One-Way Anova and Least Squares Method based on Fuzzy

16 Pages·2014·0.25 MB·English

TJFS: Turkish Journal of Fuzzy Systems (eISSN: 1309–1190)
An Official Journal of Turkish Fuzzy Systems Association
Vol.4, No.1, pp. 18-33, 2013.

One-Way Anova and Least Squares Method
*
based on Fuzzy Random Variables

Alireza Jiryaei *
Shahid Bahonar University of Kerman, Faculty of Mathematics and Computer Sciences,
Department of Statistics, Kerman, Iran
E-mail: [email protected]
*Corresponding author

Abbas Parchami
Shahid Bahonar University of Kerman, Faculty of Mathematics and Computer Sciences,
Department of Statistics, Kerman, Iran
E-mail: [email protected]

Mashaalla Mashinchi
Shahid Bahonar University of Kerman, Faculty of Mathematics and Computer Sciences,
Department of Statistics, Kerman, Iran
E-mail: [email protected]

Received: July 10, 2013 – Revised: November 18, 2013 – Accepted: November 18, 2013

Abstract

The aim of this paper is to deal with one-way analysis of variance (ANOVA) in fuzzy
environment. The traditional method of ANOVA has been generalized based on fuzzy
random variables instead of real-valued random variables. All random errors and
parameters are considered as triangular or exponential fuzzy numbers. The least squares
method is employed to estimate the fuzzy parameters and a method is discussed to
check the adequacy of the proposed model. Finally, two examples are provided to
clarify the presented discussion in this article.

Keywords: Fuzzy analysis of variance, Fuzzy least squares method, Fuzzy random
variable, Exponential and Triangular fuzzy numbers

1. Introduction and background

ANOVA is a collection of statistical models and their associated procedures, in which
the observed variance in a particular variable is partitioned into different sources of
variation. In simplest form (one-way ANOVA), it provides a statistical test of whether
or not the means of several independent groups are all equal. As a common method in
statistical inference, ANOVA has many applications in agricultural and engineering
sciences.

* This paper is an extended version of (Jiryae et al. (2012)).
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In real world, the fuzziness of an observed variable often happens in two cases. The first
case is due to technical conditions of measurements where the response variable cannot
be measured exactly and so in this case, data cannot be recorded clearly with precise
(non-fuzzy) numbers and only in linguistic terms to justify the required tolerance of the
errors in measurements. The second case is due to the fact that the response variable
will be given in terms of linguistic forms, such as linguistic report of experts or report of
a farmer about his products, which are not numeric. In both cases, the data could be
represented by the notion of fuzzy sets to analyse the experiment (Nourbakhsh at al.
(2011)). In many applied sciences the values obtained from outcomes of experiments
are fuzzy; therefore it is necessary to use fuzzy sets theory for modelling and dealing
with experimental outcomes of these sciences. Since the fuzzy sets theory was
introduced by Zadeh (Zadeh, 1965) to the scientific community, many authors used this
theory in various areas of scientific fields. In the following, we review some recent
works on ANOVA in fuzzy environment. One-way Fuzzy ANOVA (FANOVA) is
compared with other classical methods using simulation techniques in (Degaribay,
1987). Two different approaches to one-way ANOVA have been developed in
(Montenegro et al.), based on fuzzy random variables in Puri and Ralescu’s sense (Puri
and Ralescu, 1986). A processing method for ANOVA using the data including
vagueness is proposed in (Konishil et al.) which has removed the influence of
vagueness for sum of squares using the moment correction. A bootstrap approach to
ANOVA for fuzzy valued sample data is introduced in (Gil et al.). One-way and two-
way ANOVA is considered in (Buckley, 2006) using a set of confidence intervals for
variance parameter. Analysis of variance is given in (Wu, 2007) by considering the
sets of fuzzy data, introducing the pessimistic and optimistic degrees and solving
an optimization problem. An approach to ANOVA test is presented using identifying
𝛼th-e𝑐 𝑢fu𝑡zzy scale of measurement with a special subset of a functional Hilbert space in
(González-Rodríguezl et al.). One-way ANOVA based on Zadeh’s extension principle
to a case where observed data are fuzzy has been studied in (Nourbakhsh at al.). In this
paper, a different method is suggested for one-way ANOVA when all random variables
are symmetric fuzzy random variables.

The rest of this paper is organized as follows. Some basic concepts about the symmetric
triangular and exponential fuzzy numbers are presented in Section 2. In Section 3, the
classical one-way ANOVA is reviewed briefly. In Section 4, a method is described for
one-way ANOVA based on a meter on fuzzy numbers where the random variables are
considered to be fuzzy rather than crisp. Also, the least squares method is employed to
introduce the estimates of fuzzy parameters and another method is discussed to check
the adequacy of the presented model in this section. In Section 5, two examples are
provided to illustrate the idea of this article. Finally, Section 6 presents the conclusion.

