Using Cox cluster processes to model latent pulse location patterns in hormone concentration data

Abstract

Many hormones, including stress hormones, are intermittently secreted as pulses. The pulsatile location process, describing times when pulses occur, is a regulator of the entire stress system. Characterizing the pulse location process is particularly difficult because the pulse locations are latent; only hormone concentration at sampled times is observed. In addition, for stress hormones the process may change both over the day and relative to common external stimuli. This potentially results in clustering in pulse locations across subjects. Current approaches to characterizing the pulse location process do not capture subject-to-subject clustering in locations. Here we show how a Bayesian Cox cluster process may be adapted as a model of the pulse location process. We show that this novel model of pulse locations is capable of detecting circadian rhythms in pulse locations, clustering of pulse locations between subjects, and identifying exogenous controllers of pulse events. We integrate our pulse location process into a model of hormone concentration, the observed data. A spatial birth-and-death Markov chain Monte Carlo algorithm is used for estimation. We exhibit the strengths of this model on simulated data and adrenocorticotropic and cortisol data collected to study the stress axis in depressed and non-depressed women.

1. Introduction

Regulation of the human stress system is maintained by signaling between the hormones in the hypothalamic–pituitary–adrenal (HPA) axis (Walker and others, 2010, 2012; Lightman, 2008). The primary hormones in this axis, adrenocorticotropic hormone (ACTH) and cortisol, are intermittently secreted in boluses, called pulses (McMaster and others, 2011; Lightman and Conway-Cambell, 2010; Spiga and others, 2011). Alterations in the hormone secretion patterns have been implicated in many health conditions, e.g., depression (Carroll and others, 1976; Young and others, 2001, 2004), post-traumatic stress disorder (Yehuda, 2002), and sleep apnea (Henley and others, 2009). New treatment strategies are beginning to focus on pulsatile delivery (Russell and others, 2014; Henley and Lightman, 2014). Thus, it is clinically important to be able to adequately model the pulse location processes of HPA axis hormones.

To study pulsatile secretion, hormone concentration values are obtained every few minutes for a period up to 24 h (Figure 1). Although the observed data are hormone concentrations, biologic and clinical interest is often on the latent pulse locations. In our motivating study ACTH and cortisol time series were generated from blood samples collected every 10 min for 24 h on 52 women, 26 depressed and 26 healthy controls (Young and others, 2001). This is a typical design. In the primary analysis the pulse location model was summarized using a simple frequency count over the period of observation. This may be overly simplistic for ACTH and cortisol. The pulse locations of these hormones exhibit circadian patterns and potentially other inhomogeneities over the period of observation (see Figure S1 of supplementary material available at Biostatistics online). These patterns may differ between groups even when the overall frequencies remain similar.

Fig. 1.

Observed ACTH and cortisol hormone concentration data for one pair of control and depressed subjects (top panel). Fitted intensity functions for ACTH and cortisol (second panel) with random effects for cluster size and width and the strict repulsion prior on the cluster locations. The gray lines are the 10% and 90% pointwise credible intervals. The bottom two panels are the joint posterior distributions of the cluster centers for the controls (top histogram) and depressed subjects (bottom histogram).

Models of the pulse location process have generally focused on characterizing the inter-event distribution within subjects or 24-h circadian rhythms in the location process (Anderson and O'Sullivan, 1993; Keenan and Veldhuis, 1997; Liu and Wang, 2007). We are interested in developing a more general model of the pulse location process that is able to characterize circadian rhythms in the pulse location model when present, but also is capable of identifying and quantifying temporal clustering of pulse locations across subjects. The subject-to-subject clustering of the pulse locations is of interest because various external stimuli (e.g., feeding, light, and sleep cycles) are known to influence cortisol levels (Greenspan and Gardner, 2004) and are tightly regulated and common across subjects in typical studies (including our motivating study). More (or less) clustering in pulse locations across subjects may indicate that the regulation of the hormone is more (or less) sensitive to external stimuli.

Our new model of pulse locations is based on the Cox cluster point process (Cox, 1955; Møller and Waagepetersen, 2004). Given pulse locations are latent, we embed the pulse location model into an existing deconvolution model of hormone concentration values. A spatial birth-and-death Markov chain Monte Carlo (MCMC) algorithm (Preston, 1977; Geyer and Møller, 1994) is used for estimation.

