A transition model for fuzzy correlated longitudinal responses

Abstract

Longitudinal study is type of studies that researchers visit subject in several time. Therefore, there are observations of the same subjects that are correlated. These types of studies are widely used in medical science. On the other hand, in medical studies, we frequently face situations that response mastered by linguistic terms. A new transition model which will be able to handle correlation between fuzzy responses is introduced. In this paper we model the transition possibility by fuzzy logistic regressions, and representing how the covariates relate to changes in response. With p covariates, there are (p + 1) parameters including intercepts, which we estimate by extended least squares method. These possibilities depend on the covariates. By using a real data set, an applied example is provided to explain the applicability of the proposed model in clinical studies. In the clinical studies, the effect of hydro-alcoholic extract of Urtica Dioica on menorrhagia (for which the status is basically expressed by linguistic/fuzzy terms) is investigated also the effect of mental intervention in recovery of patients with Lichen Planus disease.

Keywords

Fuzzy data longitudinal study transition model fuzzy logistic regression possibilistic odds

1 Introduction

One of the most common medical researches is longitudinal study in which measurements of same subject are conducted repeatedly through time. In longitudinal studies, researchers are able to characterize the development or change in response variable over time [7]. Due to correlation between responses of same subject, ordinary generalized linear models are not appropriated in these cases. There are three main approaches to address the issue longitudinal data: marginal, transition and random effects models. These models are different from correlation handling and coefficient interpretation that due to different goals of inference of these models [26]. Starm et al. used a random effect model for binary correlated response and used EM algorithm for parameters estimate [5]. Stiratelli et al. [24] applied random effects model and using EM algorithm for estimated of parameters. Transition model is an extension of generalized linear regression for describing the conditional distribution of each response as an explicit function of past responses and predictors. Kalbfleisch and Lawless introduced methods for the analysis of panel data under a continuous-time Markov model. They showed the procedures for obtain maximum likelihood estimates and associated asymptotic covariance matrices for transition intensity parameters in time homogeneous models [12]. Muenz and Rubinstein proposed Markov models for covariate dependence of binary repeated response. They assumed that the sequence of response is followed a binary Markov chain. They modeled the transition probabilities for the responses by two logistic regressions, and showed how the covariates related to changes in response [15]. Diggle et al. [23] described transition model for correlative categorical data. Islam and Chowdhury suggested covariate dependence Markov model for modeling repeated responses [16].

The transition models frequently used in medical science. For example, Lee and Daniels used a Markov model for Longitudinal quality-of-life data from a colorectal cancer clinical trial study [13]. On the other hand, in medical studies, we frequently face situations that response mastered by linguistic terms. For example, to measure the severity of disease or pain in patients, using the numerical exact numbers is not usual, so that the results are usually reported as linguistic terms like low, medium, high, and very high [19].

For modeling vague responses, Tanaka et al. [9] introduced a regression model in fuzzy environment, for the first time. Fuzzy linear regression has been developed by a lot of researchers over the past decades. For example, Sekkeli et al. [8] proposed classification models based on Tanaka’s fuzzy linear regression approach and applied their model in customer satisfaction data. Nasrabadi and Nasrabadi considered a fuzzy linear regression models with fuzzy/crisp output, fuzzy/crisp input considered, and proposed an estimated method along with a mathematical programming based approach [17].

The second approach of fuzzy linear regression analysis adopts the fuzzy least squares method for minimizing errors between the given response and the estimated response, proposed by Diamond [22]. Yang and Lin proposed two estimation methods along with a fuzzy least-squares approach for linear regression with fuzzy input-output [21]. Torabi and behboodian introduced a method for least absolutes estimating of fuzzy parameters in a linear model with fuzzy input and fuzzy output, by used “Resolution Identity” [10]. In binary based fuzzy response field, Pourahmad et al. [27] proposed a new term that is named possibilistic odds and extended a possibilistic-based logistic regression in which the covariates were crisp and response variable was fuzzy. Namdari et al. [20] suggested a new possibilistic logistic regression in which the predictors were crisp and response variables were vague and were measured by linguistic terms.

