Assessing the numerical integration of dynamic prediction formulas using the exact expressions under the joint frailty-copula model

Abstract

Joint models allow survival outcomes of a patient to be dynamically predictable based on intermediate events observed after treatment. The existing dynamic prediction methods need some numerical integration over frailty or random-effects distributions since the integral is implicit and improper. For a joint frailty-copula model, the Clayton copula and the gamma frailty model have been used to derive a dynamic prediction formula based on meta-analytic data. However, the prediction formulas under the Clayton copula involve an improper integral and need to be approximated numerically, as in all the joint models. In this paper, we consider the Gumbel copula and the FGM copula to obtain the exact (true) prediction formula without any numerical integration. The proposed formula also provides a tool for assessing approximation errors occurring to the approximation by numerical integration. Our numerical assessments show some approximation errors of the true prediction formula especially when the frailty distribution is heavy-tailed. A real data example illustrates how the proposed formulas can be used for assessing the approximation error. Our study may suggest some assessments for approximation errors in other types of joint models using more complex random-effects and frailty distributions.

Appendices

Appendix 1. Derivations under the Gumbel copula

We derive the dynamic prediction formulas under the Gumbel copula given $\alpha =1$.

1.1 Derivation of Eq. (5)

By substituting $\alpha =1$ into Eq. (3), the joint survival function under the Gumbel copula is

$$C_{\theta } \left[ {S_{X} \left( {x|u} \right),S_{D} \left( {y|u} \right)} \right] = {\text{exp}}\left[ { - u\left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}}}} } \right].$$

By integrating out the unobserved frailty u, the joint survival function is

$$\begin{aligned} \mathop \int \limits_{0}^{\infty } C_{\theta } \left[ {S_{X} \left( {x|u} \right),S_{D} \left( {y|u} \right)} \right]f_{\eta } \left( u \right){\text{d}}u & = \mathop \int \limits_{0}^{\infty } {\text{exp}}\left[ { - u\left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}}}} } \right]f_{\eta } \left( u \right){\text{d}}u \\ & = \left[ {1 + \eta \left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}}}} } \right]^{{ - \frac{1}{\eta }}} , \\ \end{aligned}$$

where the last equality follows from the Laplace transform $\underset{0}{\overset{\infty }{\int }}\mathrm{e}\mathrm{x}\mathrm{p}\left(-su\right){f}_{\eta }\left(u\right)\mathrm{d}u={\left(1+\eta s\right)}^{-\frac{1}{\eta }}$.

1.2 Derivation of Eq. (6)

By integrating out the unobserved frailty u,

$$\begin{gathered} \mathop \int \limits_{0}^{\infty } uR\left( x \right)^{\theta } \left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}} - 1}} {\text{exp}}\left[ { - u\left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}}}} } \right]f_{\eta } \left( u \right){\text{d}}u \hfill \\ = R\left( x \right)^{\theta } \left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}} - 1}} \mathop \int \limits_{0}^{\infty } u\exp \left[ { - u\left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}}}} } \right]~f_{\eta } \left( u \right){\text{d}}u \hfill \\ {\text{~}} = R\left( x \right)^{\theta } \left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}} - 1}} \left[ {1 + \eta \left\{ {R\left( x \right)^{{\theta + 1}} + \Lambda \left( y \right)^{{\theta + 1}} } \right\}^{{\frac{1}{{\theta + 1}}}} } \right]^{{ - \frac{1}{\eta } - 1}} , \hfill \\ \end{gathered}$$

where the last equality follows from the derivative of the Laplace transform

$$\mathop \int \limits_{0}^{\infty } u{\text{~exp}}\left( { - su} \right)f_{\eta } \left( u \right){\text{d}}u = - \frac{\partial }{{\partial s}}\mathop \int \limits_{0}^{\infty } {\text{exp}}\left( { - us} \right)f_{\eta } \left( u \right){\text{d}}u = \left( {1 + \eta s} \right)^{{ - \frac{1}{\eta } - 1}} .$$

Appendix 2. Derivations under the FGM copula

We derive the dynamic prediction formula under the FGM copula under $\alpha =1$ and $\alpha =0$.

