Analysis of cyclic recurrent event data with multiple event types

Abstract

Recurrent event data frequently arise in practice, and in some cases, the event process has cyclic or periodic components. We propose a semiparametric rate model with multiple event types that have such features. Generalized estimating equations are used for the estimation of regression coefficients after profiling the baseline rate function with a fully nonparametric estimator. The proposed estimators are shown to be consistent and asymptotically Gaussian. Their finite-sample behavior is assessed through simulation experiments. The predictability of the model with and without the cyclic component is also compared. With the cyclic component, our model improves the predictability of a conventional model without the cyclic feature. Data on recurrent fire alarms in Blenheim, New Zealand, are used for illustration purposes.

1. Introduction

Recurrent events are commonly observed in a long follow-up period in clinical studies. For example, young children often experience multiple asthma attacks (Duchateau et al. 2007), and cystic fibrosis patients tend to have recurrent pulmonary exacerbations of respiratory symptoms (Therneau and Grambsch 2013). In these studies, it is often of interest to investigate whether treatments or covariates are associated with the risk of clinical events.

In the last 2 decades, statistical methods for the analysis of recurrent events have received much attention. Conventionally, recurrent events are modeled as the realization of an underlying counting process, where the calendar time or the time since enrollment is used as the time scale. Methods for evaluating the covariate effects in the regression analysis then fall into two categories: conditional methods and marginal methods. The conditional methods usually deal with intensity function that is conditional on the event history (Prentice et al. 1981; Andersen and Gill 1982; Wang et al. 2001; Liu et al. 2004), while the marginal methods usually deal with rate function that is conditional only on the current values of covariates (Pepe and Cai 1993; Lawless and Nadeau 1995; Lin et al. 2000; Schaubel et al. 2006; Liu et al. 2010; Su et al. 2020). In some clinical trials, the recurrent event occurrence might be stopped by a terminal event. The censoring mechanism is informative, because the recurrent event process and the terminal event process are dependent. To take this dependency into account, Cai et al. (2010) used the marginal model approach, while Huang and Wang (2004) and Rondeau et al. (2007) proposed a frailty model that is based on an intensity function. Cook and Lawless (2007) provided a comprehensive review of the statistical methods for recurrent event data.

Despite these attentions to recurrent events, most of the current literature focuses on the modeling of a single event type. In many examples, however, the recurrent events are of multiple types. Examples include bacterial, viral, and fungal infections after bone marrow transplants (Ninin et al. 2001) and repeated mucoid and non-mucoid Pseudomonas æruginosa infections in early childhood of cystic fibrosis patients (Li et al. 2005; Lin et al. 2013). Recurrent events with multiple types can also occur in environmental science. The motivating example in this work is the fire alarm data collected in the urban and rural areas of Blenheim, New Zealand, where multiple types of fire alarms occurred periodically in the area (Wang et al. 2012). Of particular interest is the association between fire incidents and various indicators such as area, alarm method, heat sources, and objects ignited. This association could potentially vary with the type of fire in the sense that the regression coefficient of a specific indicator is not necessarily identical across different types of fire.

Several statistical methods have been proposed in the literature to analyze recurrent event data with multiple types. Prentice et al. (1981) first suggested extending their conditional intensity model to multiple types of infections such as bacterial, fungal, and viral origin. To describe the dependence among different types of recurrent events, Abu-Libdeh et al. (1990) proposed a non-homogeneous mixed Poisson process. Lawless et al. (2001) proposed gap time models for failures of surgically implanted shunts in children with hydrocephalus and discussed the possibility of extension to multiple types of recurrent events. Cai and Schaubel (2004) proposed a class of proportional marginal mean and rate models for physician office visits and hospitalizations induced by asthma. Schaubel and Cai (2006a, b) then considered the analysis for the same kind of data when the even type is possibly missing. Sun et al. (2009) studied time-varying covariate effects for the recurrent events with multiple types based on a semiparametric rate model. Zhu et al. (2010) considered a joint modeling approach for regression analysis in the presence of a dependent terminal event. Zhao et al. (2012) studied a joint semiparametric frailty-based proportional intensity model with both time-dependent and time-independent covariates. Other rate models include a general semiparametric additive rate model in Chen et al. (2012) and Ye et al. (2015), where Ye et al. (2015) also proposed a remediation method when the event type is missing at random.

Moreover, one research area of interest is to investigate the correlation strength between different event outcomes. Most recently, Emura et al. (2017) and Emura et al. (2018) proposed a copula-based frailty model that is designed for meta-analysis in particular and can be further extended to recurrent event models with an informative terminal event presented. However, their model can be valid only when the terminal event process is a renewal process, and the correlation remains constant over time. Li et al. (2019) considered a copula-based frailty model under the Bayesian framework, which relaxes the renewal process assumption and quantifies the correlation between recurrent events and the terminal event. Li et al. (2020) further considered a joint hierarchical copula model that allows the correlation between recurrent events and a terminal event to vary over time.

In this article, a vital feature of the fire data is that fire occurrence is cyclical over a year, as it is highly associated with temperature and humidity. While it would be desirable to account for such seasonality in the model, the inclusion of cyclic functions in the regression analysis of recurrent events with multiple event types has not been discussed in the literature. We aim to fill this gap. To analyze cyclic recurrent events with multiple types, we propose in Sect. 2 a semiparametric rate model with baseline rate functions that renew at either random or fixed time points to account for the cyclical features of the process, mimicking the notion of modulated renewal processes discussed in Lin and Fine (2009). Proceeding along similar lines in Cai and Schaubel (2004), we use estimating equations to derive regression coefficient estimates after profiling out the cyclic baseline rate functions by a nonparametric estimator. Phenomena with the cyclic features are relatively common in studies in nature, including global warming on seasonal precipitations and floods (Vera et al. 2006), climate change on migration cycles of birds (Coppack and Both 2002), and a periodic-based model for cyclic temporal relations (Hornsby et al. 1999). The techniques developed herein could potentially be adapted to these various contexts.

The remaining sections are organized as follows. In Sect. 2, we present the proposed semiparametric rate model with cyclic baseline rate functions for multiple recurrent event types. The corresponding estimating procedures for regression coefficient parameters and cyclic rate function are also presented. The consistency and asymptotic normality of the estimators are derived in Sect. 3. The finite-sample properties of our estimators are evaluated through simulation experiments in Sect. 4, as well as predictability of the model with and without the cyclic component. In Sect. 5, we apply the proposed method to the fire prevention data from New Zealand. Concluding remarks can be found in Sect. 6. The R codes are deposited to Github (https://github.com/ChienLinSu/SL_Recurrent_Cyclic).

2. Model and estimation method

Let $N_{ij}^{*}(t)={\int }_0^{t}d{N}_{ij}^{*}(s)$ be the number of events of type $j\in \{1,\dots ,J\}$ at time t for subject $i\in \{1,\dots ,n\}$. Let C_ij and $Y_{ij}(t)=\mathbf{1}\left(C_{ij}\ge t\right)$ denote the type-specific censoring time and at-risk function, respectively. The observed event process for subject i and type j can thus be written as $N_{ij}(t)={\int }_0^t{Y}_{ij}(s)d{N}_{ij}^{*}(s)$.

