\alpha }}\\ {{x_1} - {W^T}{{\hat{\xi }}_1}}\\ {{x_2} - {W^T}{{\hat{\xi }}_2}} \end{array}} \right) \\ Var\left( {b|Y_{comp},\hat{\theta }} \right) = \hat{D} - {C_{12}}C_{22}^{ - 1}C_{12}^T \end{array} \end{aligned}$$

where

$$\begin{aligned} {C_{12}} = \left( {\begin{array}{*{20}{c}} {\hat{D}{Z^T}}&{\hat{D}{{\hat{\lambda }}_1}}&{\hat{D}{{\hat{\lambda }}_2}} \end{array}} \right) \end{aligned}$$

and

$$\begin{aligned} {C_{22}} = \left( {\begin{array}{*{20}{c}} {Z\hat{D}{Z^T} + {{\hat{\sigma }}^2}I}&{}\quad {Z\hat{D}{{\hat{\lambda }}_1}}&{}\quad {Z\hat{D}{{\hat{\lambda }}_2}}\\ {\hat{\lambda }_1^T\hat{D}{Z^T}}&{}\quad {\hat{\lambda }_1^T\hat{D}{{\hat{\lambda }}_1} + \hat{s}_1^2}&{}\quad {\hat{\lambda }_1^T\hat{D}{{\hat{\lambda }}_2}}\\ {\hat{\lambda }_2^T\hat{D}{Z^T}}&{}\quad {\hat{\lambda }_2^T\hat{D}{{\hat{\lambda }}_1}}&{}\quad {\hat{\lambda }_2^T\hat{D}{{\hat{\lambda }}_2} + \hat{s}_2^2} \end{array}} \right) \end{aligned}$$

$E\left( {b{b^T}|{Y_{comp}},\hat{\theta }} \right)$ can then be calculated as

$$\begin{aligned} E\left( {b{b^T}|{Y_{comp}},\hat{\theta }} \right) = E\left( {b|Y_{comp},\hat{\theta }} \right) {E^T}\left( {b|Y_{comp},\hat{\theta }} \right) + Var\left( {b|Y_{comp},\hat{\theta }} \right) \end{aligned}$$

In the case when either or both $x_1$ and $x_2$ are censored,$E(b | Y _{obs}, \hat{\theta })$ and $E(b b^T | Y _{obs}, \hat{\theta })$ can be calculated as follows. We have

$$\begin{aligned} E\left( {b|Y_{obs},\hat{\theta }} \right) = E\left\{ {E\left( {b|Y_{comp},\hat{\theta }} \right) |Y_{obs},\hat{\theta }} \right\} = E\left( \hat{b} |Y_{obs},\hat{\theta }\right) \end{aligned}$$

and

$$\begin{aligned} E\left( {b{b^T}|Y_{obs},\hat{\theta }} \right)= & {} E\left\{ {E\left( {b{b^T}|Y_{comp},\hat{\theta }} \right) |Y_{obs},\hat{\theta }} \right\} \\= & {} E\left\{ {E\left( {b|Y_{comp},\hat{\theta }} \right) {E^T}\left( {b|Y_{comp},\hat{\theta }} \right) {+} Var\left( {b|Y_{comp},\hat{\theta }} \right) |Y_{obs},\hat{\theta }} \right\} \\= & {} E\left( {\hat{b}{{\hat{b}}^T}|Y_{obs},\hat{\theta }} \right) + E\left\{ {Var\left( {b|Y_{comp},\hat{\theta }} \right) |Y_{obs},\hat{\theta }} \right\} \\= & {} E\left( {\hat{b}{{\hat{b}}^T}|Y_{obs},\hat{\theta }} \right) + Var\left( {b|Y_{comp},\hat{\theta }} \right) \end{aligned}$$

where $\hat{b} = E\left( {b|Y_{comp},\hat{\theta }} \right)$. The last step follows because $Var\left( {b|Y_{comp},\hat{\theta }} \right)$ is not a function of $Y_{comp}$.

