Analysis of Recurrent Event Processes with Dynamic Models for Event Counts

Abstract

Recurrent events are of interest in many research fields. The analysis of past developments of processes through dynamic covariates is useful to understand the present and future of processes generating recurrent events. In this paper, we consider modelling and estimation of effects of number of prior events and carryover effects on the evolution of recurrent event processes through dynamic models for event counts. These process features are related to monotonic trends in gap times and clustering of events together over time in recurrent event processes, and are frequently seen in biomedical and epidemiology studies involving recurrent events. Insights about the impacts of these features may provide opportunities for treatment improvements and to develop prevention strategies. We discuss issues in the parametric maximum likelihood estimation of these features through multiplicative recurrent event models for event counts. We extend our discussion to the settings in which excess variation in the rate of event occurrences across multiple individuals is present. We illustrate our methods to analyse a dataset from an asthma study involving children.

Appendices

Appendix A Explosion of Dynamic Recurrent Event Processes

In this appendix, we present details and results of a simulation study conducted to investigate the issue of dishonest process $\left\{ N_{i} (t);t\ge 0\right\}$ with the intensity function (3). For the data generation, we considered the intensity function (3) with $\lambda _{0i} (t)$ fixed at a constant value $\alpha$ $(\alpha =1)$ and without external covariates $\varvec{x}_{i}(t)$. We generated data under 36 scenarios with various values of $\gamma$, $\beta$, $\Delta$ and $\tau$ as denoted in Table 9. With every combination of $( \gamma , \beta , \Delta , \tau )$, we simulated 10000 realizations under four different models. The first model (Model 1 in Table 9) was the aforementioned dishonest process with the intensity function $\lambda _{i}[t \, | \, \mathcal {H}_{i}(t)] = \exp \left[ \gamma \, N_{i}(t^{-}) + \beta \, Z_{i}(t) \right]$. We replaced the dynamic covariate $N_{i}(t^-)$ with its trimmed version $N_{i}^{*}(t^{-})$ given in (6) with cutoff points $c=100,50$ and 20 in the second, third and fourth models (Models 2, 3, and 4 in Table 9), respectively, so that we were able to compare the results in an empirical setting. Note that $c=\infty$ in Model 1, meaning that $N_{i}^{*}(t^{-})=N_{i}(t^{-})$. In this empirical setting, we considered the process as exploded if it experienced 1000 events during its followup, and in this case, the simulation algorithm was stopped generating further events. Number of explosions and the maximum number of events in 10000 realizations for each scenario are reported in Table 9.

Table 9 Number of exploded processes and the maximum number of events in 10000 realizations

When $\gamma =0$, processes in the first twelve scenarios in Table 9 did not explode for any value of $\tau$. When $\gamma >0$, the number of explosions increased as the followup time $\tau$ increased. It is also noted that, when $\gamma >0$, any increase in the value of $\beta$ or carryover effects period $\Delta$ also resulted in an increase in the number of explosions. However, when $\gamma >0$, the number of explosions reduced as the cutoff value c decreased for any given ($\gamma$, $\beta$, $\Delta$, $\tau$) combination. Some other suggestions, e.g. modifying $N_{i}(t^-)$ with time t, to handle explosions due to $N_{i}(t^-)$ are discussed by Aalen et al. [3, Section 8.6.3]. The coefficients of those modified versions of $N_{i}(t^-)$ may not be easily interpretable. Therefore, we prefer to use $N_{i}^*(t^{-})$ given in (6) to handle dishonest processes in most of the applications especially when the subjects do not experience too many events during their followups.

Appendix B Derivation of Marginal Intensity Function in Random Effects Model

Suppose that m independent continuous time counting processes are included in a study. Let $\left\{ N_i(t),t\ge 0\right\}$, $i=1$, 2, $\ldots$, m, be the ith process with the intensity function $\lambda _i\left[ t \, | \, \mathcal {H}_i(t)\right]$, where $\mathcal {H}_i(t) = \left\{ N_i(s); 0 \le s<t\right\}$ is the history of the process. Our goal is to derive the marginal density function $\lambda _i \left[ t \, | \, \mathcal {H}_i(t)\right]$ from the conditional intensity function $\lambda _i\left[ t \, | \, \mathcal {H}_i(t),\nu _i\right]$, where the $\nu _i$ are i.i.d. random effects from a gamma distribution with mean 1 and variance $\phi$. To this end, we let $\delta N_i(t)$ denote the number of events in $[t,t+\delta t)$ for the counting process $\left\{ N_i(t),t\ge 0\right\}$. Then, as $\delta t \rightarrow 0$,

$$\begin{aligned} \begin{aligned} \lambda _i\left[ t \, | \, \mathcal {H}_i(t)\right] \, \delta t&= \Pr \left\{ \delta N_i(t)=1 \, | \, \mathcal {H}_i(t) \right\} = \dfrac{ \Pr \left\{ \delta N_i(t)=1, \mathcal {H}_i(t) \right\} }{ \Pr \left\{ \mathcal {H}_i(t) \right\} },\\&= \dfrac{ \int _{0}^{\infty } \Pr \left\{ \delta N_i(t)=1 \, | \, \mathcal {H}_i(t),\nu _i \right\} \Pr \left\{ \mathcal {H}_i(t) \,| \, \nu _i \right\} g(\nu _i) \;d\nu _i}{ \int _{0}^{\infty } \Pr \left\{ \mathcal {H}_i(t) \, | \, \nu _i \right\} g(\nu _i) \;d\nu _i}, \end{aligned} \end{aligned}$$

