Possibilistic Logistic Regression for Fuzzy Categorical Response Data Mahshid Namdari Department of Biostatistics, Student Research committee, Faculty of Paramedical Sciences, Shahid Beheshti University of Medical Sciences, Tehran, Iran
[email protected] S. Mahmoud Taheri Department of Engineering Sciences, College of Engineering, University of Tehran, Tehran, Iran, and Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 8415683111, Iran
[email protected] Alireza Abadi Department of Health and Community Medicine, School of Medicine, Shahid Beheshti University of Medical Sciences, Tehran, Iran
[email protected] Mansour Rezaei Chronic Respiratory Diseases Research Center, National Research Institute of Tuberculosis and Lung Disease (NRITLD), Shahid Beheshti University of Medical Sciences, Tehran, Iran
[email protected] Naser Kalantari Department of Community nutrition, Faculty of Nutrition Sciences and Food Technology, Shahid Beheshti University of Medical Sciences, Tehran, Iran,
[email protected] Abstract—A new possibilistic logistic regression is investigated, which can be used in cases where the explanatory variables are crisp observations but the values of the response variable are non-precise and are measured by linguistic terms. For evaluating the model, a goodness-of-fit criterion which is called the mean of capability index is employed. A numerical example in a real clinical study about child’s appetite status is given to explain the method. Keywords— Fuzzy logistic regression; possiblistic odds; linguistic variable; appetite I. INTRODUCTION AND BACKGROUND In clinical studies, due to lack of suitable and advanced measuring instruments the researchers are usually encountered with non-precise observations. Moreover, linguistic terms have a key role in clinical research and in many practical situations it is more realistic and suitable to evaluate a clinical situation by linguistic terms rather than exact numerical ones. For instance, in order to measure the severity of disease or pain in patients, the use of numerical and exact numbers is not usual and linguistic terms like low, medium, high, very high, ... are used. These terms can be viewed as fuzzy sets. Moreover, the borderline between these fuzzy sets is not crisp even if they are measured in numerical scale [1, 2]. Another source of imprecise and vague observations in clinical research is due to scientists' disagreements in defining some disease criteria which have brought out vague diagnosis. Lupus and Behcet are the examples in this field, which, there is no biological examination and the disease is diagnosed by some defined and wholly accepted criteria [3]. To distinguish patients in these diseases, cases which have some of those defined criteria (not all of them) have a vague status. Statistical regression models are not suitable for modeling the relationship between vague observations and situations which ambiguity is associated with the observed data. More flexible methods than statistical ones are needed to handle these situations and fuzzy regression method can be a suitable alternative approach. Since Zadeh [4] introduced fuzzy set theory, its application has been widely developed in the most diverse applicable medical disciplines, such as fuzzy clustering [5], discriminate fuzzy analysis [6], fuzzy regression [1, 2], fuzzy cognitive maps for disease diagnostic [7], and fuzzy modeling of dynamic events in epidemiology [8]. Among different approaches, fuzzy regression analysis is one of the most popular methods of modeling and prediction [9]. Fuzzy regression aims to provide regression models in fuzzy environments, i.e., when the relationships among variables are non-precise, when the observations are non-exact and/or the distributional assumptions of statistical regression models are not fulfilled or cannot be checked due to the issue of small sample sizes [1, 2, 9]. From the conceptual and methodological point of view, there are two general categories of fuzzy linear regression