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Multilevel cluster-weighted models for the evaluation of hospitals

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Abstract

In recent years, increasing attention has been directed toward problems inherent to quality control in healthcare services. In particular, it is necessary to measure effectiveness with respect to improving healthcare outcomes of diagnostic procedures or specific treatment episodes. The performance of hospitals is usually evaluated by multilevel models and other methods for risk adjustment. However, these approaches are not suitable for data with large unobserved heterogeneity. A potentially large source of unobserved heterogeneity comes from the variation of the regression coefficients between groups of individuals sharing similar but unobserved characteristics. To overcome such drawbacks, we propose the multilevel cluster-weighted model, a new mixture model approach for handling hierarchical data.

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Notes

  1. 1.

    MDC means major diagnostic class and identifies a way for grouping the discharges based on the diagnoses and the interventions provided to the patients.

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Acknowledgments

The authors warmly thank the Associate Editor and the anonymous reviewers for very helpful comments and suggestions that greatly improved the quality of the paper. This research has been partially supported by the Italian Ministry of University and Research (MIUR), grant FIRB 2012: Mixture and latent variable models for causal inference and analysis of socio-economic data.

Author information

Correspondence to Salvatore Ingrassia.

Appendix: Proof of Proposition 1

Appendix: Proof of Proposition 1

Assume \(\sigma ^2_{X|c}=\sigma ^2_{X}\) and \(\pi _c=\pi =1/K\), \(c=1,\ldots ,K\); the density in (8) yields

$$\begin{aligned} p(x_{ij}, y_{ij}; \varvec{\vartheta })&= \sum ^{K}_{c=1} p(y_{ij}|x_{ij}; \varvec{\xi }_c) \phi (x_{ij};\mu _{X|c},\sigma ^2_{X}) \pi \\&=p(x_{ij}; \varvec{\psi }) \sum ^{K}_{c=1} p(y_{ij}|x_{ij}; \varvec{\xi }_c) \frac{\phi (x_{ij};\mu _{X|c},\sigma ^2_{X}) \pi }{p(x_{ij}; \varvec{\psi }) } \\&= p(x_{ij}; \varvec{\psi }) \sum ^{K}_{c=1} p(y_{ij}|x_{ij}; \varvec{\xi }_c) \frac{\exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}-\mu _{X|c})^2 } \right) }{\sum _{k=1}^{K}\exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}-\mu _{X|k})^2 } \right) } , \end{aligned}$$

where

$$\begin{aligned} p(x_{ij}; \varvec{\psi }) = \sum _{c=1}^K \phi (x_{ij};\mu _{X|c},\sigma ^2_{X}) \pi = \sum _{c=1}^K \frac{1}{\sqrt{2 \pi \sigma ^2_{X}}} \exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}-\mu _{X|c})^2 } \right) \pi . \end{aligned}$$

The proposition is proven once we show that the term

$$\begin{aligned} \frac{\exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}-\mu _{X|c})^2 } \right) }{\sum _{k=1}^{K}\exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}-\mu _{X|k})^2 } \right) } \end{aligned}$$

is equivalent to the probability of latent class membership (4) for suitable parameters \(a_c\) and \(b_c\). To this end, consider that

$$\begin{aligned}&\frac{\exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}-\mu _{X|c})^2 } \right) }{\sum _{k=1}^{K}\exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}-\mu _{X|k})^2 } \right) } = \frac{\exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}^2-2\mu _{X|c}x_{ij}+\mu _{X|c}^2) } \right) }{\sum _{k=1}^{K}\exp \left( \displaystyle {-\frac{1}{2 \sigma ^2_{X}}(x_{ij}^2-2\mu _{X|k}x_{ij}+\mu _{X|k}^2) } \right) } \nonumber \\&\qquad = \frac{\exp \left( \displaystyle {-\frac{x_{ij}^2}{2 \sigma ^2_{X}} } \right) \exp \left( \displaystyle {\frac{2\mu _{X|c}x_{ij}-\mu _{X|c}^2}{2 \sigma ^2_{X}} } \right) }{\sum _{k=1}^{K}\exp \left( \displaystyle {-\frac{x_{ij}^2}{2 \sigma ^2_{X}} } \right) \exp \left( \displaystyle {\frac{2\mu _{X|k}x_{ij}-\mu _{X|k}^2}{2 \sigma ^2_{X}} } \right) } \nonumber \\&\qquad = \frac{ \exp \left( \displaystyle {\frac{2\mu _{X|c}x_{ij}-\mu _{X|c}^2}{2 \sigma ^2_{X}} } \right) }{\sum _{k=1}^{K} \exp \left( \displaystyle {\frac{2\mu _{X|k}x_{ij}-\mu _{X|k}^2}{2 \sigma ^2_{X}} } \right) } = \frac{ \exp \left( \displaystyle {\frac{\mu _{X|c}}{\sigma ^2_{X}} x_{ij}- \frac{\mu _{X|c}^2}{2 \sigma ^2_{X}} } \right) }{\sum _{k=1}^{K} \exp \left( \displaystyle {\frac{\mu _{X|k}}{\sigma ^2_{X}} x_{ij}- \frac{\mu _{X|k}^2}{2 \sigma ^2_{X}} } \right) }, \end{aligned}$$

which assumes the form (4) when \(b_c=\mu _{X|c}/\sigma ^2_{X}\) and \(a_c=\mu _{X|c}^2 / 2 \sigma ^2_X\). This completes the proof.

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Berta, P., Ingrassia, S., Punzo, A. et al. Multilevel cluster-weighted models for the evaluation of hospitals. METRON 74, 275–292 (2016). https://doi.org/10.1007/s40300-016-0098-3

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Keywords

  • Cluster-weighted models
  • Mixture models
  • Hierarchical data
  • Multilevel models