According to what is being said in non-fuzzy situation, these fuzzy regression models are not appropriate for inference about correlated binary fuzzy response too. In this study, we introduced a new method which will be able to handle correlation between fuzzy responses through a fuzzy logistic regression model.

In Section 2 we describe logistic regression and classical transition model for binary crisp response and some fuzzy consepts. In section 3 we defined new fuzzy transition model in possibility case and in section 4 we proposed fuzzy transition model for linguistic term case. In Section 5 and 6, we illustrate proposed transition model for fuzzy binary based longitudinal in two clinical studies. In section 7 is provided a conclusion.

2 Preliminary

2.1 Logistic regression

Logistic regression is a model to analyze the studies that response variable is measured with binary variable. The purpose of logistic regression is to detect the best fitting model to describe the relationship between the binary response and set of predictors. As an example, imagine how coronary heart disease (CHD) can be predicted by the level of serum cholesterol. The probability of CHD increases with the serum cholesterol level [4].

Since logistic regression computes the probability of an event happening over the probability of an event not happening, the impact of predictor variables is usually explained in terms of $\frac{π}{1 - π}$ in which π is probability of an event happening and $\frac{π}{1 - π}$ is called odds.

With logistic regression, the natural log odds is modeled as a linear function of the predictor variables. If π (x) is detected as the probability of success related to predictors, the logistic regression is defined as follow: $logit [π (x)] = ln (\frac{π (x)}{1 - π (x)}) = α + β x$ where π (x) is the probability of interested response and x is the predictor variable. The parameters of the logistic regression are α and β. This is the simple logistic model [1].

This model is applied in a wide variety of studies. It is used in biomedical studies, social science researches, marketing and genetic studies. For instance, Levinson et al. [3] applied logistic regression to analysis of the genotype data of affected sibling pair (ASPs) and their parents from several research centers.

2.2 Transition model: the classical approach

A particular case of longitudinal study is called transition (or Markov) models. Let Y_ij be the response of ith subject in jth flow up time where i = 1, . . . , n and j = 1, . . . , t. The joint distribution of the responses Y_i1, Y_i2, . . . , Y_it of the ith subject can be computed as follows: $\begin{matrix} p (y_{i 1}, y_{i 2}, . . ., y_{it}) = p (y_{it} | y_{i 1}, y_{i 2}, . . ., y_{it - 1}) \\ \times p (y_{it - 1} | y_{i 1}, y_{i 2}, . . ., y_{it - 2}) \\ \times . . . \times p (y_{i 2} | y_{i 1}) \times p (y_{i 1}), \end{matrix}$ where p (. | .) showed the transition probability. Usually, researchers focus on how predictor variables affect the transition probabilities. the transition probabilities can be represented as function of predictor variables X and model coefficients that need to be estimated [23]. This model is an aspect of generalized linear model that the conditional distribution of each response is expressed as an explicit function of the previous responses (H_ij) and the predictors (X_ij): $f (y_{ij} | H_{ij}) = exp {\frac{y_{ij} φ (θ_{ij})}{φ} + c (y_{ij}, φ)}$ for known functions φ (θ_ij) and c (y_ij, φ). According to distribution of response variable and the right choice of link function (h), model is defined by $h^{- 1} (E (Y_{ij} | x_{ij}, H_{ij})) = X β + α f (H_{ij})$ where H_ij = (Y_i1, Y_i2, . . . , Y_ij-1) denotes the history of the previous responses at the jth time, and f (H_ij) denotes some known functions of the history of the past responses. When the response variable is discrete, these models are presented as Markov chain models. There is an extensive history to the use of Markov chains to model equally spaced discrete longitudinal data with a finite number of states or categories (see e.g. [29–6]).

