## Statistical Applications Using Fuzzy SetsDespite considerable interest of statisticians of all kinds in high-dimensional, sparse, categorical data, the standard methods for dealing with this interest have specific limitations. One approach, the factor analysis of tetrachoric correlation, often falls prey to the use of incorrect approximating assumptions. Another, latent structure analysis, can become computational refractory, except for problems with fewest cases and variables. Now there's a third approach using a new strategy for resolving measure theoretic issues involving this type of data. That approach centers on the fuzzy set or fuzzy partition models generated by convex geometrical sets. Originally developed in electrical engineering, these models have been finding a growing number of applications in computer science, physics, and theoretical biology. This popularity stems from the power of fuzzy set models to vastly improve on the approximation of the infinite dimensionality and heterogeneity of the real world that arises from the use of statistical partitions, no matter how fine. In this unique book, these models are applied to concrete data from the National Long Term Care Surveys, the National Channeling Demonstration, the Social/HMO Demonstration, the California MSSP Study, and more. In each case the results are compared to the alternative, competing analytic procedures, such as latent class analysis, and are shown to fit the data better, provide substantively more meaningful results, and generate excellent predictions of external variables not used to form the basic dimensions of the model. The models are also shown to be able to predict Medicare and private health expenditures, mortality and morbidity risks, andhealth services use, as well as provide a high measure of clinical meaningfulness for medical and nursing experts. Numerous tables are also provided, showing the results of specific analyses and illustrating how the parametric structure of the models identifies critical features of the data set. By presenting a number of real world, complex analyses that use specific data, this pioneering work is able to show the robustness of the fuzzy set model approach, deal with the relevant technical issues in its successful application, and provide concrete, convincing demonstrations of the theory in practice. |

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### Contents

Crisp and Fuzzy Sets in Statistics | 1 |

The Likelihood Formulation of the Fuzzy Set Partition | 21 |

Estimation of the Parameters of the GoM Model | 50 |

Copyright | |

7 other sections not shown

### Common terms and phrases

algorithm assumed assumption Bayesian budget cancer case-mix changes Chapter coefficients cognitive constraints convex convex set correlation costs covariates crisp set curve squaring data sets defined described dimensional Dirichlet distribution disability discrete disease distribution effects empirical Bayesian episode equation evaluated extreme profiles factor analysis females fuzzy classes fuzzy partition fuzzy set models gik.t GoM model Grade of Membership hazard hazard function health and functioning heterogeneity hospital identifiability impairment increases individual kkjl latent latent class model likelihood function males Manton mathematical matrix maximizing maximum likelihood measurement space measures medical conditions Medicare mortality MTFs multiple multivariate NLTCS nonresponse nursing home observed outcomes persons pijl Poisson population probability problems procedures quadratic random variables regression represent response risk factors S/HMO sample simplex solution space specific Stallard statistical stochastic process structural survey Table Tolley trajectory transitions unobserved values vector weights Woodbury yijl