COMPARISON ON GLMM UNIRESPONSE, BIRESPONSE, AND REDUCTION WITH PCA ON LONGITUDINAL DATA Adji Achmad Rinaldo Fernandes 1 , I Nyoman Budiantara 2 , Bambang Widjanarko 2 , and Suhartono 2 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-id 2 %ecturer, !e"artment o# Statistics, Institut $ekno logi 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 by using 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%&&, a nd 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, Bire sponse, 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"" roa ch to lon git udi nal data ana lysis- $his ana lysis is don e by summari@i ng (ec tor o#re"eated measurements re"eated measurement3 #or each cross4sectional unit subject3 into the (ec tor #or m esti mators sub ject 4s"e ci#i c regress ion coe ##i cie nts in the #ir st 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
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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
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
$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
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
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-
$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-
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
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