When the response variable is binary, a first order Markov chain is presented by the transition matrix $[\begin{matrix} π_{00} & π_{01} \\ π_{10} & π_{11} \end{matrix}]$ where π_ab = P (Y_ij = b|Y_ij-1 = a), a, b = 0, 1. Note that, each row of a transition matrix sums to one since P (Y_ij = 0|Y_ij-1 = a) + P (Y_ij = 1|Y_ij-1 = a) = 1. As its name implies, the transition matrix records the probabilities of making each of the possible transition from one visit to the next. The transition probabilities are modeled as function of predictors. Therefore model is presented by $P (Y_{ij} = y_{ij} | Y_{ij - 1} = y_{ij - 1}) = β_{0} + β_{1} x_{ij} + α y_{ij - 1}$ where y_ij = 0, 1 and β₀, β₁, α are coefficient of model [1]. By such a model we can express conditional models with expressing the conditional distribution of the response at any time given the past responses and the predictors. This model frequently used in medical studies that patient condition isn’t stable.

2.3 Fuzzy numbers and fuzzy arithmetic

Definition 1. (Fuzzy Set) A fuzzy set of the universal set X is defined as a set of ordered pairs: $\tilde{A} = {(x, μ_{\tilde{A}} (x)) | x \in X},$ where, $μ_{\tilde{A}}$ is called the membership function of $\tilde{A}$ , and $μ_{\tilde{A}} (x)$ is the grade of membership of x in $\tilde{A}$ .

Definition 2. Let E be a function space, such that u ∈ E if and only if u : R → [0, 1] is a function which satisfies the following requirements:

normality: u (x₀) =1 for some x₀, $- \infty < x_{0} < + \infty;$

u is a convex fuzzy set, $u (λ x + (1 - λ) y) \geq \min {u (x), u (y)},$

and x, y ∈ R, 0 ≤ λ ≤ 1;

u is upper semicontinuous, $lim_{x \to t} sup f (x) = f (t), - \infty < t < + \infty;$

(u) ₀ = closure {t|t ∈ R, u (t) >0} is compact.

The space E is called a fuzzy number space and each u ∈ E is called a fuzzy number.

Definition 3. A (crisp) set of elements that belongs to a fuzzy set $\tilde{A}$ at least to the degree α called α-level set of $\tilde{A}$ , $A_{α} = {x \in X | \tilde{A} \geq α},$

Definition 4. (LR Fuzzy Number) A fuzzy number $\tilde{A}$ is of LR-type, if it has the membership function as follows: $\tilde{A} (x) = {\begin{matrix} L (\frac{m - x}{α}), & x \leq m, \\ R (\frac{x - m}{β}), & x > m . \end{matrix}$ Where, L and R are decreasing shape function from R⁺ to [0, 1] with L (0) =1, L (x) <1 for all x > 0, L (x) >0 for all x < 1 and L (1) =0. Similar condition hold for R. The real number m is called the mean value and α and β are called the left and rigth spreads, respectively.

In special case, for L (x) = R (x) = max {0, 1 - |x|}, $\tilde{A}$ i called triangular fuzzy number and is denoted by A = (m, α, β) _T.

Definition 5. (Extension Principle) Let X be Cartesian product of universes X₁ × X₂ × . . . × X_n and ${\tilde{A}}_{1}, {\tilde{A}}_{2}, . . ., {\tilde{A}}_{n}$ be n fuzzy set in X₁, X₂, . . . , X_n respectively. Suppose that f is a mapping from X to a universe Y as y = f (x₁, x₂, . . . , x_n). Then the extension principle allows us to define a fuzzy set $\tilde{B}$ in Y by: $\begin{matrix} \tilde{B} (y) = \\ {\begin{matrix} sup_{(x_{1}, . . ., x_{n}) \in f^{- 1}} \min {{\tilde{A}}_{1}, . . ., {\tilde{A}}_{n}}, & f^{- 1} (y) \neq 0, \\ 0, & otherwise . \end{matrix} \end{matrix}$ in which f^-1 is the inverse image of f [11, 18].