2. Methods

We first develop the pulse location model and then show how we integrate it with a hormone concentration model. Table 1 provides notation and definitions for the parameters in the model.

Table 1.

Notation

	Pulse location model notation
	Set of pulse locations for a population of subjects
	Set of pulse locations for subject
	Parameters defining the population intensity function
	Set of cluster centers common to all subjects
	Expected number of pulses in cluster
	Variance of the pulse locations in cluster
	Rate of non-clustered pulse locations/hour
	Parameters defining the subject-level intensity function
	Expected number of pulses in cluster for subject ;
	Rate of non-clustered pulse locations for subject ;
	Hormone concentration model notation for subject
	Observed hormone concentration at time
	Pulsatile secretion function,
	Pulse mass and width for pulse
	Pulse location for pulse
	Number of pulses to time
	Mean pulse mass and width, respectively
	Variance–covariance matrix for the pulse mass and width
	Elimination function
	Half-life
	Baseline concentration,
	Number of knots in b-spline
	Set of knot locations
	Set of b-spline coefficients
	Model error
	Model error for log hormone concentration at time
	Model error variance

2.1. The pulse location model

2.1.1. Overview and notation

The Cox cluster process is a Poisson process (Cox and Isham, 1984), which is defined by its intensity function, , where is time. The intensity function defines the rate at which events occur in time. In our case, the latent pulse locations for each subject serve as “the data” that the Cox cluster process is modeling.

There are two levels defining the intensity function of our Cox cluster process. Level 1 is the subject level and involves the pulse locations for each subject (see right panel of Figure S2 of supplementary material available at Biostatistics online, the x's mark pulse locations). Level 2 is the population level and is a model of locations where pulses cluster in time across subjects (see right panel of Figure S2 of supplementary material available at Biostatistics online, 's mark latent cluster locations). The Level 1 model conditions on a realization from the Level 2 model, so we first define Level 2 and then Level 1.

Level 2 (population level):The population cluster locations are modeled as a marked Strauss process (Strauss, 1975) on time interval , which is slightly larger than the observation time period to accommodate boundary issues. The Strauss process was chosen because it is a repulsive process that reduces the likelihood that cluster locations occur too close together in time. Strauss processes are defined by a temporal range of repulsion (), the strength of repulsion (), and a rate parameter . Strict repulsion results when and there is no repulsion when . The number of clusters generally increases as increases. These parameters are user-specified and sensitivity to the choices of these parameters should be investigated when fitting the data.

Notation:Let be a realization of cluster locations from the Strauss process on that is common for all subjects. For subject , , each cluster, , is defined by two marks, and , which describe the size and spread, respectively, of a Gaussian-shaped contribution to the intensity function centered at cluster center . The joint density of , , and is defined in Section 2.1.2. The and are often modeled as random effects with a mean and variance that is common for all subjects, but additional restrictions are possible (e.g., a common across subjects for each cluster).

Level 1 (subject level):For subject , the set of latent pulse locations, , is a realization of a Cox cluster process on . The are driven by random subject-specific intensity functions, which we now derive. For each , let be an inhomogeneous Poisson process with random intensity function where is the density of a Gaussian distribution with mean and variance . The expected number of pulse secretion events in is by definition . Thus, except at the boundaries, is the expected number of pulses in cluster for subject . Pulses not associated with a population cluster are also allowed. Let this process be modeled as an independent homogeneous Poisson process with random time-constant intensity function . Then, by the superposition principle (Cox and Isham, 1984), the pulse location process for subject , , is the union of the clustered and non-clustered processes across population cluster centers (i.e., ) with random intensity function, .

2.1.2. Defining the density of the pulse location model

Here we develop the density of the Cox process for a population of subjects to (1) show how to combine information across subjects and (2) develop other constraints necessary for estimation.

Combining Level 1 densities for all subjects:As developed above, let be the subject level Cox cluster process on driven by random intensity function , where “” . Conditioned on , the density of the (latent) pulse locations for subject has the form

(2.1)

where (2.1) is defined with respect to the measure of a unit-rate Poisson process instead of the standard Lebesgue measure (Cox and Isham, 1984; Møller and Waagepetersen, 2004).

