Op imal Selec ion o Mic oa ay Analysis Me hods
Using a Concep ual Clus e ing Algo i hm
C. Rubio-Escude o1, R. Rome o-Záliz1, O. Co dón1, O. Ha a i1,
C. del Val1, and I. Zwi 1,2
1 Depa men o Compu e Science and A i icial In elligence,
C/Daniel Saucedo A anda s/n, G anada 18071, Spain
{c ubio, ocio, oco don, oha a i, del al, zwi }@decsai.ug .es
2 Howa d Hughes Medical Ins i u e, Washing on Uni e si y School o Medicine,
S . Louis, MO
Abs ac . The apid de elopmen o me hods ha selec o e /unde exp essed
genes om mic oa ay expe imen s ha e no ye ma ched he need o ools ha
iden i y in o ma ional p o iles ha di e en ia e be ween expe imen al condi-
ions such as ime, ea men and pheno ype. Unce ain y a ises when me hods
de o ed o iden i y signi ican ly exp essed genes a e e alua ed: do all mic oa -
ay analysis me hods yield simila esul s om he same inpu da ase ? do di -
e en mic oa ay da ase s equi e dis inc analysis me hods?. We pe o med a
de ailed e alua ion o se e al mic oa ay analysis me hods, inding ha none o
hese me hods alone iden i ies all obse able di e en ial p o iles, no subsumes
he esul s ob ained by he o he me hods. Consequen ly, we p opose a p oce-
du e ha , gi en ce ain use -de ined p e e ences, gene a es an op imal sui e o
s a is ical me hods. These solu ions a e op imal in he sense ha hey cons i u e
pa ial o de ed subse s o all possible me hod-associa ions bounded by bo h, he
mos speci ic and he mos sensi i e a ailable solu ion.
1 In oduc ion
Ad ances in molecula biology and compu a ional echniques pe mi he sys ema ical
s udy o molecula p ocesses ha unde lie biological sys ems [1]. Pa icula ly, mi-
c oa ay echnology has e olu ionized mode n biomedical esea ch by i s capaci y o
moni o changes in RNA abundance o housands o genes simul aneously [2].
To add ess he s a is ical challenge o analyzing hese la ge da a se s, new me hods
ha e eme ged ([3], [4], [5], [6], [7] and many o he s). Howe e , he e is a dea h o
compu a ional me hods o acili a e unde s anding o di e en ial gene exp ession p o-
iles (e.g., p o iles ha change o e ime and/o o e ea men s and/o o e pa ien )
and o decide which is he mos eliable me hod o iden i y di e ences ac oss p o iles.
We in es iga ed he pe o mance o se e al commonly used s a is ical me hods, in-
cluding T-Tes s [4], Pe mu a ion Tes s [5], Analysis o Va iance [6] and Repea ed
Measu es ANOVA [7], in iden i ying di e en ial exp ession p o iles ha change o e
ime, ea men s and pheno ype. We ound ha hese me hods do no iden i y all
ob-
se able dis inc p o iles. Mo eo e , none o hem subsumes he esul s ob ained by
he o he me hods.
In iew o hese esul s, we p opose a concep ual clus e ing me hod [8], [9], [10],
de o ed o disco e op imal associa ions o mic oa ay analysis me hods in an e o
o iden i y di e en ial gene exp ession p o iles.
2 Me hods
We p opose a concep ual clus e ing app oach [8], [9], [10] de o ed o iden i y op imal
associa ions among mic oa ay analysis me hods in an e o o iden i y di e en ial
exp ession p o iles (Fig. 1). This app oach consis s o six phases: (1) p ep ocessing o
he da ase ; (2) iden i ica ion o di e en ially exp essed genes by applica ion o se -
e al s a is ical me hods; (3) a angemen o a la ice s uc u e con aining all possible
associa ions o he s a is ical me hods applied; (4) associa ion o di e en ially ex-
p essed genes in o di e en ial p o iles by clus e ing genes ha change hei exp es-
sion o e ime, pa ien and/o ea men ; (5) e alua ion o he pe o mance o he
me hod-associa ions based on hei speci ici y and sensi i i y in he iden i ica ion o
p e iously de ec ed di e en ial p o iles, using mul iobjec i e op imiza ion echniques
[11], [12]. We c ea e a se o me hod associa ion ules based on he lea ned mappings
o di e en ial p o iles in o me hod-associa ions, [13]; (6) inally, we a e able o p e-
dic op imal me hod-associa ions o iden i y di e en ial p o iles in new mic oa ay
da ase s by use o he me hod associa ion ules.