3 Fuzzy transition model: a) representing responses by degree of memberships

To defined the relationship between a correlated binary biased response variable and a set of predictor variables by transition model, for each subject, let the responses and predictors consist of the observations $({\tilde{x}}_{ij 0}, {\tilde{x}}_{ij 1}, . . ., {\tilde{x}}_{ijp - 1}, {\tilde{H}}_{ij}, {\tilde{Y}}_{ij})$ $1 \leq i \leq n, j = 1, . . ., t$ where ${\tilde{x}}_{ijk}$ , k = 0, 1, . . . , p - 1 are fuzzy predictors and ${\tilde{Y}}_{ij}$ is a fuzzy response to detect the status of each case related to binary response categories, and H_ij is fuzzy history of response, (here, ${\tilde{Y}}_{ij}$ and ${\tilde{x}}_{ij}$ are fuzzy response and fuzzy predictor of ith subject in jth follow up, respectively). Because of non-precise binary response observation, Bernoulli probability distribution cannot be applied for such as data. Therefore, the probability of success, P (Y_ij = 1) = π_ij, cannot be calculated, and so, the probability odds expression $\frac{π_{ij}}{1 - π_{ij}}$ is meaningless [10]. An alternative approach in such cases is to define the possibility of success instead of the probability of success. As it is well-known, Possibility is another aspect of uncertainty [9]. If the consistency degree with the known characteristic for each response is represented by μ_ij (0 ≤ μ_ij ≤ 1), then the “possibilistic odds” is defined as follows.

Definition 6. (Possibilistic Odds) Let μ_i,i = 1, . . . , n be the possibility of success, μ_i = poss (Y_i = 1), Then the ratio $\frac{μ_{ij}}{1 - μ_{ij}}$ is considered as possibility odds of the ith case in which detects the possibility of success related to the possibility of non-success [27]. For instance, if possibility of developing breast cancer for especial patient is 0.3 then the possibilistic odds of cancer is 0.43.

In fuzzy longitudinal study, the researchers usually ask the expert to assign a consistent degree (possibility of being patient in medical study) to each case as a number between 0 and 1 at each time. Therefore μ_ij, i = 1, . . . , n, j = 1, . . . , t is the possibility of success and the possibility transition model proposed as follows: ${\tilde{W}}_{ij} = \ln \frac{μ_{ij}}{1 - μ_{ij}} = {\tilde{b}}_{0} + {\tilde{b}}_{1} x_{ij 1} + . . . + \tilde{α} μ_{ij - 1}$ where ${\tilde{b}}_{0}, {\tilde{b}}_{1}, . . ., {\tilde{b}}_{p - 1}, \tilde{α}$ are the fuzzy coefficients of the model that are triangular fuzzy numbers as follows: ${\tilde{b}}_{k} = (a_{k}^{c}, s_{k}^{L}, s_{k}^{R})_{T}, k = 0, 1, . . ., p - 1$ and $\tilde{α} = (α^{c}, γ^{L}, γ^{R})_{T}$ .

Based on fuzzy arithmetic, it can be shown that, ${\tilde{W}}_{ij}$ is a triangular fuzzy number, so that ${\tilde{W}}_{ij} = (f_{ij}^{c} (x), f_{ij}^{L} (x), f_{ij}^{R} (x))_{T},$ where $\begin{matrix} f_{ij}^{c} (x) & = & a_{0}^{c} + a_{1}^{c} x_{ij 1} + . . . + a_{p - 1}^{c} x_{ijp - 1} + α^{c} μ_{ij - 1}, \\ f_{ij}^{R} (x) & = & s_{0}^{R} + s_{1}^{R} x_{ij 1} + . . . + s_{p - 1}^{R} x_{ijp - 1} + γ^{R} μ_{ij - 1}, \\ f_{ij}^{L} (x) & = & s_{0}^{L} + s_{1}^{L} x_{ij 1} + . . . + s_{p - 1}^{L} x_{ijp - 1} + γ^{L} μ_{ij - 1} . \end{matrix}$

