Intuitionistic Fuzzy ANOVA and its Application in Medical Diagnosis
School of Advanced Sciences, VIT University, Tamil Nadu, India.
*Corresponding Author Email: anuradhadhanapal1981@gmail.com
ABSTRACT:
A heuristic statisticalmethod for testing intuitionistic fuzzy hypotheses of onefactor and two factor ANOVA model using intuitionistic fuzzy data samples is proposed. In this method, the decision rules to accept or reject the null and alternative hypotheses are given. The idea of the proposed procedure has been clarified with the help of numerical examples using R studio.
KEYWORDS: Testing hypotheses, Intuitionistic fuzzy data, ANOVA model
INTRODUCTION:
Statistical hypothesis testing is an applied statistical analysis in which inference of population parameters are found using the numerical samples of the population. The data analysts have interested to learn tests of statistical hypothesis for analysing the population parameters. In an experiment study, various treatments are applied to test subjects and the response data is gathered for analysis. A critical tool for carrying out the analysis is the Analysis of Variance(ANOVA). It enables a researcher to differentiate treatment results based on easily computed statistical quantities from the treatment outcome. ANOVA is a technique of statistics that will enable us to test the null hypothesis that more than two population means are equal against the alternative hypothesis that they are not equal by using their sample data. This statistical technique is widely used in almost all areas of research. In conventional hypothesis testing Devore considering samples are precise and the significance test leads to the binary decision. However, in real life the sample data cannot be recorded precisely always. So, imprecise data sample may be got for testing the hypothesis.Intuitionistic fuzzy set (IFS) is one of the generalizations of fuzzy set theory.
Atanassov introduced the concept of IFS. The knowledge and semantic representation of IFS become more meaningful, resourceful and applicable since it includes the degree of membership function, degree of nonmembership function and the hesitation margin. Intuitionistic fuzzy set is a tool in modeling real life problems like sales analysis, marketing, financial services and also it is used in different fields of science.MikihikoKonishi et al. proposed the method of ANOVA for the fuzzy interval data by using the concept of fuzzy sets. Hypothesis testing of one factor ANOVA model for fuzzy data was proposed by Wu using the hlevel set and the notions of pessimistic degree and optimistic degree by solving optimization problems. Kalpanapriya and Pandian proposed a new statistical fuzzy hypothesis testing of ANOVA model for finding the significant difference among more than two population means when the data of their samples are imprecise.
Akbari et al. discussed a nonparametric method for testing the statistical hypotheses based on intuitionistic fuzzy data. Mohsen Arefi and Seyed Mahmoud Taheri proposed a least squares regression modelwhere the available data, of both explanatory variable(s) and the response variable, as well as the parameters of the model, are assumed to be Atanassov’s intuitionistic fuzzy numbers.GholamrezaHesamian and Mohamad Ghasem Akbariproposed the statistical test based on intuitionistic fuzzy hypotheses.Alireza et al. proposed an oneway ANOVA technique in fuzzy environment in which the least squares method is employed. Nourbakhsh etal. developed oneway ANOVA for a sample having fuzzy observations. In this paper, one factor and two factor ANOVA model with imprecise data is proposed. Ranking technique is adopted for ranking the imprecise data. The intuitionistic ANOVA problem has been transformed into crisp one and solved using R studio. Numerical examples are provided to illustrate the approach.
Preliminaries:
We need the following definitions of intuitionistic fuzzy set, triangular intuitionistic fuzzy number, membership and nonmembership function of an intuitionistic fuzzy set/number which can be found in Atanassov.
Let X is a nonempty set. An intuitionistic fuzzy set A in X is given by a set of ordered triples, where define respectively, the degree of membership and degree of nonmembership of the element to the set A, which is a subset of X, and for every element.
Definition: A triangular intuitionistic fuzzy number is denoted by where with the following membership function and nonmembership function is given as:
Definition:
Let and be any two triangular intuitionistic fuzzy numbers then the arithmetic operations as follows:
Definition:
The ranking of a triangular intuitionistic fuzzy number is defined as
Definition:
Let and be two triangular intuitionistic fuzzy numbers. The ranking of and by the R(.) on E , the set of triangular intuitionistic fuzzy numbers is defined as follows:
(i)if and only if
(ii) if and only if
(iii) if and only if
(iv)
(v)
One factor ANOVA Model:
The conducting of an experiment by allotting factorswhose effects are to be experimented to homogeneous experimental units by a simple random sampling design such that every unit can receive any factor with equal chance, analyzing total variation in the results into variation due to factors and variation due to chance and then testing the significance or otherwise of treatments varies over error variation. A onefactor analysis of variance method is used when the analysis involves only one factor with more than two levels and different subjects in each of the experimental conditions. In this section, we proposed the intuitionisticonefactor ANOVA model using ranking technique.Now, the one factor ANOVA model with intuitionistic fuzzy data is illustrated by the following example using Rstudio.
Example:
The department of biochemistry at virginia tech study the dietary residual zinc in the blood stream for group of rats. Five pregnant rats were randomly assigned to each diet group and each was given the diet on day 22 of pregnancy. The amount of zinc in parts per million was calculated. Since under some unexpected situations, we cannot calculate it precisely. We wish to test whether there is any significant difference in residual dietary zinc among the three diets for the following intuitionist fuzzy data.
Table 1: Intuitionistic fuzzy amount of zinc in rats
Diet 


