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Measuring the impact of apnea and obesity on circadian activity patterns using functional linear modeling of actigraphy data

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Description

Actigraphy provides a way to objectively measure activity in human subjects. This paper describes a novel family of statistical methods that can be used to analyze this data in a more comprehensive way. Methods A statistical method for testing differences in activity patterns measured by actigraphy across subgroups using functional data analysis is described. For illustration this method is used to statistically assess the impact of apnea-hypopnea index (apnea) and body mass index (BMI) on circadian activity patterns measured using actigraphy in 395 participants from 18 to 80 years old, referred to the Washington University Sleep Medicine Center for general sleep medicine care. Mathematical descriptions of the methods and results from their application to real data are presented. Results Activity patterns were recorded by an Actical device (Philips Respironics Inc.) every minute for at least seven days. Functional linear modeling was used to detect the association between circadian activity patterns and apnea and BMI. Results indicate that participants in high apnea group have statistically lower activity during the day, and that BMI in our study population does not significantly impact circadian patterns. Conclusions Compared with analysis using summary measures (e.g., average activity over 24 hours, total sleep time), Functional Data Analysis (FDA) is a novel statistical framework that more efficiently analyzes information from actigraphy data. FDA has the potential to reposition the focus of actigraphy data from general sleep assessment to rigorous analyses of circadian activity rhythms.

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Published by
Published 01 January 2011
Reads 11
Language English

Wang
etal
.
JournalofCircadianRhythms
2011,
9
:11
http://www.jcircadianrhythms.com/content/9/1/11

RESEARCH

OpenAccess

Measuringtheimpactofapneaandobesityon
circadianactivitypatternsusingfunctionallinear
modelingofactigraphydata
JiaWang
1
,HongXian
1,2
,AmyLicis
3
,ElenaDeych
1
,JiminDing
4
,JenniferMcLeland
3
,CristinaToedebusch
3
,TaoLi
1
,
StephenDuntley
3
andWilliamShannon
1*

Abstract
Background:
Actigraphyprovidesawaytoobjectivelymeasureactivityinhumansubjects.Thispaperdescribesa
novelfamilyofstatisticalmethodsthatcanbeusedtoanalyzethisdatainamorecomprehensiveway.
Methods:
Astatisticalmethodfortestingdifferencesinactivitypatternsmeasuredbyactigraphyacrosssubgroups
usingfunctionaldataanalysisisdescribed.Forillustrationthismethodisusedtostatisticallyassesstheimpactof
apnea-hypopneaindex(apnea)andbodymassindex(BMI)oncircadianactivitypatternsmeasuredusing
actigraphyin395participantsfrom18to80yearsold,referredtotheWashingtonUniversitySleepMedicine
Centerforgeneralsleepmedicinecare.Mathematicaldescriptionsofthemethodsandresultsfromtheir
applicationtorealdataarepresented.
Results:
ActivitypatternswererecordedbyanActicaldevice(PhilipsRespironicsInc.)everyminuteforatleast
sevendays.Functionallinearmodelingwasusedtodetecttheassociationbetweencircadianactivitypatternsand
apneaandBMI.Resultsindicatethatparticipantsinhighapneagrouphavestatisticallyloweractivityduringthe
day,andthatBMIinourstudypopulationdoesnotsignificantlyimpactcircadianpatterns.
Conclusions:
Comparedwithanalysisusingsummarymeasures(e.g.,averageactivityover24hours,totalsleep
time),FunctionalDataAnalysis(FDA)isanovelstatisticalframeworkthatmoreefficientlyanalyzesinformationfrom
actigraphydata.FDAhasthepotentialtorepositionthefocusofactigraphydatafromgeneralsleepassessmentto
rigorousanalysesofcircadianactivityrhythms.
Keywords:
Apnea,BMI,circadianactivitypatterns,FunctionalDataAnalysis