Let ${ℛ}_K=\left\{r_0,\dots ,r_K\right\}$ be a sequence of renewal times, the same across all event types, with $r_0=0$ and $r_K\le \tau <r_{K+1}$, where τ is the end of the observation period. For simplicity, we assume that K is finite, and the renewal times equally spaced between r₀ and r_K; i.e., $r_k=k{r}_1$ for $k=0,\dots ,K$, and $K\ge 0$. Neither assumption is necessary. The renewal times may come from a realization of a point process that induces unequal gaps between renewal times similar to the one used in Lin and Fine (2009). If K = 0, the process is simply a conventional point process without the cyclic feature.

Let ${ℛ}_t=\left\{r_k\in {ℛ}_K:r_k<t\right\}$ be the subset of renewal times smaller than $t\in (0,\tau ]$. The proposed semiparametric rate model is specified as:

$$\text{E}\left\{d{N}_{ij}^{*}(t)\mid {\mathit{Z}}_{ij}(t)\right\}=e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}(t)}d{\mu }_{0,j}\left(t-r_t\right),t\in (0,\tau ],$$ (1)

where Z_ij(t) is a p × 1 vector of covariates related to events of type $j\in \{1,\dots ,J\},\mathit{\beta }$ is a p × 1 vector of corresponding regression coefficients, the upper script ⊤ represents the transpose of a vector and $d{\mu }_{0,j}$ is the baseline rate function of event type j, which is indexed by t − r_t, the gap time between the most recent renewal time $r_t=\max {ℛ}_t$ and time t. The baseline rate function is the component that accounts for the cyclic nature of the data. In model (1), regression coefficients can be corresponding to different event types by formatting ${\mathit{Z}}_{ij}(t)$ with type-specific covariates. For example, when J = 2, one can define ${\mathit{Z}}_{i1}(t)={[{\tilde{\mathit{Z}}}_{i1}^{\top }(t),{\mathbf{0}}_2^{\top }]}^{\top }$ and ${\mathit{Z}}_{i2}(t)={[{\mathbf{0}}_1^{\top },{\tilde{\mathit{Z}}}_{i2}^{\top }(t)]}^{\top }$, where ${\tilde{\mathit{Z}}}_{ij}(t)$ is the type-specific covariate for j = 1,2, and 0_j is a zero vector. By defining $\mathit{\beta }={\left({\mathit{\beta }}_1^{\top },{\mathit{\beta }}_2^{\top }\right)}^{\top }$, the model is equivalent to:

$$\text{E}\left\{d{N}_{ij}^{*}(t)\mid {\tilde{\mathit{Z}}}_{ij}(t)\right\}=e^{{\mathit{\beta }}_j^{\top }{\tilde{\mathit{Z}}}_{ij}(t)}d{\mu }_{0,j}\left(t-r_t\right),$$

for j = 1,2, which is a marginal rate model for the individual recurrent event process of type j.

Let ${\mu }_{0,j}(u)={\int }_0^{u}d{\mu }_{0,j}(v)$ denote the jth baseline mean function, which is an accumulation of the baseline rate function up to time $u\in \left(0,r_1\right]$. The term “mean function” represents the expected value of the cumulative number of events as a function of t. It could also be called cumulative rate function (CRF) as the model (1) is defined by a rate function, and can often be seen in the recurrent event literature, e.g., Lin et al. (2000), Cai and Schaubel (2004), Chen et al. (2012), and Ye et al. (2015). We thus use the term mean function in the rest of the paper. To demonstrate the idea of cyclic mean functions, we consider t = 7, renewal times ${ℛ}_K=\{0,2,4,6,8\}$, ${ℛ}_7=\{0,2,4,6\}$, ${\mu }_{0,1}\left(s-r_s\right)=0.15\times \left(s-r_s\right)$ and ${\mu }_{0,2}\left(s-r_s\right)=0.2\times \left(s-r_s\right)$. Figure 1 shows the cyclic mean functions of event types 1 and 2. Each of the mean functions renews at ren0065wal times 2, 4, and 6, until time t =7 after the most recent renewal time t = 6.

Fig. 1.

Illustration of cyclic mean functions with t = 7, renewal times ${ℛ}_K=\{0,2,4,6,8\}$, ${ℛ}_7=\{0,2,4,6\}$, ${\mu }_{0.1}\left(s-r_s\right)=0.15\times \left(s-r_s\right)$, and ${\mu }_{0,2}\left(s-r_s\right)=0.2\times \left(s-r_s\right)$

To estimate ${\mu }_{0,j}(u)$ and 𝜷, we utilize estimating equations:

$$\sum _{i=1}^n\sum _{j=1}^J{\int }_0^{\tau }{\mathit{Z}}_{ij}(s)\{d{N}_{ij}(s)-Y_{ij}(s)e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}(s)}d{\mu }_{0,j}\left(s-r_s\right)\}={\mathbf{0}}_{p\times 1},$$ (2)

and

$$\sum _{i=1}^n\sum _{k=0}^K{\int }_0^u\{d{N}_{ij}\left(r_k+v\right)-Y_{ij}\left(r_k+v\right)e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}\left(r_k+v\right)}d{\mu }_{0,j}(v)\}=0,$$ (3)

for $j\in \{1,\dots ,J\}$ and $u\in \left(0,r_1\right]$. Eq. (3) is a type-specific estimating equation for the baseline mean function of type j. The equation is built based on model (1) and a moment estimator with $E\{d{N}_{ij}^{*}(t)-e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}(t)}d{\mu }_{0,j}\left(t-r_t\right)\mid {\mathit{Z}}_{ij}(t)\}=0$. Equation (3) leads to a Breslow-type estimator of the mean function μ_0,j(u) viz.:

$${\widehat{\mu }}_{0,j}^{\mathrm{Cyclic}}(u;\mathit{\beta })={\int }_0^{u}d{\widehat{\mu }}_{0,j}^{\text{Cyclic}}(v;\mathit{\beta }),$$ (4)

with:

$$d{\widehat{\mu }}_{0,j}^{\text{Cyclic}}(v;\mathit{\beta })=\frac{{\sum }_{i=1}^n{\sum }_{k=1}^{K}d{N}_{ij}\left(r_k+v\right)}{{\sum }_{i=1}^n{\sum }_{k=0}^K{Y}_{ij}\left(r_k+v\right)e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}\left(r_k+v\right)}},$$

which is a nonparametric cyclic function that jumps one step at every observed gap time from the renewal point and identical in every interval between renewal times. The upper script “Cyclic” denotes the cyclic property of the rate function. The term “Cyclic” is also applied to related estimators in the following sections if applicable.