2. Preliminaries

Here some required concepts of fuzzy sets theory are reviewed (Nguyen and Walker,
2005). Let be a universal set and . Any is a fuzzy
set on , the of is given by crisp set , for any .
𝑋 𝐹(𝑋) = �𝐴̃|𝐴̃:𝑋 → [0,1]� 𝐴̃ ∈ 𝐹(𝑋)

𝑋 𝛼-𝑐𝑢𝑡 𝐴̃ 𝐴̃[𝛼] = �𝑥 ∈ 𝑋|𝐴̃ ≥ 𝛼� 𝛼 ∈ [0,1]
19
Definition 2.1. is called a fuzzy number, if:
• there exi𝐴s̃ts∈ a𝐹 u(nℝiq)ue such that , and
• the of are closed and bounded sets on , for any ,
𝑥0 ∈ ℝ 𝐴̃(𝑥0) = 1
where is 𝛼th-e𝑐 𝑢s𝑡e𝑠t of a𝐴̃ll real numbers. ℝ 𝛼 ∈ (0,1]

Definitℝion 2.2. Some special cases of fuzzy numbers are defined as follows:
• is said to be a symmetric exponential fuzzy number (SEFN) and is
denoted by , if ; ,
• 𝐸� ∈ 𝐹(ℝ) is said to be a symmetric triangular fu2zzy number (STFN) and is
𝐸(𝑎,𝑠𝑎) 𝐸�(𝑥) = exp{−[(𝑥−𝑎)⁄𝑠𝑎] } 𝑥 ∈ ℝ
denoted by , if ; .
𝑇� ∈ 𝐹(ℝ)
where and 𝑇(𝑎,𝑠𝑎) 𝑇� a(𝑥re) t=he[ c1o−re (a|n𝑥d− th𝑎e| ⁄sp𝑠r𝑎e)a]d 𝐼 [v𝑎a−l𝑠u𝑎e,𝑎s+ o𝑠𝑎f] (th𝑥e)se𝑥 tw∈oℝ kinds of
fuzzy numbers, respectively. It is assumed that and are set to be as
𝑎 ∈ ℝ 𝑠𝑎 ∈ (0,+∞)
the indicator function of . For example, STFN and SEFN are
𝐸(𝑎,0) 𝑇(𝑎,0) 𝐼{𝑎}
shown in Figure 1.
𝑎 𝑇� = 𝑇(0,1) 𝐸� = 𝐸(0,2)

Figure 1. STFN and SEFN
𝑇� = 𝑇(0, 1) 𝐸� = 𝐸(0,2)

Using the extension principle presented by Zadeh (Dubois and Prade, 1980 and Nguyen
and Walker, 2005), the following interesting results for SEFNs and STFNs will be
obtained.

Theorem 2.1. Let be the set of all positive real numbers and . Suppose that
, + , and : +
ℝ 𝑘 ∈ ℝ
𝐴̃ =•𝐸 (T𝑎,h𝑠e𝑎 s)um𝐵� o=f 𝐸tw(o𝑏 ,S𝑠E𝑏F)N𝐶̃s = a𝑇n(d𝑐 ,𝑠𝑐 i)s 𝐷� = 𝑇(𝑑,𝑠𝑑) .
• The sum of two STFNs and is .
• The scalar multiplicatio𝐴ñ of SE𝐵�FN 𝐴̃⨁ b𝐵y� = is𝐸 (𝑎+𝑏,𝑠𝑎 +𝑠𝑏) .
• The scalar multiplicatio𝐷�n of ST𝐶̃FN 𝐷�⨁ by𝐶̃ = is𝑇 (𝑑+𝑐,𝑠𝑑 +𝑠𝑐) .
𝐴̃ 𝑘 𝑘⨂𝐴̃ = 𝐸(𝑘𝑎,𝑘𝑠𝑎)
𝐶̃ 𝑘 𝑘⨂𝐶̃ = 𝑇(𝑘𝑐,𝑘𝑠𝑐)
20
Definition 2.3. (Xu and Li, 2001) In order to measure the distance between two fuzzy
numbers and , the following distance is presented,
𝐴̃ ∈ 𝐹(ℝ) 𝐵� ∈ 𝐹(ℝ)
,
2 1 2
�in𝐴̃ w⊝hi𝐵c�h� = ∫0 𝑓(𝛼)�𝑑�𝐴̃[𝛼],𝐵�[𝛼]�� 𝑑𝛼
,
2 2 1⁄2
𝐿 𝐿 𝑈 𝑈
𝑑�𝐴̃[𝛼],𝐵�[𝛼]� = ��𝐴̃𝛼 −𝐵�𝛼 � +�𝐴̃𝛼 −𝐵�𝛼� �
is the distance between and , and is a real-value
and increasing function on w𝐿ith 𝑈 and 𝐿 𝑈 .
𝐴̃[𝛼] = �𝐴̃𝛼,𝐴̃𝛼� 𝐵�[𝛼] = [𝐵�𝛼,𝐵�𝛼] 𝑓(𝛼)
1
[0,1] 𝑓(0) = 0 ∫0 𝑓(𝛼)𝑑𝛼 = 1⁄2
Clearly, reflects the closeness and overlap degree between and
. Function can be considered as the weight of , and the property
𝑑�𝐴̃[𝛼],𝐵�[𝛼]� 𝐴̃[𝛼]
2
of monotone increase of means that the higher the membership of the , the more
𝐵�[𝛼] 𝑓 �𝑑�𝐴̃[𝛼],𝐵�[𝛼]��
important it is in determining the distance between and . The conditions
𝑓 𝛼-𝑐𝑢𝑡
and ensure that the distance defined here is the ordinary extension in
𝐴̃ 𝐵� 𝑓(0) = 0
def1ined by an absolute value. That is, this distance becomes an ordinary one in
∫0 𝑓(𝛼)𝑑𝛼 = 1⁄2
when the fuzzy number becomes decadent to crisp. In application, can be chosen
ℝaccording to the actual situation (Xu and Li, 2001). In the following discussion we sℝet
𝑓
, .
𝑚+1 𝑚
𝑓(𝛼) = 2 𝛼 𝑚 = 1,2,…
Theorem 2.2. Consider weighted function in Definition 2.3.
𝑚+1 𝑚
• The distance between two SEFNs 𝑓(𝛼) = 2 a𝛼nd is
𝐴̃ = 𝐸(𝑎,𝑠𝑎) 𝐵� = 𝐸(𝑏,𝑠𝑏)
.
2 2 1 2
• �T𝐴h̃e⊝ di𝐵s�t�anc=e b(e𝑎tw−e𝑏en) tw+o𝑚 S+T1F(N𝑠𝑎s −𝑠𝑏) and is
𝐶̃ = 𝑇(𝑐,𝑠𝑐) 𝐷� = 𝑇(𝑑,𝑠𝑑)
.
2 2 2 2
�𝐶̃ ⊝𝐷�� = (𝑐−𝑑) +(𝑚+2)(𝑚+3)(𝑠𝑐 −𝑠𝑑)
Proof. The proof is easy by considering Definition 2.3; see (Xu and Li, 2001). 