2.1 Iden i ica ion o Di e en ially Exp essed Genes
We pe o m he e ie al o di e en ially exp essed genes om one expe imen al
condi ion o he o he /s by applica ion o se e al s a is ical echniques [3], [14], ha -
bo ing S uden ’s T-Tes p oposed in [4], including some o he a ian s he me hod
poses o dis inguish changes in he abundance o RNA occu ing o e bo h ea men
and ime; Pe mu a ion Tes desc ibed in [5], also including a ime app oach; Analysis
o Va iance desc ibed in [6]; and Longi udinal Da a app oach by using Repea ed
Measu es Analysis o Va iance desc ibed in [7].
DIFFERENTIALLY
EXPRESSED
GENES
MICROARRAY
RAW DATA
MICROARRAY
PREPROCESSED
DATA
PREPROCESSING
(1)
(4)
IDENTIFICATION
OF DIFFERENTIAL
PROFILES
DIFFERENTIAL
EXPRESSION
PROFILES
METHOD
EVALUATION
SCALING
(2)
IDENTIFICATION
OF DIFF.
EXPRESSED
GENES
(6)
PREDICTION
QUERY PROFILES
+
MICROARRAY
DATA
(OPTIONAL)
METHOD
SELECTION
(5)
CREATION OF
METHOD ASS.
RULES
METHOD
ASSOCIATION
RULES
(3)
ASSOCIATION
OF STATISTICAL
METHODS
LATTICE
OF METHOD
ASSOCIATIONS
STATISTICAL
METHODS
CONCEPTUAL
CLUSTERING
PROFILE
IDENTIF.
PROFILE
PRUNING
NORMALIZING
Fig. 1. G aphical ep esen a ion o he me hodology. The squa ed boxes ep esen he phases o
he me hodology, he ound co ne ed boxes co espond o he inpu /ou pu da a a each s ep,
and he ellipses he ope a ions pe o med a each phase.
2.2 De ec ion o Me hod-Associa ions
We a ange a la ice con aining all po en ial associa ions o he s a is ical me hods
used o e ie e di e en ially exp essed genes (Fig. 2). The me hods a e associa ed as:
}...,...,,,,,{ 21312121 nn MMMMMMMMMMM ⊕⊕⊕⊕⊕= , (1)
whe e ⊕ is a classical se ope a o (e.g., he union ( U) o he in e sec ion (I)) ap-
plied o he se s o genes e ie ed by each me hod, and 1
M
co esponds o T-Tes ,
2
M
o T-Tes conside ing ime, 3
M
Pe mu a ion Tes , 4
M
Pe mu a ion Tes con
side ing ime, 5
M ANOVA o e ea men , 6
M ANOVA o e ime, 7
M ANOVA
o e ea men and ime, 8
M RMANOVA o e ea men , 9
M RMANOVA o e
ime and 10
M RMANOVA o e ea men and ime.
The la ice con aining all po en ial me hod-associa ions, M, is s uc u ed om op
(i.e., in e sec ion o all me hods) o bo om (i.e., union o all me hods) [15]. Each
node in he la ice ( MM i∈) is applied o he mic oa ay da ase (D) e ie ing he
se o di e en ially exp essed genes ha a e ecognized by he me hod o me hod-
associa ions in such node D))Mi((.