Based on the fuzzy arithmetic, the membership function of the fuzzy estimated response can be shown as follows: $\begin{matrix} {\tilde{W}}_{ij} (w_{ij}) = \\ {\begin{matrix} 1 - \frac{f_{ij}^{c} (x) - w_{ij}}{f_{ij}^{L} (x)}, & f_{ij}^{c} (x) - f_{ij}^{L} (x) \leq w_{ij} \leq f_{ij}^{c} (x), \\ 1 - \frac{w_{ij} - f_{ij}^{c} (x)}{f_{ij}^{R} (x)}, & f_{ij}^{c} (x) < w_{ij} \leq f_{ij}^{c} (x) - f_{ij}^{R} (x) . \end{matrix} \end{matrix}$

For estimation of the coefficients, the idea is to minimize the total fuzziness of the obtained model by minimizing the sum of the spreads of the fuzzy outputs, the subject to some constraints by which one can insure about the validity of the obtained model. In this regard, it is assumed that:

Each observation, w_ij has a membership degree as big as h in the function of the fuzzy estimated response, W_ij, ${\tilde{W}}_{ij} (w_{ij}) \geq h,$ and $w_{ij} = \ln (\frac{μ_{ij}}{1 - μ_{ij}})$ , h ∈ (0, 1).

The fuzzy coefficients are such that the fuzziness of the model is minimized. The determination of fuzzy coefficients leads to a linear programming problem, in which the objective function is the sum of the spreads of the fuzzy outputs, $\begin{matrix} Z & = & p (s_{0}^{L} + s_{0}^{R} + \sum [(s_{i}^{L} + s_{i}^{R}) \sum x_{ij} \\ + (γ_{i}^{L} + γ_{ij}^{R}) μ_{ij - 1}]) \end{matrix}$ According to first constraint can be written asfollow:

$\begin{matrix} 1 - \frac{{(f_{ij} (x))}^{c} - w_{ij}}{{(f_{ij} (x))}^{L}} \geq h \\ \Rightarrow (1 - h) s_{0}^{L} + (1 - h) \sum s_{i}^{L} x_{ij} + (1 - h) γ^{L} \\ μ_{ij - 1} - a_{0}^{c} - \sum a_{i}^{c} x_{ij} - α^{c} μ_{ij - 1} \geq - w_{i}, \\ 1 - \frac{w_{ij} - {(f_{ij} (x))}^{c}}{{(f_{ij} (x))}^{R}} \geq h \\ \Rightarrow (1 - h) s_{0}^{R} + (1 - h) \sum s_{i}^{R} x_{ij} + (1 - h) γ^{R} \\ μ_{ij - 1} - a_{0}^{c} - \sum a_{i}^{c} x_{ij} - α^{c} μ_{ij - 1} \geq w_{i} . \end{matrix}$

Finally, one can minimize the objective function by using a linear programming algorithm (such as the Simplex method) to estimate the center, and the left and the right spreads of each coefficient. In this research, we use the Lingo 8.0 [14] software.

4 Fuzzy transition model: b) representing responses by linguistic terms

In the following, we defined a transition model with fuzzy binary based observations and possibility of success as a linguistic term such as: Very Low, Low, Medium, High, and Very High.

Definition 7. (possibilistic odds for linguistic term) Let μ_i, i = 1, . . . , n be the possibility of success, μ_i = poss (Y_i = 1) which is represented by a linguistic term, μ_i = {low, medium, high}. 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 $\frac{μ_{ij}}{1 - μ_{ij}}, i = 1, . . ., n$ is considered as possibility odds of the ith case which detects the possibility of success related to the possibility of nonsuccess [28].