Definition 2.4. Let . Let be a conventional random variable and
, then +
𝑘,𝑑 ∈ {0}∪ℝ 𝑋
𝑆𝑋 =• 𝑘𝑋+𝑑 is a symmetric exponential fuzzy random variable (SEFRV) and,
• is a symmetric triangular fuzzy random variable (STFRV).
𝑋� = 𝐸(𝑋,𝑆𝑋)
𝑋� = 𝑇(𝑋,𝑆𝑋)

21
3. Classical one-way ANOVA

The classical ANOVA model is given in any textbook on the linear models (Cochran
and Cox, 1957 and Montgomery, 1991). In this section a brief review is presented for
classical one-way ANOVA. Let be the number of independent levels of the factor
under the study. Each level is denoted by index , . The number of cases
for th level is represented by 𝑎and the total number of the cases under the study is
. The th value of the response variab𝑖le 𝑖in= t1h, 2le,v…e,l 𝑎is represented by , so
𝑖 𝑛𝑖
𝑎 . For instance can be the tensional resistance of th thread piece that
𝑁 = ∑𝑖=1𝑛𝑖 𝑗 𝑖 𝑌𝑖𝑗
can be made from th quantity of cotton percent. Now one-way ANOVA model can be
𝑗 = 1,2,…,𝑛𝑖 𝑌𝑖𝑗 𝑗
stated as:
𝑖
; and ,
w𝑌𝑖𝑗he=re𝜇 𝑖 + is𝜖 𝑖t𝑗he𝑖 v=al1u,e2 ,o…f ,r𝑎espons𝑗e= va1r,i2a,b…le, i𝑛n𝑖 th experiment for the th level. ’s are
independent random variables having the normal distribution and is the
𝑌𝑖𝑗 𝑗 𝑖 𝜖𝑖𝑗
mean parameter of th factor level, such that is the to2tal mean. From
, it is derived that ’s are the indep1en𝑎dent rand𝑁o(m0 ,v𝜎ar)iables 𝜇w𝑖ith the
𝑖 𝜇 = 𝑁∑𝑖=1𝑛𝑖𝜇𝑖
distribution for any and .
𝑌𝑖𝑗 = 𝜇𝑖 +𝜖𝑖𝑗 𝑌𝑖𝑗
2
The total sum𝑁 (o𝜇f𝑖 ,s𝜎qu)ares ( 𝑖),= th1e, 2tr,e…at,m𝑎ent su𝑗m= o1f, 2sq,…ua,r𝑛es𝑖 ( ) and the error sum
of squares ( ) are defined as:
𝑆𝑆𝑇 𝑆𝑆𝑇𝑟
𝑆𝑆𝐸
,
𝑎 𝑛𝑖 2
𝑆𝑆𝑇𝑌 = ∑𝑖=1∑𝑗=1�𝑌𝑖𝑗 −𝑌�..�

𝑎 2
𝑆𝑆𝑇𝑟 𝑌 = ∑𝑖=1𝑛𝑖(𝑌�𝑖. −𝑌�..)
and
,
𝑎 𝑛𝑖 2
𝑆𝑆𝐸𝑌 = ∑𝑖=1∑𝑗=1�𝑌𝑖𝑗 −𝑌�𝑖.�
where and for .
1 𝑎 𝑛𝑖 1 𝑛𝑖
𝑌�.. = 𝑁∑𝑖=1∑𝑗=1𝑌𝑖𝑗 𝑌�𝑖. = 𝑛𝑖∑𝑗=1𝑌𝑖𝑗 𝑖 = 1,2,…,𝑎
Note that we have the relation . It can be shown that
has chi-square distribution with degrees of freedom. When all ’s become equal2,
and have ch𝑆i𝑆-s𝑇q𝑌ua=re𝑆 d𝑆i𝑇st𝑟r𝑌ib+ut𝑆io𝑆n𝐸s𝑌 with and d𝑆e𝑆gr𝐸e𝑌e⁄s 𝜎of
freedom2, respectively. 2Also the𝑁 s−ta𝑎tistic has 𝜇F𝑖isher distribution
𝑆w𝑆it𝑇h𝑌 ⁄𝜎 an𝑆d𝑆 𝑇𝑟𝑌⁄𝜎 degrees of freedom which is de𝑁no−ted1 by 𝑎 −1 , where
𝐹𝑌 = 𝑀𝑆𝑇𝑟𝑌⁄𝑀𝑆𝐸𝑌
and are “treatment mean square”
and “e𝑎rr−or 1mean sq𝑁ua−re𝑎”, respectively. 𝐹𝑎−1,𝑁−𝑎
N𝑀o𝑆w𝑇 𝑟𝑌ba=se𝑆d𝑆 𝑇o𝑟n𝑌 ⁄o(b𝑎se−rv1e)d rand𝑀om𝑆𝐸 𝑌sa=mp𝑆l𝑆e𝐸 𝑌w⁄e( 𝑁w−an𝑎t )to decide on testing the null
hypothesis , against the alternative hypothesis
is rejected if the observed value of is greater than
;𝐻 w0:e𝜇 1fa=il 𝜇to2 r=eje⋯ct = 𝜇,𝑎 w=he𝜇re is th quantile of Fi𝐻sh1er∶
𝑛𝑜𝑡 𝑎𝑙𝑙 𝜇𝑖’𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙. 𝐻0 𝐹𝑌
distribution with and degrees of freedom.
𝐹1−𝛼;𝑎−1,𝑁−𝑎 𝐻0 𝐹1−𝛼 ;𝑎−1,𝑁−𝑎 (1−𝛼)
𝑎−1 𝑁−𝑎
22
4. One-way ANOVA based on fuzzy random variables

With regard to the methodology of classical one-way ANOVA, the following model is
presented for the one-way ANOVA when the variables under the study are fuzzy
random variables:
, and ,
𝑌w�𝑖𝑗he=re𝜇 �𝑖⨁ i𝜖s 𝑖̃𝑗th𝑖e =nu1m,2b,e…r ,o𝑎f fact𝑗or= l1ev,2e,ls…, ,𝑛𝑖’s are fuzzy random variables, is the
number of cases and is the fuzzy mean of th factor level, . Note
𝑎 𝜖𝑖̃𝑗 𝑛𝑖
that is the total fuzzy mean. Now testing the following
hypotheses1 is of i𝑎nter𝜇e�s𝑖t∈ in𝐹 o(nℝe)-way ANOVA based o𝑖n fuzzy random 𝑖v=ari1a,b2l,e…s: ,𝑎
𝜇� = 𝑁⨂[⊕𝑖=1(𝑛𝑖⨂𝜇�𝑖)]
,