M
3
M
1

M
2

M
3


M
n
M
1
M
2
M
3
M
2
M
3
M
4
M
n-2
M
n-1
M
n
M
1
M
2
M
3
…M
k
M
n-2
M
n-1
M
n
M
2
M
3
M
4
M
1
M
2
M
3
M
n-1
M
n
M
2
M
3
M
1
M
3
M
1
M
2
M
n
M
1
M
2
M
1
M
2
M
1
M
3
M
2
M
3
M
n-1
M
n
M
3
……
……
…
……
……
Fig. 2. La ice s uc u e con aining all s a is ical me hods po en ial associa ions
2.3 Iden i ica ion o Di e en ial P o iles
The se o genes p e iously iden i ied in Sec ion 2.2 se es as a means o c ea e di -
e en ial exp ession p o iles (i.e., se s o genes wi h coo dina e changes in RNA
abundance) be ween ea men T
P, con ol C
P and subjec . The applied ep esen a ion
(Fig. 3) allows us o iden i y di e en pa e n beha io among pa ien s inside he
same expe imen al g oup, since his in o ma ion may be missed i pa ien s in he same
expe imen al g oup we e no plo ed indi idually.
We clus e ed sepa a ely genes in ea men and con ol g oups. The e o e, genes
belonging o a clus e in ea men , T
P, can i in mo e han one clus e in con ol, C
P,
and ice e sa. We apply he K-means clus e ing algo i hm [16] and iden i y di e en-
ial p o iles deno ed as )( CT PP , which a e pai wise ela ionships be ween p o iles, T
P
T ea men - 6
Con ol - 3
TREATMENT CONTROL
HOUR 0 2 4 6 9 24 0 2 4 6 9 24 0 2 4 6 9 24 0 2 4 6 9 24
PATIENT 1 2 3 4
HOUR 0 2 4 6 9 24 0 2 4 6 9 24 0 2 4 6 9 24 0 2 4 6 9 24
PATIENT 5 6 7 8
Fig. 3. The exp ession p o iles ha e been ep esen ed sepa a ely o each expe imen al g oup
and pa ien s a anged indi idually
and C
P, om ea men and con ol expe imen s, espec i ely. This ela ionship is de-
ined as he signi ican in e sec ion o genes be ween T
P and C
P, which is cons ained
by a h eshold based on he ypical s a is ical powe o 80%.
2.4 C ea ion o Me hod Associa ion Rule
We c ea e a se o me hod associa ion ules ha , gi en a se o di e en ial
p o iles que ied by he use , sugges s he mos app op ia e me hod-associa ions
capable o e ie e hem. The me hod associa ion ules a e c ea ed based on he
la ice s uc u e om Sec ion 2.2, con aining all po en ial me hod-associa ions, and
he se o all possible di e en ial p o iles P om Sec ion 2.4 de ined as
})(,....,){( 1lCTCT PPPPP =whe e PPP jCT ∈)( ep esen s each o he di e en ial p o-
iles p esen in P.
2.4.1 Me hod-Associa ion Pe o mance E alua ion
We e alua e he pe o mance o he me hod-associa ions MM i∈ o he que y p o-
iles ),,..,(1s
SxxX = o e wo objec i es: speci ici y and sensi i i y
)/( FNTPTNySpeci ici += )/( FNTPTPySensi i i += , (2)
whe e TP s ands o T ue Posi i es (i.e., genes exhibi ing p o ile S
uXx ∈, which
ha e been success ully e ie ed by he applied me hod-associa ion i
M), TN s ands
o T ue Nega i es (i.e., genes exhibi ing p o ile S
uXx ∉ and no e ie ed by i
M),
FP s ands o False Posi i es (i.e., genes exhibi ing p o ile S
uXx ∉and e ie ed by
i
M) and FN s ands o False Nega i es (i.e., genes exhibi ing p o ile S
uXx ∈and
no e ie ed by i
M). These ou ac o s a e calcula ed as:
u
iu
TP
ϕ
ηϕ
I
=)(D
)η(D)(D
TN u
iu
ϕ
ϕ
−
−−
=I
u
iu
u
iu )η(D
FN
)(D
η)(D
FP
ϕ
ϕ
ϕ
ϕ
−
=
−
−
=II , (3)
whe e u
ϕ
ep esen s he genes in he mic oa ay se D ha exhibi he que ied p o ile
S
uXx ∈, and (D)Mii =
η
, he genes om D e ie ed by he me hod-associa ion i
M.
