Markov-modulated marked Poisson processes for modelling disease dynamics based on medical claims data

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Markov-modulated marked Poisson processes

for modelling disease dynamics based on

medical claims data

Sina Mews1,∗

, Bastian Surmann2, Lena Hasemann2,

and Svenja Elkenkamp2

1Department of Business Administration and Economics,

Bielefeld University, Germany

2Department for Health Economics and Health Care Management,

Bielefeld University, Germany

Abstract

We explore Markov-modulated marked Poisson processes (MMMPPs) as a nat-

ural framework for modelling patients’ disease dynamics over time based on medi-

cal claims data. In claims data, observations do not only occur at random points

in time but are also informative, i.e. driven by unobserved disease levels, as poor

health conditions usually lead to more frequent interactions with the healthcare sys-

tem. Therefore, we model the observation process as a Markov-modulated Poisson

process, where the rate of healthcare interactions is governed by a continuous-time

Markov chain. Its states serve as proxies for the patients’ latent disease levels and

further determine the distribution of additional data collected at each observation

∗Corresponding author; email: sina.mews@uni-bielefeld.de.

arXiv:2210.13133v2 [stat.AP] 3 Nov 2022

time, the so-called marks. Overall, MMMPPs jointly model observations and their

informative time points by comprising two state-dependent processes: the observa-

tion process (corresponding to the event times) and the mark process (corresponding

to event-speciﬁc information), which both depend on the underlying states. The ap-

proach is illustrated using claims data from patients diagnosed with chronic obstruc-

tive pulmonary disease (COPD) by modelling their drug use and the interval lengths

between consecutive physician consultations. The results indicate that MMMPPs

are able to detect distinct patterns of healthcare utilisation related to disease pro-

cesses and reveal inter-individual diﬀerences in the state-switching dynamics.

Keywords: chronic obstructive pulmonary disease (COPD), continuous time, disease pro-

cess, hidden Markov model (HMM), informative observation times, maximum likelihood

1 Introduction

Whenever a patient interacts with the healthcare system, claims data are routinely col-

lected from all providers caring for the patient (such as physicians, pharmacies, or hospi-

tals), containing information on costs, patient-speciﬁc diagnoses, and received treatments

like medication. These rich databases on real-life healthcare provisions are increasingly

prominent in public health research as well as decision-making processes of diﬀerent stake-

holders. For example, claims data are used to optimise health service provisions (e.g. Dutta

et al., 2022), estimate the prevalence and incidence of diseases (e.g. Nerius et al., 2017), or

identify patients at risk of hospital readmission (e.g. Min et al., 2019). While claims data

have been extensively used for disease prediction (see, e.g., Nielsen et al., 2017; Christensen

et al., 2018; Hossain et al., 2019), much less attention has been paid to their vast amount

of (implicit) information on disease dynamics over time, such as (changes in) medication,

hospital stays, or the frequency of physician consultations (but see Ploner et al., 2020).

In this contribution, we thus draw on these comprehensive data sets to model the tempo-

ral courses of diseases and to learn about patients’ health conditions over time. Gaining

insights into the latter is crucial to assess treatment eﬀects on disease progression and

to distinguish health conditions associated with diﬀerent demands of care. These results,

in turn, can be used to evaluate individual as well as economic consequences of speciﬁc

diseases.

For modelling patients’ disease processes over time, medical claims data pose two main

challenges, namely (1) that patients’ disease stages are not (directly) observed and (2) that

patients’ interactions with the healthcare system are driven by the disease stages them-

selves, with more frequent interactions when the patient’s condition is poor. Regarding

(1), while claims data provide extensive records on patients’ received services and diag-

noses, they lack information on the actual health condition, speciﬁcally on distinct disease

stages (such as mild, moderate, and severe) graded according to disease-speciﬁc practice

guidelines. Therefore, the unobserved evolution of patients’ disease activity over time

needs to be inferred from available medical data. A popular approach for estimating prop-

erties of the disease process underlying the observed data are (continuous-time) hidden

Markov models (HMMs; e.g. Bureau et al., 2003; Jackson et al., 2003). Although these

latent-state models have been successfully applied in diﬀerent studies on disease progres-

sion, for example based on monthly MRI scans (Altman and Petkau, 2005) or screening

data (Amoros et al., 2019), they are not appropriate for modelling routinely collected

claims data due to the second (2) challenge: whereas (continuous-time) HMMs assume

that observation times are non-informative (i.e. independent of the underlying disease

process), observation times of claims data are informative (i.e. dependent on a patient’s

disease activity), as healthcare interactions are predominantly initiated on demand by the

patient. Speciﬁcally, a patient’s poor health condition usually leads to more frequent and

hence clustered interactions over time — a key characteristic of claims data that HMMs

cannot account for.