1.Introduction
todaytimeactivityortotalactivity,[7,8]standarddevia-
Activitymeasuredbywristactigraphyhasbeenshowntionofsleeponsettime,[9]andintra-dailyvariability
tobeavalidmarkerofentrainedPolysomnography[10].Morecomplexmodelingofactigraphyincludes
(PSG)sleepphaseandisstronglycorrelatedwithspectralanalysis,[7]cosinoranalysis[7]andwaveform
entrainedendogenouscircadianphase[1].Actigraphyeductioncalculatedasan

averagewaveform

forsome
dataisrecordeddensely,suchaseveryminuteoreveryperiod[11].
15seconds,foreachpatientovermultipledays.ThisInthispaperweproposeanovelstatisticalframework,
dataisgenerallyanalyzedbyreducingthetimeseriesFunctionalLinearModeling(FLM),asubsetofFunc-
activityvaluestosummarystatisticssuchassleep/waketionalDataAnalysis(FDA),foranalyzingactigraphy
ratios,[2,3]totalsleeptime,[2,4]sleepefficiency,[5,6]datatoextractandanalyzecircadianactivityinforma-
wakeaftersleeponset,[2,3,6]ratioofnighttimeactivitytionthroughdirectanalysisofrawactivityvalues[12].
FLMextendsstandardlinearregressiontotheanalysis
*Correspondence:wshannon@wustl.edu
offunctions,whichinthiscaserepresentcircadian
1
Dept.ofMedicine,WashingtonUniversitySchoolofMedicine,(660South
activitypatterns.FLMisperformedby1)convertinga
EuclidAvenue),St.Louis,(63110),USA
subject

srawactigraphydatatoafunctionalform(i.e.,
Fulllistofauthorinformationisavailableattheendofthearticle
©2011Wangetal;licenseeBioMedCentralLtd.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommons
AttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproductionin
anymedium,providedtheoriginalworkisproperlycited.

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JournalofCircadianRhythms
2011,
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http://www.jcircadianrhythms.com/content/9/1/11

continuouscurveovertime),and2)analyzingsetsof
functionstoseeiftheydifferstatisticallyacrossgroups.
OurFLM-basedanalysisshowswhereandwithwhat
levelthedifferencebetweengroupsoccursalongthe
time,whichprovidesvaluablereferenceforclinicalana-
lysisandtreatments,anddistinguishesourmethods
fromexistingcircadiananalysisworks(see[13]fora
review).Moreover,weadoptedanon-parametricper-
mutationFtesttodetectthedifferencebetweengroups,
whichmakestheresultsrobusttotheuncertaintyin
rawdatadistribution.UsingFLM,weshowthatthe
apnea-hypopneaindex(apnea)hasastatisticallysignifi-
cantimpactoncircadianactivitypatterns,whilebody
massindex(BMI)inthisdatasethaslittleimpact.
2.Methods
2.1ParticipantsandMeasures
Participantswererecruitedprospectivelyfromtheclinic
atWashingtonUniversityinSt.LouisSleepMedicine
Center.Thesleepcenterisamultidisciplinaryclinicata
tertiarymedicalfacility.Clinicpatientswithasuspected
diagnosisofobstructivesleepapnea(OSA),insomnia,or
restlesslegssyndrome(RLS)wereinvitedtoparticipate.
Pregnantwomen,individualsunderageof18,and
patientswhoreportworkinganeveningorovernight
shiftwereexcludedfromparticipationduetoknown
biologicallydifferentcircadianclocks.Clinicalcovariates
suchasBMI,co-morbidities,concomitantmedications,
andpresentingsleepcomplaintswerecollected.Partici-
pantsunderwentanovernightPSGwhenclinicallyindi-
cated.Thesedatawerecollectedinaccordancewiththe
standardsoftheAmericanAcademyofSleepMedicine
(AASM)andwerereviewedbyaboardcertifiedsleep
physician.PSGdatawerescoredaccordingtothe
AASMManualfortheScoringofSleepandAssociated
Events.Thisongoingstudyhasbeenapprovedbythe
WashingtonUniversitySchoolofMedicineInstitutional
ReviewBoard.
ActivitywasmeasuredusingActicaldevices(Philips
RespironicsInc.)whichwerepositionedonthenon-domi-
nantwristofsubjectsattheinitialsleepcentervisitand
settomeasureactivityeveryminutefor7days.Three
hundredandninetyfivepatientshavebeenrecruited,of
which305haveapneaand/orBMImeasured.Thissub-
groupcomesfromalargerNIHfundedstudycurrently
recruitingacrosssectionof750patientsreferredtothe
WashingtonUniversitySleepMedicineCenterforthepur-
poseofdevelopingandvalidatingfunctionaldataanalysis
methodsforactigraphydata(HL092347).
2.2.FunctionalDataAnalysis(FDA)
FDAisanemergingfieldinstatisticsthatextendsclassi-
calstatisticalmethodsforanalyzingsetsofnumbers
(scalarsforunivariateanalyses,andvectorsfor