𝐻0:𝜇�1 = ⋯ = 𝜇�2 = 𝜇�𝑎 = 𝜇� �
�
Re𝐻g1a:r𝑛d𝑜in𝑡g 𝑎 t𝑙o𝑙 𝜇t�h𝑖’e𝑠 p𝑎r𝑟o𝑒p o𝑒s𝑞e𝑢d𝑎 i𝑙d.ea in Definition 2.3, the statistics , and
can be calculated through the following formulas:
𝑆𝑆𝑇𝑌� 𝑆𝑆𝑇𝑟𝑌� 𝑆𝑆𝐸𝑌�
, (1)
𝑎 𝑛𝑖 2
𝑆𝑆𝑇𝑌� = ∑𝑖=1∑𝑗=1�𝑌�𝑖𝑗 ⊝𝑌��..�
(2)
2
𝑎
𝑆𝑆𝑇𝑟 𝑌� = ∑𝑖=1𝑛𝑖�𝑌��𝑖. ⊝𝑌��..�
and
(3)
𝑎 𝑛𝑖 2
𝑆𝑆𝐸𝑌� = ∑𝑖=1∑𝑗=1�𝑌�𝑖𝑗 ⊝𝑌��𝑖.�
where and for .
Note that it 1can be 𝑎easily p𝑛𝑖roved that 1 𝑛𝑖 . The test statistic in one-
𝑌��.. = 𝑁⨂�⊕𝑖=1�⊕𝑗=1 𝑌�𝑖𝑗�� 𝑌��𝑖. = 𝑛𝑖⨂�⊕𝑗=1 𝑌�𝑖𝑗� 𝑖 = 1,2,…,𝑎
way ANOVA based on fuzzy random variables can be calculated as a real number by
, in which 𝑆𝑆𝑇𝑌� = 𝑆𝑆𝑇𝑟𝑌� +𝑆𝑆 a𝐸n𝑌�d
are the fuzzy square means of the treatments and errors, respectively.
𝐹𝑌� = 𝑀𝑆𝑇𝑟𝑌�⁄𝑀𝑆𝐸𝑌� 𝑀𝑆𝑇𝑟𝑌� = 𝑆𝑆𝑇𝑟 𝑌�⁄(𝑎−1) 𝑀𝑆𝐸𝑌 = 𝑆𝑆𝐸𝑌�⁄(𝑁−𝑎)

4.1. One-way ANOVA based on SEFRVs

In this subsection, we consider the following constraint for the fuzzy elements of the
model :
𝑌�𝑖𝑗 = 𝜇�𝑖⨁𝜖𝑖̃𝑗
• and for and . (4)
𝜖𝑖̃𝑗 = 𝐸�𝜖𝑖𝑗,𝑆𝜖𝑖𝑗� 𝜇�𝑖 = 𝐸�𝜇𝑖,𝑆𝜇𝑖� 𝑖 = 1,2,…,𝑎 𝑗 = 1,2,…,𝑛𝑖
In addition we consider the following assumptions on ’s:
• ’s are SERFVs such that . 𝜖 𝑖̃ 𝑗 (5)
𝜖𝑖̃𝑗 𝑆𝜖𝑖𝑗 = 𝑘𝜖𝑖𝑗 +𝑑
23
• Core values of ’s are independent random variables with normal distribution
, i.e. . (6)
𝜖𝑖̃𝑗
2 2
𝑁(0,𝜎 ) 𝜖𝑖𝑗~𝑁(0,𝜎 )
Note 4.1.1. In fact, constraint (4) and assumptions (5) and (6), have unified the concepts
of the randomness and fuzziness. Since probabilistic and possibilistic errors of the
model are modeled by assumption (6) and constraint (4), respectively.

𝑌�𝑖𝑗 = 𝜇�𝑖⨁𝜖𝑖̃𝑗
Theorem 4.1.1. Consider constraint (4) for the model . Then
𝑌�𝑖𝑗 = 𝜇�𝑖⨁𝜖𝑖̃𝑗
,
1
𝑆𝑆𝑇𝑌� = 𝑆𝑆𝑇𝑌 +𝑚+1𝑆𝑆𝑇𝑆𝑌
,
1
𝑆𝑆𝑇𝑟𝑌� = 𝑆𝑆𝑇𝑟𝑌 +𝑚+1𝑆𝑆𝑇𝑟𝑆𝑌

1
𝑆𝑆𝐸𝑌� = 𝑆𝑆𝐸𝑌 +𝑚+1𝑆𝑆𝐸𝑆𝑌
and
, (7)
𝑀𝑆𝑇𝑟𝑌� (𝑚+1)𝑀𝑆𝑇𝑟𝑌+𝑀𝑆𝑇𝑟𝑆𝑌
𝐹𝑌� = 𝑀𝑆𝐸𝑌� = (𝑚+1)𝑀𝑆𝐸𝑌+𝑀𝑆𝐸𝑆𝑌
in which
• , and
𝑎 𝑛𝑖 2 𝑎 2
𝑆𝑆𝑇𝑌 = ∑𝑖=1∑𝑗=1�𝑌𝑖 𝑗a−re 𝑌t�h..�e sum o𝑆f𝑆 𝑇sq𝑟u𝑌a=res∑ f𝑖o=r1 t𝑛h𝑖e( c𝑌�o𝑖.r−e v𝑌�a..l)ues of ’s, 𝑆𝑆𝐸𝑌 =
𝑎 𝑛𝑖 2
• ∑𝑖=1∑𝑗=1�𝑌𝑖𝑗 −𝑌�𝑖.� , 𝑌a�𝑖n𝑗d
𝑎 𝑛𝑖 2 𝑎 2
𝑆𝑆𝑇𝑆𝑌 = ∑𝑖=1∑𝑗=1�𝑆𝑌𝑖𝑗 a−re𝑆 t𝑌̅h..e� sum𝑆 𝑆o𝑇f 𝑟s𝑆q𝑌u=are∑s 𝑖o=f1 t𝑛h𝑖e� s𝑆p𝑌̅ r𝑖e. −ad𝑆 v𝑌̅a..�lues of 𝑆𝑆’s𝐸, 𝑆𝑌 =
2
• 𝑎 𝑛𝑖 and are the mean squares of
∑𝑖=1∑𝑗=1�𝑆𝑌𝑖𝑗 −𝑆𝑌̅ 𝑖.� 𝑌�𝑖𝑗
the core values of ’s,
𝑀𝑆𝑇𝑟𝑌 = 𝑆𝑆𝑇𝑟𝑌⁄(𝑎−1) 𝑀𝑆𝐸𝑌 = 𝑆𝑆𝐸𝑌⁄(𝑁−𝑎)
• and are the mean squares
𝑌�𝑖𝑗
of the spread values of ’s.
𝑀𝑆𝑇𝑟𝑆𝑌 = 𝑆𝑆𝑇𝑟𝑆𝑌⁄(𝑎−1) 𝑀𝑆𝐸𝑆𝑌 = 𝑆𝑆𝐸𝑆𝑌⁄(𝑁−𝑎)