1 
2 
3 
1 
(0.30,0.50,0.70)(0.10,0.50,0.90) 
(0.40,0.42,0.44)(0.38,0.42,0.46) 
(1.04,1.06,1.08)(1.02,1.06,1.10) 
2 
(0.40,0.42,0.44)(0.38,0.42,0.46) 
(0.38,0.40,0.42)(0.36,0.40,0.44) 
(0.80,0.82,0.84)(0.78,0.82,0.86) 
3 
(0.63,0.65,0.67)(0.61,0.65,0.69) 
(0.71,0.73,0.75)(0.69,0.73,0.77) 
(0.70,0.72,0.74)(0.68,0.72,0.76) 
4 
(0.45,0.47,0.49)(0.43,0.47,0.51) 
(0.45,0.47,0.49)(0.43,0.47,0.51) 
(0.70,0.72,0.74)(0.68,0.72,0.76) 
5 
(0.42,0.44,0.46)(0.40,0.44,0.48) 
(0.67,0.69,0.71)(0.65,0.69,0.73) 
(0.80,0.82,0.84)(0.78,0.82,0.86) 
Now, using the ranking technique for the above measure, we get the following table:
Table 2 :Intuitionistic fuzzy amount of zinc in rats after ranking method
Diet 


1 
2 
3 
1 
0.50 
0.42 
1.06 
2 
0.42 
0.40 
0.82 
3 
0.65 
0.73 
0.72 
4 
0.47 
0.47 
0.72 
5 
0.44 
0.69 
0.82 
Null hypothesis for the above data is considered as against the alternative hypothesis, . Using R studio, we obtained the following result for table 2
Table 3: Summary