2.1 Derivation of Eq. (9)

By substituting $\alpha =1$ into Eq. (7), the joint survival function under the FGM copula is

$$\begin{gathered} C_{\theta } \left[ {S_{X} \left( {x|u} \right),S_{D} \left( {y|u} \right)} \right] \hfill \\ \quad = {\text{exp}}\left[ { - \left\{ {uR\left( x \right) + \Lambda \left( y \right)} \right\}} \right]\left[ {1 + \theta \left\{ {1 - {\text{exp}}\left( { - uR\left( x \right)} \right)} \right\}\left\{ {1 - {\text{exp}}\left( { - \Lambda \left( y \right)} \right)} \right\}} \right]. \hfill \\ \end{gathered}$$

By integrating out the unobserved frailty u, the joint survival function is

$$\begin{gathered} \mathop \int \limits_{0}^{\infty } C_{\theta } \left[ {S_{X} \left( {x|u} \right),S_{D} \left( {y|u} \right)} \right]f_{\eta } \left( u \right){\text{d}}u \hfill \\ = \mathop \int \limits_{0}^{\infty } {\text{exp}}\left[ { - u\left\{ {R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]\left[ {1 + \theta \left\{ {1 - {\text{exp}}\left( { - uR\left( x \right)} \right)} \right\}\left\{ {1 - {\text{exp}}\left( { - u\Lambda \left( y \right)} \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u \hfill \\ = \left( {1 + \theta } \right)\mathop \int \limits_{0}^{\infty } {\text{exp}}\left[ { - u\left\{ {R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u - \theta \mathop \int \limits_{0}^{\infty } {\text{exp}}\left[ { - u\left\{ {2R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u \hfill \\ - \theta \mathop \int \limits_{0}^{\infty } {\text{exp}}\left[ { - u\left\{ {R\left( x \right) + 2\Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u + \theta \mathop \int \limits_{0}^{\infty } {\text{exp}}\left[ { - 2u\left\{ {R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u \hfill \\ = \frac{{\left( {1 + \theta } \right)}}{{\left[ {1 + \eta \left\{ {R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]^{{\frac{1}{\eta }}} }} - \frac{\theta }{{\left[ {1 + \eta \left\{ {2R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]^{{\frac{1}{\eta }}} }} \hfill \\ - \frac{\theta }{{\left[ {1 + \eta \left\{ {R\left( x \right) + 2\Lambda \left( y \right)} \right\}} \right]^{{\frac{1}{\eta }}} }} + \frac{\theta }{{\left[ {1 + 2\eta \left\{ {R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]^{{\frac{1}{\eta }}} }}, \hfill \\ \end{gathered}$$

where the last equality follows from the Laplace transform $\underset{0}{\overset{\infty }{\int }}\mathrm{e}\mathrm{x}\mathrm{p}\left(-su\right){f}_{\eta }\left(u\right)\mathrm{d}u={\left(1+\eta s\right)}^{-\frac{1}{\eta }}$.

2.2 Derivation of Eq. (10)

By integrating out the unobserved frailty u,

$$\begin{gathered} \left[ {\mathop \int \limits_{0}^{\infty } u~{\text{exp}}\left[ { - u\left\{ {R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u + \theta \mathop \int \limits_{0}^{\infty } u~{\text{exp}}\left[ { - u\left\{ {R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u} \right. \hfill \\ - 2\theta \mathop \int \limits_{0}^{\infty } u~{\text{exp}}\left[ { - u\left\{ {2R\left( x \right) + \Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u - \theta \mathop \int \limits_{0}^{\infty } u~{\text{exp}}\left[ { - u\left\{ {R\left( x \right) + 2\Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u \hfill \\ \left. { + 2\theta \mathop \int \limits_{0}^{\infty } u~{\text{exp}}\left[ { - u\left\{ {2R\left( x \right) + 2\Lambda \left( y \right)} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u} \right] \hfill \\ = \left[ {\frac{{1 + \theta }}{{\left\{ {1 + \eta \left( {R\left( x \right) + \Lambda \left( y \right)} \right)} \right\}^{{\frac{1}{\eta } + 1}} }} - \frac{{2\theta }}{{\left\{ {1 + \eta \left( {2R\left( x \right) + \Lambda \left( y \right)} \right)} \right\}^{{\frac{1}{\eta } + 1}} }}} \right. \hfill \\ - \left. {\frac{\theta }{{\left\{ {1 + \eta \left( {R\left( x \right) + 2\Lambda \left( y \right)} \right)} \right\}^{{\frac{1}{\eta } + 1}} }} + \frac{{2\theta }}{{\left\{ {1 + 2\eta \left( {R\left( x \right) + \Lambda \left( y \right)} \right)} \right\}^{{\frac{1}{\eta } + 1}} }}} \right], \hfill \\ \end{gathered}$$

