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Multilevel cluster-weighted models for the evaluation of hospitals

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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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  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.


  1. 1.

    AHRQ: Agency for healthcare research and quality. Technical report, US Department of Health and Human Services. Rockville, Guide to Inpatient Quality Indicators (2003). http://www.ahrq.gov/dat/hcup

  2. 2.

    Ash, A.S., Fienberg, S.F., Louis, T.A., Normand, S.-L.T., Stukel, T.A., Utts, J. (2012). Statistical issues in assessing hospital performance. Technical report, Committee of Presidents of Statistical Societies. http://imstat.org/news/2012/03/05/1330972991833.html

  3. 3.

    Asparouhov, T., Muthén, B.: Advances in latent variable mixture models. In: Hancock, G., Samuelson, K. (eds.) Advances in Latent Variable Mixture Models, pp. 27–51. Information Age Publishing, Charlotte (2008)

  4. 4.

    Bagnato, L., Punzo, A.: Finite mixtures of unimodal beta and gamma densities and the \(k\)-bumps algorithm. Comput. Stat. 28(4), 1571–1597 (2013)

  5. 5.

    Biernacki, C., Celeux, G., Govaert, G.: Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models. Comput. Stat. Data Anal. 41(3–4), 561–575 (2003)

  6. 6.

    Böhning, D., Dietz, E., Schaub, R., Schlattmann, P., Lindsay, B.: The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family. Ann. Inst. Stat. Math. 46(2), 373–388 (1994)

  7. 7.

    Dayton, C.M., Macready, G.B.: Concomitant-variable latent-class models. J. Am. Stat. Assoc. 83(401), 173–178 (1988)

  8. 8.

    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the em algorithm. J. R. Stat. Soc. B 39(1), 1–38 (1977)

  9. 9.

    Dubois, R., Brook, R., Rogers, W.: Adjusted hospital death rates: potential screen for quality of medical care. Am. J. Publ. Health 77, 1162–1167 (1987)

  10. 10.

    Fusco, D., Barone, A.P., Sorge, C., D’Ovidio, M., Stafoggia, M., Lallo, A., Davoli, M., Perucci, C.A., Re Val, P.E.: outcome research program for the evaluation of health care quality in lazio, Italy. BMC Health Serv. Res. 12(1), 25 (2012)

  11. 11.

    Geiser, C.: Data Analysis with MPlus. Guilford Press, New York (2013)

  12. 12.

    Gershenfeld, N.: Nonlinear inference and cluster-weighted modeling. Ann. N. Y. Acad. Sci. 808(1), 18–24 (1997)

  13. 13.

    Goldstein, H.: Multilevel Statistical Models, 4th edn. Wiley, London (2010)

  14. 14.

    Goldstein, H., Spiegelhalter, D.: League table and their limitations: statistical issues in comparisons of institutional performance (with discussion). J. R. Stat. Soc. 159(5), 385–443 (1996)

  15. 15.

    Iezzoni, L.I.: Risk Adjustment for Measuring Healthcare Cutcomes. Health Administration Press, USA (2003)

  16. 16.

    Ingrassia, S., Punzo, A.: Decision boundaries for mixtures of regressions. J. Kor. Stat. Soc. 45(2), 295–306 (2016)

  17. 17.

    Ingrassia, S., Minotti, S.C., Vittadini, G.: Local statistical modeling via the cluster-weighted approach with elliptical distributions. J. Classif. 29(3), 363–401 (2012)

  18. 18.

    Ingrassia, S., Minotti, S.C., Punzo, A.: Model-based clustering via linear cluster-weighted models. Comput. Stat. Data Anal. 71, 159–182 (2014)

  19. 19.

    Ingrassia, S., Punzo, A., Vittadini, G., Minotti, S.C.: The generalized linear mixed cluster-weighted model. J. Classif. 32(1), 85–113 (2015)

  20. 20.

    Jones, A.M., Lomas, J., Moore, P., Rice, N.: A quasi-Monte carlo comparison of developments in parametric and semi-parametric regression methods for heavy tailed and non-normal data: with an application to healthcare costs. Technical report, HEDG, c/o Department of Economics, University of York (2013)

  21. 21.

    Karlis, D., Xekalaki, E.: Choosing initial values for the EM algorithm for finite mixtures. Comput. Stat. Data Anal. 41(3–4), 577–590 (2003)

  22. 22.

    Krumholz, H.M., Wang, Y., Mattera, J.A., Wang, Y., Han, L.F., Ingber, M.J., Roman, S., Normand, S.-L.T.: An administrative claims model suitable for profiling hospital performance based on 30-day mortality rates among patients with an acute myocardial infarction. Circulation 113(13), 1683–1692 (2006)

  23. 23.

    Leyland, A., Boddy, F.: League tables and acute myocardial infarction. Lancet 351, 555–558 (1998)

  24. 24.

    Lilford, R., Mohammed, M., Spiegelhalter, D., Thomson, R.: Use and misuse of process and outcome data in managing performance of acute medical care: avoiding institutional stigma. Lancet 364, 1147–1154 (2004)

  25. 25.

    Martini, G., Berta, P., Mullahy, J., Vittadini, G.: The effectiveness-efficiency trade-off in health care: the case of hospitals in Lombardy, Italy. Reg. Sci. Urban Econ. 49, 217–231 (2014)

  26. 26.

    McLachlan, G.J., Peel, D.: Finite Mixture Models. Wiley, New York (2000)

  27. 27.

    McNicholas, P.D., Murphy, T.B., McDaid, A.F., Frost, D.: Serial and parallel implementations of model-based clustering via parsimonious Gaussian mixture models. Comput. Stat. Data Anal. 54(3), 711–723 (2010)

  28. 28.

    Muthén, B., Asparouhov, T.: Multilevel regression mixture analysis. J. R. Stat. Soc. Ser. A (Stat. Soc.) 172(3), 639–657 (2009)

  29. 29.

    Normand, S.-L.T., Glickman, M.E., Gatsonis, C.A.: Statistical methods for profiling providers of medical care: issues and applications. J. Am. Stat. Assoc. 92(439), 803–814 (1997)

  30. 30.

    Opit, L.: The Measurement of Health Service Outcomes. Oxford, London (1993)

  31. 31.

    Punzo, A.: Flexible mixture modeling with the polynomial Gaussian cluster-weighted model. Stat. Model. 14(3), 257–291 (2014)

  32. 32.

    R Core Team.: R: a Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2015)

  33. 33.

    Rice, N., Leyland, A.: Multilevel models: applications to health data. J. Health Serv. Res. 1(3), 154–164 (1996)

  34. 34.

    Snijders, T.A., Bosker, R.J.: Multilevel Analysis, 2nd edn. SAGE Publications, London (2012)

  35. 35.

    Wedel, M.: Concomitant variables in finite mixture models. Statistica Neerlandica 3, 362–375 (2002)

  36. 36.

    Zaslavsky, A.: Statistical issues in reporting quality data: small samples and casemix variation. Int. J. Qual. Health Care 13(6), 481–488 (2001)

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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.

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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

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


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

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