Expressed equivalently, one can write the mean function in calendar time, namely:

$${\widehat{\mu }}_{0,j}^{\text{Cyclic}}(t;\mathit{\beta })={\int }_0^{t}d{\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(s-r_s;\mathit{\beta }\right),t\in (0,\tau ],$$ (5)

with

$$d{\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(s-r_s;\mathit{\beta }\right)=\frac{{\sum }_{i=1}^n{\sum }_{k=0}^{K}d{N}_{ij}\left(r_k+s-r_s\right)}{{\sum }_{i=1}^n{\sum }_{k=0}^K{Y}_{ij}\left(r_k+s-r_s\right)e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}\left(r_k+s-r_s\right)}}.$$

The estimator is an accumulation of the cyclic rate function estimator up to t. Obviously, without the cyclic feature of the baseline rate function in the model (1), i.e., K=0, the corresponding baseline mean function estimator is a special case of the cyclic estimator (5) that can be expressed as:

$${\tilde{\mu }}_{0,j}^{\text{CS}}(t;\mathit{\beta })={\int }_0^{t}d{\tilde{\mu }}_{0,j}^{\text{CS}}(s;\mathit{\beta }),$$ (6)

where

$$d{\tilde{\mu }}_{0,j}^{\text{CS}}(s;\mathit{\beta })=\frac{{\sum }_{i=1}^{n}d{N}_{ij}(s)}{{\sum }_{i=1}^n{Y}_{ij}(s)e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}(s)}},$$

which is the baseline mean function estimator proposed by Cai and Schaubel (2004). We use the upper script “CS” to denote the estimator. The same term is also applied to related estimators in the following sections when applicable. As one can see, the difference between the cyclic mean function estimator (5) and non-cyclic mean function estimator (6) is that the cyclic estimator has more jumps in each renewal interval than the non-cyclic estimator. The size of the jump $d{\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(s-r_s;\mathit{\beta }\right)$ in the cyclic estimator is a weighted average of the size of the jumps $d{\tilde{\mu }}_{0,j}^{\text{CS}}(s;\mathit{\beta })$ in the non-cyclic estimator over the K renewal periods.

Substituting (4) into (2) yields the following estimating equation for β, which are free of ${\mu }_{0,1},\dots ,{\mu }_{0,J}$:

$${\mathit{U}}_n(\mathit{\beta })=\sum _{i=1}^n\sum _{j=1}^J{\int }_0^{\tau }\{{\mathit{Z}}_{ij}(s)-{\overline{\mathit{Z}}}_j(s;\mathit{\beta })\}d{N}_{ij}(s)={\mathbf{0}}_{p\times 1},$$ (7)

where ${\overline{\mathit{Z}}}_j(t;\mathit{\beta })={\mathit{S}}_j^{(1)}(t;\mathit{\beta })/S_j^{(0)}(t;\mathit{\beta })$ with:

$${\mathit{S}}_j^{(d)}(t;\mathit{\beta })=n^{-1}\sum _{i=1}^n{\mathit{Z}}_{ij}{(t)}^{\otimes d}{Y}_{ij}(t)e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}(t)},$$

${\mathit{Z}}_{ij}^{\otimes 0}=1$ and ${\mathit{Z}}_{ij}^{\otimes 1}={\mathit{Z}}_{ij}$ Let ${\widehat{\mathit{\beta }}}_n$ denote our estimator for β solving Eq. (7). One can see that Eq. (7) is identical to the estimating equations of the partial likelihood function for a proportional hazards model, stratified by the event type. Hence, one can utilize the coxph function in the R library survival (Therneau 2017) to solve for the coefficient estimates with the “strata” option to indicate the stratification. The estimator for μ_0,j(u) can then be obtained by plugging ${\widehat{\mathit{\beta }}}_n$ into (4). The resulting estimator is denoted by ${\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(u;{\widehat{\mathit{\beta }}}_n\right)$. Note that the estimating equation (7) for β is the same as Eq. (5) in Cai and Schaubel (2004) with $g(x)=\exp (x)$, regardless whether the baseline rate function is cyclical. Theorem 1 in the following section shows that the asymptotic variance of ${\widehat{\mathit{\beta }}}_n$ obtained from Eq. (7) is also identical to the one in Cai and Schaubel (2004). It does not surprise the estimation is identical, since the baseline rate function in Cai and Schaubel (2004) needs no specification, including the cyclic function as a special case. However, when the baseline rate function is cyclic, the Breslow-type estimator used in Cai and Schaubel (2004) for the baseline mean function is not efficient, since less information is utilized. Theorem 2 in Sect. 3 presents the asymptotic behavior of our cyclical baseline mean function estimator in Eq. (4). The merit of considering a cyclic feature in the baseline mean function estimation is the efficiency gain through the average of information across multiple cycles. Therefore, a more precise estimation for the cyclic-specific rate function is anticipated. We explore the improvement of efficiency in Sect. 4 through comprehensive simulation studies.

3. A symptotic properties and prediction

3.1. Asymptotic properties

Set ${\mathit{V}}_j(t;\mathit{\beta })={\mathit{S}}_j^{(2)}(t;\mathit{\beta })/S_j^{(0)}(t;\mathit{\beta })-{\overline{\mathit{Z}}}_j{(t;\mathit{\beta })}^{\otimes 2}$, where ${\mathit{Z}}_{ij}^{\otimes 2}={\mathit{Z}}_{ij}{\mathit{Z}}_{ij}^{\top }$. For $d\in \{0,1,2\}$, let ${\mathit{s}}_j^{(d)}(s;\mathit{\beta }),{\overline{z}}_j(s;\mathit{\beta })$ and ${\mathit{v}}_j(s;\mathit{\beta })$ denote the limiting values, as $n\to \infty$, of ${\mathit{S}}_j^{(d)}(t;\mathit{\beta })$, ${\overline{\mathit{Z}}}_j(t;\mathit{\beta })$ and ${\mathit{V}}_j(s;\mathit{\beta })$, respectively. We have ${\mathit{v}}_j(t;\mathit{\beta })={\mathit{s}}_j^{(2)}(t;\mathit{\beta })/s_j^{(0)}(t;\mathit{\beta })-{\overline{z}}_j{(t;\mathit{\beta })}^{\otimes 2}$. The asymptotic properties of ${\widehat{\mathit{\beta }}}_n$ and ${\widehat{\mu }}_{0j}\left(t;{\widehat{\mathit{\beta }}}_n\right)$ given below are proved in Appendix A of the online supplementary materials.

Theorem 1

Under the regular conditions listed in Appendix A, ${\widehat{\mathit{\beta }}}_n$ is consistent and asymptotically Gaussian, i.e., ${\widehat{\mathit{\beta }}}_n\stackrel{a.s.}{\to }{\mathit{\beta }}_{0}and\sqrt{n}\left({\widehat{\mathit{\beta }}}_n-{\mathit{\beta }}_0\right)⇝{\mathcal{N}}_p\left[0,\mathit{\Sigma }\left({\mathit{\beta }}_0\right)\right]$, where $\mathit{\Sigma }(\mathit{\beta })=\mathit{A}{(\mathit{\beta })}^{-1}\mathit{B}(\mathit{\beta })\mathit{A}{(\mathit{\beta })}^{-1}$:

$$\begin{gathered} \mathit{A}(\mathit{\beta })=\sum _{j=1}^J{\int }_0^{\tau }{\mathit{v}}_j(s;\mathit{\beta })s_j^{(0)}(s;\mathit{\beta })d{\mu }_{0,j}\left(s-r_s\right), \\ \mathit{B}(\mathit{\beta })=\text{E}\left\{{[\sum _{j=1}^J{\mathit{U}}_{1,j}(\mathit{\beta })]}^{\otimes 2}\right\}, \\ {\mathit{U}}_{i,j}(\mathit{\beta })={\int }_0^{\tau }\{{\mathit{Z}}_{ij}(s)-{\overline{z}}_j(s;\mathit{\beta })\}d{M}_{ij}(s;\mathit{\beta }), \end{gathered}$$

with $d{M}_{ij}(t;\mathit{\beta })=d{N}_{ij}(t)-Y_{ij}(t)e^{{\mathit{\beta }}^{\top }{\mathit{Z}}_{ij}(t)}d{\mu }_{0,j}\left(t-r_t\right)$.

