joint modelling of competing risks and current status data...

© 2019 Royal Statistical SocietyThis article has been contributed to by US Government employees and their work is in the public domain in the USA.

0035–9254/19/68000

Appl. Statist. (2019)

Joint modelling of competing risks and currentstatus data: an application to a spontaneous labourstudy

Youjin Lee and Mei-Cheng Wang

Johns Hopkins School of Public Health, Baltimore, USA

and Katherine L. Grantz and Rajeshwari Sundaram

Eunice Kennedy Shriver National Institute of Child Health and HumanDevelopment, Bethesda, USA

[Received September 2018. Revised February 2019]

Summary. The second stage of labour begins when the cervix is fully dilated and pushing be-gins until the fetus is delivered. A Caesarean delivery (CD) or operative vaginal delivery (OVD)is typically encouraged after the recommended time set by ‘expert consensus’. This recom-mended time has been set out of concern for an increased chance of maternal and neonatalmorbidities due to a prolonged second stage of labour, but without thorough consideration ofheterogeneous risks for spontaneous vaginal delivery (SVD) and morbidities among women.To provide quantitative evidence for the recommendation, the first step is to compare the risksfor SVD, CD or OVD, and the risks of maternal or neonatal morbidities simultaneously acrossthe duration of the second stage of labour. To address such risk comparisons statistically, oneneeds to study the joint distribution for the time to delivery due to each mode and time to ma-ternal or neonatal morbidity given information provided for each individual. We introduce a jointmodel which combines the competing risks data for delivery time and current status data forany type of maternal or neonatal morbidity given each woman’s baseline characteristics.Thesetwo processes are assumed dependent through individual-specific frailty under the joint model.Our numerical studies include a simulation that reflects the structure of observed real data anda detailed real data analysis based on nearly 12000 spontaneous labours. Our finding indicatesthe necessity to incorporate maternal characteristic such as age or body mass index in assess-ing the probability for delivery due to SVD, CD or OVD and the onset of morbidities across thesecond stage of labour.

Keywords: Competing risks model; Current status data; Random effect; Shared parametermodel

1. Introduction

The process of labour by which a pregnant woman gives birth is complex comprising mul-tiple stages of progression; initiated by the contractions heralding the first stage, the secondstage of labour begins with 10 cm of cervical dilation (Hellman and Prystowsky, 1952), andthe third stage starts right after the birth. Understanding the progression of labour in preg-nant women has received considerable attention by the obstetric community and the public ingeneral considering the escalating rate of Caesarean delivery (CD) world wide (Betran et al.,

Address for correspondence: Rajeshwari Sundaram, Division of Intramural Population Health Research, EuniceKennedy Shriver National Institute of Child Health and Human Development, 6710B Rockledge Drive, Bethesda,MD 20892, USA.E-mail: [email protected]

2 Y. Lee, M.-C. Wang, K. L. Grantz and R. Sundaram

2016). Our inability to reduce the CD rate has been attributed in part to the incomplete un-derstanding of a normal labour process motivating a closer look at practices by the obstetriccommunity.

Quantification of the duration of the first stage of labour started with the seminal work ofFriedman (1955). Recently, motivated by evolving obstetrical practices and population overthe past 50 years, the labour curve assessing the first stage of labour has been updated byZhang et al. (2002, 2010a). Ma and Sundaram (2018) have studied this issue with a rigor-ous statistical approach that accounted for informative examinations and unmeasured start oflabour that were not considered before, allowing identification of distributions for per-centimetredilations.

Regarding the duration of the second stage, the American College of Obstetrics and Gy-necology and the Society for Maternal Fetal Medicine provide recommendations to identify‘second-stage arrest’ after which the obstetricians may intervene to deliver. The College andSociety have provided recommendations (Caughey et al., 2014) for diagnosing second-stagelabour arrest as at least 3 h of pushing in nulliparous women and 2 h of pushing in multiparouswomen without an epidural, although they also allow some consideration of longer durationsin women making progress to be determined individually.

However, it is unclear whether the prolongation of the second stage of labour beyond thecurrently accepted windows to achieve a spontaneous vaginal delivery (SVD) results in a higherchance of serious neonatal or maternal complications. In fact, the conventional guidance forthe duration of the second stage does not take into account the individual heterogeneity in risk.In addition, baseline information which is likely to be correlated with the risk of SVD and risksof neonatal or maternal morbidities (Zhang et al., 2002; Laughon et al., 2014) has not beenconsidered. There have been several studies to reveal the association of maternal and neonataloutcomes to the length of the second stage of labour (Myles and Santolaya, 2003; Cheng et al.,2004), or to the mode of delivery during the second stage of labour (Murphy et al., 2001; Mylesand Santolaya, 2003; Allen et al., 2009), but none of them thoroughly investigated both of theevents jointly. Recently, Grantz et al. (2018) assessed the recommendations under a competingrisks framework for multiple modes of delivery with or without morbidities. However, they didnot simultaneously assess the competing risks that are associated with each mode of deliveryand risks for the onset of morbidities given baseline maternal characteristics. In this paper, weintroduce a joint model which incorporates available data that were obtained during the secondstage of labour to provide useful risk comparisons for obstetricians. The goal of our method’sdevelopment and analysis is

(a) to evaluate the risks of SVD and for onset of any morbidities jointly over the second stageof labour,

(b) to identify whether the recommendations should be dependent on known risk factors,like maternal age at birth or maternal body mass index (BMI) and

(c) to quantify the risks of medical interventions for delivery vis-a-vis the risk of maternal orneonatal morbidities.

The remainder of this paper is organized as follows. We briefly introduce the motivatingConsortium of Safe Labor (CSL) study in Section 2 and detailed data structure, models andestimation procedure are discussed in Section 3. A simulation study and the application resultsfor the CSL study are discussed in Section 4 and Section 5 respectively. We conclude withdiscussion in Section 6.