So the next step is to calculate $E\left( {\hat{b}{{\hat{b}}^T}|Y_{obs},\hat{\theta }} \right)$. Since

$$\begin{aligned} \hat{b}{\hat{b}^T} = {C_{12}}C_{22}^{ - 1}\left( {\begin{array}{*{20}{c}} {y - T\hat{\alpha }}\\ {{x_1} - {W^T}{{\hat{\xi }}_1}}\\ {{x_2} - {W^T}{{\hat{\xi }}_2}} \end{array}} \right) {\left( {\begin{array}{*{20}{c}} {y - T\hat{\alpha }}\\ {{x_1} - {W^T}{{\hat{\xi }}_1}}\\ {{x_2} - {W^T}{{\hat{\xi }}_2}} \end{array}} \right) ^T}C_{22}^{ - 1}{C_{21}} \end{aligned}$$

we need to calculate $E( {{x_1}|Y_{obs},\hat{\theta }})$, $E( {{x_2}|Y_{obs},\hat{\theta }})$, $E( {{x_1 ^2}|Y_{obs},\hat{\theta }})$, $E( {{x_2 ^2}|Y_{obs},\hat{\theta }})$ and $E( {{x_1 x_2}|Y_{obs},\hat{\theta }})$. See next section for the details of how to calculate these expectations under different scenarios. The calculation of $E(\epsilon \epsilon ^T | Y_{obs}, \hat{\theta })$, $E( {r_j^2|{Y_{obs}},\hat{\theta }} ),j=1,2$ can be done in a similar fashion.

To calculate $E( x_jb|{Y_{obs}},\hat{\theta })$, $j=1,2$, we have

$$\begin{aligned} E(x_jb|Y_{obs},\hat{\theta })= & {} E \left\{ E(x_j b |Y_{comp}, \hat{\theta })| Y_{obs}, \hat{\theta }\right\} \\= & {} E \left\{ x_j {C_{12}}C_{22}^{ - 1}\left( {\begin{array}{*{20}{c}} {y - T\hat{\alpha }}\\ {{x_1} - {W^T}{{\hat{\xi }}_1}}\\ {{x_2} - {W^T}{{\hat{\xi }}_2}} \end{array}} \right) | Y_{obs}, \hat{\theta }\right\} \end{aligned}$$

Again see next section for the calculation of $E(x_1^2|Y_{obs}, \hat{\theta })$, $E(x_2^2|Y_{obs}, \hat{\theta })$ and $E(x_1x_2|Y_{obs}, \hat{\theta })$.

To summarize, here are the formulas to calculate these quantities.

• $E(b_i|Y_{obs}, \hat{\theta }) = C_{12}C_{22}^{-1}E(O_i)$

• $E(b_i b_i ^T|Y_{obs}, \hat{\theta }) = C_{12}C_{22}^{-1}E(O_iO_i^T)C_{22}^{-1}C_{12}^T+ \hat{D} - C_{12}C_{22}^{-1}C_{12}^T$

• $E(\epsilon _i \epsilon _i ^T|Y_{obs}, \hat{\theta }) = F_{12}C_{22}^{-1}E(O_iO_i^T)C_{22}^{-1}F_{12}^T+ \hat{\sigma }^2 I - F_{12}C_{22}^{-1}F_{12}^T$

• $E(r_{i,1}^2 |Y_{obs}, \hat{\theta }) = G_{12}C_{22}^{-1}E(O_iO_i^T)C_{22}^{-1}G_{12}^T+ \hat{s}_1^2 - G_{12}C_{22}^{-1}G_{12}^T$

• $E(r_{i,2}^2 |Y_{obs}, \hat{\theta }) = H_{12}C_{22}^{-1}E(O_iO_i^T)C_{22}^{-1}H_{12}^T+ \hat{s}_2^2 - H_{12}C_{22}^{-1}G_{12}^T$