analysis, i.e. possibilistic and fuzzy least squares methods [10]. In this paper, the possibilistic approach is used to develop a new method for estimating regression parameters in logistic regression. Possibilistic regression analysis was initially proposed by Tanaka et al. [11]. It uses a fuzzy linear system as a regression model whereby the objective was to minimize the total spread of the fuzzy parameters subject to the support of the estimated values to cover the support of the observed values for a certain h-level. One of the advantages of Tanaka’s possibilistic model is in its simplicity in programming and computation. Some of its disadvantages can be stated as its sensitivity to outliers and resulting in too wide ranges in estimation. In order to solve the mentioned shortcomings some studies have been done [12, 13, 14]. In contrast to fuzzy linear regression, there have been only a few articles on fuzzy logistic regression. Logistic regression analysis is one of the famous non-linear methods which is used to model a categorical response variable based on ordinary explanatory variables [15]. It is widely used in studies in the health sciences. Like other statistical models, logistic regression is heavily depended on its assumptions [2], and, non-precise or vague observations, which occur frequently in practice, may cause some difficulties. Sometimes, due to lack of suitable instruments or well defined criteria, we may have suspicion in determining the state of the response variable (0 or 1) and therefore, cannot categorize the individual samples in one of two response categories or in some practical situations it is more flexible and common to express the amount of response variable by linguistic terms such as “very low, low, average, high, very high” instead of crisp numbers. So, in these situations due to the vague status of cases relative to response categories, a probability distribution cannot be considered for the response variable. Therefore, the statistical logistic regression is not appropriate and fuzzy logistic regression could be an alternative choice [1, 2]. There have been few studies on logistic regression in the fuzzy environment [1, 2, 16, 17, 18, 19]. The present work investigates a possibilistic approach to fuzzy logistic regression, which was initially proposed by Taheri and Mirzaei Yeganeh [19]. Our focus in this paper is on the situation which the observations of explanatory variables are crisp (exact) observations and the dependent variable is non-precise (fuzzy number). II. METHODOLOGY OF FUZZY LOGISTIC REGRESSION A. Logistic Regression In traditional statistics, in order to regress a binary response variable with two categories on a set of explanatory variables 1 2( , , ..., )pX x x x= , a binary logistic regression model can be used. Since, in binary logistic regression the response variable 0, 1y = (Failure / Success) has binomial distribution, with E(Y) = P(Y = 1) = ; π 0 < < 1,π therefore, a function of mean response named “logit” ( )ln( 1 )π π− is considered for modeling a linear combination of explanatory variables [15]. The relationship between the explanatory and response variables is in the following form 0 1 1ln( ) ....1 p p b b x b xπ π = + + + − (1) in which the expression ( )1π π− is called the probabilistic odds of characteristic 1 (probabilistic odds of success), and j=0, 1, ..., p,jb are the model’s parameters. In practice, sometimes, due to lack of suitable instruments or well defined criteria, we may have suspicion in determining the state of the response variable (0 or 1) and therefore, cannot categorize the individual samples in one of two response categories. Or in many practical situations it is more natural and feasible to measure the response variable by the means of linguistic variables rather than in exact ones. So, in these situations due to the vague status of cases relative to response categories, Bernoulli probability distribution cannot be considered for the response variable. Therefore, the probability of