The data that are analysed in the paper and the programs that were used to analyse them canbe obtained from

Joint Modelling of Competing Risks and Current Status Data 3

https://onlinelibrary.wiley.com/hub/journal/14679876/series-c-datasets

2. Consortium on Safe Labor data

The CSL, which is supported by the Eunice Shriver Kennedy National Institute of Child Healthand Human Development, National Institutes of Health, collected electronic medical recordsfrom a total of 228668 deliveries from 2002 to 2008. These electronic medical records includematernal or neonatal outcomes, their demographic characteristics, a summary of the deliveryprocedures and medical history. Accuracy of the data was validated, showing a rate of concor-dance of greater than 95% with medical charts in key variables (Zhang et al., 2002, 2010b). Wedefined the duration of the second stage of labour as the difference between the date and timeof birth from the date and time of 10-cm cervical dilation as recorded in the maternal medicalrecord. We used the same analytic cohort corresponding to the studies of the second stage oflabour, using the same exclusion and inclusion criteria as presented in the primary papers onsecond-stage labour for the CSL (Laughon et al., 2014; Grantz et al., 2018). In particular, womenwho had an antepartum stillbirth, prior uterine incision or failed to reach the second stage oflabour were excluded; deliveries which had recorded date and time of birth in the maternalmedical record were included. In our analysis, we had an additional restriction of using datawhich had complete information on maternal BMI at the time of admission and maternal age.In this paper, we refer to maternal or neonatal morbidity as any serious maternal or neonataloutcome to be avoided. Consistent with Grantz et al. (2018), any maternal or neonatal morbiditycomprised of serious maternal complications including postpartum haemorrhage, blood trans-fusion, Caesarean hysterectomy and intensive care unit admission or death, as well as neonatalserious complications including shoulder dystocia with fetal injury (clavicular fracture, Erbspalsy, Klumpkes palsy or hypoxic ischaemic encephalopathy), a need for continuous positiveairway pressure resuscitation or higher, neonatal intensive care unit length of stay greater than72 h, sepsis, pneumonia, hypoxic–ischaemic encephalopathy or periventricular leukomalacia,seizure, intracranial haemorrhage or periventricular haemorrhage, asphyxia or neonatal death.

Considering all the deliveries that entered the second stage of labour, our primary interest liesin understanding the risk that is associated with allowing the woman to prolong her second stageof labour especially comparing risks for various modes of delivery—SVD, CD and operativevaginal delivery (OVD), with respect to risks for neonatal and maternal morbidities. Because itis well known in the obstetrics literature that the use of an epidural during labour and a woman’sprevious live birth status (first time in labour or not) change the profile of the progression oflabour, we focus on the progression of labour of first-time labour (nulliparous) without use ofan epidural to consider only patterns of normal labours.

(a) (b)

Fig. 1. Delivery time stratified by parity and by delivery methods for (a) nulliparous women and (b) multi-parous women without an epidural: , recommended duration of the second stage; , SVD; , CD; , OVD


Table 1. Descriptive statistics of the three delivery methods for nulliparous and multiparous women in theCSL data

Parity Delivery time (min) Results for SVD Results for CD Results for OVD Total

Nulliparous [0, 670) 5510 (90.8%) 110 (1.8%) 448 (7.4%) 6068Multiparous [0, 651) 12000 (98.1%) 37 (0.3%) 199 (1.6%) 12236

Fig. 1 illustrates delivery time over the second stage separately for three different modesor methods of delivery (SVD, CD and OVD), which are stratified by whether nulliparous ormultiparous women. A vertical line in Fig. 1 indicates the historical recommended time forpursuing SVD, differently defined for nulliparous and multiparous women. The descriptivestatistics of delivery times for different delivery methods by parity are presented in Table 1.Along with time to delivery by different modes, maternal and neonatal morbidities observed atthe time of delivery are available. Each woman’s baseline characteristics such as age and BMI arealso available. We aim to incorporate all of these available data on time to delivery, morbiditystatus at delivery and baseline covariates in the model to assess risks for each mode of deliveryand for morbidities simultaneously.

3. Methodology

To investigate the maternal or neonatal outcomes during the second stage of labour, previouswork has predominantly used a logistic regression (Murphy et al., 2001; Cheng et al., 2004;Burrows et al., 2004; Allen et al., 2009), or run logistic regressions on the delivery time stratifiedby a few time intervals (Cheng et al., 2007). Recently, Grantz et al. (2018) have combined themode of delivery with morbidity status at delivery into multiple competing causes to analyse thedata. Though this latest analytical strategy accounts for the competing causes, it is still limitedin its ability to compare changes in risks for competing causes of delivery with changes in risksof morbidities over time. Additionally, none of the aforementioned approaches have ascertainedthe effect of covariates on the risks for these concurrent events. To study our objective, we needto infer the joint distribution of time in the second stage of labour and time to development ofany neonatal or maternal morbidity. Here, the main challenge comes from the type of data thatare available; in particular, morbidity status is ascertained only at the time of delivery, leadingto time to morbidity being a current status type of data whereas the duration of the second stageof labour is a competing risks type of data with competing causes being SVD, CD or OVD.Furthermore, the two survival times of interest are dependent not just on measured covariates ofinterest but also by unmeasured factors. Our proposed joint model consequently acknowledgesthese issues which we discuss in detail next.

