FUZZY LOGISTIC REGRESSION: A NEW POSSIBILISTIC MODEL AND ITS APPLICATION IN CLINICAL VAGUE STATUS

Document Type: Research Paper

Authors

1 Department of Biostatistics, School of Medicine, Shiraz University of Medical Sciences, Shiraz, 71345-1874, Iran

2 Department of Biostatistics, School of Medicine, Shiraz University of Medical Sciences, Shiraz, 71345-1874, Iran

3 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran

Abstract

Logistic regression models are frequently used in clinical
research and particularly for modeling disease status and patient
survival. In practice, clinical studies have several limitations
For instance, in the study of rare diseases or due ethical considerations, we can only have small sample sizes. In addition, the lack of suitable and
advanced measuring instruments lead to non-precise observations and disagreements among scientists in defining disease
criteria have led to vague diagnosis. Also,
specialists often
report their opinion in linguistic terms rather than numerically. Usually, because of these  limitations, the assumptions of the statistical model do not hold and hence their use is questionable. We therefore need to develop new methods for
modeling and analyzing the problem.
In this study, a model called the  `` fuzzy logistic model '' is
proposed for the case when the explanatory variables are
crisp and the value of the binary response variable is reported
as a number between zero and one (indicating the possibility of
having the property). In this regard, the concept of `` possibilistic odds '' is also
introduced. Then, the methodology and formulation
of this model is explained in detail and a linear programming approach is use to estimate the model parameters. Some goodness-of-fit criteria are proposed and a numerical example is given as an example.

Keywords


bibitem{Ag:cda}
A. Agresti, {it Categorical data analysis}, Wiley, New york,
2002.

bibitem{ArTa:Epflrm}
A. R. Arabpour and M. Tata, {it Estimating the parameters of a
fuzzy linear regression model}, Iranian Journal of Fuzzy Systems,
{bf 5} (2008), 1-19.


bibitem{BaWhGo:Lrmlsur}
S. C. Bagley, H. White and B. A. Golomb, {it Logistic regression
in the medical literature: standards for use and reporting with
particular attention to one medical domain}, Journal of Clinical
Epidemiology, {bf 54} (2001), 979-985.

bibitem{Cel:Lsmffvd}
A. Celmins, {it Least squares model fitting to fuzzy vector
data}, Fuzzy Sets and Systems, {bf 22} (1987), 260-269.

bibitem{CoDuGiSa:Lselrmfr}
R. Coppi, P. D'Urso, P. Giordani and A. Santoro, {it Least
squares estimation of a linear regression model with LR fuzzy
response}, Computational Statistics and Data Analysis, {bf
51} (2006), 267-286.

bibitem{Di:Lsfsfv}
P. Diamond, {it Least squares fitting of several fuzzy
variables}, Proc. of the Second IFSA Congress, Tokyo, (1987),
20-25.

bibitem{DuKeMePr:Fia}
D. Dubois, E. Kerre,  R. Mesiar and  H. Prade, {it Fuzzy interval
analysis,  In: D. Dubois,  H. Prade, eds.}, Fundamentals of Fuzzy
Sets, Kluwer, 2000.

bibitem{FaBrKaHa:Haprinme}
A. S. Fauci, E. Braunwald, D. L. Kasper, S. L. Hauser, D. L.
Longo, J. L. Jameson and J. Loscalzo, {it Harrison's principals
of internal medicine}, Wiley, New York, {bf II} (2008), 2275-2279.

bibitem{GAMs:G}
GAMS (General Algebraic Modeling System), {it A high-level modeling
system for mathematical programming and optimization}, GAMS
Development Corporation, Washington, DC, USA, 2007.

bibitem{HaMaYa:Aeflr}
H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, {it A note on
evaluation of fuzzy linear regression models by comparing
membership functions}, Iranian Journal of Fuzzy Systems, {bf
6} (2009), 1-6.

bibitem{HoSoDo:Fulelireanus}
D. H. Hong, J. Song and H. Y. Do, {it Fuzzy least-squares linear
regression analysis using shape preserving operation}, Information
Sciences, {bf 138} (2001), 185-193.


bibitem{Lingo:A}
LINGO 8.0, {it A linear programming, integer programming, nonlinear
programming and global optimization solver}, Lindo System
Inc, 1415 North Dayton Str., Chicago, 2003.

bibitem{MATLAB:A}
MATLAB R., {it A technical computing environment for high-performance
numeric computation and Visualization}, The Math Works Inc.,
2007.


bibitem{MiTa:Plrlpa}
S. Mirzaei Yeganeh and S. M. Taheri, {it Possibilistic logistic
regression by linear programming approach}, Proc. of the 7th
Seminar on Probability and Stochastic Processes, Isfahan
University of Technology, Isfahan, Iran, (2009), 139-143.

bibitem{MoTa:Pedomodel}
J. Mohammadi and S. M. Taheri, {it Pedomodels fitting with fuzzy
least squares regression}, Iranian Journal of Fuzzy Systems, {bf
1} (2004), 45-61.

bibitem{Pe:Alfm}
G. Peters, {it A linear forecasting model and its application in
economic data}, Journal of  Forecasting, {bf 20} (2001), 315-328.

bibitem{RoPe:Mtfgrfr}
S. Roychowdhury and W. Pedrycz, {it Modeling temporal functions
with granular regression and fuzzy rule}, Fuzzy Sets and Systems,
{bf 126} (2002), 377-387.

bibitem{SaGi:A}
B. Sadeghpour and  D. Gien, {it A goodness of fit index to
reliability analysis in fuzzy model, In: A. Grmela, ed.,
Advances in Intelligent Systems, Fuzzy Systems, Evolutionary
Computation}, WSEAS Press, Greece, (2002), 78-83.

bibitem{Sh:Frm}
A. F. Shapiro, {it Fuzzy regression models}, ARC, 2005.

bibitem{TaHe:Amu}
B. D. Tabaei, and W. H. Herman, {it A multivariate logistic
regression equation to screen for Diabetes}, Diabetes Care, {bf
25} (2002), 1999-2003.

bibitem{Ta:Trends}
S. M. Taheri, {it Trends in fuzzy statistics}, Austrian Journal
of Statistics, {bf 32} (2003), 239-257.

bibitem{TaKe:Flar}
S. M. Taheri and M. Kelkinnama, {it Fuzzy least absolutes
regression}, Proc. of 4th International IEEE Conference on
Intelligent Systems, Varna, Bulgaria, {bf 11} (2008), 55-58.

bibitem{TaUeAs:Lraw}
H. Tanaka, S. Uejima, K. Asai, {it Linear regression analysis
with fuzzy model}, IEEE Trans. Systems Man Cybernet., {bf
12} (1982), 903-907.

bibitem{VaBa:Fast}
E. Van Broekhoven and B. D. Baets, {it Fast and accurate of
gravity defuzzification of fuzzy systems outputs defined on
trapezoidsal fuzzy partitions}, Fuzzy Sets and Systems, {bf
157} (2006), 904-918.

bibitem{Za:Fs}
L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf
8} (1965), 338-353.

bibitem{Zimm:Fu}
H. J. Zimmermann, {it Fuzzy set theory and its applications}, 3rd
ed., Kluwer, Dodrecht, 1996.