2.4.2 Me hod-Associa ion Selec ion
We e alua e he me hod-associa ions in M based on hei speci ici y and sensi i i y.
These wo objec i es a e always con lic ing, so we use a mul iobjec i e op imiza ion
echnique o maximize hem, allowing us o de ec all op imal me hods-associa ions in
M o he que y p o iles S
X [11], [12]. We de ine objec i es )( 2,1 OO co esponding
o speci ici y and sensi i i y espec i ely.
2.4.3 C ea ion o a Se o Me hod Associa ion Rules
We use he non-domina ed me hod-associa ions desc ibed in Sec ion 2.4.2 o c ea e
he me hod associa ion ules R},...,{ 1k
RR= whe e
R
R
∈is de ined as:
R
: IF 1
xIS
CT PP 1
)( AND , . . . , AND s
x IS
sCT PP )( THEN
z IS i
MWITH
C, (4)
whe e )xx s
,...,( 1 a e he p o iles S
X que ied by he use ;
CT PP 1
)(,…, PPP
sCT ∈)( ;
Mz ∈ is he app op ia e me hod-associa ion o e ie e S
X acco ding o ule
R
;
and
C deno es a measu e o he speci ici y/sensi i i y le els o
z, de ined as:
21
2211 ))(*())(*(
ww
MOwMOw
C
ii
+
+
=,
(5)
whe e 1
w and 2
w a e he weigh s associa ed o ),( 21 OO espec i ely. These alues
a e p o ided by he use based on he ele ance o each o hese objec i es o he
pa icula s udy. I no alues a e gi en, he s anda d (0.5, 0.5) a e used.
2.5 P edic ion Using Me hod Associa ion Rules
The p edic ion phase wo ks a wo le els depending on he gi en inpu . I he inpu is
a mic oa ay da a se D’, ou me hodology will p o ide he di e en ial exp ession
p o iles P’ in he da a se along wi h he op imal me hod-associa ions o e ie e such
p o iles. I migh be he case ha some o he di e en ial p o iles P’ unco e ed om
D’ we e no included in he se o di e en ial p o iles P al eady lea ned by he me h-
odology. Consequen ly, he in o ma ion p o ided as inpu will be used o upda e P
and R. I he inpu is a se o que y p o iles S
X, he ou pu will consis o he op imal
me hod-associa ion h
M o S
X a a ce ain
C alue. To ob ain hese ou pu s, we
apply ma ching and in e ence ope a ions o he me hod associa ion ule se [17].
Gi en an associa ion ule se },...,{ 1k
RRR =, o he di e en ial p o iles p o-
ided as he que y se ),...,( 1s
SxxX =, we de ine he ma ching deg eeQo
S
uXx ∈wi h he i -pa o he associa ion ule
R
as:
uCTu
uCTu PPxPPxQ )(1))(( ,−−= , (6)
wi h being he Euclidean dis ance, and )( CT PP he cen oids o he p o iles.
The e o e, gi en a se o que y p o iles S
X, we de ine he s eng h o ac i a ion o
he i -pa o he ule
R
as:
)))(,(,...,))(,(min()( 1
1
sCTs
CT
S PPxQPPxQXR =. (7)
Le )),(( S CXRh deno e he deg ee o associa ion o he que y p o iles S
X
wi h he me hod-associa ion i
Macco ding o ule
R
and he speci ici y/sensi i i y

le el
C. This deg ee is ob ained by applying a p oduc ope a o be ween )(S XR
and
C. The op imal me hod-associa ion o he que ied p o iles S
X is de ined as:
i
M/(
i
hk
iSi CXR ∈
=max)),((
h)),( S CXR . (8)
3 Resul s
We apply ou p ocedu e o a da a se de i ed om longi udinal blood exp ession p o-
iles o human olun ee s ea ed wi h in a enous endo oxin compa ed o placebo.