𝑌�𝑖𝑗
Proof. With respect to and using the presented arithmetic operations in
Theorem 2.1, we have for
𝑌�𝑖𝑗 = 𝜇�𝑖⨁𝜖𝑖̃𝑗
and Therefore by (1-3) and using Theorem 2.2 the proof follows. 
𝑌�𝑖𝑗 = 𝐸�𝜇𝑖 +𝜖𝑖𝑗, 𝑆𝜇𝑖 +𝑆𝜖𝑖𝑗� = 𝐸�𝑌𝑖𝑗,𝑆𝑌𝑖𝑗� 𝑖 = 1,2,…,𝑎

Theo𝑗r=em1 ,24,.…1.2,𝑛. 𝑖.Under constraint (4) and assumptions (5) and (6) for the model
, the test statistic in (7) has Fisher distribution with and
degrees of freedom, under the null hypothesis .
𝑌�𝑖𝑗 = 𝜇�𝑖⨁𝜖𝑖̃𝑗 𝐹𝑌� 𝑎 −1 𝑁−𝑎

𝐻0:𝜇�1 = 𝜇�2 = ⋯ = 𝜇�𝑎 = 𝜇�
Proof. With respect to (4-6) and using Theorem 2.1, we have
, where and for
𝑌�𝑖𝑗 = 𝐸�𝑌𝑖𝑗,𝑆𝑌𝑖𝑗� =
2 2 2
𝐸�𝜇𝑖 +𝜖𝑖𝑗, 𝑆𝜇𝑖 +𝑘𝜖𝑖𝑗 +𝑑� 𝑌𝑖𝑗~𝑁(𝜇𝑖,𝜎 ) 𝑆𝑌𝑖𝑗~𝑁�𝑆𝜇𝑖 +𝑑,𝑘 𝜎 �
24
and . Therefore , since
. So, under we have
𝑖 = 1,2,…,𝑎 𝑗 = 1,2,…,𝑛𝑖 𝑆𝑌𝑖𝑗 = 𝑆𝜇𝑖 +𝑘𝑌𝑖𝑗 −𝑘𝜇𝑖 +𝑑 𝑌𝑖𝑗 =
𝜇𝑖 +𝜖𝑖𝑗 𝐻0:𝜇�1 = 𝜇� 2 = ⋯ = 𝜇� 𝑎 = 𝜇 � (8)
a𝑆n𝑌𝑖d𝑗 = 𝑆𝜇 +𝑘𝑌𝑖𝑗 −𝑘𝜇 +𝑑
. (9)
𝑌F𝑖r𝑗o=m 𝜇(8+) a𝜖n𝑖d𝑗 (9), it could be concluded that:

2
𝑎
𝑆𝑆𝑇𝑟𝑆𝑌 = ∑𝑖=1𝑛𝑖�𝑆𝑌̅ 𝑖. −𝑆𝑌̅ ..�

2
𝑎 1 𝑛𝑖 1 𝑎 𝑛𝑖
𝑆𝑆𝑇𝑟𝑆𝑌 = ∑𝑖=1𝑛𝑖�𝑛𝑖∑𝑗=1𝑆𝑌𝑖𝑗–𝑁∑𝑖=1∑𝑗=1𝑆𝑌𝑖𝑗�
+ + + +
2
𝑎 1 𝑛𝑖 1 𝑎 𝑛𝑖
𝑆𝑆𝑇𝑟𝑆𝑌 = ∑𝑖=1𝑛𝑖�𝑛𝑖∑𝑗=1�𝑆𝜇 𝑘𝑌𝑖𝑗 −𝑘𝜇 𝑑�–𝑁∑𝑖=1∑𝑗=1�𝑆𝜇 𝑘𝑌𝑖𝑗 −𝑘𝜇 𝑑��

2
2 𝑎 1 𝑛𝑖 1 𝑎 𝑛𝑖
𝑆𝑆𝑇𝑟𝑆𝑌 = 𝑘 ∑𝑖=1𝑛𝑖�𝑛𝑖∑𝑗=1𝑌𝑖𝑗 −𝑁∑𝑖=1∑𝑗=1𝑌𝑖𝑗�
. (10)
2
𝑆S𝑆im𝑇i𝑟l𝑆a𝑌rl=y, 𝑘it c𝑆a𝑆n𝑇 b𝑟e𝑌 shown that
. (11)
2
𝑆𝑆𝐸𝑆𝑌 = 𝑘 𝑆𝑆𝐸𝑌
Hence, by considering (7), (10) and (11) we can see that under :
𝐻0

𝑀𝑆𝑇𝑟𝑌�
𝐹𝑌� = 𝑀𝑆𝐸𝑌�

(𝑚+1)𝑀𝑆𝑇𝑟𝑌+𝑀𝑆𝑇𝑟𝑆𝑌
𝐹𝑌� = (𝑚+1)𝑀𝑆𝐸𝑌+𝑀𝑆𝐸𝑆𝑌

𝑆𝑆𝑇𝑟𝑌 𝑆𝑆𝑇𝑟𝑆𝑌
(𝑚+1) 𝑎−1 + 𝑎−1
𝐹𝑌� = 𝑆𝑆𝐸𝑌 𝑆𝑆𝐸𝑆𝑌
(𝑚+1)𝑁−𝑎+ 𝑁−𝑎

𝑀𝑆𝑇𝑟𝑌
𝐹𝑌� = 𝑀𝑆𝐸𝑌
.
T𝐹h𝑌�e=ref𝐹o𝑌re, similar to the test statistic of classical ANOVA, the test statistic has
Fisher distribution with and degrees of freedom. 
𝐹𝑌�
𝑎−1 𝑁−𝑎
25
Theorem 4.1.3. where stands for mathematical
expectation.
𝐸𝑥(𝑀𝑆𝑇𝑟𝑌�) ≥ 𝐸𝑥(𝑀𝑆𝐸𝑌�) 𝐸𝑥(.)
Proof. By (7), we have
(12)
a𝐸n𝑥d( 𝑀𝑆𝑇𝑟𝑌�) = (𝑚+1)𝐸𝑥(𝑀𝑆𝑇𝑟𝑌)+𝐸𝑥�𝑀𝑆𝑇𝑟𝑆𝑌�
. (13)
𝐸𝑥(𝑀𝑆𝐸𝑌�) = (𝑚+1)𝐸𝑥(𝑀𝑆𝐸𝑌)+𝐸𝑥�𝑀𝑆𝐸𝑆𝑌�
Since and we have:
2
𝑌𝑖𝑗 = 𝜇𝑖 +𝜖𝑖𝑗 𝜖𝑖𝑗~𝑁(0,𝜎 )

