ejsr i a letter 8622

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COMPARISON ON GLMM UNIRESPONSE, BIRESPONSE, AND REDUCTION

WITH PCA ON LONGITUDINAL DATA

Adji Achmad Rinaldo Fernandes1, I Nyoman Budiantara2, Bambang Widjanarko2, and Suhartono2

1

h! Student, !e"artment o# Statistics, Institut $eknologi Se"uluh No"ember Surabaya%ecturer, !e"artment o# &athematics, Statistics Study rogram, 'ni(ersitas Bra)ijaya

*mail+ #ernandesub-ac-id2%ecturer, !e"artment o# Statistics, Institut $eknologi Se"uluh No"ember Surabaya

ABSTRACT

In the .uantitati(e research "articularly in the #ield o# health research, it o#ten uses

longitudinal data using re"eated measurements on some indi(iduals )ithin some "eriod o#

time- /ne method used #or longitudinal data )ith .uantitati(e res"onse is the 0eneral %inear

&ied &odel 0%&&3- Study using t)o res"onse (ariables can be sol(ed by using three

methods+ #irst by using both res"onse (ariables at once )ith Bi4res"onse 0%&&, both byusing both (ariables 'nires"on 0%&& "artial res"onse, and the third uses 5A4reduction

0%&& 6ac.min40adda , 2777, and 8ermanussen, 27793- $his research uses the "rimary

data and the simulation data- $hus, in this study )e )ill com"are )hich o# the three best

methods #or the analysis o# longitudinal data )ith bi4res"onse 0%&& using unires"onse,

bires"onse 0%&&, and 5A40%&&- From the results o# the research conducted, it can be

concluded as #ollo)s+ a3 In the lo) correlation condition correlations bet)een 7-77 to 7-:73,

unires"onse 0%&& is more #easible to be used- b3 'nder conditions moderate correlation

correlation bet)een 7-:1 to 7-;73, and the 0%&& Bires"onse and 0%&& 5A reduction

are #easible to be used, and c3 'nder conditions o# high correlation correlation bet)een 7-;1

to 7-<73, 0%&& Bires"onse is the best choice in sha"ing the model 0%&& on longitudinal

data-

Keywords: GLMM, Uniresponse, Biresponse, and Reduction PCA

1. INTRODUCTION

!e(elo"ment o# longitudinal data analysis as one o# the grou"s in the statistical

sciences has been increasing in the use mainly in the #ield o# health research- $hrough the

incor"oration o# cross4sectional data and time series data, the use o# longitudinal data is more

in#ormati(e, (aried and su"erior in studying the dynamic changes =1>- According to ?erbeke

and &olenberghs =2>, the analysis o# t)o4stage t)o4stage analysis3 constitutes an alternati(e

a""roach to longitudinal data analysis- $his analysis is done by summari@ing (ector o#

re"eated measurements re"eated measurement3 #or each cross4sectional unit subject3 intothe (ector #orm estimators subject4s"eci#ic regression coe##icients in the #irst stage and

connect the "robe to the inde"endent (ariables are kno)n to use the techni.ues in the second

stage regression multi"eubah- &erging these t)o stages into a single statistical model is

called the 0eneral %inear &ied &odel 0%&&3-

In the #ield o# health research, it is o#ten #ound more than one res"onse (ariable on the

result o# interrelated obser(ation and a set o# inde"endent (ariables deri(ed #rom "atients

studied in some "eriod o# time )ith .uantitati(e res"onse- 6ac.min40adda, et al- 27773,

analy@ing longitudinal data in the #orm o# t)o res"onse (ariables using the 0eneral %inear

&ied &odel 0%&&3 simultaneously bires"onse3 and com"are them i# is done in "artial

unires"onse3- 8ermanussen & =:> uses a reduction #rom bires"onse to unires"onse (ariables

mailto:[email protected]

mailto:[email protected]



using rinci"al 5om"onent Analysis 5A3 as a res"onse to the 0eneral %inear &ied &odel