where the last equality follows from the derivative of the Laplace transform

$$\mathop \int \limits_{0}^{\infty } u{\text{~exp}}\left( { - su} \right)f_{\eta } \left( u \right){\text{d}}u = - \frac{\partial }{{\partial s}}\mathop \int \limits_{0}^{\infty } {\text{exp}}\left( { - us} \right)f_{\eta } \left( u \right){\text{d}}u = \left( {1 + \eta s} \right)^{{ - \frac{1}{\eta } - 1}} .$$

2.3 Derivation of Eq. (11)

By substituting $\alpha =0$ into Eq. (7), the joint survival function under the FGM copula is

$$\begin{gathered} C_{\theta } \left[ {S_{X} \left( {x|u} \right),S_{D} \left( {y|u} \right)} \right] \hfill \\ = {\text{exp}}\left[ { - \left\{ {uR\left( x \right) + \Lambda \left( y \right)} \right\}} \right]\left[ {1 + \theta \left\{ {1 - {\text{exp}}\left( { - uR\left( x \right)} \right)} \right\}\left\{ {1 - {\text{exp}}\left( { - \Lambda \left( y \right)} \right)} \right\}} \right]. \hfill \\ \end{gathered}$$

By integrating out the unobserved frailty u, the joint survival function is

$$\begin{gathered} \mathop \int \limits_{0}^{\infty } C_{\theta } \left[ {S_{X} \left( {x|u} \right),S_{D} \left( {y|u} \right)} \right]f_{\eta } \left( u \right){\text{d}}u \hfill \\ = \mathop \int \limits_{0}^{\infty } {\text{exp}}\left[ { - \left\{ {uR\left( x \right) + \Lambda \left( y \right)} \right\}} \right]\left[ {1 + \theta \left\{ {1 - {\text{exp}}\left\{ { - uR\left( x \right)} \right\}} \right\}\left\{ {1 - {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}} \right\}} \right]f_{\eta } \left( u \right){\text{d}}u \hfill \\ = {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}\mathop \int \limits_{0}^{\infty } {\text{exp}}\left\{ { - uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u + \theta {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}\mathop \int \limits_{0}^{\infty } {\text{exp}}\left\{ { - uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u \hfill \\ - \theta {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}\mathop \int \limits_{0}^{\infty } {\text{exp}}\left\{ { - 2uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u - \theta {\text{exp}}\left\{ { - 2\Lambda \left( y \right)} \right\}\mathop \int \limits_{0}^{\infty } {\text{exp}}\left\{ { - uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u \hfill \\ + \theta {\text{exp}}\left\{ { - 2\Lambda \left( y \right)} \right\}\mathop \int \limits_{0}^{\infty } {\text{exp}}\left\{ { - 2uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u \hfill \\ = {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}\left[ {\frac{{1 + \theta \left\{ {1 - {\text{exp}}\left( { - \Lambda \left( y \right)} \right)} \right\}}}{{\left\{ {1 + \eta R\left( x \right)} \right\}^{{\frac{1}{\eta }}} }} + \frac{{\theta \left\{ {{\text{exp}}\left( { - \Lambda \left( y \right)} \right) - 1} \right\}}}{{\left\{ {1 + 2\eta R\left( x \right)} \right\}^{{\frac{1}{\eta }}} }}} \right], \hfill \\ \end{gathered}$$

where the last equality follows from the Laplace transform

$$\mathop \int \limits_{0}^{\infty } {\text{exp}}\left( { - su} \right)f_{\eta } \left( u \right){\text{d}}u = \left( {1 + \eta s} \right)^{{ - \frac{1}{\eta }}} .$$