For all $j\in \{1,\dots ,J\}$, set $W_{j:n}(u)=\sqrt{n}\left\{{\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(u;{\widehat{\mathit{\beta }}}_n\right)-{\mu }_{0,j}(u)\right\}$ and write ${\mathit{W}}_n(u)={\left(W_{1:n}(u),\dots ,W_{J:n}(u)\right)}^{\top }$ and $\mathit{W}(u)={\left(W_1(u),\dots ,W_J(u)\right)}^{\top }$. We define an appropriate metric on the pertinent function space. Let $f_j(t)\in D\left(0,r_1\right]$, where $D\left(0,r_1\right]$ is the space of cadlag functions, such that $f_j(\cdot )\to ℝ$. Set $\mathit{f}(t)={\left(f_1(t),\dots ,f_J(t)\right)}^{\top }$, such that $\mathit{f}(\cdot ):\left[0,r_1\right]\to {ℝ}^J$. Define $D{\left(0,r_1\right]}^J$ to be the space consisting of functions $\mathit{f}(\cdot )$ and, following the approach of Spiekerman and Lin (1998), equip $D{\left(0,r_1\right]}^J$ with the metric, $\rho (g,h)=\max {\{{\sup }_{t\in \left[0,r_1\right]}|g_j(t)-h_j(t)|\}}_{j=1}^J$ for $g(\cdot ),h(\cdot )\in D{\left(0,r_1\right]}^J$. After defining an appropriate metric space with respect to which convergence can be discussed, the asymptotic behavior of ${\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(u;{\widehat{\mathit{\beta }}}_n\right)$. is given in the next theorem.

Theorem 2

The baseline mean function estimator ${\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(u;{\widehat{\mathit{\beta }}}_n\right)$ converges almost surely to ${\mu }_{0,j}(u)$ uniformly in $u\in \left(0,r_1\right]$ and ${\mathit{W}}_n(u)$ converges weakly to W(u), which is a multivariate zero-mean Gaussian process in the space $D{\left(0,r_1\right]}^J$ with covariance function $Cov(W_j(u),W_{\ell }(v))$ given by:

$${\mathit{\psi }}_{j\ell }\left(u,v;{\mathit{\beta }}_0\right)=\text{E}\left\{{\mathit{\Psi }}_{ij}\left(u;{\mathit{\beta }}_0\right){\mathit{\Psi }}_{i\ell }\left(v;{\mathit{\beta }}_0\right)\right\},$$

where

$${\mathit{\Psi }}_{ij}\left(u;{\mathit{\beta }}_0\right)={\int }_0^u\frac{{\sum }_{k=0}^{K}d{M}_{ij}\left(r_k+v;{\mathit{\beta }}_0\right)}{{\sum }_{k=0}^K{s}_j^{(0)}\left(r_k+v;{\mathit{\beta }}_0\right)}+{\mathit{c}}_j{\left(u;{\mathit{\beta }}_0\right)}^{\top }\mathit{A}{\left({\mathit{\beta }}_0\right)}^{-1}{\mathit{U}}_i\left({\mathit{\beta }}_0\right),$$

and

$${\mathit{c}}_j\left(u;{\mathit{\beta }}_0\right)={\int }_0^u\frac{{\sum }_{k=0}^K{\mathit{s}}_j^{(1)}\left(r_k+v;{\mathit{\beta }}_0\right)}{{\sum }_{k=0}^K{s}_j^{(0)}\left(r_k+v;{\mathit{\beta }}_0\right)}d{\mu }_{0,j}\left(v;{\mathit{\beta }}_0\right)$$

for all $u,v\in \left(0,r_1\right]$ and $j,\ell \in \{1,\dots ,J\}$.

3.2. Variance estimation

In the previous section, we derived the asymptotic variance of the regression coefficient estimator ${\widehat{\mathit{\beta }}}_n$, and cyclic baseline mean function estimator ${\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(u;{\widehat{\mathit{\beta }}}_n\right)$. In this section, we describe how one can estimate the variance of these estimators empirically. Specifically, the covariance matrix of ${\widehat{\mathit{\beta }}}_n$ can be empirically estimated by $\widehat{\mathit{\Sigma }}\left({\widehat{\mathit{\beta }}}_n\right)={\widehat{\mathit{A}}}_n{\left({\widehat{\mathit{\beta }}}_n\right)}^{-1}{\widehat{\mathit{B}}}_n\left({\widehat{\mathit{\beta }}}_n\right){\widehat{\mathit{A}}}_n{\left({\widehat{\mathit{\beta }}}_n\right)}^{-1}$, where:

$$\begin{array}{l}{\widehat{\mathit{A}}}_n\left({\widehat{\mathit{\beta }}}_n\right)=n^{-1}\sum _{i=1}^n\sum _{j=1}^J{\int }_0^{\tau }{\mathit{V}}_j\left(s;{\widehat{\mathit{\beta }}}_n\right)d{N}_{ij}(s),\\ {\widehat{\mathit{B}}}_n\left({\widehat{\mathit{\beta }}}_n\right)=n^{-1}\sum _{i=1}^n{\widehat{\mathit{U}}}_i{\left({\widehat{\mathit{\beta }}}_n\right)}^{\otimes 2},\\ {\widehat{\mathit{U}}}_i\left({\widehat{\mathit{\beta }}}_n\right)=\sum _{j=1}^J{\int }_0^{\tau }\left\{{\mathit{Z}}_{ij}(s)-{\overline{\mathit{Z}}}_j\left(s;{\widehat{\mathit{\beta }}}_n\right)\right\}d{\widehat{M}}_{ij}\left(s;{\widehat{\mathit{\beta }}}_n\right),\end{array}$$

with $d{\widehat{M}}_{ij}\left(s;{\widehat{\mathit{\beta }}}_n\right)=d{N}_{ij}(s)-Y_{ij}(s)e^{{\widehat{\mathit{\beta }}}_n^{\top }{\mathit{Z}}_{ij}(s)}d{\widehat{\mu }}_{0j}^{\text{Cyclic}}\left(s-r_s;{\widehat{\mathit{\beta }}}_n\right)$. The standard error (SE) of ${\widehat{\beta }}_j$ is the square root of the jth diagonal element of $n^{-1}\widehat{\mathit{\Sigma }}\left({\widehat{\mathit{\beta }}}_n\right)$, which is a sandwich estimator that be computed by the coxph function with a robust standard error estimate. Moreover, the covariance function ${\mathit{\psi }}_{j\ell }\left(u,v;{\mathit{\beta }}_0\right)$ in Theorem 2 can be empirically estimated by:

$${\widehat{\mathit{\psi }}}_{j\ell }\left(u,v;{\widehat{\mathit{\beta }}}_n\right)=n^{-1}\sum _{i=1}^n{\widehat{\mathit{\Psi }}}_{ij}\left(u;{\widehat{\mathit{\beta }}}_n\right){\widehat{\mathit{\Psi }}}_{i\ell }\left(v;{\widehat{\mathit{\beta }}}_n\right),$$

where

$${\widehat{\Psi }}_{ij}\left(u;{\widehat{\mathit{\beta }}}_n\right)={\int }_0^u\frac{{\sum }_{k=0}^{K}d{\widehat{M}}_{ij}\left(r_k+v;{\widehat{\mathit{\beta }}}_n\right)}{{\sum }_{k=0}^K{S}_j^{(0)}\left(r_k+v;{\widehat{\mathit{\beta }}}_n\right)}+{\widehat{\mathit{C}}}_j{\left(u;{\widehat{\mathit{\beta }}}_n\right)}^{\top }{\widehat{\mathit{A}}}_n{\left({\widehat{\mathit{\beta }}}_n\right)}^{-1}{\widehat{\mathit{U}}}_i\left({\widehat{\mathit{\beta }}}_n\right),$$

with

$${\widehat{\mathit{C}}}_j\left(u;{\widehat{\mathit{\beta }}}_n\right)={\int }_0^u\frac{{\sum }_{k=0}^K{\mathit{S}}_j^{(1)}\left(r_k+v;{\widehat{\mathit{\beta }}}_n\right)}{{\sum }_{k=0}^K{S}_j^{(0)}\left(r_k+v;{\widehat{\mathit{\beta }}}_n\right)}d{\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(v;{\widehat{\mathit{\beta }}}_n\right).$$

The standard error of ${\widehat{\mu }}_{0,j}^{\text{Cyclic}}(u;{\widehat{\mathit{\beta }}}_n)$ can be estimated by taking the square root of $n^{-2}{{\sum }_{i=1}^n\left[{\widehat{\mathit{\Psi }}}_{ij}(u;{\widehat{\mathit{\beta }}}_n)\right]}^2$.

3.3. Prediction

Under the proposed semiparametric rate model (1), the mean number of type j events by time C₀ for a subject with covariate ${\mathit{Z}}_0(t)$ can be expressed as:

$$Q_j\left(C_0,{\mathit{Z}}_0\right)\equiv {\int }_0^{C_0}{e}^{{\mathit{\beta }}^{\top }{\mathit{Z}}_0(s)}d{\mu }_{0,j}\left(v_s\right),$$ (8)

which can be estimated by:

$${\widehat{Q}}_j^{\text{Cyclic}}\left(C_0,{\mathit{Z}}_0\right)={\int }_0^{C_0}{e}^{{\widehat{\mathit{\beta }}}_n^{\top }{\mathit{Z}}_0(s)}d{\widehat{\mu }}_{0,j}^{\text{Cyclic}}\left(v_s;{\widehat{\mathit{\beta }}}_n\right),$$ (9)

where ${\widehat{\mu }}_{0,j}^{\text{Cyclic}}(v_s;{\widehat{\mathit{\beta }}}_n)$ and ${\widehat{\mathit{\beta }}}_n$ are obtained from our estimates (4) and (7), respectively.

Let ${\widehat{Q}}_j^{\text{CS}}\left(C_0,{\mathit{Z}}_0\right)={\int }_0^{C_0}{e}^{{\widehat{\mathit{\beta }}}_n^{\top }{\mathit{Z}}_0(s)}d{\tilde{\mu }}_{0,j}^{\text{CS}}\left(s;{\widehat{\mathit{\beta }}}_n\right)$ be the prediction for the number of type j events by time C₀ with the baseline mean function estimated by (6), as in Cai and Schaubel (2004). The comparison in prediction performance between ${\widehat{Q}}_j^{\text{Cyclic}}(C_0;{\mathit{Z}}_0)$ and ${\widehat{Q}}_j^{\text{CS}}(C_0;{\mathit{Z}}_0)$ is shown via simulation studies in the following section. Remarkably, the prediction comparison here is based on the difference between the observed and predicted number of events in the follow-up time from a new subject. Our prediction is, in fact, more flexible than the one from Cai and Schaubel (2004), since our method can be used for prediction for periods that are not observed in the current cohort. For example, in the fire alarm data in New Zealand, we can predict the number of fire alarms in the year of 2008, even though the data were observed only from 2004 to 2007. Because of the periodic component in our model, our prediction for future events of the same subject needs no extrapolation of the baseline rate function to the extended period, unlike the one used in Cai and Schaubel (2004). We illustrate this superior prediction idea by predicting the number of fire alarms in the year of 2008 in Sect. 5.

4. Simulation study

In this section, we report the results of simulations conducted to assess the finite-sample performance of the proposed estimators. Recurrent events were generated from an intensity model defined, for all $i\in \{1,\dots ,n\}$ and $j\in \{1,2\}$, by:

$$\text{E}\left\{d{N}_{ij}^{*}(t)\mid Z_{ij},U_i\right\}=U_i{e}^{{\mathit{\beta }}^{\top }{Z}_{ij}}d{\mu }_{0,j}\left(t-r_t\right),$$ (10)

where U_i is a frailty term that accounts for possible dependency between different types of events. We assume that the random effect U_i follows a Gamma distribution with E(U_i) = 1 and var(U_i) = η, inducing a positive correlation between event times within a subject. We let $\eta \in \{0,0.5,1\}$, where a larger η corresponds to higher correlation and η = 0 indicates independence between event types. The marginal rate model of the simulation model (10) satisfies the proposed model (1) since:

$$E\left[d{N}_{ij}^{*}(t)\mid Z_{ij}\right]=E\left(E\left[d{N}_{ij}^{*}(t)\mid Z_{ij},U_i\right]\right)=E\left[U_i\right]e^{{\mathit{\beta }}^{\top }{Z}_{ij}}d{\mu }_{0j}\left(t-r_t\right)=e^{{\mathit{\beta }}^{\top }{Z}_{ij}}d{\mu }_{0j}\left(t-r_t\right).$$

We set the baseline mean functions as ${\mu }_{0,1}\left(t-r_t\right)=0.15\times \left(t-r_t\right)$ and ${\mu }_{0,2}\left(t-r_t\right)=0.2\times \left(t-r_t\right)$, where the nearest renewal times r_t is from a set of even numbers between 0 and 22. We consider one time-independent covariate Z_i, where Z_i follows a Bernoulli distribution with mean 0.5. Regression coefficients are set to be β₁ = −0.3 and β₂ = −0.2 for the first and second types of recurrent events, respectively. In the notation of model (1), $\mathit{\beta }={\left({\beta }_1,{\beta }_2\right)}^{\top },Z_{i1}={\left(Z_i,0\right)}^{\top }$, and $Z_{i2}={\left(0,Z_i\right)}^{\top }$. The censoring time C_i is generated uniformly between 0 and τ = 22 and independent from recurrent event processes. Note that, given the random effect $U_i,N_{ij}^{*}(t)$ is a non-homogeneous Poisson process over calendar time t, but a homogeneous Poisson process between renewal times. Therefore, with the ℓth event time $T_{ij,\ell }$ fixed, we can compute the (ℓ + 1) st event time $T_{ij,\ell +1}$ for event type j in subject i by solving the equation:

$$-\ln \left({\Upsilon }_{ij,\ell +1}\right)={\int }_{T_{ij,\ell }}^{T_{ij,\ell +1}}{U}_i\exp \left({\beta }_j{Z}_i\right)d{\mu }_{0,j}\left(u-r_u\right),$$

where ${\Upsilon }_{ij,\ell +1}$ follows a uniform distribution over (0, 1). In our setting, the average number of events per subject is 2.48 and 3.14 for the first and second types of events, respectively, when η = 0.