0%&&3-

$he "ur"oses o# the research to be obtained are as #ollo)s+ 0eneral %inear &odel

0etting &ied &odel 0%&&3 among the best in the res"onse unires"onse, multi(ariate and

(ariable reduction using 5A

Bene#its o# the research are as #ollo)s+ 13 As an alternati(e to sol(ing "roblems inlongitudinal data analysis )ith multi"le res"onse, 23 Selection o# the best models in the

0eneral %inear &ied &odel 0%&&3 is e"ected to be used as an alternati(e #or

researchers in the #ield o# longitudinal data analysis-

2. THEORY REVIEW

2.1.General Linear Mie! M"!el #GLMM$

?erbeke and &olenberghs =2>, longitudinal data on "ractice uses the linear regression

#unction on each subject subject4s"eci#ic3- Analysis o# t)o4stage combination into a single

statistical model called the 0eneral %inear &ied &odel 0%&&3- !iggle, et- al- =>, the

0eneral %inear &ied &odel 0%&&3 )as obtained #rom t)o4stage analysis, so the analysis

a""roach uses linear regression #unction on each subject subject4s"eci#ic3- &odel 0eneral%inear &ied &odel 0%&&3 is obtained+

% & & 13

)here matri ni"3 inde"endent (ariables are kno)n- $he model assumes that (ector o#

re"eated measurement re"eated measurements3 #ollo)s linear regression model )ith

"o"ulation4s"eci#ic "arameter, ie, the same #or all subjects3 and subject4s"eci#ic "arameter,

assumed to be random so called random e##ects &olenbergh and ?erbeke =2>3-

0%&& )ith $)o ?ariables Res"onse

$hiebaut, et al- =C>, de#ines the 0eneral %inear &ied &odel on t)o res"onse(ariables )ith 0aussian miture models o# the com"onents are random, the 1st order o# the

auto4regressi(e, AR 13 and residual com"onents-

Su""ose , is the res"onse (ector #or subject i, )ith as the (ector measurement, then

k k 1-23 )ith - I# t)o longitudinal data are #ree, it can be used the #ollo)ing

t)o models+

% & & & 23

% & & & :3

)here

' N #(, $ and ' N #(, $

' N#(, $ and ' N#(, $

' N#(, $ and ' N#(, $

% matri nDi " E k3 inde"endent (ariables that are kno)n

% 4dimensional (ector contains #ied e##ects fixed effect 3

% matri o# kno)n inde"endent (ariables, modeling the res"onse (ariable that is

arranged based on the time #or the i4th subject-



% (ector o# random e##ect random effect 3 . dimension, )ith

3

% 2ni 4 t3, stochastic "rocess that allo)s the relationshi" bet)een measurements

order41 (ector reali@ation o# the auto4regressi(e3 AR 13-

% dimensional identity matri -

According to Weiss =;>, the general model o# the 0eneral %inear &ied &odel 0%&&3 on

t)o res"onse (ariables, as #ollo)s+

% & & 3

)ith+

' N#(, $, ' N#(, $, !an ' N#(, G$

G % co(ariance matri on the t)o res"onse (ariables-

)here+

% , % , % , % , and

% i) 2 )hich is the reali@ation o# the t)o4(ector res"onse (ariable-

% i) residual (ector com"onents are assumed #ree-

i % i) co(ariance matri o# the residual com"onent-

#*$ % i) co(ariance #unction o# t)o (ariables41 res"onse to the order o# the auto4

regressi(e deri(ed #rom %

(alue , so that (alue r , and r , is (alue e"

.Symbol re#ers to notation o# Gronecker-

2.2. Sele+*i"n " *-e Be)* M"!el

I# se(eral (ariable models are #ound a""ro"iately in the results o# diagnostic studies, it

can be used then the best models that )ill be used on the data- Selection o# the best model can

be done by calculating the (alue o# AI5 AkaikeHs In#ormation 5riterion3, )ith the #ormula+