DF 
Sum Sq 
Mean Sq 
F Value 
Pvalue 
Diet 
2 
0.3236 
0.16178 
9.33 
0.000359 
Residuals 
12 
0.02081 
0.01734 
From table 3 it was analyzed to reject the null hypothesis for the given intuitionistic fuzzy data since, the probability value is less than the significance level 0.05. Therefore the alternative hypothesis of the given intuitionistic fuzzy data is accepted. Thus we conclude that there is a significant difference in residual dietary zinc among the three diets.
Two factor ANOVA Model:
The conducting of an experiment on experimental units, which differ in quality with respect to one character and hence stratified with respect to such changing character into different withinhomogeneous blocks and then allotting treatments, whose effects are to be experimented, to homogenized units within each block by simple random sampling design independently without repetitions and thereafter splitting the total variation in the results into blocks variation, treatments variation, residual /error variation and lastly testing the significance of these variations over error variation. In this section, we concentrate on the case of two factors of interest with respect to Intuitionistic fuzzy set such that each one of factors occurs with its own number of levels. When we interest to study the effects of two factors with respect to Intuitionistic fuzzy set, it is much more advantageous to perform a twoway analysis of variance, as opposed to two separate onefactor ANOVAs. The intuitionistic twofactor ANOVA model using ranking method is proposed.Now, the two factor ANOVA model with intuitionistic fuzzy data is illustrated by the following example using Rstudio.
Example:
Twenty four sheep were used for personnel in the department of animal Science to experiment at Virginia tech to study area and aqueous ammonia treatment of wheat straw. The purpose was to improve nutritional value for male sheep. The diet treatments were controlled, urea at feeding, ammoniatreated straw, and ureatreated straw and they were separated according to relative weight.
There were four sheep in each homogeneous group (by weight) and each of them was given one of the four diets in random order. Since, under some unexpected situations, we cannot measure the weight of sheep precisely. For each of the 24 sheep, the percent dry matter digested was measured. We wish to test whether the diets are homogeneous or not for the following intuitionistic fuzzy data.
Table 4: Intuitionistic fuzzy weight of the sheep under nutritional systems

Diet 



Control 
Urea at feeding 
Ammonia treated 
Urea treated 
Group by Weight 
1 
(30,32,34)(28,32,36) 
(31,35,39)(27,35,43) 
(47,49,51)(45,49,53) 
(44,46,48)(42,46,50) 
2 
(34,36,38)(32,36,40) 
(36,38,40)(34,38,42) 
(47,53,59)(41,53,65) 
(38,42,46)(34,42,50) 

3 
(34,36,38)(32,36,40) 
(33,37,41)(29,37,45) 
(49,52,55)(46,52,58) 
(41,45,49)(37,45,53) 

4 
(38,40,42)(36,40,44) 
(32,34,36)(30,34,38) 
(41,45,49)(37,45,53) 
(41,45,49)(37,45,53) 

5 
(32,34,36)(30,34,38) 
(33,37,41)(29,37,45) 
(45,47,49)(43,47,51) 
(41,43,45)(39,43,47) 

6 
(31,33,35)(29,33,37) 
(32,34,36)(30,34,38) 
(47,49,51)(45,49,53) 
(45,47,49)(43,47,51) 
Now, using the ranking technique for the imprecise weight, we get the following table 5:
Table 5: Weight of the sheep under nutritional systems after ranking

Diet 



Control 
Urea at feeding 
Ammonia treated 
Urea treated 
Group by Weight 
1 
32 
35 
49 
46 
2 
36 
38 
53 
42 

3 
36 
37 
52 
45 

4 
40 
34 
45 
45 

5 
34 
37 
47 
43 

6 
33 
34 
49 
47 
Null hypothesis for columns: against
the alternative hypothesis, Null hypothesis for rows against
the alternative hypothesis,
Using R studio, we obtained the following result for table 6
Table 6: Summary

DF 
Sum Sq 
Mean Sq 
FValue 
Pvalue 
Diet 
3 
844.1 
281.37 
42.154 
0.00000151 
Weight 
5 
17.7 
3.54 
0.531 
0.75 
Residuals 
15 
100.1 
6.67 

Since probability value is less than the significance level 0.05, the null hypothesis is rejected for the given intuitionistic fuzzy data. Therefore the alternative hypothesis of the given intuitionistic fuzzy data is accepted. Thus we conclude that the diets are not homogeneous.
CONCLUSION:
In this paper, we consider one factor and two factor ANOVA problems with intuitionistic fuzzy data. The procedure of the proposed approach is explained by using real life examples. Decision makers may use the proposed ANOVA techniques in the real life issues based intuitionistic fuzzy set in real life problem involving samples for taking appropriate decisions in an easy and effective manner.
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Received on 05.07.2017 Modified on 26.09.2017
Accepted on 01.11.2017 © RJPT All right reserved
Research J. Pharm. and Tech 2018; 11(2):653656.
DOI: 10.5958/0974360X.2018.00122.1