Table 1 shows the simulation results of the regression coefficient estimation under different sample sizes n and variance η based on 500 replications. The coefficient estimates can be obtained using the coxph function in the R library survival. One can specify the subject variable as “cluster” and the event type variable as “strata.” To obtain the sandwich-type variance estimation, one shall choose the “robust=T” option to obtain the robust standard error estimates. We report bias, empirical standard deviation (SD), average of standard error estimates (SE), empirical coverage rate (CR) of the 95% confidence interval, and root-mean-square error (RMSE). The results corroborate the theoretical findings. One can see that the regression coefficient estimators are consistent. The bias decreases toward zero when the sample size is large. The SD and SE are close, and their difference decreases as the sample size increases, which results in nominal 95% empirical coverage rates. Moreover, the RMSE decreases as the sample size increases. As η increases and two recurrent event processes are more dependent, all SD, SE, and RMSE increase under such a situation.

Table 1.

Simulation results of the regression coefficient estimation

n	η	Est.	Bias	SD	SE	CR	RMSE
50	0	${\widehat{\beta }}_1$	– 0.009	0.263	0.270	0.960	0.270
		${\widehat{\beta }}_2$	0.051	0.256	0.238	0.926	0.243
	0.5	${\widehat{\beta }}_1$	– 0.003	0.375	0.333	0.916	0.333
		${\widehat{\beta }}_2$	– 0.029	0.375	0.331	0.896	0.332
	1	${\widehat{\beta }}_1$	– 0.022	0.650	0.494	0.860	0.494
		${\widehat{\beta }}_2$	– 0.035	0.622	0.499	0.868	0.500
100	0	${\widehat{\beta }}_1$	– 0.003	0.208	0.192	0.932	0.192
		${\widehat{\beta }}_2$	0.035	0.178	0.167	0.930	0.170
	0.5	${\widehat{\beta }}_1$	– 0.014	0.250	0.240	0.942	0.240
		${\widehat{\beta }}_2$	– 0.029	0.245	0.241	0.928	0.242
	1	${\widehat{\beta }}_1$	0.005	0.422	0.376	0.896	0.376
		${\widehat{\beta }}_2$	– 0.014	0.433	0.381	0.906	0.381
200	0	${\widehat{\beta }}_1$	0.002	0.139	0.136	0.942	0.136
		β^2	0.015	0.119	0.120	0.938	0.120
	0.5	${\widehat{\beta }}_1$	– 0.012	0.179	0.174	0.940	0.174
		${\widehat{\beta }}_2$	– 0.034	0.172	0.172	0.946	0.175
	1	${\widehat{\beta }}_1$	– 0.022	0.266	0.276	0.948	0.276
		${\widehat{\beta }}_2$	– 0.027	0.289	0.277	0.938	0.278
2000	0	${\widehat{\beta }}_1$	0.002	0.085	0.083	0.946	0.083
		${\widehat{\beta }}_2$	0.008	0.071	0.072	0.950	0.072
	0.5	${\widehat{\beta }}_1$	– 0.002	0.053	0.055	0.958	0.055
		${\widehat{\beta }}_2$	– 0.006	0.048	0.050	0.962	0.050
	1	${\widehat{\beta }}_1$	– 0.002	0.068	0.066	0.940	0.066
		${\widehat{\beta }}_2$	0.007	0.059	0.058	0.940	0.058

Table 2 presents the simulation results of cyclic baseline mean function ${\widehat{\mu }}_{0,1}^{\text{Cyclic}}(t)$, and ${\widehat{\mu }}_{0,2}^{\text{Cyclic}}(t)$ calculated at t = 2, compared with the one without cyclical components ${\tilde{\mu }}_{0,1}^{\text{CS}}(t)$, and ${\tilde{\mu }}_{0,2}^{\text{CS}}(t)$ in (6). Overall, the performance of cyclical estimator ${\widehat{\mu }}_{0,1}^{\text{Cyclic}}(t)$ and ${\widehat{\mu }}_{0,2}^{\text{Cyclic}}(t)$ are consistent as the bias shrinks when the sample size increases. It is also reassuring that SD and SE are close even under a small sample size, such as n = 50. As η increases and the association strength between event types becomes larger, SD and SE of the estimators increase as expected. The performance of both point and variance estimation improves as n increases, which leads to a 0.95 nominal level of coverage rates (CR). On the contrary, without acknowledging the cyclic structure, the estimators ${\tilde{\mu }}_{0,j}^{\text{CS}}(t)$ have a larger variation than our estimator in both event types. Our cyclic baseline mean function, indeed, takes advantage of utilizing information across all cycles for better efficiency.

Table 2.

Simulation results of the cyclic baseline mean function estimation and a conventional counterpart without cyclic components

n	η		Estimators
			Type 1		Type 2
			${\widehat{\mu }}_{0,1}^{\text{Cyclic}}$	${\widehat{\mu }}_{0,1}^{\text{CS}}$	${\widehat{\mu }}_{0,2}^{\text{Cyclic}}$	${\widehat{\mu }}_{0,2}^{\text{CS}}$
50	0	Bias	− 0.007	− 0.001	− 0.005	− 0.010
		SD	0.055	0.092	0.069	0.114
		SE	0.059	0.090	0.077	0.101
		CR	0.940	0.916	0.928	0.898
	0.5	Bias	0.037	− 0.002	0.079	0.014
		SD	0.082	0.098	0.121	0.125
		SE	0.081	0.097	0.119	0.121
		CR	0.926	0.910	0.940	0.914
	1	Bias	0.070	0.008	0.094	− 0.003
		SD	0.108	0.116	0.140	0.154
		SE	0.103	0.107	0.144	0.155
		CR	0.940	0.897	0.925	0.920
100	0	Bias	− 0.000	− 0.001	− 0.003	− 0.002
		SD	0.038	0.065	0.045	0.077
		SE	0.044	0.064	0.051	0.074
		CR	0.962	0.935	0.960	0.944
	0.5	Bias	0.032	0.006	0.056	− 0.011
		SD	0.064	0.083	0.088	0.089
		SE	0.063	0.071	0.087	0.084
		CR	0.900	0.880	0.930	0.900
	1	Bias	0.064	0.012	0.078	− 0.008
		SD	0.081	0.093	0.094	0.114
		SE	0.080	0.089	0.097	0.114
		CR	0.925	0.906	0.931	0.936
200	0	Bias	− 0.004	− 0.003	0.000	− 0.010
		SD	0.028	0.044	0.036	0.053
		SE	0.028	0.044	0.037	0.052
		CR	0.952	0.945	0.956	0.945
	0.5	Bias	0.020	0.009	0.041	0.008
		SD	0.042	0.052	0.058	0.059
		SE	0.041	0.050	0.057	0.061
		CR	0.936	0.932	0.940	0.938
	1	Bias	0.034	0.004	0.059	0.003
		SD	0.048	0.059	0.067	0.074
		SE	0.047	0.059	0.067	0.073
		CR	0.938	0.942	0.940	0.944