AI5 nln 2m C3

)here+

n number o# obser(ation

residual range estimator-

m number o# sus"ected "arameter in the modelBest model is the model that has the smallest AI5 (alue $hiebaut, et al - =C>3-



. METHODS

$he data obtained are the #irst "rimary data o# "atients )ith $y"e 2 !iabetes &ellitus

listed in hos"itali@ed "atients in RSSA &alang- In the health sector, le(els o# Fasting lasma

0lucose F03 and hemoglobin le(els 8bA1c3 is kno)n to correlate )ith each other-!iabetes &ellitus $y"e 2 mainly occurs in adults but sometimes in adolescence and most

"eo"le )ith $y"e 2 !iabetes &ellitus obese- $his study conducted t)o drug thera"ies, using

oral anti4diabetic thera"y /A!3 and insulin thera"y- $hera"y )as "er#ormed )ith !iabetes

&ellitus $y"e 2 aims to reduce le(els o# Fasting lasma 0lucose F03 bet)een <74

1:7mgJdl and hemoglobin le(els 8bA1c3 o# less than 9K- 8bA1c le(els indicate the amount

o# sugar that is bound by the "rotein in red blood cells- Because red blood cells li(e u" to :

months, the 8bA1c test sho)s blood sugar a(erage o(er the last : months-

$he second data are simulated data generation3 )ith a (ariety o# conditions the

correlation bet)een the t)o res"onse (ariables, namely 13 a lo) correlation absolute (alue

o# r correlation coe##icient bet)een the t)o res"onse (ariables3 in the range o# 7-747-:73, 23

moderate correlation absolute (alue o# r in the range 7-:147-;73, and :3 a high correlation r absolute (alues in the range 7-;147-<73-

1- *"loration data #rom both res"onse (ariables simultaneously and "artially+

a- *"loration o# indi(idual "ro#iles-

b- *"loration marginal models+ the a(erage structure #ied e##ects3, the (ariety and

structure o# the correlation structure o# t)o res"onse (ariables-

2- $entati(e model o# #orming )ith the initial determination o# #ied e##ects number and

random e##ects Σ*$tentati# and Σ*Atentati# 33 as )ell as the correlation o# t)o res"onse

(ariables $-

:- *amination o# signi#icance o# #ied e##ects "arameter in early model using Maximum

Lieli!ood method &%3- I# gained (alue 42loglieli!ood 42ln%&%3 )hich is con(ergen

, then go to net "hase, but i# this condition is not

#ul#illed then back to early stages o# model building hase 23-

- *amination o# the signi#icance o# #ied e##ects "arameter in early model using the F test,

other than the time #ied e##ects3 )hich is not signi#icant then go back to hase 2 and

resha"ing early models )ithout including #ied e##ects into the model in addition to the

time is not signi#icant-

C- Formation o# marginal models+

a- Structure o# (ariance random e##ect3+

b- A(erage structure #ied e##ects3+c- correlation structure

;- *stimation o# marginal models+

a- *stimate o# the range o# a""ro"riate com"onents using R*&% estimators-

b-arameter estimation using the corres"onding #ied e##ects estimator &aimum

%ikelihood &%3-

L- Formation o# the #inal model-

9- 5alculating the (alue o# AI5-

<- Inter"retation o# the model-

Formation o# longitudinal models )ith t)o res"onse (ariables using the 0eneral %inear

&ied &odel 0%&&3 using SAS <-1-: so#t)are assistance-

/. RESULTS AND DISCUSSION



/.1. Da*a e0l"ra*i"n

In the "rimary data o# "atients )ith !iabetes &ellitus $y"e 2, there are t)o res"onses

the research+ Fasting "lasma glucose le(els F03 and hemoglobin le(els 8bA1c3

*"loration o# Indi(idual ro#ile illustrates ho) changes in le(els o# res"onse Fasting

lasma 0lucose F03 against time on each subject )as obser(ed, )hile the conclusion o#

the di(ersity changes in res"onse to the subject and inter4subject is other in#ormation that can be obtained #rom this e"loration- Indi(idual "ro#iles are #ormed is "resented in Figure 1-