belonging to category 1 ( )( ) 1P Y = cannot be computed and so the probabilistic odds cannot be modeled [1, 2]. In these situations there exist some aspects of uncertainty that measure the vagueness of the phenomena (due to their inconsistency to the existent criteria or the vagueness in their definition) which cannot be summarized in random terms of regression models. This kind of uncertainty is evaluated by a measure called possibility [20, 21]. In this research we will focus on the situation which the observations of explanatory variables are crisp numbers but the dependent variable is non-precise and is measured by linguistic terms. In our strategy, the possibility of success is considered instead of the probability. The concept of “possibilistic odds”, which was firstly proposed by Taheri and Mirzaei Yeganeh [19] and Pourahmad et al. [1, 2], will be given in the following definition. Definition 1: Let , i=1, 2, ..., niμ be the possibility of characteristic 1 or success, for the ith case, ( 1).ii Poss Yμ = = The possibility of success for the preferred characteristic is defined by a linguistic term, {..., low, average, high, ...}.iμ ∈ Proper fuzzy numbers can be defined by an expert for each terms of the linguistic variable. These terms should be defined in such a way that the union of their supports cover the whole range of (0, 1). Then the ratio 1i iμ μ− is considered as possibilistic odds of the ith case, which detects the possibility of success relative to the possibility of non success. Example 1: Suppose that child’s appetite is measured by linguistic terms. Definitions of linguistic terms which detect the possibility of mentioned characteristic (appetite status) are suggested by the authors. Our definition is in the form of triangular fuzzy numbers and is given in (2). Triangular fuzzy numbers that are defined for the possibility of having a good appetite (Very little, Little, Average, Much, Very much),μ = are shown in Fig. 1. Fig. 1. Membership functions of the linguistic terms for child’s appetite 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 X m em be rs hi p fu nc tio n Very Little Little Average Much Very Much 0.021 0.01 0.02 0.01Very little(x)= 0.021 0.02 0.25 0.23 0.251 0.02 0.25 0.23 little(x)= 0.251 0.25 0.50 0.25 0.501 Average(x)= x x x x x x x x −⎧ − ≤ ≤⎪⎪ ⎨ −⎪ − < ≤ ⎪⎩ −⎧ − ≤ ≤⎪⎪ ⎨ −⎪ − < ≤ ⎪⎩ − 0.25 0.50 0.25 0.501 0.50 0.75 0.25 0.751 0.50 0.75 0.25 Much(x)= 0.751 0.75 0.98 0.23 0.981 0.75 0.98 0.23Very much(x)= 0.981 0.98 0.99. 0.01 x x x x x x x x x x x x −⎧ ≤ ≤⎪⎪ ⎨ −⎪ − < ≤ ⎪⎩ −⎧ − ≤ ≤⎪⎪ ⎨ −⎪ − < ≤ ⎪⎩ −⎧ − ≤ ≤⎪⎪ ⎨ −⎪ − < ≤ ⎪⎩ (2) For modeling the possibilistic odds based on a set of crisp explanatory variables, the observations for the ith case are considered as )1 , ..., , ,( 21 iiipii xxx μμ − . In order to be able to use the Tanaka's possibilistic approach for estimating the regression parameters, the logarithm transformation of the possibilistic odds (ln ( 1 ))i iμ μ− can be regressed on the explanatory variables. The fuzzy logistic regression model is, therefore, as follows 0 0 1 1ln ....1 i=1, 2, ..., n. i i i i p ip i W x x x μ β β β μ = = + + + − (3) For simplicity in calculation and interpretation, we assume that the regression parameters ( , ) ,j j j Tc sβ = 0,1, 2, ...,j p= are symmetric triangular fuzzy numbers. B. Estimation of Model Parameters For estimating the regression parameters we will use the methodology that is described in Tanaka et al. [22]. The basic concept for formulating the possibilistic linear regression is to use the mutual relation between the observed and estimated intervals which are obtained by h-level sets of the observed and the estimated fuzzy numbers respectively. Our idea is to obtain fuzzy parameters by minimizing the total spread of the fuzzy parameters subject to [ ] [ ] ,i h i hw W⊂ for i = 1, 2, …, n. It means that we must find ( , ) , 0,1, 2, ...,j j j Tc s j pβ = = such that 0 0 1 1ln ....1 i i i i p ip i W x x x μ β β β μ = = + + + − be the smallest fuzzy set which satisfies