We expec o iden i y molecula pa hways ha p o ide insigh in o he hos esponse
o e ime o sys emic in lamma o y insul s, as pa o a La ge-scale Collabo a i e Re-
sea ch P ojec sponso ed by he Na ional Ins i u e o Gene al Medical Sciences
(www.glueg an .o g) [18].
The da a we e acqui ed om blood samples collec ed om eigh no mal human
olun ee s, ou ea ed wi h in a enous endo oxin (i.e., pa ien s 1 o 4) and ou wi h
placebo (i.e., pa ien s 5 o 8) [18]. Complemen a y RNA was gene a ed om ci cula -
ing leukocy es a 0, 2, 4, 6, 9 and 24 hou s a e he i. . in usion and hyb idized wi h
GeneChips® HG-U133A 2.0 om A yme yx Inc., con aining a se o 22283 genes.
3.1 Iden i ica ion o Di e en ially Exp essed Genes
The s a is ical me hods ha bo ed ha e been applied using he s anda d p- alue
05.0
=. The numbe o di e en ially exp essed genes e ie ed by each o he
me hods om he o iginal se o genes is 1
M-10942 genes, 2
M-7841, 3
M-3904,
4
M-8023, 5
M-13151, 6
M-4588, 7
M-6070, 8
M-8557, 9
M-3995, 10
M-3367.
These alues show he numbe o signi ican genes e ie ed by each o he s a is ical
me hods anges in a wide ank. Mo eo e , he conco dance a es also a y widely, in-
Table 1. Coincidence be ween me hods in he e ie al o genes. The numbe in each cell
ep esen s a a io o coincidence be ween genes e ie ed by he s a is ical me hod in ha col-
umn and he genes e ie ed by he s a is ical me hod in ha ow ela i e o he o al numbe o
genes e ie ed by he me hod in he ow ( RowColumnRow /)( I).
% 1
M2
M3
M 4
M 5
M6
M7
M 8
M9
M10
M
1
M -- 92.20 52.29 75.05 96.48 69.23 85.55 70.06 61.33 50.52
2
M56.06 -- 34.07 57.84 85.27 59.54 71.11 62.64 50.57 42.98
3
M 82.19 88.07 -- 96.24 94.77 57.35 78.75 72.87 56.86 46.73
4
M67.22 85.19 54.84 -- 95.16 55.49 73.65 70.20 51.49 42.83
5
M55.20 77.80 33.45 58.94 -- 50.28 66.72 66.38 46.42 38.93
6
M59.04 83.51 31.11 52.84 77.30 -- 89.63 56.56 60.64 49.38
7
M58.36 79.79 34.18 56.10 82.05 71.70 -- 62.34 57.23 49.07
8
M 57.36 84.34 37.96 64.17 95.96 54.30 74.80 -- 49.62 40.51
9
M62.10 84.21 36.63 58.21 84.74 72.00 84.95 61.36 -- 72.31
10
M 59.56 83.34 35.05 56.37 82.72 68.26 84.80 58.34 84.19 --
σ
dica ing ha none o he me hods subsumes he o he s (Table 1)(e.g., om he genes
e ie ed by 3
M, only 31.11% a e also e ie ed by 5
M, and 52.29% by 1
M).
3.2 Associa ion o S a is ical Me hods
The la ice a anged in his pa icula wo k con ains all po en ial combina ions o un-
ion and in e sec ion o he en s a is ical me hods applied. Thus, M’ is de ined as
}...,...,,...,
,...,,,.,,..,{ 10932132132
31211021
MMMMMMMMMM
MMMMMMMM ⊕⊕⊕⊕⊕⊕⊕⊕ ⊕⊕=
We ound ha he e is a ela ionship be ween he s a is ical me hods and he di e -
en ial p o iles hey a e able o iden i y (see Sec ion 2.2), ha ing di e en ial p o iles
iden i ied by some me hods and no by o he s. Fo example, he di e en ial p o ile in
(Fig. 4(a)) ha bo s 29 genes in ou da ase D and is only e ie ed by hose s a is ical
me hods ha ake in o accoun he ime ac o (e.g., 2
M, which e ie es mo e han
90% o hese genes). This happens because he s a is ical me hods ha conside he
ea men s. con ol ac o make an a e age o he exp ession alues om pa ien s 1
and 2 wi h hose o pa ien s 3 and 4 by conside ing hem as eplicas. Consequen ly,
he di e en ial beha io be ween hem is los .