Figure 1- Indi(idual ro#ile Res"onse F0

From Figure 1, it sho)s the changes in le(els o# Fasting lasma 0lucose F03 di##erentin "atients obser(ed in measurements- Indi(idual "ro#iles are #ormed also sho)s the in#luence

o# the change o# time months3 to changes in le(els o# Fasting lasma 0lucose F03 is

di##erent #or each "atient- Bet)een obser(ations on each "atient did not sho) high (ariability,

it is seen #rom the gra"h that is #ormed #or each "atient has a "attern o# relati(ely constant

o(er time-

&arginal !istribution e"loration is carried through the e"loration o# the a(erage

structure, the structure and the (ariety o# correlation structures- 5onclusions on the e##ects o#

tentati(e models )ill remain on the e"loration results obtained #rom the a(erage structure,

)hile the structure o# the range "ro(ide initial conclusions about )hether or not to include

random e##ects in addition to the #ied e##ects model o# tentati(e-

i3re 2 S*r3+*3re " A4erae Re)0"n)e PG

Result o# a(erage structure o# data e"loration in Figure 2 sho)s the gra"h changes in

time months3 to changes in le(els o# Fasting lasma 0lucose F03 sho)ed a linear "attern-

$hus the #ied e##ects linear time structure )ill be considered in the #ormation o# tentati(e

models at a later stage-

In contrast to the res"onse le(els Fasting lasma 0lucose F03, in res"onse

8emoglobin le(els 8bA1c3 seen #rom the results o# e"loration o# indi(idual "ro#iles in

Figure : sho)s the irregularity o# the line #ormed as a result o# the use o# the unit o# time- A

change in hemoglobin le(els 8bA1c3 e(ery time obser(ations lead to the conclusion o# the

in#luence o# the change o# time months3 to changes in hemoglobin le(els 8bA1c3 in

"atients-



Figure :- Indi(idual ro#ile Res"onse 8bA1c

*"loration results a(erage structure in Figure : sho)s a gra"h #ormed decreased linearly,

indicating there are signi#icant changes in time months3 on le(els o# hemoglobin 8bA1c3-

Figure Structure o# a(erage 8bA1c Res"onse

/.2. E)*a5li)-6en* " M"!el Unire)0"n)e a)*in Pla)6a Gl3+")e Le4el) 7 PG

Results o# unires"onse model building res"onse Fasting lasma 0lucose %e(els J F0

are "resented in $able 1, the results sho)ed "artial #ied e##ects testing using t4test statistic

#or #ied e##ects in the treatment o# oral anti4diabetic /A!3 and insulin-

Ta5le 1 Para6e*er E)*i6a*i"n " ie! Ee+*) M"!el Re)0"n)e PGPara6e*er re)0"n)e Std.Erro

r

*calculatin P-value

Interse" 1:9-7: 2;-L27< C-1L M7-7771

$ime ij 4:-;L<7 7-7C; 4<-7L M7-7771

Agei 1-1L;1 7-C1:1 2-2< 7-72<

S"eci#ication+ sign stating signi#icant at CK le(el-

$he test results in $able 1 sho)s that signi#icant time and negati(e slo"e- $his

indicates that changes o(er time, the "atient had a change in F0 res"onse that tends to #all-

$ests on concomitant (ariables that age has a signi#icant e##ect on the res"onse, it sho)s that

"atients )ith older age had higher F0 res"onse than "atients )ith a younger age- Finalmodel is gi(en as #ollo)s+ 1:9-7: 3 4:-;L<7 3Waktuij 1-1L;1