Given that the Level 1 processes are independent across subjects conditioned on , the joint density of the pulse locations for all subjects is the product of the subject-level densities from (2.1):

and , and

The identifiable parameters are , , and because each subject contributes only a small number of pulses to each cluster and a small number of non-clustered pulses. In other words, the population intensity function is better defined than the subject-level intensity functions. To move between the population and subject models, we further assume that both and are equivalent for all subjects. Thus, and , and with these assumptions the subject-level intensity function is a scaling of the population-level intensity function ().

2.2. Integrating the pulse location model with the hormone concentration model

2.2.1. Deconvolution model of hormone concentration

We use an existing deconvolution model (Veldhuis and Johnson, 1992; Johnson, 2003, 2007; Carlson and others, 2009); however, integration with other hormone concentration models (Keenan and others, 1998; O'Sullivan and O'Sullivan, 1988) is also plausible.

As in Johnson (2007), let be the observed hormone concentration for subject at time , where and . The times of observations are generally similar for all subjects and so to simplify notation we use for all . The convolution model for hormone concentration for subject at time is as follows:

(2.2)

where is a slowly changing baseline component representing non-pulsatile hormone secretion and is the pulsatile secretion rate function. By is denoted the hormone elimination function, and is the model error at time consisting of both biological and technical components. Each of these components also depends on a set of parameters whose notation is shown in Table 1 but has been suppressed here for brevity. We model hormone concentration on the log scale because hormone concentrations are positive and the error structure is likely multiplicative on the natural scale (Rodbard and others, 1970). The one is added to aid in model fitting.

The pulse secretion rate function is defined as

with The function is the pulse shape function and is assumed Gaussian in shape, i.e., (Johnson, 2003, 2007; Carlson and others, 2009). Each pulse is defined by a location, , a mass , and a duration . The number of pulses up to time for subject is denoted as and is modeled by the counting process derived from the Cox process pulse location model (Section 2.1).

To integrate the pulse location and secretion models, we extend the Cox pulse location process to be a marked process (Cox and Isham, 1984). Thus, for the subject-level process the marked Cox process includes both the population level parameters in the Cox process and the pulse mass and width parameters (and their priors).

The hormone elimination function is modeled as a single exponential decay, i.e., where is the decay rate for subject and is the half-life. For this application, the baseline concentration function, , is modeled by a b-spline as described in Johnson (2007) and defined by b-spline basis coefficients and corresponding knot locations ; however, a constant baseline could also be used when appropriate.

2.3. Parameter priors

2.3.1. Prior factorization

The priors for a population of subjects factor as follows:

In this factorization, (1) and (2) are the components defining the marked Cox process for the pulse location model. (1a) and (1b) are the Cox process developed in Section 2.1 and (1c) are the prior distributions for the parameters in the Cox process. The parameters in (1b and c) are the new features that are characterized using this approach. (2) defines the priors on the parameters in the pulsatile secretion function (the marks of the Cox process); (3) defines the priors on the other hormone concentration parameters (baseline, half-life, and model error). Together (2) and (3) defined the deconvolution model for concentration.

2.3.2. Priors for parameters defining the Cox cluster process (1c)

We implemented two models for the Cox cluster process. In the first model, the priors on the expected number of pulses in each cluster in the population, , and the variance of the pulse locations in each cluster, , vary from cluster to cluster and are defined hierarchically as follows:

Values for and could also be estimated and have corresponding priors, but for this work are fixed and set to reflect that there is similarity in the expected number and spread across clusters.

In the second model, the expected number of points in each cluster and the variance of each cluster were common across clusters (i.e., and ) and were defined as follows:

and and are set by the user. When was estimated, the prior was . The specific values chosen for these parameters can be found in Section 3.2. We used the same priors for (2) and (3) as in previous work (Johnson, 2007). For brevity, the exact prior distributions and values of the parameters in (2) and (3) can be found in Table S1 of supplementary material available at Biostatistics online.