2.4 Derivation of Eq. (12)

By integrating out the unobserved frailty u,

$$\begin{gathered} {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}\left[ {\left( {1 + \theta } \right)\mathop \int \limits_{0}^{\infty } u{\text{exp}}\left\{ { - uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u - 2\theta \mathop \int \limits_{0}^{\infty } u{\text{exp}}\left\{ { - 2uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u} \right. \hfill \\ - \left. {\theta {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}\mathop \int \limits_{0}^{\infty } u{\text{exp}}\left\{ { - uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u + 2\theta {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}\mathop \int \limits_{0}^{\infty } u{\text{exp}}\left\{ { - 2uR\left( x \right)} \right\}f_{\eta } \left( u \right){\text{d}}u} \right] \hfill \\ = {\text{exp}}\left\{ { - \Lambda \left( y \right)} \right\}\left[ {\frac{{1 + \theta \left\{ {1 - {\text{exp}}\left( { - \Lambda \left( y \right)} \right)} \right\}}}{{\left\{ {1 + \eta R\left( x \right)} \right\}^{{\frac{1}{\eta } + 1}} }} + \frac{{2\theta \left\{ {{\text{exp}}\left( { - \Lambda \left( y \right)} \right) - 1} \right\}}}{{\left\{ {1 + 2\eta R\left( x \right)} \right\}^{{\frac{1}{\eta } + 1}} }}} \right], \hfill \\ \end{gathered}$$

where the last equality follows from the derivative of the Laplace transform

$$\mathop \int \limits_{0}^{\infty } u{\text{~exp}}\left( { - su} \right)f_{\eta } \left( u \right){\text{d}}u = - \frac{\partial }{{\partial s}}\mathop \int \limits_{0}^{\infty } {\text{exp}}\left( { - us} \right)f_{\eta } \left( u \right){\text{d}}u = \left( {1 + \eta s} \right)^{{ - \frac{1}{\eta } - 1}} .$$

Appendix 3. R functions to compute prediction formulas

To implement dynamic prediction, we introduce our original R functions that compute $\hat{F}\left( {t,t + w|.,\boldsymbol{Z}} \right)$ under the Gumbel copula (Sect. 3.1) and the FGM copula (Sect. 3.2). The R source codes are given in Electric Supplementary Material.

Users who wish to calculate $\widehat{F}\left(t,t+w|.,\boldsymbol{Z}\right)$ under the Gumbel copula should use the R functions called F.windows.Gumbelcop1(.) or F.windows.Gumbelcop(.). The former can calculate the exact value of $\widehat{F}\left(t,t+w|.,\boldsymbol{Z}\right)$ under $\alpha =1$ while the latter calculate the approximate value under any $\alpha$. Hence, "cop" and "cop1" are related to $\alpha$. Under the FGM copula, users should use F.windows.FGMcop1(.), F.windows.FGMcop0(.), or F.windows.FGMcop(.), where the first two gives the exact values while the last one gives the approximate value.

The above functions are workable when their subroutine functions are defined in the workspace (global environments). The names of subroutines are without “s”, such as F.window.Gumbelcop1(.), F.window.Gumbelcop(.), and others.

All the functions have the following arguments to be supplied by users:

• prediction time = time (scalar)

• widths of prediction interval = widths (scalar or vector)

• event time = X (scalar)

• covariates for TTP = Z1 (scalar or vector)

• covariates for OS = Z2 (scalar or vector)

• regression coefficients for TTP = beta1 (scalar or vector)

• regression coefficients for OS = beta2 (scalar or vector)

• variance of frailty = eta (scalar)

• copula parameter = theta (scalar)

• power parameter of frailty = alpha (scalar)

• parameters related to the baseline hazard for TTP = g (vector of length 5)

• parameters related to the baseline hazard for OS = h (vector of length 5)

• lower bound for time-to-event = xi1 (scalar)

• upper bound for time-to-death = xi3 (scalar)

• logical value to decide if the plot is shown = Fplot (TRUE or FALSE).

Below, we provide a short example of using our R functions. Suppose we wish to calculate F under the Gumbel copula given $\alpha =1$ using the exact method. After loading all the R functions into the R console, we enter the following commands (red parts) to obtain the results (blue parts):

The output shows the values of $\widehat{F}\left(t,t+w|X=x,\boldsymbol{Z}\right)$ and $\widehat{F}\left(t,t+w|X>t,\boldsymbol{Z}\right)$ for given values of $t$ and $w$ under $X=300$. Below is the interpretation of the last line of the output.

• (i). Given that a patient had no tumour progression at time $t=1000$ (i.e. $\mathrm{X}>1000$), the probability of death between $t=1000$ and $t+w=2000$ is 0.539.

• (ii). Given that a patient had tumour progression at time $x=300$ (i.e. $\mathrm{X}\le 1000$), the probability of death between $t=1000$ and $t+w=2000$ is 0.962.

Further examples for the input and output are given in Electric Supplementary Material.