3.1. Data structureLet T ∈ .0, τ ] be the duration of the second stage of labour, the time between reaching 10 cmdilation, i.e. the end of the first stage, and the time of delivery. Note that T is subject to com-peting risks due to different modes of delivery, e.g. SVD, CD or OVD. It is reasonable to assumethat the observed time is bounded, i.e. T � τ < ∞. We analyse the time to the second stage oflabour with competing risks having J ∈ N types of delivery mode, denoted by an indicatorΔ∈{1, 2, : : : , J}. In contrast, the time of morbidity during the second stage of labour, denotedby M ∈ .0, τ ], can be observed only through current status data, i.e. whether there has been a


morbidity during the second stage of labour or not until the time of delivery; thus we observe M

only through a binary value of U := I.M �T/∈{0, 1}. As the rate of incidence of both maternaland neonatal morbidities is relatively low, we focus on morbidity of either type, i.e. neonatal ormaternal. Our primary interest is in understanding the risks of the competing causes of deliveryunder the presence of risks for any morbidities. In addition, we are interested in assessing theassociation of baseline covariates like maternal age and BMI with these two events. For this, weintroduce a random vector X∈Rq for the baseline covariate. We denote by {Oi = .Ti, Δi, Ui, Xi/ :i=1, 2, : : : , n}, independent and identically distributed (IID) observations for n subjects.

3.2. Joint frailty modelA frailty model in general has been widely used for its ability to incorporate correlations withinindividuals or between individuals for recurrent event models (Henderson and Shimakura, 2003;Huang and Wang, 2004; Box-Steffensmeier and De Boef, 2006) and clustered survival data(Sastry, 1997; Li and Ryan, 2002; Glidden and Vittinghoff, 2004). We introduce a frailty modelparticularly to capture dependence between time to delivery and time to onset of morbidity inour context. Let Z ∈R be a random variable for individual level frailty. Then, conditionally on.X, Z/, we assume that

(a) ‘quasi-independence’ holds for competing risks in the observed region and(b) the event time T follows the event-specific proportional hazards model

fT ,Δ.t, δ|X =x, Z = z/=J∏

j=1{zλj0.t/ exp.β′

jx/}I.δ=j/ exp{

−J∑

j=1zΛj0.ti/ exp.β′

jx/

}, .1/

where β′j are the cause-specific regression coefficients that are associated with the covari-

ates x.

Next, given X and Z, we model the time to onset of any morbidity M, also by using aproportional hazards model with frailty so that it has the hazard Zh0.t/ exp.α′X/, where h0.t/

denotes the baseline hazard for the time to any morbidity. Then, under the assumption that T

is independent of M given .X, Z/, we derive the density for current status data:

fU.u|T = t, X =x, Z = z/= [exp{−zH0.t/ exp.α′x/}]1−u[1− exp{−zH0.t/ exp.α′x/}]u: .2/

Combining the density for the time to delivery (equation (1)) and for the current status ofany morbidity at delivery (equation (2)) with a common frailty Z, their joint density can beexpressed as

fT ,Δ,U.t, δ, u|X =x, Z = z/=J∏

j=1{zλj0.t/ exp.β′

jx/}I.δ=j/ exp{−

J∑j=1

zΛj0.ti/ exp.β′jx/

}

× [exp{−zH0.t/ exp.α′x/}]1−u[1− exp{−zH0.t/ exp.α′x/}]u, .3/

given frailty Z and covariates X. On the basis of this joint density, we can express the observedlikelihood based on n IID observations {Oi = .Ti, Δi, Ui, Xi/ : i=1, 2, : : : , n} with an unknownset of parameters as follows:

L.T, Δ, U|X =x;Θ/=n∏

i=1

∫ [J∏

j=1{ziλj0.ti/ exp.β′

jxi/}I.δi=j/ exp{

−J∑

j=1zi Λj0.ti/ exp.β′

jxi/

}]

× [exp{−zi H0.ti/ exp.α′xi/}]1−ui [1− exp{−zi H0.ti/ exp.α′xi/}]ui

×fZ.zi;σ/dzi, (4)


where {zi : i=1, 2, : : : , n} denotes the unobserved individual level frailties and fZ.·,σ/ denotesa continuous density with unknown parameter σ, e.g. if fZ.·;σ/ is a gamma density withparameters σ := .γ1,γ2/. We denote by Θ= .α′, {β′

j : j =1, 2, : : : , J}, {λj0.t/ : j =1, 2, : : : , J ; t ∈.0, τ ]}, {h0.t/ : t ∈ .0, τ ]},σ/ a set of unknown parameters in the model formulation that are tobe estimated on the basis of the observed data.

3.3. Estimation methodBesides the flexibility that is earned from the frailty, we also allow baseline hazards with flexiblefunctional forms for cause-specific hazards and baseline hazard for the time of onset of anymorbidity. We estimate the baseline hazards by using cubic B-spline models (Rosenberg, 1995)so that cause-specific baseline hazards for J sources of risk, {λ0j.t/ :j =1, 2, : : : , J ; t ∈ .0, τ ]} andbaseline hazards for the time of onset of any morbidity M, {h0.t/ : t ∈ .0, τ ]}, can be estimatedthrough the coefficients of B-spline functions. Both in the simulation and in the real data analysis,we have K = 5 knots for splines at the (10%, 25%, 50%, 75%, 90%) quantiles of delivery timeT = {T1, T2, : : : , Tn}. Then, assuming J competing risks and one type of morbidity, a total ofJ ×9+9 parameters specify the baseline splines in our observed likelihood (equation (4)).