$he model describes the o(erall a(erage rate le(els Fasting lasma 0lucose F03 in

:C "atients be#ore the measurement )as 1:9-7: mg J dl and the reduction or additional le(els

o# Fasting lasma 0lucose F03 )as in#luenced by the e##ects o# changes in "atient time

months3- $he addition o# 1 year o# age )ith $y"e 2 !iabetes &ellitus "atients, based on -2

models can raise le(els o# #asting "lasma glucose as 1-1L;1 mg J dl-

/.. E)*a5li)-6en* " *-e Unire)0"n)e M"!el #8a!ar He6"l"5in7H5A1+$

Results o# unires"onse model building res"onse hemoglobin le(els 8bA1c3 is

"resented in A""endi :- Final model are "resented in $able 2, the results sho)ed "artialtesting #ied e##ects using t4test statistics #or #ied e##ects on insulin and /A! thera"y-



Ta5le 2. Para6e*er E)*i6a*i"n " ie! Ee+*) M"!el Re)0"n)e H5A1+

Para6e*er Pen!3a Std.Erro

r

*calculation P-value

Interse" ;-<91 7-;CC2 17-;; M-7771

$imeij 47-2772 7-721L 4<-:: M-7771

Sei 47-;9;L 7-29L2 42-:< 7-71<C

Agei 7-7211 7-71:L7 :-7L 7-77:7




Final model is gi(en as #ollo)s+

;-<91 47-2772 3 $imeij 4 7-;9;L Sei

$he model describes the o(erall a(erage le(el o# hemoglobin le(els 8bA1c3 in

"atients :C "atients be#ore the measurement is ;-<91K and the reduction or the addition o#

hemoglobin le(els 8bA1c3 )as in#luenced by the e##ects o# changes in "atient timemonths3- $he addition o# 1 year o# age )ith $y"e 2 !iabetes &ellitus "atients, based on the

abo(e model can im"ro(e hemoglobin le(els o# 7-7211K- $ests on concomitant (ariables

namely gender, the in#luence o# gender on the res"onse o# hemoglobin le(els 8bA1c3 is

signi#icant and negati(e- Se doll )ith a "euabh 7 is #emale and 1 is male, indicating that

#emale "atients had a better res"onse than the male "atients-

/./. E)*a5li)-6en* " M"!el " Bire)0"n)e Re!3+*i"n 9i*- PCA re)3l*)

Results o# unires"onse modeling o# the results o# res"onse reduction )ith 5A is

"resented in A""endi :- Final models are "resented in $able :, the results sho)ed "artial

testing #ied e##ects using t4test statistics #or #ied e##ects on insulin and /A! thera"y-

Ta5le Para6e*er E)*i6a*i"n " ie! Ee+*) M"!el Re!3+*i"n 9i*- PCA re)3l*)


r

*-i*3n P-value

Interse" 1-:<:< 7-L< 2-<1 7-77:

$imeij 47-71:1 7-72C 47-C: 7-C<2

Sei 47-72<9 7-77<L 4:-7; 7-772L

'siai 7-L2; 7-27: 2-:1 7-7227


Final model is gi(en by the #ollo)ing e.uation+

1-:<:< 47-71:1 3$imeij 4 7-72<9Sei 7-L2;Agei eij

$he model describes the o(erall a(erage rate o# reduction in 5A "atients results in

:C "atients be#ore the measurement is 1-:<:<K and the reduction or reduction )ith the

addition o# the results o# 5A "atients is in#luenced e##ect change in time months3- $he

addition o# 1 year o# age )ith $y"e 2 !iabetes &ellitus "atients, based on the abo(e model

can im"ro(e the outcome o# Res"onse Reduction )ith 5A at 7-L2;K- $ests on concomitant

(ariables namely gender, the in#luence o# gender on the res"onse o# the result is a signi#icant

reduction )ith 5A and negati(e- Se doll )ith a "euabh 7 is #emale and 1 is male,

indicating that #emale "atients had a better res"onse than the male "atients-

/.:. Bire)0"n)e M"!el

Results o# the bires"onse model building res"onse Fasting lasma 0lucose le(els

F03 and hemoglobin le(els 8bA1c3 is "resented in A""endi :- Final model are "resented

in $able , the results sho) the e##ect o# "ermanent "artial testing using t4test statistics #or