When evaluating the prediction performance of ${\widehat{Q}}_j^{\text{Cyclic}}(\cdot ,\cdot )$ and ${\widehat{Q}}_J^{\text{CS}}(\cdot ,\cdot )$, we generate n subjects, split them equally into a training set (M = n/2 subjects) and validation set (M subjects), fit our proposed model with and without cyclic baseline rate function to the training set, and then use the parameter estimates to predict the number of recurrent events of subjects in the validation set. Table 3 summarizes the average prediction bias and the predicted mean square error (PMSE) for ${\widehat{Q}}_j^{\text{Cyclic}}(\cdot ,\cdot )$ and ${\widehat{Q}}_j^{\text{CS}}(\cdot ,\cdot )$, which are defined by $M^{-1}{\sum }_{m=1}^M{\{{\widehat{Q}}_j^{\text{Cyclic}}\left(C_m,Z_m\right)-Q_m^j\}}^2$ and $M^{-1}{\sum }_{m=1}^M{\{{\widehat{Q}}_j^{\text{CS}}\left(C_m,Z_m\right)-Q_m^j\}}^2$, respectively, where $Q_m^j$ is the number of observed type j event of subject m in the validation set with censoring time C_m and covariate Z_m. We also calculate the mean square error (MSE) based on training data set. The components C_m, Z_m, and $Q_m^j$ in the PMSE are replaced by those obtained from training data set. As one can see in Table 3, our estimator ${\widehat{Q}}_j^{\text{Cyclic}}(\cdot ,\cdot )$ has smaller PMSE and MSE than ${\widehat{Q}}_i^{\text{CS}}(\cdot ,\cdot )$ in each scenario. As expected, the MSE is smaller than the PMSE in each case, since the MSE is calculated based on the training data set where the regression coefficient estimators and rate function are estimated. The PMSE and MSE increase for both ${\widehat{Q}}_j^{\text{Cyclic}}(\cdot ,\cdot )$ and ${\widehat{Q}}_j^{\text{CS}}(\cdot ,\cdot )$ when η is larger. Our estimator ${\widehat{Q}}_j^{\text{Cyclic}}(\cdot ,\cdot )$ has improved on prediction accuracy than ${\widehat{Q}}_j^{\text{CS}}(\cdot ,\cdot )$ under a larger η. As expected, PMSE and MSE decrease as the sample size n increases, showing that prediction accuracy improves when the sample size is large.

Table 3.

Simulation results for prediction based on 500 replicates

n	η		Estimators
			Type 1		Type 2
			${\widehat{Q}}_1^{\text{Cyclic}}$	${\widehat{Q}}_1^{\text{CS}}$	${\widehat{Q}}_2^{\text{Cyclic}}$	${\widehat{Q}}_2^{\text{CS}}$
100	0	Bias	0.013	0.010	− 0.000	− 0.011
		PMSE	2.204	2.221	3.481	3.513
		MSE	1.950	1.968	3.114	3.165
	0.5	Bias	0.038	0.035	0.042	0.036
		PMSE	4.731	4.777	9.599	9.655
		MSE	4.310	4.340	8.182	8.194
	1	Bias	− 0.020	− 0.022	− 0.054	− 0.061
		PMSE	8.368	8.427	15.341	15.399
		MSE	7.278	7.437	14.385	14.623
200	0	Bias	0.020	0.011	0.046	0.027
		PMSE	2.161	2.171	3.401	3.414
		MSE	1.918	1.936	3.048	3.074
	0.5	Bias	0.033	0.020	0.068	0.045
		PMSE	4.722	4.736	9.253	9.282
		MSE	4.267	4.282	8.029	8.078
	1	Bias	− 0.009	− 0.022	− 0.0138	− 0.035
		PMSE	7.960	7.983	14.994	15.003
		MSE	6.839	6.943	13.136	13.293
400	0	Bias	0.009	− 0.006	0.052	0.013
		PMSE	2.138	2.145	3.324	3.367
		MSE	1.898	1.917	2.954	2.992
	0.5	Bias	0.016	− 0.009	0.010	− 0.031
		PMSE	4.338	4.343	9.047	9.052
		MSE	3.815	3.839	7.911	7.954
	1	Bias	0.050	0.024	0.044	0.006
		PMSE	7.558	7.561	12.385	12.393
		MSE	6.413	6.448	11.271	11.289

5. Real data analysis

The fire prevention data from the New Zealand Fire Service contain information on all fire incidents that occurred within and around the town of Blenheim between January 1, 2004, and December 31, 2007. The dataset can be downloaded from https://projecteuclid.org/euclid.aoas/1346418578#supplemental. The purpose of the study was to prevent fires at an early stage and to distinguish between the events (Wang et al. 2012). During the 4 years, there were 704 fire incidents in total, which can be grouped into two major types (J =2 ), namely, structure and vegetation fire incidents. The structure fire includes fires involving the structural components of various types of residential, commercial, and industrial buildings, such as condors, stores, and barns. The vegetation fire includes open fires of various vegetation surfaces, such as savannahs, forest, and agricultural residues. A fire triggered by peat that is made by humans or occurs naturally, e.g., by lightning, is also considered as a vegetation fire. Each fire event has information such as fire area (urban or rural), alarm method (by emergency call or other methods), heat source (initiated by hot objects, cigarette/smoking materials, or other sources), and objects ignited (by outdoor items or other objects). We treat this information as covariates in model (1) for the event occurrence. To work with complete covariates, we ignored fire events with missing values in heat sources and objects ignited, resulting in 130 structure and 159 vegetation fires in 30 gridded areas. Figure 2 shows the event distribution of the two fire types in calendar time. One can see that structure fires occur more frequently during wintertime (June–August), while vegetation fires occur more frequently during summertime (December–February). Both fire types have their own cyclic patterns over the 4 years.

Fig. 2.

Distribution of fire types among urban area and objects ignited (outdoor item)

Table 4 shows the frequency, total exposure in days (TED), and annual occurrence rate (AOR) of the two fire types at each covariate level. The TED is the total number of days between events from all of the areas exposed to the level of the covariate. The AOR is derived by 365 times a ratio of the event frequency to the TED and can be interpreted as a constant annual rate per area. From the table, one can see that the urban area is associated with a higher annual rate of structure fire (1.66 vs. 1.12) than the rural area, while the rural area has a higher annual rate of vegetation fire (1.47 vs. 2.14) than the urban area. The frequency of alarms received by emergency call dominates in both fire types. However, the AOR is smaller when compared with other alarms due to longer exposure time. Hot objects as heat sources occur more frequently in the structure fire, but the cigarette/smoking material has the highest AOR. Outdoor items are the main objects ignited in vegetation fires with AOR 1.89. The corresponding p values are derived from the AOR comparison between covariate levels using Poisson regression with TED as the offset variable.