However, since we cannot observe the frailties but observe only the marginalized outcomes,the observed likelihoods must involve integration with respect to the unknown frailty. We useGaussian quadrature for numerical integration to obtain the (approximated) likelihood. Denotethe nodes and their corresponding weights for Gaussian quadratures as {yq : q = 1, 2, : : : , Q}and {wq : q=1, 2, : : : , Q}. The pairs of quadrature nodes and weights {.yq, wq/; q=1, 2, : : : , Q}depend on the distribution of the frailty. We assume log-normal and standard Weibull dis-tributions as a frailty model in our numerical experiment and real data analysis. Using theGaussian–Hermite quadrature formula applied on the log-scale of frailty, we can formalize ourapproximated likelihood when frailties follow a log-normal distribution as follows:

n∏i=1

Q∑q=1

wq

[J∏

j=1{exp.−yq/λj0.ti/ exp.β′

jxi/}I.δi=j/ exp{

−J∑

j=1exp.−yq/Λj0.ti/ exp.β′

jxi/}]

× [exp{− exp.−yq/H0.ti/ exp.α′xi/}]1−ui [1− exp{− exp.−yq/H0.ti/ exp.α′xi/}]ui

× .2πσ2/−1=2 exp.y2q −y2

q=2σ2/: (5)

For a Weibull-distributed frailty model, we apply Gauss–Laguerre quadrature having a supportof .0, ∞/. In that case we add the following likelihoods:

n∏i=1

Q∑q=1

wq

[J∏

j=1{yqλj0.ti/ exp.β′

jxi/}I.δi=j/ exp{

−J∑

j=1yq Λj0.ti/ exp.β′

jxi/

}]

× [exp{−yq H0.ti/ exp.α′xi/}]1−ui [1− exp{−yq H0.ti/ exp.α′xi/}]uiσyσ−1q exp.−yσq /: .6/

For either case we should evaluate n×Q values for every iteration when maximizing the aboveapproximated likelihood with respect to Θ. This requires heavy computational time in R butintegration of R and C++ via the Rcpp package accelerates the speed.

Instead of a log-normal or Weibull distribution, we may assume that Z follows a gamma distri-bution with mean 1, i.e. fZ.zi;σ/=z

−1+1=σi exp.−zi=σ/{Γ.1=σ/σ1=σ}−1, and then the likelihood

function is


L.T, Δ, U|X;Θ/=n∏

i=1

∫ [J∏

j=1{ziλj0.ti/ exp.β′

jxi/}I.δi=j/ exp{

−J∑

j=1zi Λj0.ti/ exp.β′

jxi/

}]

× [exp{−zi H0.ti/ exp.α′xi/}]1−ui [1− exp{−zi H0.ti/ exp.α′xi/}]ui

× z−1+1=σi exp.−zi=σ/{Γ.1=σ/σ1=σ}−1dzi: (7)

In this case, however, a parameter σ and the other parameters are not identifiable as they areequivalent to a scaling factor c within exp.−zic/, e.g. c=ΣJ

j=1σ−1Λj0.ti/ exp.β′

jxi/.

4. Simulation study

We examine the performance of the proposed joint model through a numerical experimentthat simulates the real data set-up. We present the performance of the joint model under twodifferent frailty models. In the simulation set-up, the number of competing risks type is set toJ =2, representing two modes of delivery: SVD and CD. We generate IID data {.Ti, Δi, Ui, Xi/ :i=1, 2, : : : , n}, assuming that Δi =1 if a subject i delivered through SVD and Δi =2 if subjecti experienced CD; Ui = 1 if there had been any morbidities during i’s second-stage labour andUi =0 otherwise. Time to delivery, Ti, is generated from Cox proportional hazards with covariateXi ∼IID N.0, 1/ under the Weibull-distributed baseline hazards. More details can be found inAppendix A.

We first assume a log-normal distribution for the frailty without any administrative censoring(as in our real data scenario). Table 2 shows estimation results when we fix the number of nodesin Gauss–Hermite quadrature as Q=20 for σ=0:5 and Q=10 for σ=1:0, where σ denotes thevariance of ln.Z/. When σ= 0:5, the quadrature nodes are sporadically distributed compared

Table 2. Log-normal distribution simulation estimation results†

n Q Parameter Bias SE SEM CP

500 20 β1 (1.0) 0.0243 0.1017 0.1050 0.9119β2 (−2:0) −0:0585 0.1673 0.1762 0.9019α (1.5) 0.0718 0.2083 0.2284 0.9139σ (0.5) 0.0573 0.1823 0.1783 0.9439

1000 20 β1 (1.0) 0.0080 0.0665 0.0705 0.9373β2 (−2:0) −0:0252 0.1118 0.1161 0.9206α (1.5) 0.0323 0.1483 0.1490 0.9060σ (0.5) 0.0222 0.1296 0.1246 0.9457

500 10 β1 (1.0) 0.0032 0.1108 0.1108 0.9289β2 (−2:0) −0:0132 0.1734 0.1702 0.9239α (1.5) 0.0048 0.1898 0.1958 0.9349σ (1.0) −0:0384 0.2195 0.2092 0.9429

1000 10 β1 (1.0) −0:0019 0.0756 0.0777 0.9358β2 (−2:0) −0:0049 0.1194 0.1195 0.9418α (1.5) −0:0041 0.1380 0.1378 0.9348σ (1.0) −0:0260 0.1561 0.1587 0.9488

†The number of replicates is nr = 1000 for each simulation scheme;SE stands for the sampling standard error of each parameter estimatebased on these nr = 1000 replicates; SEM is the sampling mean ofthe standard error estimate from nb=100 bootstrap samples per eachreplicate; CP is the coverage probability of the empirical 95% confi-dence interval. The number of quadrature nodes is Q=20 for σ=0:5and Q=10 for σ=1:0 following a Gauss–Hermite quadrature rule.