Page2of10

multivariateanalyses)toanalyzingsetsoffunctions[13]
[15].FDAisasubsetofthelargerfieldcalled

object
dataanalysis

or

objectorienteddataanalysis

thatuses
statisticalmethodstoanalyzedatathatareinnon-
numericformsuchasimages,graphs(e.g.,trees),or
functions[14,15].Thegoalofobjectorienteddataana-
lysisistoanalyzeobjectsintheirnaturalform(e.g.,
functions,graphs)toextractmoreinformationthan
generallycanbeextractedwhentheobjectsarecon-
vertedintosimplersummarymeasures(e.g.,average
activitylevel,totalsleeptime)wherestandardstatistical
methodscanbeapplied.
2.2.1Functionalsmoothing
Functionaldataanalysis(FDA)beginsbyreplacingdis-
creteactivityvaluesmeasuredateachtimeunit(e.g.,
minute)byafunctiontomodelthedataandreduce
variability.Thefunctionrepresentstheexpectedactivity
valueateachtimepointmeasured.Sincetheactigraphy
hasequidistantdata,toallowflexibilityinrepresenting
thedataasafunction,aFourierexpansionmodelis
used,thoughanysmoothingmethodcouldbeused.Let
y
kj
bethediscreteactivitycountforpatientkattime
point
t
kj
,thenthemodel
y
kj
=
Activity
k
(
t
kj
)+
ε
k
(
t
kj
)
(1)
representsactivity,where
k
=1,2,...
,N
,
N
istotalnum-
berofpatients,
j
=1,2,...,
T
k
,
T
k
isthetotalnumberof
timepointsforpatient
k
.Inourdataset,observation
timesareminutesfrommidnighttomidnightin24
hours,soallsubjectshavethesamenumberofmeasure-
ments
T
k
.
Weconverttherawactigraphydatatoafunctional
formusingabasisfunctionexpansionfor
Activity
k
(
t
j
)
Activity
k
(
t
j
)=
a
1
k

1
(
t
j
)+
a
2
k

2
(
t
j
)
2()+
···
+
a
nk

n
(
t
j
)
where
{
a
ik
}
in
=1
arescalarcoefficientsforpatientkand
{

i
(
·
)
}
in
=1
arebasisfunctions.Possiblebasisfunctions
includepolynomials(
f
(
t
)=
a
1
t
+
a
2
t
2
+...+
a
n
t
n
),Four-
(
f
(
t
)=
a
1
+
a
2
sin(
ω
t
)+
a
3
cos(
ω
t
)+
ierbasis,splines,
a
4
sin(2
ω
t
)+
a
5
cos(2
ω
t
)+
···
+
a
n
ϕ
n
)
andwavelets.
Experimentalresults(unpublished)showmostbasis
functionsworkequallywellandwehavefoundaFour-
ierexpansionwithn=9basisfunctionscapturethe
majortrendofactivitypatternwithreducednoise.Let

1
(
t
)=1,

2
(
t
)=cos(
ω
t
),

3
(
t
)=
2πsin(
ω
t
),
...
,

8
(
t
)=cos(4
ω
t
),

9
(
t
)=sin(4
ω
t
),
ω
=
TwhereTistheperiod,inourcaseT=1440(number
ofminutesin24hours).Equation1becomes