#ied e##ects on insulin and /A! thera"y-

Ta5le / Para6e*er E)*i6a*i"n " ie! Ee+*) M"!el Bire)0"n)e


r

*-i*3n P-value

Interse" 1-;1<1 7-971; 41-2L 7-7212

Waktuij 47-1<:7 7-71<< 4-;7 M-7771



'siai 7-7:<:: 7-71C97 2-< 7-71C2


Reduction o# Fasting lasma 0lucose le(els F03 and hemoglobin le(els 8bA1c3 in

:C "atients a##ected e##ect change in time months3- $he addition o# 1 year o# age )ith $y"e 2

!iabetes &ellitus "atients, based on the abo(e model can raise le(els o# #asting "lasma

glucose F03 and hemoglobin le(els 8bA1c3 o# 7-7:<::-

/n t)o res"onse (ariables bires"onse3 by inserting t)o random slo"e e##ect can be

gi(en to the #ollo)ing e.uation+

F03 47-:9<1 417-;<<23 29-722 7-77:1;C3

8bA1c3 47-712 47-;273 1-<:CC 7-71LL3

/.;. C"60ari)"n " T9" M"!el) Unire)0"n)e an! Bire)0"n)e M"!el) In O4erall

Results o# "arameter estimation and standard error o# t)o unires"onse and bires"onsemodels to the data o# "atients )ith !iabetes &ellitus $y"e 2 is "resented in $able C

Ta5le :. C"60ari)"n Para6e*er Re)0"n)e an! Standard Error

Res"ons

e

F0

/ne Res"on3

8bA1c

/ne Res"on3

Reduction )ith 5A Bires"onse

$)o Res"on3

estimated S-*-

estimated S-*- estimated S-*-

estimated S-*-

$hera"hy 41-77 41-77 7-72< 7-:;9

1-:< 7-L<47-77 7-17

Agei 1-1L; 7-C1: 7-72 7-71 7-7:7 7-717 7-7:< 7-77<

$imei 4:-;L< 7-7; 47-277 7-721 7-71: 7-72C 47-1< 7-721

AI5 L;<-: 7;-2 :1;-

Based on $able C it can be said that the model o# t)o res"onse (ariables bires"onse3

has a (alue o# "arameter estimators and standard error (alues are likely to be small- $he best

model selection can be indicated by the (alue o# AI5 Akaike In#ormation 5riterion3 in

$able C-

From $able C it can be seen that #or the o(erall com"arison o# the model, 0%&& )ith

bires"onse model is the most a""ro"riate #or use in longitudinal data o# :C "atients )ith $y"e

2 !iabetes &ellitus )ith t)o res"onse (ariables are correlated-

/.<. C"60ari)"n GLMM M"!el in Si63la*i"n Da*a

$he second data is simulated data generation3 )ith a (ariety o# conditions the

correlation bet)een the t)o res"onse (ariables simulation 13 lo) correlation absolute (alue

o# r correlation coe##icient bet)een the t)o res"onse (ariables3 in the range o# 7-747-:73,

simulation 23 moderate correlation (alues absolute r in the range 7-:147-;73, and

simulation :3 a high correlation r absolute (alues in the range 7-;147-<73- $)o res"onse

(ariables used are the same as the #irst data that F0 le(els and 8bA1c le(els- $able ; o# AI5

(alues #or all three models are unires"onse, reduction )ith 5A, and bires"onse simulated

data at 17 1 to :- From $able ; sho)s that the correlation bet)een the res"onse conditions on

simulated data 1 on condition that the lo) correlations ranged #rom 7-7 to 7+:7, )hich is the

best model that "roduces the smallest AI5 (alue is unires"onse 0%&& models, namely the#ormation o# "artial models o# both res"onse obser(ations-