Table 3. Weibull distribution simulation estimation results†


500 20 β1 (1.0) −0:0231 0.1187 0.1179 0.9470β2 (−2:0) 0.0158 0.1893 0.1955 0.9570α (1.5) 0.0483 0.2096 0.2217 0.9420γ (1.5) 0.5419 1.0720 1.1341 0.9750

1000 20 β1 (1.0) −0:0340 0.0803 0.0836 0.9209β2 (−2:0) 0.0268 0.1308 0.1437 0.9399α (1.5) 0.0347 0.1481 0.1558 0.9349γ (1.5) 0.1531 0.5542 0.7053 0.9369

500 20 β1 (1.0) 0.0102 0.1022 0.1110 0.9580β2 (−2:0) −0:0693 0.1637 0.1776 0.9210α (1.5) 0.1274 0.2011 0.2195 0.8830γ (2.0) −0:3839 0.4285 0.6669 0.7790

1000 20 β1 (1.0) 0.0012 0.0858 0.0820 0.9453β2 (−2:0) −0:0212 0.1411 0.1371 0.9565α (1.5) 0.0251 0.1601 0.1585 0.9464γ (2.0) 1.0771 1.9772 1.8170 0.9808

†The number of replicates is nr=1000 for each simulation scheme. Thenumber of quadrature nodes is Q = 20 following a Gauss–Laguerrequadrature rule when frailties truly follow a Weibull distribution withshape parameter γ. In the case of Weibull-distributed frailties, theestimates for γ are less stable than the estimates for σ in a log-normaldistribution.

with the distribution of frailties, so there is a modest bias in the coefficients but increased samplesize from n=500 to n=1000 decreases the bias. In Appendix A, we present additional simulationresults for log-normally distributed frailty but different numbers of quadrature nodes for eachof σ.∈{0:5, 1:0}/ in Table 8 there.

For the second scenario, we generate frailties from a Weibull distribution with a shape param-eter γ. The rest of the simulation set-up is the same as above. Under this scenario, we observean extremely right-skewed duration of labour for a small percentage of samples. This resultedin numerical instability in those instances. However, we observed in our various simulations(which are not presented here) that trimming the data at the 95th percentile (approximately val-ues larger than 240) of the durations provided much stable results. Table 3 shows the estimationresults when Q= 20 quadrature nodes are spread out on .0, ∞/. We required a larger numberof quadrature nodes for γ=2:0 compared with γ=1:5 in general. In some cases from Table 3,a coverage probability for γ is larger than 0.95 because of inflated sampling standard error orsampling mean of the standard error estimate from a few extreme estimates. The first, secondand the third quartiles under γ= 1:5 are .1:2547, 1:3610, 1:5059/ and .1:2944, 1:3740, 1:4836/

when n=500 and n=1000 respectively; when the true value of γ is 2:0 the three quartiles go from.1:3159, 1:4516, 1:6244/ to .1:6367, 1:8941, 2:0466/ as the sample size increases from n=500 ton = 1000. Overall, our estimation procedure works well, with improving performance as thesample size increases.

5. Analysis of the second stage of labour and neonatal and maternal morbidities

In this section, we present a detailed analysis of the CSL data. We focus on labours without theuse of an epidural in this paper as it is well accepted in the obstetrics literature that an epiduralconsiderably changes the progression of labour. We present the analysis for the nulliparous


women in the main part of the paper. A detailed analysis of multiparous women without anepidural is presented in the on-line supplementary material.

5.1. Data descriptionWe begin this section by presenting some descriptives regarding the underlying covariates andmode of delivery that dictate our analytic approach. Recall that M denotes the time to neonatalor maternal morbidity and U is its current status indicator. In addition, T denotes the duration ofthe second stage of labour and Δ denotes the cause of competing risks data with Δ=1 implyingSVD, Δ = 2 implying CD and Δ = 3 implying OVD. We present the cross-tabulation of thefrequency distribution of labours beyond the recommended time of 2 h or not and by morbidityor not in Table 4. Observe that around 13% of cases delivered beyond the recommended timeof 2 h. On the basis of obstetrics literature, we consider two important covariates of interest:mother’s BMI at admission time categorized into three groups—underweight or normal (lessthan 25.0), overweight (25.0 or greater and less than 30.0) and obese (30.0 or greater)—as well asmothers’ age, standardized to mean 0 and a standard deviation of 1. In Fig. 2, we illustrate thedistribution of age across delivery modes and BMI classification for nulliparous women. We notethat the representation of underweight or normal weight mothers compared with overweight orobese mothers is very limited in the CD and younger mothers are more in SVD compared withCD or OVD.

5.2. Estimation resultsWe fitted the proposed joint model described in Section 3 under two different choices of frailtydistribution: a log-normal and a Weibull distribution. We evaluated the choices of frailty dis-tribution by using a negative log-likelihood as well as gradient function approach of Verbekeand Molenberghs (2013) as shown in the on-line supplemental section S3. Based on the modeldiagnostics, the log-normal distribution for frailty showed a better fit to the data so we presentbelow the results based on the log-normal distribution. The corresponding results assuming aWeibull-distributed frailty can be found in the supplemental section S2.

We present the nulliparous women’s results based on the joint model in Table 5: regressioncoefficients for the Cox model corresponding to competing risks of SVD, CD and OVD aswell as the Cox model of morbidities and the coefficient from the frailty model (σ) when wehave mother’s age, quadratic effect of age, and mother’s BMI together in the model. We haveconfirmed that the functional form of age is nearly linear by using splines in the Cox model fittedto the second stage of labour. Table 5 shows that there is a significant decrease in risk for SVDand OVD for overweight and obese women and with advancing maternal age, whereas obese

Table 4. Frequency of SVD and other modes of delivery foreach stratum classified by delivery time (within 2 h of the his-torical recommended time or not) and onset of morbidities fornulliparous women without an epidural

Morbidity SVD CD/OVD

Delivery within Without 4571 (75.3%) 312 (5.1%)2 h (87.2%) With 379 (6.2%) 32 (0.5%)

Delivery beyond Without 516 (8.5%) 180 (3.0%)2 h (12.8%) With 44 (0.7%) 34 (0.6%)


(a) (b) (c)

Fig. 2. Histogram of mother’s age according to BMI classification for nulliparous women without an epiduralunder (a) SVD, (b) CD and (c) OVD: , underweight or normal; , overweight; , obese

