# Clustering of recurrent events data applied to the re-admission of elderly people at hospital

Vincent Vandewalle, Genia Babykina, Jean-Baptiste Beuscart, Jesus A. Carretero Bravo, Fabien Visade

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Vincent Vandewalle, Genia Babykina, Jean-Baptiste Beuscart, Jesus A. Carretero Bravo, Fabien Visade. Clustering of recurrent events data applied to the re-admission of elderly people at hospital. European Conference on Data Analysis 2022, Sep 2022, Naples, Italy. hal-03899727

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Submitted on 15 Dec 2022

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# Clustering of recurrent events data applied to the re-admission of elderly people at hospital

Vincent Vandewalle

Joint work with Génia Babykina, Jean-Baptiste Beuscart, Jesús Á. Carretero Bravo, Fabien Visade

European Conference on Data Analysis 2022
Thursday, September 15, 2022

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1. Re-admission of elderly people data

2. Model-based clustering of counting process

3. Results on simulated data

4. Results on re-admission data

5. Conclusion and perspectives
1. Re-admission of elderly people data

2. Model-based clustering of counting process

3. Results on simulated data

4. Results on re-admission data

5. Conclusion and perspectives
## DAMAGE cohort (Deschasse et al., 2021)

* Prospective multi-centric cohort of patients of 75 years old or more

* Hospitalised in France (Hauts de France and Normandy regions)

* 3081 patients of whom 1531 patients have had at least one re-hospitalisation

* Socio-demographic, health, geriatric variables

## Recurrent event considered

Successive readmission to the hospital

## Medical questions

* Is the readmission process correlated with some clinical variables?

* Is the intensity of the readmission process correlated with survival?
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Correlation of the event with other variables (1/2)

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sex

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charlson

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Correlation of the event with other variables (2/2)

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## Identify cluster of patients

* Adapt the treatment to each cluster

* Avoid therapeutic obstinacy

## Assumptions

* Patients in a same cluster should follow the same recurrent events distribution

* The effect of covariates can be different in each cluster

## Proposal

Assume that the data come from a *mixture of counting processes* with *intensity depending on covariates*
1. Re-admission of elderly people data

2. Model-based clustering of counting process

3. Results on simulated data

4. Results on re-admission data

5. Conclusion and perspectives
Mixtures in survival analysis on time to first event: Cox and extensions

* Different censoring schemes (Peng, 2000)

* Censored data with high-dimensional covariates (Bussy et al., 2019)

Many applications in the analysis of recurrent events

Hospital readmissions, disease recurrences, repeated failures of industrial
systems

Extensions of Cox model with recurrence: counting point process setting

* Andersen-Gill model (Andersen and Gill, 1982)

* Frailty models (Aalen, 1988; Oakes, 1992)

Mixtures of recurrent events

* Discrete-time duration model (McDonald and Rosina, 2001; Dean, Bauer, and Shanahan, 2014)

* Frailty-mixture models for recurrent event of *cure models* in health applications (Xu et al., 2012; Tawiah, McLachlan, and Ng, 2020).
Fully parametric mixture model based on the Andersen-Gill model with covariates

Notations

• $(T_{ij})_{j \ge 1}$ : the times of successive events for $i$ ($i = 1, \dots, n$)

• $N_i(t)$ the number of events occurred to individual $i$

• $Z_i \in \{0, 1\}^K$ the cluster of $i$ in binary coding

• $\{N_i(t)\}_{t \ge 0}$ given $Z_{ik} = 1$ characterised by intensity $\lambda_{ik}(t)$

$$ \lambda_{ik}(t)dt = \mathbb{E}[dN_i(t)|\mathcal{F}(t-), \mathbf{X}_i(t), Z_{ik} = 1], $$

$\mathbf{X}_i(t)$ the covariate process and $\mathcal{F}(t-)$ the filtration generated by the process

Intensity of the process in cluster $k$

$$ \lambda_{ik}(t) = \lambda_{0k}(t) \exp(\mathbf{X}_i \beta_k^T) $$

with baseline Weibull intensity $\lambda_{0k}(t) = \gamma_{1k}\gamma_{2k}t^{\gamma_{2k}-1}$
Likelihood of path i

$$p(\mathbf{t}_i | \mathbf{X}_i; \boldsymbol{\theta}) = \sum_{k=1}^{K} \pi_k \frac{\prod_{j=1}^{n_i} [\lambda_{ik}(t_{ij}, \boldsymbol{\theta}_k)] \times \exp\left(-\int_{0}^{\tau_i} \lambda_{ik}(u, \boldsymbol{\theta}_k) du\right)}{p(\mathbf{t}_i | \mathbf{X}_i, Z_{ik}=1; \boldsymbol{\theta})}$$

EM algorithm

Starting from $\boldsymbol{\theta}^{(0)}$, alternate the following two steps until convergence

* **E-step**: update the class membership matrix at iteration $h$

$$\rho_{ik}^{(h)} = \frac{\pi_k^{(h-1)} p(\hat{\boldsymbol{\theta}}_k^{(h-1)})}{\sum_{l=1}^{K} \pi_k^{(h-1)} p(\hat{\boldsymbol{\theta}}_k^{(h-1)})},$$