T ea men - 8
T ea men - 21
Con ol - 9
Con o l - 7
TREATMENT CONTROL
a)
b
)
Fig. 4. Examples o di e en ial p o iles only iden i ied by some o he s a is ical me hods
3.3 Iden i ica ion o Di e en ial P o iles
The exp ession p o iles ha e been ep esen ed sepa a ely o each expe imen al g oup
(Sec ion 2.3), and pa ien s a anged indi idually. In ou cu en p oblem, wi h eigh
pa ien s, ou ea ed wi h in a enous endo oxin (i.e., pa ien s 1 o 4) and ou wi h
placebo (i.e., pa ien s 5 o 8), and da a e ie ed o e ime a hou s 0, 2, 4, 6, 9 and 24,
each p o ile is ep esen ed by 24 consecu i e ime poin s (see Fig. 5).
The di e en ial p o iles ex ac ed om he ea men g oup show di e en le els o
exp ession change. Fo example, he e a e se s o genes sha ing e y high a ia ions
in he le els o exp ession (e.g., p o iles 15, 19, 21, and 22 in Fig. 5). In addi ion,
some o he p o iles show di e en ial cha ac e is ics o he pa ien s (e.g., p o iles 8
and 16 in Fig. 5). In he con ol g oup, he p o iles a e mo e homogeneous han in he
ea men g oup.
Typically, es ing he coincidence among di e en da a sou ces and clus e ing
me hods se es as a ool o in es iga e he alidi y o he iden i ied g oupings [19].
We ollow his guideline o inc ease he con idence in he ob ained di e en ial p o-
iles. The e o e, we calcula e he coincidence be ween ou e ie ed di e en ial p o i-
T ea men - 1
T ea men - 4
T ea men - 7
T e a me n - 10
T ea men - 2
T ea men - 5
T ea men - 8
T ea men - 11
T e a me n - 3
T e a me n - 6
T e a me n - 9
T ea men - 12
0
5000
10000
15000
20000
25000
0
5000
10000
15000
20000
25000
0
5000
10000
15000
20000
25000
0
5000
10000
15000
20000
25000
T e a men - 13
T e a men - 16
T e a men - 19
T e a men - 22
T e a men - 14
T e a men - 17
T e a men - 20
T e a men - 23
T e a men - 15
T e a men - 18
T e a men - 21
T e a men - 24
Co n ol - 1
Co n ol - 4
Co n ol - 7
Con ol - 10
Co n ol - 2
Co n ol - 5
Co n ol - 8
Con ol - 11
Con ol - 3
Con ol - 6
Con ol - 9
Con ol - 12
0
5000
10000
15000
20000
25000
0
5000
10000
15000
20000
25000
0
5000
10000
15000
20000
25000
0
5000
10000
15000
20000
25000
T ea men - 1
TREATMENT CONTROL
Fig. 5. Rep esen a ion o he di e en ial p o iles ob ained sepa a ely o he ea men and con-
ol g oups using he s a is ical me hods applied in he cu en wo k
les and ex e nal in o ma ion p o ided by he Gene On ology da abase [20]. To ad-
d ess his p oblem we de eloped an e olu iona y mul iobjec i e concep ual clus e ing
me hodology (R.R.Z., C.R.E., O.C., J.P.C., and I.Z., manusc ip in p epa a ion) ha
ex ac s clus e s composed o ea u es such as biological p ocesses, molecula unc-
ions and cellula componen s de ined a di e en speci ici y le els, and compa e
hese clus e s wi h ou di e en ial p o iles by using a coincidence index es based on
he hype geome ic dis ibu ion [9], [10], [19].