• $E(x_{i,1}b_i|Y_{obs},\hat{\theta }) = C_{12}C_{22}^{-1}E(x_{i,1}O_i)$

• $E(x_{i,2}b_i|Y_{obs},\hat{\theta }) = C_{12}C_{22}^{-1}E(x_{i,2}O_i)$

where

• ${C_{12}} = ( {\begin{array}{*{20}{c}} {\hat{D}{Z^T}}&{\hat{D}{{\hat{\lambda }}_1}}&{\hat{D}{{\hat{\lambda }}_2}} \end{array}} )$

• ${F_{12}} = ( {\begin{array}{*{20}{c}} {{{\hat{\sigma }}^2} I}&\mathbf {0}&\mathbf {0} \end{array}} )$

• ${G_{12}} = ( {\begin{array}{*{20}{c}} \mathbf {0}&{\hat{s}_1^2}&0 \end{array}} )$

• ${H_{12}} = ( {\begin{array}{*{20}{c}} \mathbf {0}&0&{\hat{s}_2^2} \end{array}} )$

• ${C_{22}} = \left( {\begin{array}{*{20}{c}} {Z\hat{D}{Z^T} + {{\hat{\sigma }}^2}I}&{}\quad {Z\hat{D}{{\hat{\lambda }}_1}}&{}\quad {Z\hat{D}{{\hat{\lambda }}_2}}\\ {\hat{\lambda }_1^T\hat{D}{Z^T}}&{}\quad {\hat{\lambda }_1^T\hat{D}{{\hat{\lambda }}_1} + \hat{s}_1^2}&{}\quad {\hat{\lambda }_1^T\hat{D}{{\hat{\lambda }}_2}}\\ {\hat{\lambda }_2^T\hat{D}{Z^T}}&{}\quad {\hat{\lambda }_2^T\hat{D}{{\hat{\lambda }}_1}}&{}\quad {\hat{\lambda }_2^T\hat{D}{{\hat{\lambda }}_2} + \hat{s}_2^2} \end{array}} \right)$

• $O_i=\left( {\begin{array}{*{20}{c}} {y_i - T\hat{\alpha }}\\ {{x_{i,1}} - {W^T}{{\hat{\xi }}_1}}\\ {{x_{i,2}} - {W^T}{{\hat{\xi }}_2}} \end{array}} \right)$

Note that $E(\epsilon _i ^T \epsilon _i |Y_{obs}, \hat{\theta })$ can be calculated by summing over the diagonal elements of $E(\epsilon _i \epsilon _i ^T|Y_{obs}, \hat{\theta })$, i.e., $E(\epsilon _i ^T \epsilon _i |Y_{obs}, \hat{\theta })=trace\{E(\epsilon _i \epsilon _i ^T|Y_{obs}, \hat{\theta })\}$.

Appendix 3: Calculation of the First Two Moments of x Given $Y_{obs}$

There are three types of individuals in the observed data: 1) those whose $x_1$ are observed but not $x_2$, i.e., $x_1 < c, x_2 > c$; 2) those whose $x_2$ are observed but not $x_1$, i.e., $x_1 > c, x_2 < c$; and 3) neither $x_1$ nor $x_2$ is observed, i.e, $x_1> c, x_2 > c$. We now derive $E( {{x_1}|Y_{obs},\hat{\theta }})$, $E( {{x_2}|Y_{obs},\hat{\theta }})$, $E( {{x_1 ^2}|Y_{obs},\hat{\theta }})$, $E( {{x_2 ^2}|Y_{obs},\hat{\theta }})$ and $E( {{x_1 x_2}|Y_{obs},\hat{\theta }})$ for each of the three scenarios.

1.1 Scenario I: $x_2$ Alone is Censored

For the first scenario in which $x_1$ is observed and $x_2$ is censored, we have $Y_{obs}=(\mathbf {y},x_1,x_2>c)$ and

• $E( {{x_1}|Y_{obs},\hat{\theta }})=x_1$

• $E( {{x_1 ^2}|Y_{obs},\hat{\theta }})=x_1^2$

• $E( {{x_2}|Y_{obs},\hat{\theta }}) = E(x_2|\mathbf {y},x_1,x_2>c, \hat{\theta })$