Table 5. Estimated parameters and their 95% confidence intervals in parentheses based on nb D 1500bootstrapped sample for nulliparous women†

Underweight Overweight Obese Age Age2

or normal

SVD Reference group −0:18 [−0:30,−0:06] −0:15 [−0:26,−0:01] −0:39 [−0:45,−0:34] 0.02 [−0:01, 0.06]CD Reference group −0:08 [−0:91, 0:54] 0.74 [−0:13, 1.30] −0:07 [−0:38, 0:27] −0:07 [−0:24, 0:15]OVD Reference group −0:48 [−0:71,−0:18] −0:56 [−0:87,−0:28] −0:24 [−0:39,−0:14] −0:10 [−0:18, 0:01]M Reference group −0:23 [−0:45, 0:14] 0.01 [−0:24, 0.35] −0:23 [−0:36,−0:13] −0:04 [−0:13, 0:07]

†The estimate of the coefficient in the frailty model is σ=1:10 [1:02, 1:19].

nulliparous women had an increased risk for CD and a risk of developing neonatal or maternalmorbidity, but not at the 0:05 level of significance. We also found that age was protective forneonatal or maternal morbidity. However, we do not have enough representation of older womenin the nulliparous part of the data and thus do not overinterpret the association with age. Wereplicated the above analyses including only age or BMI in the joint model and found similarresults for the associations.

5.3. Probability table for nulliparous womenFor decision making in the labour process, rather than reporting the cause-specific hazardsfor each delivery mode, the probability of observing the events in a short time interval mayprove to be more informative and straightforward from the perspective of both the mothersand obstetricians in guiding their decision making. We are particularly motivated by interest inanswering questions like ‘What is the chance that we can still achieve SVD without increasing


(a) (b)

Fig. 3. Estimated cause-specific hazards of SVD, CD and OVD along with estimated hazards of morbidityfor nulliparous women (a) age 23 years and BMI less than 25 and (b) age 28 years and BMI 30 or more (weobserve a noticeable increase in hMorbidity.t/ from 120 min (2 h), but not substantially in (a), whereas thereis a noticeable increase in λCD.t/ from 100 min and the three types of hazard are crossing over around 125min in (b)): , λSVD.t/; , λCD.t/; , λOVD.t/; , hMorbidity(t )

the probability of morbidities too much?’ or ‘The chance of having CD or OVD would increasesharply beyond a certain time?’. The cumulative incidence function (Kalbfleisch and Prentice,2011) fits to this demand as it shows cumulative probabilities over time due to a particular deliv-ery mode. The cumulative incidence functions were first derived from the estimated coefficientsas elaborated in the previous section, and then probabilities of delivering via SVD (Δ = 1),CD (Δ= 2) and OVD (Δ= 3) in the next 20 min (for the nulliparous case) or in the next 10min (for the multiparous case, presented in the on-line supplementary section) are calculated.In addition, probabilities of having any morbidity in the next time interval are also calculatedbased on the cumulative hazards of M. We incorporate the simultaneous effect of maternalBMI and age in calculating the probability table so that we have distinct probabilities at eachtime point depending on maternal BMI and age. This means that we might provide a differenttime to change the delivery mode depending on the maternal characteristics rather than havehomogeneous results regardless of any maternal characteristics.

We consider two scenarios of age and BMI and illustrate how our proposed models can beused to infer about the risks that are associated in prolonging the second stage of duration. Thetwo examples include a normal weight woman aged 23 years and an obese woman aged 28 years.


Tab

le6.

Est

imat

ion

ofpr

obab

ilitie

sfo

rob

serv

ing

each

even

tdu

ring

the

next

20m

inan

dea

ch95

%co

nfide

nce

inte

rval

for

nulli

paro

usun

derw

eigh

tor

norm

alw

eigh

t(B

MIl

ess

than

25)

wom

enof

age

23ye

ars

with

outa

nep

idur

alba

sed

onth

efit

ted

resu

ltsin

Tabl

e5†

Tim

e(m

in)

Res

ults

forΔ

=1

(SV

D)

Res

ults

forΔ

=2

(CD

)R

esul

tsfo

rΔ

=3

(OV

D)

M

(0,2

0]0.

3944

[0.3

656,

0.42

45]

0.00

07[0

.000

3,0.

0016

]0.

0167

[0.0

117,

0.02

46]

0.08

09[0

.057

5,0.

1029

](2

0,40

]0.

2476

[0.2

353,

0.25

61]

0.00

03[0

.000

1,0.

0009

]0.

0136

[0.0

101,

0.02

02]

0.03

74[0

.024

6,0.

0626

](4

0,60

]0.

1295

[0.1

209,

0.13

86]

0.00

02[0

.000

1,0.

0007

]0.

0067

[0.0

049,

0.01

10]

0.06

38[0

.042

1,0.

0890

](6

0,80

]0.

0693

[0.0

624,

0.07

56]

0.00

02[0

.000

1,0.

0005

]0.

0060

[0.0

042,

0.00

82]

0.07

45[0

.049

9,0.

0956

](8

0,10

0]0.

0353

[0.0

308,

0.03

95]

0.00

01[0

.000

0,0.

0003

]0.

0045

[0.0

031,

0.00

60]

0.06

24[0

.035

7,0.

0825

](1

00,1

20]

0.02

27[0

.019

3,0.

0257

]0.

0001

[0.0

001,

0.00

04]

0.00

36[0

.002

5,0.

0047

]0.

0570

[0.0

317,

0.07

29]

(120

,140

]0.

0147

[0.0

121,

0.01

71]

0.00

02[0

.000

1,0.

0005

]0.

0027

[0.0

019,

0.00

36]

0.05

04[0

.030

2,0.

0622

](1

40,1

60]

0.00

84[0

.006

7,0.