the condition that the support of all given observed outputs { }iw are covered by the support of the estimated fuzzy outputs { }iW for a certain h-value. The h-level set of the observed logarithmic transformation of observed odds [ ]i hw can be calculated with the aim of extension principle. As an example consider the following membership function for the ith observation 1 (x)= 1 i i Li i Li i i i i Ui Ui x s x s x x s s α α α μ α α α −⎧ − − ≤ ≤⎪ ⎪ ⎨ −⎪ − < ≤ + ⎪⎩ (4) In order to derive [ ] ,i hw according to the extension principle, we have )) )1( )1(1 ln( ), )1( )1(1 ln(( Uii Uii Lii Lii sh sh sh sh −+ −−− − −− −+− − α α α α (5) The following interval will be derived for the h-level set of the logarithmic transformation of estimated odds ))1( ,)1(( 0000 ∑∑∑∑ ==== −+−− p j ijj p j ijj p j ijj p j ijj xshxcxshxc (6) Example 2: In Example 1, Consider iμ = ‘‘Much’’ for the ith observation. The membership function of the logarithmic transformation of observed odds for this case is calculated from the defined membership function of iμ in (2) and extension principle, as follows exp( )0.75 exp( )1 exp( )1 0.75 0.25 1 exp( ) =ln = exp( )1 0.75 exp( )1 exp( )1 0.75< 0.98 0.23 1 exp( ) 0.50 . i i i x xx x w x xx x μ μ ⎧ −⎪ +⎪ − ≤ +⎪ ⎨− ⎪ − ⎪ +− ≤ ⎩ ≤ ⎪ + And the h-level set of the logarithmic transformation of observed odds for this case is calculated as follows )) 23.0)1(75.0 23.0)1(75.01ln(), 25.0)1(75.0 25.0)1(75.01ln(( h h h h −+ −−−− −− −+−− (7) For instance, for h=0.5, (0.5, 1.86) is the desired interval for this case. In order to minimizing the total spread of the fuzzy parameters subject to [ ] [ ] ,i h i hw W⊂ the following Linear Programming (LP) problem must be solved ) )1( )1(1ln()1( ) )1( )1(1 ln()1( .. ... 00 00 1100 Uii Uii p j ijj p j ijj Lii Lii p j ijj p j ijj ippii sh shxshxc sh sh xshxc ts xsxsxsJ(c)Min −+ −−−−>−+ −− −+− − The above LP problem can be solved by an efficient software such as Lingo [23]. III. GOODNESS- OF- FIT CRITERIA In order to evaluate the goodness-of-fit of the model (3), the following criterion, which is based on the capability index between two fuzzy sets [24], can be used 1 1 min( ,1 ( )) n i i i i i MCI w W w W n = = − •∑ (9) where, sup{min[ ( ), ( )]} inf{max[ ( ), ( )]} i i i i x R i i i ix R w W w x W x w W w x W x ∈ ∈ = • = The MCI’s maximum value is 1 and its minimum value is 0, a larger MCI confirms that the model supports the data well. IV. A REAL CLINICAL EXAMPLE Poor appetite in young children is a major concern of parents and reduced appetite and low food intake are often reported in children [25, 26, 27]. Some pediatricians in Iran prescribe folic acid empirically as an appetite-enhancing drug for underweight children with poor appetite [26]. In this cross sectional study we want to assess the relationship between the level of folic acid and appetite status in children. The study sample included 49 children aged 5-6 years, who were randomly selected from 20 preschools in the city of Tehran, the capital of the I.R. Iran, during the year 2011. Anthropometric measurements were taken by trained staff using standardized methods. Information on demographic characteristics was obtained through interviewing parents at enrolment in the study. A 2-ml non-fasting blood sample was drawn from each child in the morning and it was analyzed for serum folate by the RIA method (DRG kit, United States). Child’s appetite was assessed through asking their mothers by the following single question: "How do you describe the amount of food that normally has been eaten by your child in the last few days: “very little”, “little”, “average”, “much” or “very much”. A part of participants’ data is shown in Table 1. The relationship between folic acid serum level and appetite was evaluated by the means of regression modeling, through adjusting the effect of some covariates that may have an impact on child’s appetite, including child’s age, body mass index (BMI) and Birth order, as well as mother’s working status (housewife/unemployed=1 or