* **M-step**: update model parameter by maximizing the expectation of the log-likelihood

$$\hat{\boldsymbol{\theta}}_k^{(h)} = \underset{\boldsymbol{\theta}}{\operatorname{argmax}} \sum_{i=1}^{n} \rho_{ik}^{(h)} \log p(\mathbf{t}_i | \mathbf{X}_i, Z_{ik} = 1; \boldsymbol{\theta})$$
BIC criterion

$$ \text{BIC}(K) = \ell(\hat{\theta}; K) - \frac{\nu_K}{2} \log n $$

with $\nu_K = K(d+2)$, and n the number of paths

Remark

The choice of the covariates could also be performed using BIC
1. Re-admission of elderly people data

2. Model-based clustering of counting process

3. Results on simulated data

4. Results on re-admission data

5. Conclusion and perspectives
Consistency of the parameters estimation

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20 replicates

Bayes error:
14.5%

Consistency of
the parameters
estimation
1. Re-admission of elderly people data

2. Model-based clustering of counting process

3. Results on simulated data

4. Results on re-admission data

5. Conclusion and perspectives
The variables *sex* and *age* are taken into account as covariates

BIC selects 2 clusters

Parameters of the clusters

<table><thead><tr><th></th><th>π&#x0302;<sub>k</sub></th><th>γ&#x0302;<sub>1k</sub></th><th>γ&#x0302;<sub>2k</sub></th><th>β&#x0302;<sub>1k</sub></th><th>β&#x0302;<sub>2k</sub></th></tr></thead><tbody><tr><th>Class 1</th><td>0.34</td><td>2.56</td><td>0.97</td><td>-0.16</td><td>-0.09</td></tr><tr><th>Class 2</th><td>0.66</td><td>0.58</td><td>0.85</td><td>-0.43</td><td>0.23</td></tr></tbody></table>

Distribution of the number of events given to the cluster (%)

<table><thead><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th></tr></thead><tbody><tr><th>Class 1</th><td>0</td><td>29</td><td>44</td><td>16</td><td>7.3</td><td>1.9</td><td>1.2</td><td>0.6</td><td>0.2</td></tr><tr><th>Class 2</th><td>70</td><td>30</td><td>0.1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

Interpretation with respect to other covariates

<table><thead><tr><th></th><th>age.x</th><th>sex</th><th>death</th></tr></thead><tbody><tr><th>Class 1</th><td>30%</td><td>63%</td><td>54%</td></tr><tr><th>Class 2</th><td>31%</td><td>67%</td><td>25%</td></tr></tbody></table>
<figure><img src="image_13.png" /></figure>
1. Re-admission of elderly people data

2. Model-based clustering of counting process

3. Results on simulated data

4. Results on re-admission data

5. Conclusion and perspectives
Conclusion

* Able to detect different recurrence types in the data

* Not uniquely based on the number of event but also on the variation of the intensity

* Additional interpretation of the re-admission process

Perspectives

* Make depend the prior probability on baseline covariates

* Perform prognosis of the patient cluster

* Study how to integrate the information on the patient cluster at baseline to medical practices
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Aalen, Odd O (1988). "Heterogeneity in survival analysis". In: *Statistics in medicine* 7.11, pp. 1121-1137.

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Andersen, Per Kragh and Richard D Gill (1982). "Cox's regression model for counting processes: a large sample study". In: *The annals of statistics*, pp. 1100-1120.

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Bussy, Simon et al. (2019). "C-mix: A high-dimensional mixture model for censored durations, with applications to genetic data". In: *Statistical methods in medical research* 28.5, pp. 1523-1539.

<figure><img src="image_17.png" /></figure>

Dean, Danielle O, Daniel J Bauer, and Michael J Shanahan (2014). "A discrete-time Multiple Event Process Survival Mixture (MEPSUM) model". In: *Psychological methods* 19.2, p. 251.

<figure><img src="image_18.png" /></figure>

Deschasse, G. et al. (2021). "Development of a predictive score for mortality at 3- and 12-month after discharge from an acute geriatric unit as a trigger for advanced care planning". In.

<figure><img src="image_19.png" /></figure>

McDonald, John W and Alessandro Rosina (2001). "Mixture modelling of recurrent event times with long-term survivors: Analysis of Hutterite birth intervals". In: *Statistical Methods and Applications* 10.1, pp. 257-272.
<figure><img src="image_20.png" /></figure>

Oakes, David (1992). "Frailty models for multiple event times". In:

Survival analysis: State of the art. Springer, pp. 371–379.

<figure><img src="image_21.png" /></figure>

Peng, Fengchun (2000). "A mixture-model approach to the analysis of survival data". In: *Generalized Linear Models: A Bayesian Perspective*. CRC Press. Chap. 15.

<figure><img src="image_22.png" /></figure>

Tawiah, Richard, Geoffrey J McLachlan, and Shu Kay Ng (2020).
"Mixture cure models with time-varying and multilevel frailties for
recurrent event data". In: *Statistical methods in medical research*
29.5, pp. 1368–1385.

<figure><img src="image_23.png" /></figure>

Xu, Ying et al. (2012). "Estimation of summary protective efficacy using a frailty mixture model for recurrent event time data". In:

*Statistics in medicine* 31.29, pp. 4023–4039.