3. Estimation

We developed a spatial birth-and-death MCMC (SBDMCMC) algorithm to estimate our posterior distribution. Convergence of this sampler follows from arguments similar to Geyer and Møller (1994) and Stephens (2000). Implementation details are provided in Section 3.2. The steps of the algorithm are provided in Section 1 of supplementary material available at Biostatistics online.

3.1. The simulated data

We simulated data to impose either strong () or weak () clustering of pulse locations across subjects. The number of pulses for each individual for each cluster was modeled as a Bernoulli with a success probability of 0.8. Thus, the expected number of pulses in each population cluster, , was 20.8. The underlying cluster center process, , was simulated as a multiscale Strauss process (Penttinen, 1984); i.e., , where . We set and , and , and . In a third simulation, we simulated pulse locations using a renewal process, which resulted in no clustering in pulse locations across subjects. Figure 2 shows representative pulse location intensity functions for the three simulations. When generating data with clustered pulse locations, 100 sets of 26 subjects' pulse locations were simulated. When generating from the renewal process, 20 sets of 26 subjects' pulse locations were simulated. Hormone concentration profiles were generated using the deconvolution model (Equation (2.2)) with sampling every 10 min for 24 h. Information on the pulse and hormone-specific parameters can be found in Section 2 of supplementary material available at Biostatistics online.

Fig. 2.

Simulated and fitted intensity function for a randomly selected simulation for each model: (a) Strong clustering, (b) Weaker clustering, and (c) No clustering. Each simulated dataset had 26 subjects. The top panel represents the true intensity function (solid line) and the estimated intensity function (dashed line). The gray lines are the 10% and 90% pointwise credible intervals. The height of the peaks is . The width of the peaks is The peak center locations make up . The second panel shows the posterior distributions of the estimated cluster centers.

3.2. Implementation and summarization for simulated and experimental data

The prior on the cluster locations imposed strict repulsion between cluster locations (, , , no clusters could be within 36 min of each other). This increased the chance of identifying distinct cluster regions rather than slight deviations from the Gaussian assumption. These parameters were chosen to correspond to the expected number of pulses over the period of observation for each subject based on previous clinical knowledge. Another approach to choosing is to assume no repulsion and visually assess the distance between distinct clustering regions. We investigated sensitivity to the repulsion assumption by also fitting a no-repulsion model (). The sensitivity to the rate parameter in the cluster center prior was investigated by doubling and halving it. The shape of the intensity function was not dependent on ; however, when there was no repulsion and was much larger than truth, clusters were occasionally modeled by two smaller clusters closer in time. In practice it may be useful to set lower than might be expected.

When fitting the simulated data, we assumed that cluster parameters were common across clusters (i.e., and , model 2 described in Section 2.3.2). The prior on was set to a mean of with a variance of 100 log pulses and the prior mean of was with a variance of 10 log hours. When fitting the experimental data, we fitted a variety of models where and varied from cluster to cluster (model 1 in Section 2.3.2) and where and were common across clusters (model 2 in Section 2.3.2). The priors on the means of the cluster size and spread were as above. The variances of and were set to 1.0 and 0.5, respectively, and not estimated. In addition, we fitted models where the non-clustered process rate parameter, , was fixed and estimated. The prior mean for was set at pulses/24 h with a variance of 10 when was estimated. The values in these priors were chosen based on visual assessment of previous analyses. In general, there is limited sensitivity to the priors when they are vague. One exception is if the mean of the prior on is quite large compared with the distance between the clusters. This makes it more likely that extremely large variances may be simulated early in the algorithm and quick convergence of the MCMC algorithm becomes questionable.

Parameters were initialized based on the estimates from the primary analyses of these data (Young and others, 2001). In addition, subjects were initialized to have one randomly located pulse for each subject, and the Cox cluster model was initialized to have one randomly located cluster. The starting value of the variance of the pulse locations in each cluster in the Cox process () was sometimes important. When the starting value of the variance is high (e.g., larger than the typical distance between clusters), often becomes extremely large, making it difficult to identify the cluster centers. We recommend starting the variance at a small value (e.g., 0.05–) allowing for convergence from below and did this for both the simulations and the experimental data fits. Model fit was assessed graphically by generating a sample from the posterior predictive distribution and plotting the mean and 80% credible interval with the observed hormone concentration data.