Table 4.

Frequency, total event duration in days (TED), and annual occurrence rate (AOR) at each covariate level by fire types in the Blenheim fire data

Covariate	Structure				Vegetation
	Freq	TED	AOR	p value	Freq	TED	AOR	p value
Area
Urban	111	24454	1.66	0.116	83	20675	1.47	0.017
Rural	19	6181	1.12		76	12960	2.14
Alarm
111 emergency	114	27593	1.51	0.366	146	31438	1.70	0.403
Other	16	3042	1.92		13	2197	2.16
Heat source
Hot object	53	13911	1.39	0.688	16	1931	3.02	0.215
Cigarette/Smoking materials	30	5343	2.05	0.189	88	22262	1.44	0.024
Other	47	11381	1.51		55	9442	2.13
Object ignited
Outdoor	41	7904	1.89	0.136	151	31624	1.74	0.615
Other	89	22731	1.43		8	2011	1.45

In the notation of model (1), we let $\mathit{\beta }={\left({\mathit{\beta }}_1^{\top },{\mathit{\beta }}_2^{\top }\right)}^{\top },{\mathit{Z}}_{i1}(t)={\left[{\mathit{Z}}_i{(t)}^{\top },{\mathbf{0}}_2^{\top }\right]}^{\top }$ and ${\mathit{Z}}_{i2}(t)={\left[{\mathbf{0}}_1^{\top },{\mathit{Z}}_i{(t)}^{\top }\right]}^{\top }$, where ${\mathit{Z}}_i(t)$ denotes the column vector of covariates in Table 4. All of the covariates in Table 4 are considered piecewise constant functions in t between events. We set the renewal times as r₀ = 0, r₁ = 366, r₂ = 731, and r₃ = 1096 with K = 3. The estimation of the unknown parameter β was implemented using coxph() function in the R package survival (Therneau 2017), as described in Sect. 4.

Table 5 summarizes the coefficient estimation and statistical testing results. As expected, the urban area is significantly associated with a higher rate of structure fire (p value = 0.023), but is not associated with the vegetation fire (p value = 0.635). The alarm method is not associated with either fire type. Compared with other heat sources contributing to fires, hot objects are negatively associated with both fire types, although only structure fire reaches statistical significance (p value=0.020). Interestingly, cigarette/smoking material is positively associated with the increased rate of structure fire, but negatively associated with vegetation fire, although only the vegetation fire reaches the significance (p value = 0.047). The coefficient estimation of the objects ignited by outdoor items is significant in structure fire (p value<0.001), suggesting that outdoor objects are more likely to initiate the structure fire.

Table 5.

Analysis results of the Blenheim fire data

Covariate	Structure			Vegetation
	β	SE	p value	β	SE	p value
bf Area [ref: Rural]
Urban	0.523	0.230	0.023	− 0.107	0.225	0.635
Alarm [ref: Other]
111 Emergency	− 0.035	0.291	0.904	0.013	0.313	0.966
Heat source [ref: Other]
Hot object	− 0.503	0.216	0.020	− 0.037	0.311	0.905
Cigarette/smoke material	0.081	0.238	0.735	− 0.452	0.227	0.047
Object ignited [ref: Other]
Outdoor	0.675	0.190	<0.001	− 0.112	0.255	0.660

Figure 3 illustrates the baseline mean function estimations for the structure and vegetation fire incidents using our cyclic estimator (4) and non-cyclic estimator (6) from Cai and Schaubel (2004). To have a more direct comparison between the estimators, we break the baseline mean function estimation from Cai and Schaubel (2004) into four pieces, on the same scale as our proposed estimator. As the figure shows, the vegetation fire has higher baseline mean function estimates than the structure fire, which is also observed in the unadjusted frequency of the events. Summer in Blenheim, New Zealand, is typically warm and dry, making vegetation more vulnerable and flammable. Surprisingly, it appears that the structure fire occurs more periodically than the vegetation fire, since our estimate is close to the one from Cai and Schaubel (2004). It is a general conception that the vegetation fire is more dependent on the season and weather change. The vegetation fire jumps higher than expected during the summertime in 2006 and 2007. The cyclic feature of the mean/rate function appears in this fire alarm data. To illustrate the superiority of our proposed model, we predict the number of structure and vegetation fires in both urban and rural areas for the year of 2008. Figure 4 shows the prediction result. One can see that the vegetation fire tends to have a higher rate of events in the rural area than the urban area, as opposed to the structure fire. Not surprisingly, the vegetation fire occurs more often in the summer (December–February) and fall (March–May) when the average temperature is higher in New Zealand. The structure fire increases more rapidly in the winter (June–August) and spring (September–November) when the average temperature drops.

Fig. 3.

The dotted dash lines are non-cyclic baseline mean functions, and the solid lines are cyclic baseline mean functions for structure and vegetation fire type

Fig. 4.

The predicted cumulative fire events for structure and vegetation fire types in the urban area of the year 2018

6. Discussion

We proposed a semiparametric marginal rate model with cyclic baseline rate functions for recurrent event data with multiple event types. Estimating equations were used to estimate the regression coefficients. We showed that the estimators are consistent and asymptotically Gaussian. We also proposed a fully nonparametric estimator for the cyclic baseline mean function, and showed its uniform consistency and weak convergence. A prediction formula for the number of recurrent events over a specific follow-up period was also developed. Numerical studies, which include simulation studies and a real data analysis, illustrated the practical use of our model.

Our semiparametric rate model (1) with a cyclic baseline rate function dμ_0,j assumes the same set of renewal times across all types of recurrent events. One may consider different renewal times specific for each type. Under different renewal times, one can obtain the same estimates for the regression coefficients using estimating equations (7). The cyclic mean function for type j can be estimated by (4) with specific renewal times r_jk for $k=0,\dots ,K_j$. Moreover, to test the periodicity of the baseline mean function, one may develop a goodness-of-fit test that compares the difference between the cyclic mean function estimator (4) and the non-cyclic estimator (6) from Cai and Schaubel (2004). To explore the cyclic pattern, one can also utilize the rate function instead of the mean function. It is possible to adopt a smoother for the baseline rate function and estimate the regression coefficients using a full likelihood function instead of a partial likelihood function. Such smoothers include piecewise constant (Dörre 2020), smoothing splines (Emura et al. 2017), and kernel smoothing for either a rate model or a hazard model (Emura et al. 2017). Also, there may exist a spatial correlation among 30 gridded areas in the fire alarm data. Thus, incorporating spatial point pattern in the rate/intensity model may improve the prediction and forecasting.

In this work, we consider a cyclic rate function in the model (1) to describe the periodic fire occurrence. As the purpose of the fire alarm data is to prevent the fire incidents, there remain other open questions to answer. For example, one could ask for the optimal strategy to prevent the fire. Our data analysis result shows patrolling prioritization may be the answer. The police shall patrol more in the urban areas or areas with a high crime rate if an arson causes the fire. Similar suggestions can be found in Santitissadeekorn (2020).