1 𝑎 1 2 1 2
𝐸𝑥(𝑀𝑆𝑇𝑟𝑌) = 𝐸𝑥�𝑎−1�∑𝑖=1𝑛𝑖𝑌𝑖. −𝑁𝑌.. ��

2 2
1 𝑎 1 𝑛𝑖 1 𝑎 𝑛𝑖
𝐸𝑥(𝑀𝑆𝑇𝑟𝑌) = 𝑎−1�𝐸𝑥�∑𝑖=1𝑛𝑖�∑𝑗=1𝜇𝑖 +𝜖𝑖𝑗� �−𝑁𝐸𝑥��∑𝑖=1∑𝑗=1𝜇𝑖 +𝜖𝑖𝑗� ��
(14)
2 1 𝑎 2
𝐸𝑥(𝑀𝑆𝑇𝑟𝑌) = 𝜎 +𝑎−1∑𝑖=1𝑛𝑖(𝜇𝑖 −𝜇)
Where , and is the core value of . Similarly,
𝑛𝑖 𝑎 𝑛𝑖
𝑌𝑖. = ∑𝑗=1. 𝑌 𝑖 𝑗 𝑌 .. = ∑ 𝑖 = 1 ∑ 𝑗 = 1 𝑌 𝑖 𝑗 𝜇 𝜇 � (15)
2
𝐸𝑥(𝑀𝑆𝐸𝑌) = 𝜎
Therefore by (4-6), it is obtained that , and similar to above one
can conclude that
𝑆𝑌𝑖𝑗 = 𝑆𝜇𝑖 +𝑘𝜖𝑖𝑗 +𝑑
(16)
2 2 1 𝑎 2
𝐸𝑥�𝑀𝑆𝑇𝑟𝑆𝑌� = 𝑘 𝜎 +𝑎−1∑𝑖=1𝑛𝑖�𝑆𝜇𝑖 −𝑆𝜇�
and
, (17)
2 2
𝐸𝑥�𝑀𝑆𝐸𝑆𝑌� = 𝑘 𝜎
where is the spread value of . Considering (12), (14) and (16)
𝑆𝜇 𝜇�
2 2
𝐸𝑥(𝑀𝑆𝑇𝑟𝑌�) = 𝜎 [(𝑚+1)+𝑘 ]+
(18)
1 𝑎 2 𝑎 2
𝑎−1�(𝑚+1)∑𝑖=1𝑛𝑖(𝜇𝑖 −𝜇) +∑𝑖=1𝑛𝑖�𝑆𝜇𝑖 −𝑆𝜇� �
So, by (13), (15) and (17)
(19)
2 2
𝐸𝑥(𝑀𝑆𝐸𝑌�) = 𝜎 [(𝑚+1)+𝑘 ]
Finally, the proof is obvious by comparing (18) and (19). 
26
Decision rule 4.1.1. At the given significance level , the null hypothesis
is rejected if observed value of the test statistic in (7) becomes
greater than , the quantile of th𝛼e Fisher distribution wi𝐻th0 :𝜇�1 =
𝜇�2 = ⋯ = 𝜇�𝑎 = 𝜇� 𝐹𝑌�
and degrees of freedom; otherwise we accept .
𝐹1−𝛼;𝑎−1,𝑁−𝑎 (1−𝛼)th 𝑎 −1
Note𝑁 4−.1.𝑎1. As a result of Theorem 4.1.3, it is very i𝐻m0portant to note that the presented
decision rule is reasonable since we prove that one can expect that, if the observed value
of the test statistic is high, then it indicates not all ’s are equal. Also in Theorem
4.1.2, it is shown that under the null hypothesis the test
statistic has Fish𝐹e𝑌r� distribution similar to under th𝜇�e𝑖 null hypothesis
, therefore the critical value 𝐻0:𝜇�1 = is𝜇� 2c=alc⋯ula=te𝜇d�𝑎 f=ro𝜇m� Fisher
distribut𝐹io𝑌�n. 𝐹𝑌 𝐻0:𝜇1 = 𝜇2 =
⋯ = 𝜇𝑎 = 𝜇 𝐹 1−𝛼;𝑎−1,𝑁−𝑎

4.2. One-way ANOVA based on STFRVs

Similar to Subsection 4.1, here we consider the following constraint for the fuzzy
elements of the model :
𝑌�𝑖𝑗 = 𝜇�𝑖⨁𝜖𝑖̃𝑗
• and for and . (20)
𝜖𝑖̃𝑗 = 𝑇�𝜖𝑖𝑗,𝑆𝜖𝑖𝑗� 𝜇�𝑖 = 𝑇�𝜇𝑖,𝑆𝜇𝑖� 𝑖 = 1,2,…,𝑎 𝑗 = 1,2,…,𝑛𝑖
In addition we consider the following assumptions on ’s:
• ’s are STRFVs such that . 𝜖 𝑖̃ 𝑗 (21)
• Core values of ’s are independent random variables with normal distribution
𝜖𝑖̃𝑗 𝑆𝜖𝑖𝑗 = 𝑘𝜖𝑖𝑗 +𝑑
i.e. . (22)
𝜖𝑖̃𝑗
2 2
𝑁(0,𝜎 ) 𝜖𝑖𝑗~𝑁(0,𝜎 )
Theorem 4.2.1. Consider constraint (20) for the model . Then
𝑌�𝑖𝑗 = 𝜇�𝑖⨁𝜖𝑖̃𝑗
,
2
𝑆𝑆𝑇𝑌� = 𝑆𝑆𝑇𝑌 +(𝑚+2)(𝑚+3)𝑆𝑆𝑇𝑆𝑌
,
2
𝑆𝑆𝑇𝑟𝑌� = 𝑆𝑆𝑇𝑟𝑌 +(𝑚+2)(𝑚+3)𝑆𝑆𝑇𝑟𝑆𝑌
,
2
𝑆𝑆𝐸𝑌� = 𝑆𝑆𝐸𝑌 +(𝑚+2)(𝑚+3)𝑆𝑆𝐸𝑆𝑌
and
. (23)
𝑀𝑆𝑇𝑟𝑌� (𝑚+2)(𝑚+3)𝑀𝑆𝑇𝑟𝑌+2𝑀𝑆𝑇𝑟𝑆𝑌
𝐹𝑌� = 𝑀𝑆𝐸𝑌� = (𝑚+2)(𝑚+3)𝑀𝑆𝐸𝑌+2𝑀𝑆𝐸𝑆𝑌
Proof. With respect to and using the presented arithmetic operations in
Theorem 2.1, we have for
𝑌�𝑖𝑗 = 𝜇�𝑖⨁𝜖𝑖̃𝑗
and . Therefore by (1-3) and using Theorem 2.2 the proof follows. 
𝑌�𝑖𝑗 = 𝑇�𝜇𝑖 +𝜖𝑖𝑗, 𝑆𝜇𝑖 +𝑆𝜖𝑖𝑗� = 𝑇�𝑌𝑖𝑗,𝑆𝑌𝑖𝑗� 𝑖 = 1,2,…,𝑎
𝑗 = 1,2,…,𝑛𝑖
27