For the simulated data, one chain of 150 000 iterations was run for each of the 100 sets of data. For the experimental data, one chain of 225 000 iterations was run for each model estimated. The first 20 000 iterations of each chain were treated as burn-in and discarded. Thereafter, every 50th iteration was saved and used for summarizing the posterior distributions. We assessed convergence and mixing visually using trace plots of the draws (see Figure S4 of supplementary material available at Biostatistics online). Each simulation took 16 h to run on a single 2.93 GHz processor. Run times increased the more parameters that varied across clusters.

4. Results

4.1. Simulation

The bias and coverage of the individual parameters in the Cox process were small under strong clustering (Table 2) and the estimates of the individual cluster centers were unbiased [average bias of (), data not shown]. The slight biases in the parameters resulted in an upward bias in the expected number of pulses over the day (Table 2). The false-positive and -negative rates for finding individual pulse locations were both 8%. When the clustering was weaker, the parameters were unbiased with the exception of the number of clusters, which was biased high. The estimates of the individual cluster centers were unbiased [average bias of (), data not shown] and the bias in the expected number of pulses over the day was similar to the strong clustering model. This suggests that the additional clusters estimated partially in the boundary regions. The fits of the intensity functions were good for both models (Figure 2). The results for the individual-level parameters can be found in Table S2 of supplementary material available at Biostatistics online. The results were similar under a no-repulsion model of the cluster centers (see Table S3 of supplementary material available at Biostatistics online).

Table 2.

Estimation properties of intensity function parameters for simulated data assuming strict repulsion in the cluster locations and common cluster parameters. Values are medians and interquartile ranges. Coverage was obtained using 95% equal-tails credible intervals

	Truth	PM	Width 95% CI	Bias of PM	Coverage of
Parameter	(IQR)	(IQR)	(IQR)	(IQR)	95% CI
	Strong clustering
Exp. number of pulses	10.1 (9.8, 10.8)	11.9 (10.9, 12.9)	3.1 (2.9, 3.3)	1.6 (1.1, 2.2)	39
Number of clusters ()	13 (12, 14)	13 (12, 14)	3 (3, 4)	0 (0, 1)	99
Number of pulses/cluster ()	0.80	0.88 (0.85, 0.92)	0.31 (0.29, 0.33)	0.08 (0.05, 0.12)	85
(hours)	0.10	0.10 (0.09, 0.12)	0.07 (0.06, 0.07)	0.00 (, 0.02)	91
	Weaker clustering
Exp. number of pulses	10.0 (9.4, 10.9)	11.1 (10.2, 12)	3.0 (2.7, 3.1)	0.94 (0.6, 1.3)	80
Number of clusters ()	12 (11, 13)	14 (13, 14)	8 (7, 9)	2 (1, 3)	96
Number of pulses/cluster ()	0.80	0.80 (0.72, 0.88)	0.53 (0.44, 0.77)	0.00	100
(hours)	0.50	0.46 (0.39, 0.78)	0.94 (0.56, 6.20)		88
	No clustering
Exp. number of pulses	14 (13,14)	14.4 (14.2, 14.6)	3.3 (3.3, 3.4)	0.55 (0.46, 0.77)	–
Number of clusters ()	–	13 (13, 13)	7 (7, 7)	–	–
Number of pulses/cluster ()	–	1.4 (1.3, 1.5)	6.2 (4.5, 8.5)	–	–
(hours)	–	14.5 (11.2, 17.4)	3380 (1979, 6516)	–	–

PM, posterior median; IQR, interquartile; CI, credible interval; Exp., expected.

We assessed the models' ability to differentiate between strong and weak clustering. These two models only differed in the cluster variance parameter. The posterior distributions were nearly completely separated in all 100 simulations, with the posteriors for the strong clustering simulations concentrating around higher values in every case.

Assuming a clustering model when the data were not clustered resulted in an essentially flat estimated intensity function (Figure 2). The posterior of the locations of the cluster centers was uniform over the period of observation (bottom panel of Figure 2(c)). The posterior distributions of the number of events in a cluster and the variance of the clusters often covered implausibly large values (e.g., interquartile ranges containing values that were longer than the period of observation). These findings show that assuming clustering when fitting does not induce clustering when it is not present. Further, the estimated intensity function provides an expected number of pulses that are consistent with truth, and the fits have low false-positive (8%) and -negative (1%) rates.