Model with this considerations is represented as follows:

$\begin{matrix} {\tilde{W}}_{ij} & = & \ln \frac{μ_{ij}}{1 - μ_{ij}} = b_{0} + b_{1} {\tilde{x}}_{ij 1} \\ + . . . + b_{p - 1} {\tilde{x}}_{ijp - 1} + α {\tilde{W}}_{ij - 1} \end{matrix}$ i = 1, . . . , n and j = 1, . . . , t. Where, b₀, b₁, . . . , b_p-1 indicating crisp coefficient and ${\tilde{x}}_{ijk} = (x_{ij}, s_{ij})_{T}$ . According to Extension Principle, we can transform back the estimated $({\tilde{W}}_{ij})$ into the possibility of success $({\tilde{μ}}_{ij})$ . In this model, linguistic terms such as μ_i = {low, medium, high} were expressed by expert as the possibility of success. Then, logarithm transformation of possibility odds, ${\tilde{w}}_{ij} = \ln \frac{{\tilde{μ}}_{ij}}{1 - {\tilde{μ}}_{ij}}, i = 1, . . ., n, j = 1, . . ., t$ is considered as the observation response. Using the Extension Principle [29], the membership function of observation is calculated from the membership function of ${\tilde{μ}}_{ij}$ as follows; ${\tilde{w}}_{ij} (y) = sup_{\forall x : \ln \frac{x}{1 - x} = y} {\tilde{μ}}_{ij} (x)$

Since, $f (x) = \ln \frac{x}{1 - x}, 0 < x < 1$ is a one-to-one function, so, there is one and only one x ∈ (0, 1) such that $\ln \frac{x}{1 - x} = y$ . Therefore, we can rewrite ${\tilde{w}}_{ij} (y = \ln \frac{x}{1 - x}) = {\tilde{μ}}_{ij} (\frac{\exp (x)}{1 + \exp (x)})$

Note that, this means that by placement x with $\frac{\exp (x)}{1 + \exp (x)}$ in the definition of possibility term, the membership function of logarithm transformation of possibilistic odds $({\tilde{w}}_{ij})$ is obtained. To estimate the parameters of the model, the sum of squared errors between $({\tilde{w}}_{ij})$ and $({\tilde{W}}_{ij})$ should be minimized. To do this, we require to define a function which describe well the distance between two fuzzy numbers [22]. A well-known definition, suggested by Xu and Li [25], is as follows:

Definition 8. For arbitrary u, v E, their distance based on a function f (α) is $d (u, v) = {[\int_{0}^{1} f (α) d^{2} ((u)_{α}, (v)_{α}) d α]}^{1 / 2},$ in which d² ((u) _α, (v) _α) = [a₁ (α) - b₁ (α)] ² + [a₂ (α) - b₂ (α)] ², (u) _α = [a₁ (α) - a₂ (α)] and (v) _α = [b₁ (α) - b₂ (α)] are α - cuts of u and v respectively. Also, f (α) is an increasing function on [0,1] for which f (0) =0 and $\int_{0}^{1} f (α) d α = 0.05$ . Usually f (α) = α is selected.

In the following, we apply the proposed transition models to fuzzy binary based longitudinal data in two clinical studies.

5 Application in medical study 1: lichen planus data

In this section, we use the fuzzy transition model for longitudinal fuzzy response that was proposed in Section 3 to describe the efficacy of drug therapy of psychiatric disorders in oral lichen planus. In this study, the researchers fill out special examination forms for 45 patients referring to the Oral Medicine Department in Mashhad Dental School from October 2004 to December 2005 and oral lichen planus was confirmed through clinical examination, biopsy and histopathological evaluation. Then the patients were evaluated by a psychologist [30]. Forty five patients with oral Lichen planus completed 6 months of treatment with either conventional treatment or conventional treatment + drug therapy for their diagnosed psychological disorder. The primary outcome measure was response to the treatment which was recorded 3 times (see, Table 1).