(B1)

where $g(\nu _i)$ denotes the p.d.f. of the random effect $\nu _i$, $i=1$, 2, $\ldots$, m, and any probability notation with $\mathcal {H}_i(t)$ denotes the probability of observing a given sample path of the counting process $\left\{ N_i(t),t\ge 0\right\}$ over [0, t) . Thus, if we let $N_i(t^-) = n_i$, we observe $n_i$ events over the interval [0, t) at times $t_{i1}$, $t_{i2}$, $\ldots$, $t_{in_{i}}$, where $t_{i1}< t_{i2}< \cdots <t_{in_i}$. We next specify the conditional intensity function $\lambda _i\left[ t \, | \, \mathcal {H}_i(t),\nu _i \right]$ as follows:

$$\begin{aligned} \lambda _i \left[ t \, | \, \mathcal {H}_{i} (t), \nu _{i} \right] = \nu _{i} \, \exp [\varvec{\psi }^{\prime } \varvec{W^*}_i(t)], \qquad t \ge 0, \end{aligned}$$

(B2)

where $\varvec{W^*}_{i}(t)$ is a $q \times 1$ vector that is allowed to contain functions of t and the event history $\mathcal {H}_i (t)$, as well as external covariates $\varvec{x}_{i} (t)$ and $\varvec{\psi }$ is a $q \times 1$ vector of parameters. Let ${\eta }_i(t) = \exp [\varvec{\psi }^{\prime } \varvec{W^*}_i(t)]$. The integrand in the denominator of (B1) can be written as

$$\begin{aligned} \begin{aligned} \Pr \left\{ \mathcal {H}_i(t) \,| \, \nu _i \right\} \, g(\nu _i)&= \left\{ \displaystyle \prod _{j=1}^{N_i(t^-)} \nu _i \, {\eta }_i(t_{ij}) \; \exp \left( - \nu _i \int \limits _0^t {\eta }_i(u) \, du \right) \right\} \frac{\nu _i^{\phi ^{-1}-1} \exp (-\nu _i/\phi )}{\phi ^{\phi ^{-1}} \Gamma (\phi ^{-1})},\\&= \left\{ \nu _i^{ [N_i(t^-)+\phi ^{-1}-1] } \; \exp \left( - \nu _i \left[ \int \limits _0^t {\eta }_i(u) \, du+\dfrac{1}{\phi }\right] \right) \right\} \frac{\displaystyle \prod _{j=1}^{N_i(t^-)} {\eta }_i(t_{ij}) }{\phi ^{\phi ^{-1}} \Gamma (\phi ^{-1})}. \end{aligned} \end{aligned}$$

(B3)

After taking the integral of (B3) with respect to $\nu _i$, the denominator of (B1) is given by

$$\begin{aligned} \begin{aligned} \int _{0}^{\infty } \Pr \left\{ \mathcal {H}_i(t) \, | \, \nu _i \right\} g(\nu _i) \, d\nu _i&= \frac{\displaystyle \prod _{j=1}^{N_i(t^-)} {\eta }_i(t_{ij}) }{\phi ^{\phi ^{-1}} \Gamma (\phi ^{-1})} \times \dfrac{\Gamma (N_i(t^-)+\phi ^{-1})}{\left[ \int \limits _0^t {\eta }_i(u) \, du+\dfrac{1}{\phi }\right] ^{(N_i(t^-)+\phi ^{-1})} }. \end{aligned} \end{aligned}$$

(B4)

Similarly, from the result in (B3), the numerator of (B1) is equal to

$$\begin{aligned} \begin{aligned} \int _{0}^{\infty } \nu _i \, {\eta }_i(t) \, \delta t \, \Pr \left\{ \mathcal {H}_{i}(t) \, | \, \nu _i \right\} g(\nu _i) \, d\nu _i =&\frac{{\eta }_i(t) \, \delta t \displaystyle \prod _{j=1}^{N_i(t^-)} {\eta }_i(t_{ij}) }{\phi ^{\phi ^{-1}} \Gamma (\phi ^{-1})} \times \\&\qquad \dfrac{\Gamma (N_i(t^-)+\phi ^{-1}+1)}{\left[ \int \limits _0^t {\eta }_i(u) \, du+\dfrac{1}{\phi }\right] ^{(N_i(t^-)+\phi ^{-1}+1)} }. \end{aligned} \end{aligned}$$

(B5)

From the results given in (B4) and (B5) the marginal intensity function for the process $\left\{ N_i(t),t\ge 0\right\}$, $i = 1,\ldots ,m$, is then given by

$$\begin{aligned} \lambda _i\left[ t\mid \mathcal {H}_i(t)\right] = \dfrac{(1+\phi \, N_i(t^-))}{\left[ 1+\phi \int \limits _0^t {\eta }_i(u) \, du\right] } {\eta }_i(t), \qquad t \ge 0. \end{aligned}$$

(B6)