• $E( {{x_1 x_2}|Y_{obs},\hat{\theta }})=x_1E( {{x_2}|\mathbf {y},x_1,x_2>c,\hat{\theta }})$

• $E( {{x_2 ^2}|Y_{obs},\hat{\theta }}) = E^2(x_2|\mathbf {y},x_1,x_2>c, \hat{\theta }) + Var(x_2|\mathbf {y},x_1,x_2>c, \hat{\theta })$

Because

$$\begin{aligned} E\left( {\begin{array}{*{20}{c}} {{x_2}}\\ {{x_1}}\\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{W^T}{\xi _2}}\\ {{W^T}{\xi _1}}\\ {T\alpha } \end{array}} \right) \end{aligned}$$

and

$$\begin{aligned} Cov\left( {\begin{array}{*{20}{c}} {{x_2}}\\ {{x_1}}\\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\lambda _2^TD{\lambda _2} + s_2^2}&{}\quad {\lambda _2^TD{\lambda _1}}&{}\quad {\lambda _2^TD{Z^T}}\\ {\lambda _1^TD{\lambda _2}}&{}\quad {\lambda _1^TD{\lambda _1} + s_1^2}&{}\quad {\lambda _1^TD{Z^T}}\\ {ZD{\lambda _2}}&{}\quad {ZD{\lambda _1}}&{}\quad {ZD{Z^T} + {\sigma ^2}I} \end{array}} \right) \end{aligned}$$

we have

$$\begin{aligned} E(x_2|x_1,\mathbf {y}, \hat{\theta }) = W^T \xi _2 + K_{12} K_{22} ^ {-1} \left( \begin{array}{c} {x_1 - W^T \hat{\xi }_1} \\ {y - T \hat{\alpha }} \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} Var(x_2|x_1,y,\hat{\theta }) = \hat{\lambda }_2 ^ T\ D \hat{\lambda }_ 2 + \hat{s}_2^2 - K_{12}K_{22} ^ {-1}\ K_{12}^T \end{aligned}$$

where $K_{12} = (\hat{\lambda }_2 ^T D \hat{\lambda }_1, \hat{\lambda }_2 ^T D Z^T )$ and $K_{22} = \left( {\begin{array}{*{20}{c}} {\hat{\lambda }_1^T \hat{D}{\hat{\lambda }_1} + \hat{s}_1^2}&{}{\hat{\lambda }_1^T \hat{D}{Z^T}}\\ {Z \hat{D}{\hat{\lambda }_1}}&{}{Z \hat{D}{Z^T} + {\hat{\sigma }^2}I} \end{array}} \right)$.

So the calculation of $E(x_2|\mathbf {y},x_1,x_2>c, \hat{\theta })$ and $Var(x_2|\mathbf {y},x_1,x_2>c, \hat{\theta })$ can be done based on the truncated normal distribution property. In R, this can be done easily using the truncnorm package [31].

1.2 Scenario II: $x_1$ Alone is Censored

Similarly, for the second scenario in which $x_2$ is observed and $x_1$ is censored, we have $Y_{obs}=(\mathbf {y},x_1>c,x_2)$ and

• $E( {{x_2}|Y_{obs},\hat{\theta }})=x_2$

• $E( {{x_2 ^2}|Y_{obs},\hat{\theta }})=x_2^2$

• $E( {{x_1}|Y_{obs},\hat{\theta }}) = E(x_1|\mathbf {y},x_1>c,x_2, \hat{\theta })$

• $E( {{x_1 x_2}|Y_{obs},\hat{\theta }})=x_2E( {{x_1}|\mathbf {y},x_1>c,x_2,\hat{\theta }})$

• $E( {{x_1 ^2}|Y_{obs},\hat{\theta }}) = E^2(x_1|\mathbf {y},x_1>c,x_2, \hat{\theta }) + Var(x_1|\mathbf {y},x_1>c,x_2, \hat{\theta })$