The proposed fuzzy transition model for Lichen planus data is as follows: $\begin{matrix} {\tilde{W}}_{ij} & = & {\tilde{b}}_{0} + {\tilde{b}}_{1} {group}_{ij} + {\tilde{b}}_{2} {age}_{ij} + {\tilde{b}}_{3} {sex}_{ij} \\ + {\tilde{b}}_{1} {time}_{ij} + \tilde{α} μ_{ij - 1} . \end{matrix}$

The coefficients are estimated as follows: $\begin{matrix} {\tilde{W}}_{ij} & = & - 1.75 + 0.34 {group}_{ij} \\ + (0.023, 0.02)_{T} {age}_{ij} - 0.5 {sex}_{ij} \\ + (- 0.35, 0.13)_{T} {time}_{ij} + 3.1 μ_{ij - 1} \end{matrix}$

In this model, after controlling for the potential confounders, a positive relation between the group and pain was detected.

Prediction for a new case. To see the applicability of the above model, consider for example, in a new case we have the following information: previous response = 0.35, age = 57 intervention group in 2th visit. Based on such information, the possibility odds predicted by the model would be (-0.064, 1.4) _T. According to Extension Principle [25], we have $\begin{matrix} \exp ({\tilde{W}}_{new} (x)) = {\begin{matrix} {\tilde{W}}_{new} (\ln (x)), & x > 0; \\ 0 & otherwise . \end{matrix} \\ = {\begin{matrix} 1 - \frac{- \ln (x) - 0.064}{1.4}, & - 1.46 \leq \ln (x) \leq - 0.064; \\ 1 - \frac{\ln (x) + 0.064}{1.4}, & - 0.064 \leq \ln (x) \leq 1.34 . \end{matrix} \end{matrix}$

Figure 1 represents the membership function of the predicted possibilistic odds.

6 Application in medical study 2: Menorrhagia data

Menorrhagia is a common gynecological problem and leading causes of poor quality of life and iron deficiency anemia in women of reproductive age.

As there is no research on the effect of hydroalcoholic extract of Urtica Dioica on menorrhagia, this study is conducted to determine the effect of hydroalcoholic extract of Urtica Dioica on menorrhagia in Babol Azad University students on 2012-13. A randomized triple blind clinical trial was carried out on 100 women affected by menorrhagia, selected by convenience sampling, which had inclusion criteria. Data collection tools were data form, weight, meter and PLBAC chart. All samples after a control cycle of primary bleeding and dividing with random allocation to trial or control group were subjected to mefenamic acid treatment (500 mg every 8 hours) and Urtica Dioica 5 caps. Per day, from first to end of bleeding, up to 7 days, for two consecutive cycles, for trial group and for the control group, mefenamic acid and placebo as the same way of trial group, was prescribed [2]. Usually menorrhagia status was measured by PLBAC chart, but menorrhagia status was basically expressed by verbal reports, linguistic variables and borderlines of categories of linguistic variables [8]. In point of view, the reported borderline between subcategories of menorrhagia status that has been reported is vague (see,Table 2). Because fuzziness must be considered in modeling systems that human estimation is influential [9]. The membership functions of fuzzy numbers are defined as: $Without (x) = {\begin{matrix} 1 - \frac{0.02 - x}{0.01}, & 0.01 \leq x < 0.02; \\ 1 - \frac{x - 0.02}{0.03}, & 0.02 < x \leq 0.33 . \end{matrix}$ $Low (x) = {\begin{matrix} 1 - \frac{0.33 - x}{0.33}, & 0.01 \leq x < 0.33; \\ 1 - \frac{x - 0.33}{0.33}, & 0.33 < x \leq 0.66 . \end{matrix}$ $Medium (x) = {\begin{matrix} 1 - \frac{0.66 - x}{0.33}, & 0.33 \leq x < 0.66; \\ 1 - \frac{x - 0.66}{0.33}, & 0.66 < x \leq 0.99 . \end{matrix}$ $High (x) = {\begin{matrix} 1 - \frac{0.98 - x}{0.33}, & 0.66 \leq x < 0.98; \\ 1 - \frac{x - 0.98}{0.01}, & 0.98 < x \leq 0.99 . \end{matrix}$

The membership functions of responses are shown in Fig. 2.