4.2. Example: ACTH and Cortisol in depression

4.2.1. Differences in depressed and non-depressed

The posterior distributions for the parameters defining a depressed and non-depressed groups pulse locations model largely overlapped, indicating the parameters defining the intensity functions for the depressed and non-depressed groups were similar (Table 3). Visually both depressed and non-depressed groups exhibited an increase in the number of pulses per hour during the early morning hours (Figure 1). For ACTH, there was some visual clustering in the depressed subjects. A more diffuse intensity pattern was seen in the non-depressed. For Cortisol, the depressed and non-depressed subjects had essentially identical patterns in the estimated intensity function. (Figure 1).

Table 3.

Summary of posterior distributions of parameters in the Cox cluster process for ACTH and cortisol for control and depressed subjects assuming strict repulsion and allowing cluster parameters to vary across clusters

	Pulse location model parameters
	ACTH				Cortisol
	Control		Depressed		Control		Depressed
Parameter	PM	95% CI	PM	95% CI	PM	95% CI	PM	95% CI
Expected number of
Pulse events	21.3	(17.2, 25.8)	26.6	(21.8, 32.2)	31.9	(27.0, 37.7)	30.4	(25.8, 35.5)
Number of clusters ()	13.3	(10, 16)	14.4	(12, 17)	12.7	(12, 14)	12.7	(12, 15)
	3.2	(2.4, 4.0)	3.5	(2.9, 4.2)	3.9	(3.3, 4.5)	3.9	(3.3, 4.5)
	-0.7	(, 0.5)	-1.1		-2.3		-2.3
	Individual-level hormone concentration and pulse parameters
	ACTH				Cortisol
	Control		Depressed		Control		Depressed
Parameter	MPM	SE	MPM	SE	MPM	SE	MPM	SE
Number of pulses	22.6	0.5	27.8	0.5	32.4	0.7	31.5	0.7
¶	3.0	0.3	2.8	0.3	5.2	0.4	5.1	0.5
	19.5	1.5	19.1	1.3	23.8	1.5	28.3	1.9

PM, posterior mean; CI, equal-tails credible interval; MPM, mean of the posterior means; SE, standard error of the posterior means.

Events per subject per day.

Log of the mean # of secretion events per cluster.

Log of the mean of the variances of the clusters.

¶Mean pulse mass.

Half-life.

4.2.2. Differences in ACTH and cortisol patterns

The parameters defining the intensity functions were similar for ACTH and cortisol, with the exception of the mean of the cluster widths (; Table 3). The posterior distributions for the mean cluster widths were nearly separate for ACTH and cortisol. The posterior for cortisol concentrated around a smaller cluster width (posterior medians: for cortisol vs. and for ACTH controls and depressed, respectively). The average cluster width for cortisol was more consistent with strong clustering in pulse times across subjects. For cortisol, pulses for each subject were more likely to occur around 10 am and approximately every 2 h afterwards (Figure 1). For ACTH, the average cluster width was more consistent with a weak to no-clustering model. Both ACTH and cortisol exhibited circadian rhythms in the intensity function with more events occurring per hour in the early morning hours. For ACTH, peak intensities were 3 events/hour/subject in the early morning surge and fell to 0.25 events/hour/subject in the evening period. For cortisol, peak intensities were also in the early morning surge (4.5 events/hour/subject) and fell to 2 events/hour/subject in the evening hours.

4.2.3. Sensitivity analyses

We fitted models where the cluster parameters were (1) common across clusters (Model 2 in Section 2.3.2) and (2) varied from cluster to cluster (Model 1 in Section 2.3.2) to investigate the necessity of less versus more flexibility in the cluster-specific features. We also investigated strict repulsion and no repulsion in the cluster center prior.

All of the models fitted resulted in a similar shape of the intensity function defining the pulse location model (see Figure S5 of supplementary material available at Biostatistics online). However, the parameters defining the intensity functions differed. The overall expected number of pulses per day per subject for cortisol ranged from 29.5 () to 45.4 () pulses/day/subject. Further inspection of the estimated pulse locations revealed that the models differed in the number of Gaussian components used to model each pulse rather than an increase in biologically independent pulse events. Pulses being modeled with more than one Gaussian component were more frequent for models with restrictions on and (e.g., the same and for all clusters).