3.4 C ea ion o Me hod Associa ion Rules
We ha e a bi a ily selec ed six p o iles (i.e., 1
)( CT PP ,…, 6
)( CT PP ) iden i ying a o al
o 1395 genes in ou da ase D and plo ed as ea men clus e s 2, 3, 4, 5, 10 and 12 in
Fig. 5. These p o iles ep esen genes exhibi ing non-uni o m beha io o dis inc pa-
ien s in he ea men g oup, and genes wi h changes in a le el o exp ession smalle
han 5000. We applied ou me hodology o ind he op imal me hod-associa ions i
M
o e ie e hem.
3.4.1 Me hod Associa ion Pe o mance E alua ion
The esul s o he e alua ion o he me hod-associa ions con ained in he la ice M’ o
he di e en ial p o iles a e shown in Table 2, whe e he in o ma ion ela i e o he
sensi i i y and speci ici y le els o he applica ion o he mos ep esen a i e me hod-
associa ions o e D is also speci ied. On he one hand, we obse e ha he union se
o he genes ob ained by se en o he s a is ical me hods e alua ed (i.e., me hods
10876532 ,,,,,, MMMMMMM ) con ains he 1395 genes desi ed (i.e., sensi i i y
alue o 1) bu wi h a low le el o speci ici y (i.e., alue o 0.369). On he o he hand,
he in e sec ion se o genes ob ained by he same se en s a is ical me hods has a e y
low le el o sensi i i y (i.e., only 95 ou o he 1395 genes we e e ie ed), whe eas
he alue o speci ici y is e y high. In be ween hese wo ex emes we see some
o he me hod-associa ions which e alua ion e eal ade-o solu ions be ween he
speci ici y and sensi i i y objec i es (Table 2).
3.4.2 Me hod Associa ion Selec ion
Once he me hod-associa ions M ha e been e alua ed, we sea ch o he non-
dominance ela ions in hei applica ions o he mic oa ay da ase D. The decision is
based on he le els o speci ici y and sensi i i y in Table 2. The Pa e o op imal on
con o med by his se o non-domina ed me hod-associa ions is ep esen ed in Fig. 6.
Table 2. Speci ici y and sensi i i y alues o he me hod-associa ions. The non-domina ed so-
lu ions a e poin ed ou wi h a s a .
Me hods Speci ici y Sensi i i y
2
M0.611 0.707
3
M0.826 0.205
5
M0.448 0.785
* 6
M0.813 0.447
* 7
M0.747 0.587
8
M0.625 0.537
* 10
M0.859 0.322
2
M∩3
M0.803 0.432
*2
M
∪
3
M0.618 0.866
*Union o ( ),,,,,, 10876532 MMMMMMM 0.3690 1
*In e sec ion o ( ),,,,,, 10876532 MMMMMMM 0.983 0.066
1
0
0,2
0,4
0,6
0,8
1
00,2 0,4 0,6 0,8 1
Sensi i i y
Speci ici y
(0.066, 0.983)
(0.322, 0.859)
(0.447, 0.813)
(0.587, 0.747)
(0.866, 0.618)
(1, 0369)
Fig. 6. Resul s o he e alua ion o he me hod-associa ions con ained in he la ice M’ o he
six selec ed di e en ial p o iles
3.4.3 C ea ion o Me hod Associa ion Rules
The se o me hod associa ion ules is c ea ed based on he e alua ed p o iles (i.e.,
1
)( CT PP ,…, 6
)( CT PP ), and he me hod-associa ions i
Mp esen in he Pa e o op imal
on o non-domina ed solu ions. The weigh s ),( 21 ww associa ed o he objec i es
),( 21 OO a e se o (0.5, 0.5) o calcula e he speci ici y/sensi i i y measu e
C. We il-
lus a e wo associa ion ules ex ac ed om he e alua ion o M’ o e he o me p o-
iles, which ha e he ollowing o m:
1
R
: IF 1
x IS 1
1
)( CT PP AND ,…, AND 6
x IS 1
6
)( CT PP THEN 1
Z
IS 6
MWITH 1
C
whe e
Cis calcula ed based on he speci ici y/sensi i i y le els ob ained on he ap-
plica ion o such me hod o e 1
)( CT PP ,…, 6
)( CT PP p o iles (Table 2):