In order to fit a fuzzy transition model, we should model logarithm transformation of possibility odds, in each time, with linearity dependent to predictors and previous responses as follows: $\ln \frac{μ_{ij}}{1 - μ_{ij}} = b_{0} + b_{1} {group}_{ij} + b_{2} time ij + b_{3} {\tilde{y}}_{ij - 1} .$

Using fuzzy least squares method, the model coefficients are estimated and the model is presented as follows: ${\tilde{W}}_{ij} = - 5.13 + 0.21 group + 0.45 time + 4.74 {\tilde{μ}}_{ij} .$

Positive sign of group coefficient shows that intervention is effective.

Prediction for a new case. To see the applicability of the above model, consider for example, in a new case we have the following information, previous response: medium, control group in first visit. In such a case, the output predicted by the model is: $\begin{matrix} {\tilde{W}}_{new} & = & - 5.13 + 0.21 (0) + 0.45 (1) \\ + 4.74 (0.66, 0.33)_{T} \\ = & (- 1.55, 1.56)_{T} . \end{matrix}$

So, using the Extension Principle, we get $\begin{matrix} \exp ({\tilde{W}}_{new} (x)) = {\begin{matrix} {\tilde{W}}_{new} (\ln (x)), & x > 0; \\ 0 & otherwise . \end{matrix} \\ = {\begin{matrix} 1 - \frac{- 1.55 - \ln (x)}{1.56}, & - 3.11 \leq \ln (x) \leq - 1.55; \\ 1 - \frac{\ln (x) + 1.55}{1.56}, & - 1.55 \leq \ln (x) \leq 0.01 . \end{matrix} \end{matrix}$

Accordingly, Fig. 3 represents the membership function of the predicted possibilistic odds.

7 Conclusion

Longitudinal study is an observational research method in which data is gathered for the same subjects repeatedly over a period of time. These studies frequently are used in medical researches. Three types of models are considered for studying the longitudinal data:

the marginal or population averaged models,

the random-effects or subject-specific models and

the transition or response conditional models.

Relation to regression approach makes that the transition model is as the simplest model between them. On the other hand the non-precise observations are usually seen in these studies. Vague observation is one of the situations in which fuzzy modeling methods are suggested. Although there are a lot of studies in fuzzy Markov object, but in a few one of them, considering covariate dependence as a fuzzy Markov model is seen. In this article, we suggested a fuzzy transition (Markov) model in which the possibility of being a special status is not only related to previous responses, but also related to predictor variables. When response variable is reported as the possibility of having disease or linguistic terms, the fuzzy approaches could be provided suitable models. For estimating of coefficients of model, linear programming is used for possibility cases and least square method is the used for the linguistic term in our paper.

We applied our suggested model, in two real clinical studies. In lichen planus study based on fuzzy transition model, it can estimate the possibilitic odds of lichen planuse disease recuperating for each case. And in menorraghia study, the responses collected as linguistic term, therefore, we used least square method for estimation of model coefficients. Hereupon this paper, we show the different appearance of fuzzy transition model in comparison with Markov model in fuzzy environment. Further studies for extending this model for conditions with fuzzy coefficients are recommended.

Acknowledgements

We would like to show our gratitude to the Dr. Dalirsani and Dr. Kariman for allowing us to apply the data of their studies in this paper.

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