As expected, the mean number of clusters, , was higher for the no-repulsion priors (see Table S4 of supplementary material available at Biostatistics online). There were also slightly fewer pulses per cluster, , and a slightly narrower spread of the pulses in a cluster, . These patterns held for both hormones and for both groups of subjects. Visual assessment of the intensity functions and posterior distributions of cluster locations indicated that the additional clusters were modeling slight deviations from the assumed normal distribution shape of the clusters. Thus, the strict repulsion prior was more useful for identifying distinct regions of clustering.

5. Discussion

We developed a new more flexible model of the latent pulse location process governing pulsatile hormone data. Although we focused on Cox cluster processes, other Cox processes (e.g., log Gaussian Cox processes) should produce similar results. This approach is unique in that it is flexible enough to capture both circadian changes and strong-to-weak clustering in pulse release times in the population.

A strength of our approach is that it integrates the latent pulse-generating process with the deconvolution model. This approach incorporates estimation uncertainty of all parameters in the estimation of all other parameters. One challenge in jointly modeling the pulse location and hormone concentration processes is that the deconvolution model (Equation (2.2)) does not restrict a pulse to be modeled by only one secretion event. While this is advantageous in that any shape and size pulse can be accurately modeled, it presents an identifiability challenge when estimating the pulse location model. Careful investigation of the subject pulse locations and the intensity function is necessary to examine whether differences in patterns are more likely due to changes in the intensity versus just differences in the number of events used to model each pulse.

It is possible to impose a firm time constraint between pulses by changing the distribution of the number of events in a cluster from a Poisson to a Bernoulli. The intensity function would be an independent cluster model (Lawson and Denison, 2002). This approach is being investigated in future work.

We uncovered a common temporal component in the pulse location times where subjects often have pulses at similar clock times, particularly in cortisol. This pattern has not been previously characterized. We interpret these results as evidence of the existence of one or more external regulators such as feeding or waking, which were common across subjects by the study protocol. A competing explanation may be that subjects are just tightly regulated in their inter-pulse intervals, although this explanation requires a mechanism to coordinate pulse times, not just frequencies. We investigated the plausibility of this second explanation via simulation. We simulated hormone release patterns on 25 subjects. Each subject had the same first pulse location and subsequent locations were generated using a common inter-pulse interval distribution with a mean and variance similar to the original analysis of these data (Young and others, 2001). Within 2 h, the pulse locations quickly disperse in clock time, and thus lack the visual clustering evident in simulated pulse locations from our Cox model (data not shown), and in the real data. This brief exercise provides further support to our interpretation that clustering in the locations indicates an influence of external factors on the pulse regulator.

Previous analyses of the pulse location model were performed in Liu and Wang (2007). Their intensity function for ACTH were similar; however, their intensity function for cortisol was smoother and had limited evidence of clustering in pulse locations compared with ours. There were several differences in the hormone concentration models that could explain this difference. Of importance, our model incorporates a changing baseline for cortisol, while the previous analysis assumes a constant baseline. Ignoring the circadian baseline pattern may result in additional pulses being added to achieve the correct hormone level. This could disrupt the clustering signaling because additional pulses may be added in more random locations.

We fitted each group and hormone separately. Gains in posterior precision of the parameters of interest may be possible by modeling both groups together in a single model. This is because parameters that are similar across groups could be estimated by all available data. Implementation of this extension is being investigated as future work.

We have presented a new approach to modeling pulsatile hormone data that is capable of characterizing complex temporal and clustering patterns in the pulse locations. This new approach uncovered previously unseen patterns in cortisol secretion that may be informative as biomedical investigators consider future hypotheses and treatments involving the stress axis.

6. Software

Software in the form of C code, together with a sample dataset and documentation is available as Supplementary Material.

Supplementary material

Supplementary material is available at http://biostatistics.oxfordjournals.org.

Funding

This work was supported by National Institutes of Health (NCRR KL2 RR025779, NIMH R21MH094994). Contents are the authors' sole responsibility and do not necessarily represent official NIH views.