To calculate $E(x_1|\mathbf {y},x_2,x_1>c, \hat{\theta })$ and $Var(x_1|\mathbf {y},x_2,x_1>c, \hat{\theta })$, we have

$$\begin{aligned} E(x_1|x_2,\mathbf {y}, \hat{\theta }) = W^T \hat{\xi }_1 + P_{12} P_{22} ^ {-1} \left( \begin{array}{c} {x_2 - W^T \hat{\xi }_2} \\ {y - T \hat{\alpha }} \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} Var(x_1|x_2,\mathbf {y},\hat{\theta }) = \hat{\lambda }_1 ^ T\ D \hat{\lambda }_ 1 + \hat{s}_1^2 - P_{12}P_{22} ^ {-1}\ P_{12}^T \end{aligned}$$

where $P_{12} = (\hat{\lambda }_1 ^T D \hat{\lambda }_2, \hat{\lambda }_1 ^T D Z^T )$ and $P_{22} = \left( {\begin{array}{*{20}{c}} {\hat{\lambda }_2^T \hat{D}{\hat{\lambda }_2} + \hat{s}_2^2}&{}{\hat{\lambda }_2^T \hat{D}{Z^T}}\\ {Z \hat{D}{\hat{\lambda }_2}}&{}{Z \hat{D}{Z^T} + {\hat{\sigma }^2}I} \end{array}} \right)$.

1.3 Scenario III: Both $x_1$ and $x_2$ are Censored

When both $x_1$ and $x_2$ are censored, i.e, $x_1>c$ and $x_2>c$,

• $E( {{x_1}|Y_{obs},\hat{\theta }}) = E(x_1|\mathbf {y},x_1>c,x_2>c, \hat{\theta })$

• $E( {{x_1 ^2}|Y_{obs},\hat{\theta }}) = E^2(x_1|\mathbf {y},x_1>c,x_2>c, \hat{\theta }) + Var(x_1|\mathbf {y},x_1>c,x_2>c, \hat{\theta })$

• $E( {{x_2}|Y_{obs},\hat{\theta }}) = E(x_2|\mathbf {y},x_1>c,x_2>c, \hat{\theta })$

• $E( {{x_2 ^2}|Y_{obs},\hat{\theta }}) = E^2(x_2|\mathbf {y},x_1>c,x_2>c, \hat{\theta }) + Var(x_2|\mathbf {y},x_1>c,x_2>c, \hat{\theta })$

• $E( {{x_1 x_2}|Y_{obs},\hat{\theta }})=Cov( {{x_1x_2}|\mathbf {y},x_1>c,x_2>c,\hat{\theta }}) + E(x_1|\mathbf {y},x_1>c,x_2>c, \hat{\theta })*E(x_2|\mathbf {y},x_1>c,x_2>c, \hat{\theta })$

We have

$$\begin{aligned} E\left( \begin{array}{c} x_1 \\ x_2 \end{array} |\mathbf {y}, \hat{\theta }\right) = \left( \begin{array}{c} W^T \hat{\xi }_1 \\ W^T \hat{\xi }_2 \end{array} \right) + Q_{12} Q_{22} ^ {-1} \left( {y - T \hat{\alpha }} \right) \end{aligned}$$

and

$$\begin{aligned} Var\left( \begin{array}{c} x_1 \\ x_2 \end{array} |y,\hat{\theta }\right) = \left( \begin{array}{cc}\hat{\lambda }_1^T \hat{D} \hat{\lambda }_1 + \hat{s}_1^2 &{}\quad \hat{\lambda }_1 ^ T \hat{D} \hat{\lambda }_2 \\ \hat{\lambda }_2 ^ T \hat{D} \hat{\lambda }_1 &{}\quad \hat{\lambda }_2^T \hat{D} \hat{\lambda }_2 + \hat{s}_2^2 \end{array} \right) - Q_{12}Q_{22} ^ {-1} Q_{12}^T \end{aligned}$$

where $Q_{12} = (\hat{\lambda }_1 ^T D Z^T, \hat{\lambda }_2 ^T D Z^T )^T$ and $Q_{22} = Z \hat{D}{Z^T} + {\hat{\sigma }^2}I$. R package tmvtnorm [39] can then be used to calculate the mean and covariance of the truncated variables.