0100

]0.

0002

[0.0

001,

0.00

05]

0.00

17[0

.001

2,0.

0023

]0.

0391

[0.0

255,

0.04

94]

(160

,180

]0.

0057

[0.0

044,

0.00

69]

0.00

02[0

.000

1,0.

0006

]0.

0012

[0.0

009,

0.00

17]

0.03

62[0

.025

2,0.

0476

]

†Pro

babi

litie

sar

eba

sed

onth

ecu

mul

ativ

ein

cide

nce

func

tion

for

com

peti

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P.t

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=j/

,j=

1,2,

3,an

dth

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e.g.

P.t

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�t+

20/.

Tab

le7.

Est

imat

ion

ofpr

obab

ilitie

sfo

rob

serv

ing

each

even

tdur

ing

the

next

20m

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ese

(BM

I30

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ithou

tan

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ural

base

don

the

fitte

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ble

5

Tim

e(m

in)

Res

ults

forΔ

=1

(SV

D)

Res

ults

forΔ

=2

(CD

)R

esul

tsfo

rΔ

=3

(OV

D)

M

(0,2

0]0.

1018

[0.0

689,

0.15

00]

0.00

06[0

.000

1,0.

0037

]0.

0015

[0.0

005,

0.00

49]

0.01

82[0

.005

2,0.

0576

](2

0,40

]0.

1348

[0.1

010,

0.17

44]

0.00

08[0

.000

1,0.

0053

]0.

0024

[0.0

009,

0.00

77]

0.00

96[0

.003

2,0.

0314

](4

0,60

]0.

1242

[0.1

020,

0.14

58]

0.00

10[0

.000

1,0.

0067

]0.

0022

[0.0

008,

0.00

69]

0.01

81[0

.005

8,0.

0491

](6

0,80

]0.

1041

[0.0

906,

0.11

29]

0.00

10[0

.000

1,0.

0069

]0.

0031

[0.0

011,

0.00

82]

0.02

42[0

.008

1,0.

0568

](8

0,10

0]0.

0764

[0.0

685,

0.08

07]

0.00

10[0

.000

2,0.

0066

]0.

0033

[0.0

011,

0.00

84]

0.02

30[0

.007

6,0.

0508

](1

00,1

20]

0.06

68[0

.058

3,0.

0705

]0.

0017

[0.0

003,

0.00

99]

0.00

36[0

.001

2,0.

0093

]0.

0235

[0.0

078,

0.04

89]

(120

,140

]0.

0576

[0.0

466,

0.06

21]

0.00

32[0

.000

6,0.

0174

]0.

0035

[0.0

011,

0.00

93]

0.02

31[0

.008

2,0.

0452

](1

40,1

60]

0.04

22[0

.031

0,0.

0471

]0.

0046

[0.0

008,

0.02

25]

0.00

29[0

.000

9,0.

0075

]0.

0195

[0.0

075,

0.03

65]

(160

,180

]0.

0356

[0.0

232,

0.04

12]

0.00

64[0

.001

1,0.

0274

]0.

0026

[0.0

008,

0.00

69]

0.01

96[0

.007

8,0.

0357

]


Fig. 3 presents the cause-specific hazards with hazard for any morbidity under the two sce-narios of age and BMI. Table 6 and Table 7 provide the chances of delivery and that of anymorbidity per 20-min increase in duration of labour, for assessing the ‘cost’ of prolonging theduration of labour.

Table 6, which is based on the estimates in Table 5, presents the probabilities of observing eachevent within the interval for underweight or normal weight nulliparous women of age 23 yearswhose second-stage time has been recommended as 2 h. Here women have about a 40% chanceof SVD within the first 20 min on the second stage of labour, and about a 65% chance within 40min. These women have a slightly increased chance of morbidities until 80 min but the chanceof CD does not increase during this window, so the burden from prolonging the second stagemight not be critical. In contrast, as another random example, Table 7 shows the same resultsfor obese nulliparous women of age 28 years. Compared with Table 6, obese women of age 28years show a more radically changing chance of each event; first of all, the probability of havingCD increases sharply beyond 100 min whereas the probability of morbidities reaches the highestvalue between 100 and 120 min. Therefore, an older woman with a higher BMI might have toend her second stage of labour about 20 min earlier as the chance of CD increases beyond 100min. In the case of Table 7, once the second stage goes beyond 100 min women are faced with anincreasing possibility of morbidities and beyond 100 min they should also be concerned aboutincreasing possibility of CD. A sharp increase in hazards for CD after 100 min is also illustratedin Fig. 3(b). We present only these two cases for brevity, but as long as the estimates from thejoint model, Θ, are available, the probability table for each set of covariates can be derived.

6. Concluding remarks

In this study, we propose a joint model to analyse the time to delivery due to different modesand the time to neonatal or maternal morbidities during the second stage of labour givencompeting-risks-type data and current status data. This statistical framework was motivated byan important scientific question in the obstetrics literature and we envisage that the applicationof this statistical method will go beyond obstetric applications. To be specific, during surgery ingeneral, it is often unknown whether patients have experienced any fatal complications until thesurgery ends whereas surgeons may need to change the surgical operation method to preventthe prolongation of the surgery procedure. In this case surgeons should make their decisions bycomparing the risks for the end of surgery due to each surgical operation method and risks forthe complications given the current time. Like a spontaneous labour process, a decision madeduring surgery at every minute or every second becomes critical to individuals so reliable risksestimation provides an important reference for decision makers.

Our joint model includes an individual level frailty that accounts for the expected heteroge-neous distributions among individuals. Even though we have assumed a homogeneous multi-plicative effect of frailty on the baseline hazards, further heterogeneity can be introduced byputting different power terms on the frailty across different hazards. In this case, the computa-tional burden would increase but, to reduce this burden, we may employ a specific parametricassumption with a smaller number of parameters, e.g. a Weibull distribution, on the baselinehazards for the competing risk as well as current status baseline hazard.