employed=0). The consideration of confounding variables in the analysis was primarily determined a priori. Definitions of linguistic terms which detect the possibility of having a good appetite are given in (2). To develop the relationship between possibility odds of having a good appetite and the above-mentioned covariates, the proposed model is fitted: TABLE I. CHARACTERISTICS OF THE SAMPLE DATA No. Agea BMI Folic acid Birth order Mother's jobb Appetite status 1 61 15.0 2.8 2 1 Average 2 61 14.3 4.9 1 1 Little 3 61 13.9 7 1 0 Average 4 61 14.2 7 2 0 Very Little 5 61 16.3 7.7 1 1 Much 6 62 14.0 2.1 1 0 Average 7 62 24.7 3.5 3 1 Very Much 8 64 13.6 2.8 1 1 Little 9 64 16.0 3.7 1 0 Much 10 64 13.9 4.4 2 0 Average 11 64 14.5 6.9 1 1 Very Much 12 64 19.4 7.1 1 1 Very Much 13 65 14.0 5.5 1 1 Average 14 65 17.2 7.3 1 1 Much 15 66 15.1 3 3 1 Much 16 66 14.5 4.3 2 1 Average 17 66 14.2 5 2 1 Average 18 66 14.5 5.4 1 0 Average 19 67 13.5 3.9 1 0 Average 20 67 17.8 6.6 2 1 Much 21 67 15.0 8 1 1 Much 22 67 15.6 25.7 1 0 Average 23 68 15.6 3.6 1 0 Average 24 68 19.7 4.6 1 0 Much 25 68 14.6 5.7 1 1 Average 26 68 15.0 6.3 2 1 Average 27 68 15.1 9.4 1 0 Very Much 28 69 14.3 2.2 1 1 Much 29 69 13.7 2.8 1 0 Average 30 69 20.5 5.4 2 1 Very Much 31 69 16.3 7.5 1 0 Average 32 69 15.6 12.3 1 1 Much 33 70 16.2 5.8 1 0 Average 34 70 15.8 5.9 1 0 Much 35 70 14.3 6.6 1 1 Much 36 70 13.8 7 2 0 Much 27 71 16.8 1.1 2 1 Very Much 38 71 13.9 2.7 1 0 Average 39 71 15.5 7 1 0 Average 40 71 14.8 8.8 3 0 Average 41 71 13.8 9 1 0 Much 42 71 15.7 15.4 1 1 Much 43 72 21.2 1.6 1 0 Much 44 72 14.7 2.9 1 1 Little 45 72 14.1 4.4 1 1 Average 46 72 15.2 5.3 2 1 Average 47 72 11.9 6.8 3 1 Very Much 48 72 14.3 7.3 1 0 Much 49 72 14.0 8.9 1 0 Average a. Age in month b. Mothers working status (housewife) 0 1 2 3 4 5 ln 1 ' i=1, 2, ..., 49 i i i i i i i age BM I birth order folic acid m other s job μ β β β μ β β β = + + + − + + (10) To estimate ,jβ using the LP problem (8), a suitable value for h must be selected. To decide about the h value, the model can be fitted with different h values. Changing h values affects the spreads and the value of the objective function (J), such that the vagueness of the fuzzy outputs are increased by the increase in h value. By comparing the effect of different h values on the amount of objective function, a suitable value can be selected. By comparing the effect of different h values on the amount of the objective function (see Fig. 2), the amount of h = 0.5 was selected and the optimal model was obtained as follows ln ( 11.43,0) (0.023,0.03) 1 (0.58,0) ( 0.21,1.69) order (0.24,0.28) (1.36,0.13) ' i i T T i i T i T i T i T i W age BMI birth folic acid mother s job μ μ = = − + − + + − + + (11) Fig. 2. The objective function for different values of h. In summary, the estimated positive coefficient indicates that the corresponding variable is related to the increase in possibilistic odds of having a better appetite. From the obtained model (11), we understand that having a higher level of folic acid was related to the increase in possibilistic odds of having a better appetite. Child’s appetite was positively related to the body mass index and the risk for poor appetite in children was decreased if their mother was at home. To evaluate the model based on the proposed goodness-of-fit index we get MCI=0.78, which reveals good fitting. Now, suppose there is a new person with the following data: age=61 month, BMI=14 Kg/m2, serum folate=3, birt order=1, mother’s job= 0. 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Pazhand, “An interval-based approach to fuzzy regression for fuzzy input-output data,” IEEE International Conference on Fuzzy Systems, pp. 2859 - 2863, 2011. /ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 200 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 2.00333 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00167 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing false /UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams > setpagedevice