Ta5le ;. Val3e AIC "n T-ree Si63la*i"n Da*a

Da*a

Si63la*i"n 1 #r (.((=(.($ Si63la*i"n 2 #r (.1=(.;($ Si63la*i"n #r (.;1=(.>($

Unire)

0"n)e

Re!3+*i

"n PCA

Bire)0"

n)e

Unire)0"

n)e

Re!3+*i

"n PCA

Bire)0"

n)e

Unire)0"

n)e

Re!3+*i

"n PCA

Bire)0"

n)e

1 <LL-< 11:L-2 277-1 <C2-9 <::-2 9<9- 9<:-: 9:-7 L;7-L

2 <C:-< 11:C-7 272C-: <;1-; <22-; <1C-; <1:-1 971- L;7-7

: <;-< 11C-; 1<91-2 <;:-1 <;;- <<-9 <77-L 9C:-7 979-7

<;L-: 117-7 27:L-9 <1- 9<L-9 <;-; <11-9 1712- 9<C-2

C <C2-: 11::-9 1<9L-1 <9-C <2-7 9<L-9 <7;-7 927-9 9C:-L

; <C;-9 1122-2 271<-1 <C:- <;:-7 <27-L 99L-C 91<-7 L:9-

L <C;-9 11:1-2 1<9L-9 <:9-: <72-9 <:2-9 <12-C 9;;-2 9:C-7

9 <;2-C 11:;-; 2771-2 <:7-< 9<<-; <19-9 9<-: 9:- 9:2-C

< <C-2 11::- 1<<C-; <C9-9 <97- <CC-< <12-L 91<-7 9C<-C

17 <;1-C 11:;-9 1<<1-2 <C;-: <:1-; <;L-1 9<;-C 177;- L<-

&ean <;7-9 11:C-2 277;-; <C7-C <::-< <:7- <72-9 9;L-; 97<-2

In the second simulation data, namely the condition o# moderate correlation ranged

bet)een 7+:1 to 7-;7, it is seen that the 0%&& models and 0%&& Bires"onse 5A

reduction o(erall had a better AI5 (alue com"ared 'nires"onse 0%&& models- 5an be said

on the condition o# moderate correlation, 0%&& models and 0%&& Bires"onse 5A

reduction as good, because it has a (alue o# AI5 )hich tend to be almost the same-

In the simulated data :, )ith high correlations ranged bet)een 7-;1 to 7-<7, gi(ing

almost the same results )ith simulated data 2, but it )as clear that the model 0%&&Bires"onse ha(e AI5 (alues are much smaller than the 0%&& reduction o# 5A- It can be

concluded, on the condition o# lo) correlation, unires"onse 0%&& is more #easible to use-

At moderate correlation condition, 0%&& Bires"onse and 0%&& same 5A reduction un#it

#or use, and the high correlation condition, 0%&& Bires"onse is the best choice in sha"ing

the 0%&& models in longitudinal data-

:. CONCLUSIONS AND RECOMMENDATIONS

From the results o# research conducted, it can be concluded as #ollo)s+ on simulated

data a3 In the lo) correlation condition correlations bet)een 12+77 to 12+:73, unires"onse

0%&& is more #easible to use- b3 'nder conditions moderate correlation correlation

bet)een 7+:1 to 7-;73, and the 0%&& Bires"onse same 0%&& 5A reduction un#it #or use,

and c3 'nder conditions o# high correlation correlation bet)een 7-;1 to 7-<73, 0%&&

Bires"onse is the best choice in sha"ing the model 0%&& on longitudinal data-

From the results o# this study, it is suggested some o# the #ollo)ing+

1- 0%&& 'nires"onse, Bires"onse, and reduction o# 5A can be used as a settlement o# the

"roblem in the analysis o# longitudinal data )ith multi"le res"onses, the correlation

bet)een the res"onse to (arious conditions-

2- /n #urther research it is recommended to use a multi(ariate res"onses are res"onses that

use more than t)o- Because some research in the areas o# health, not least the use o# more

than t)o res"onses-



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