A natural approach to model the occurrence and clustering of events over a continuous

time interval is provided by Markov-modulated Poisson processes (MMPPs). MMPPs

generalise homogeneous Poisson point processes in that they assume the event rate to

depend on a latent ﬁnite-state Markov process in continuous time. According to this un-

derlying discrete-valued Markov process, the event rates switch between diﬀerent levels

over time, thus resulting in clustering of events. These events can, for example, correspond

to mouse clicks (i.e. Web page requests; Scott and Smyth, 2003), the surfacing of whales

(Langrock et al., 2013), or the occurrence of earthquakes (Lu, 2012). While MMPPs

have been applied in various areas, applications in the medical context and especially to

disease modelling are rare. However, Lange et al. (2015) use MMPPs to jointly model

patients’ visit times and their observed (though possibly misclassiﬁed) disease process

based on electronic health records (EHR) data. Although such information on patients’

health condition is unavailable in claims data, it is reasonable to assume that an increased

disease activity results in higher healthcare utilisation (see, e.g., Lange et al., 2015; Alaa

et al., 2017; Gasparini et al., 2020; Su et al., 2021). Consequently, claims data contain

information on the underlying disease process by accurately reﬂecting (the frequency of)

patients’ interactions with the healthcare system, which often occur in clusters (see Fig-

ure 1). These healthcare interactions can thus be modelled as an MMPP, where the rate

of interactions is governed by a continuous-time Markov chain, whose states serve as a

proxy for the patients’ unobserved disease levels.

At each interaction time with the healthcare system, claims data provide additional

observations that can help to improve inference on the underlying disease process. To

infer a patient’s disease level using both the clustered interaction times and additional

data observed at these interactions, we propose to analyse patients’ disease activity over

time using Markov-modulated marked Poisson processes (MMMPPs). MMMPPs extend

MMPPs by jointly modelling the observation times as well as data collected at these ob-

servation times, the so-called marks. These marks — just like the event rates — depend

on the underlying discrete-valued Markov process, whose latent states determine the dis-

tribution of the marks. In summary, MMMPPs thus consist of three stochastic processes:

the state process, the observation process, and the mark process. The state process,

corresponding to the unobserved Markov process, governs both the observation process,

consisting of the observation times, and the mark process, consisting of the data collected

at each observation time (i.e. the marks). The latter may correspond to treatments such

as (the amount of) drug use or costs associated with each healthcare interaction. To our

knowledge, only Alaa et al. (2017) use a similar modelling approach in the medical con-

text, namely the semi-Markov-modulated marked Hawkes process, to model hospitalised

patients’ latent clinical states over time. The main diﬀerence to our approach is that Alaa

et al. (2017) focus on risk prognosis based on EHR data, particularly patients’ vital signs

and lab tests, whereas we aim to extract information on patients’ health conditions over

time from claims data. While claims data pose particular challenges as outlined above,

these can be addressed naturally by MMMPPs, which allow inference on the latent disease

process based on observations occurring at informative and clustered points in time.

2 Markov-modulated marked Poisson processes

2.1 Basic model formulation

We consider (claims) data containing information on the random observation times T0,

T1, . . . , Tn, 0 = T0< T1< . . . < Tn, which occur at irregularly spaced points in time,

as well as additional data Yt1, . . . , Ytncollected at the realised observation times. These

sequences of random variables are referred to as the observation process and the mark

process, respectively, and depend on an underlying, unobserved state process {St}t≥0.

From now on, let the integer τ= 1,2, . . . , n denote the index of the observation in the

sequence, such that Ytτand Stτshorten to Yτand Sτ, respectively.

The state process is modelled as an N-state continuous-time Markov chain. Transi-

tions between the states are governed by a transition intensity matrix Q= (qij )i,j=1,...,N ,

whose oﬀ-diagonal elements qij ≥0, i, j = 1, . . . , N,i6=j, can be interpreted as the rates

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Markov-modulatedmarkedPoissonprocessesformodellingdiseasedynamicsbasedonmedicalclaimsdataSinaMews1;*,BastianSurmann2,LenaHasemann2,andSvenjaElkenkamp21DepartmentofBusinessAdministrationandEconomics,BielefeldUniversity,Germany2DepartmentforHealthEconomicsandHealthCareManagement,BielefeldUniversity,Ge...

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Markov-modulated marked Poisson processes for modelling disease dynamics based on medical claims data

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