To the best of our knowledge, this is the first paper to use a rigorous statistical framework ona well-defined study designed to evaluate quantitatively the American College of Obstetrics andGynecology recommendations for the second stage of labour duration. Our findings on risksof different concurrent events indicate that the guidelines or recommendations for the durationof the second stage of labour are reasonable but adapting them to be dependent on maternal


age and/or BMI might be important. This also brings to attention the need for identificationof subgroups that are at high risk of medical intervention during the second stage of labourand/or increased risk of maternal and/or neonatal morbidities. This is something that we shallbe focusing on in the future. Our probability tables will be of considerable assistance in weighingthe likelihood of SVD against the risk of serious maternal or neonatal morbidity or mortality.In particular, given each woman’s baseline characteristic, we provide data on the changes inthe probability of SVD and that of other modes of delivery as well as serious morbidity ormortality for increasing the duration of the second stage of labour. We believe that these willserve as a reference and will prove to be very useful to clinicians in their decision making andin providing counselling to women on the cost of extending the duration of the second stage oflabour.

Acknowledgements

The authors thank Dr Guoqing Diao for insightful comments during the Conference on Life-time Data Science. The research was partly supported by the intramural research programmeof the Eunice Kennedy Shriver National Institute of Child Health and Human Development.The authors also acknowledge that this study utilized the high performance computational ca-pabilities of the Biowulf Linux cluster at the National Institutes of Health, Bethesda, Maryland(http://biowulf.nih.gov).

Appendix A: Simulation models and results

For simplicity in generating data, we assumed a shared shape parameter κ for both cause-specific baselinehazards but with different shape parameters φ1 and φ2:

λ10.t;φ1,κ/=φκ1κtκ−1, Λ10.t;φ1,κ/= .φ1t/κ,

Table 8. Log-normal distribution simulation results†


500 10 β1 (1.0) 0.0273 0.0992 0.1004 0.8989β2 (−2:0) −0:0637 0.1577 0.1634 0.8969α (1.5) 0.0758 0.1977 0.2099 0.9089σ (0.5) 0.0779 0.1300 0.1245 0.9439

1000 10 β1 (1.0) 0.0126 0.0649 0.0687 0.9307β2 (−2:0) −0:0350 0.1069 0.1101 0.8985α (1.5) 0.0433 0.1399 0.1397 0.8995σ (0.5) 0.0499 0.0882 0.0869 0.9377

500 20 β1 (1.0) 0.0185 0.1158 0.1194 0.9228β2 (−2:0) −0:0412 0.1944 0.1878 0.8975α (1.5) 0.0385 0.2067 0.2120 0.9168σ (1.0) 0.0430 0.2925 0.2974 0.9360

1000 20 β1 (1.0) 0.0069 0.0803 0.0820 0.9067β2 (−2:0) −0:0076 0.1320 0.1285 0.8995α (1.5) 0.0125 0.1489 0.1471 0.9109σ (1.0) 0.0071 0.2126 0.2055 0.9254

†The number of replicates is nr = 1000 per each simulation schemeassuming a log-normal distribution for frailty. The number of quadra-ture nodes is Q=10 when σ=0:5 and Q=20 when σ=1:0 followinga Gauss–Hermite quadrature rule.


Table 9. Weibull distribution simulation results†


500 15 β1 (1.0) −0:0250 0.1136 0.1105 0.9340β2 (−2:0) 0.0231 0.1691 0.1700 0.9460α (1.5) 0.0408 0.2046 0.2048 0.9190γ (1.5) 0.0824 0.4246 0.3801 0.9280

1000 15 β1 (1.0) −0:0402 0.0769 0.0770 0.9090β2.−2:0/ 0.0404 0.1237 0.1212 0.9390α (1.5) 0.0195 0.1466 0.1435 0.9290γ (1.5) 0.0590 0.3479 0.3173 0.9110

500 15 β1 (1.0) 0.0000 0.1018 0.1078 0.9510β2.−2:0/ −0:0455 0.1633 0.1705 0.9300α (1.5) 0.0937 0.2043 0.2134 0.8930γ (2.0) −0:1629 0.6193 0.5262 0.9100

1000 15 β1 (1.0) −0:0142 0.0770 0.0752 0.9320β2 (−2:0) −0:0167 0.1236 0.1224 0.9400α (1.5) 0.0689 0.1567 0.1511 0.8940γ (2.0) −0:0910 0.6309 0.5076 0.9070

†The number of replicates is nr = 1000 per each simulation schemeassuming a Weibull distribution for frailty. The number of quadraturenodes is Q=15 following a Gauss–Laguerre quadrature rule.

Fig. 4. Baseline cause-specific hazards λ10.t/ ( ) and λ20.t/ ( ) and baseline hazard for mor-bidity, h0.t/ ( ) specified as equations (8) with respect to time to event t 2 [0, 160]: note that, around 60min, λ10.t/ and h10.t/ cross over

λ20.t;φ2,κ/=φκ2κtκ−1, Λ20.t;φ2,κ/= .φ2t/κ,

h0.t;ψ,ω/=ψωωtω−1, H0.t;ψ,ω/= .ψt/ω:.8/

For the simulation of which results are presented in Table 2 and Table 3, .φ1,φ2,κ/ = .0:01, 0:005,0:01/ and .ψ,ω/ = . 2

3 , 2:5/. Table 8 and Table 9 present additional simulation results and Fig. 4 showsthe baseline hazards in expression (8).

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Supporting informationAdditional ‘supporting information’ may be found in the on-line version of this article:

‘Supplementary materials for “Joint modeling of competing risks and current status data: an application to sponta-neous labor study”’.

joint modelling of competing risks and current status data...

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