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2011,
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Page3of10


9
Therawdatadoesnotneedtobenormalizedsinceall
Activity
k
(
t
j
)=
a
ik

i
(
t
j
)
analyzesaredoneonthefunctionalformofthedata.
i
=1
Toavoidintroducingvariationbetweenweekdayand
Wewillusethisfunctionalrepresentationforallana-weekendactivitypatterns,onlydatafrommidnight
MondaytomidnightFridaywasusedinthispaper,
lysesinthispaper.althoughthissimplificationisnotrequiredforanalysis.
Smoothcoefficientsoftheexpansion
{
a
ik
}
i
9=
i
areesti-Thefiveweekdaysofactigraphydatawereaveragedinto
matedbyminimizingtheunweightedleastsquarescri-asingle24hourprofileandasmoothFourierexpansion
terionSMSSE[12]:functionwasfittedusinga24hourperiodicityand9

1440

9

2
basisfunctions.Thisproducedasingle24hourcirca-
SMSSEy
k
|
a
k
=[
y
jk

a
ik

i
t
j
]
(3)dianactivitypatternforeachsubjectthatcanbeusedto
j
=1
i
=1
estimatepatient

sactivitylevelatanytimepoint
where
y
k
=(
y
1
k
,
y
2
k
,...,
y
1440
k
)

,
a
k
=(
a
1
k
,
a
2
k
,...,
a
9
k
)

.throughouttheday.Wearedevelopingandpreparingto
Inmatrixterms,thiscriterionbecomes:publishfunctionallinearmixedmodelswhichwillana-

lyzeeveryday

sactivitydatatoincorporatedayeffects,
SMSSEy
k
|
a
k
=(
y
k


a
k
)

(
y
k


a
k
).
(4)weekday/weekendeffects,andpre/posttreatmenteffects
whichwillprovidemoreinsightintocircadianrhythm
where
F
isa1440×9matrixwithcolumnsforbasispatternsandwithin-subjectvariability.
functionsandrowsforbasisvalueateachminute.ThisdatasmoothingmethodisillustratedinFigure1
Takingthederivativeofthecriterion
SMSSE
(
y
k
|
a
k
)foratypicalsubject.Plot(a)showsweekdaysordered
withrespectto
a
,gives2
F

F
a
k
-2
F

y
k
,andsettingthisMondaythroughFridayfromtoptobottom,withthe
equalto0andsolvingforaprovidestheestimate
a
ˆ
timeofdayindicatedontheXaxisrunningfrommid-
thatminimizestheleastsquaresolution,nighttomidnight,andtheheightofthespikeindicating




1

therawactivitylevelontheYaxiscollectedbytheacti-
ˆ
a
k
=

yk
.
(5)graphywatchateachminuteinterval.Plot(b)showsthe
Then,thevector
ˆ
y
ofsmoothedactivityfittedvaluesactivityaveragedateachminuteoverthe5days(black
points)andtheFourierexpansionrepresentingthis
ispatient

scircadianactivitypattern(redsolidline).
1

(6)
2.2.2FunctionalLinearModels
y
ˆ
k
=

a
ˆ
k
=
y
k
Reducingactigraphydatatoasummarystatisticcan
maskdifferencesacrossgroups.Forexample,ifone

−Figure1
Dataflowforonesubject
.Plot(a)showsweekdaysorderedMondaythroughFridayfromtoptobottom,withthetimeofday
indicatedontheXandtheheightofthespikeindicatingtherawactivitylevelontheYaxis.Theplot(b)showstheactivityaveragedateach
minuteoverthe5days(blackpoints)andtheFourierexpansionrepresentingthispatient

scircadianactivitypattern(redsolidline).

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JournalofCircadianRhythms
2011,
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groupofpatientshashighactivityinthemorningand
lowactivityintheafternoon,andanothergrouphasa
reversedpatternwiththesamemagnitudeofactivity,
lowactivityinthemorningandhighactivityinthe
afternoon,theiraverageactivitymaybesimilar,anda
significantdifferenceincircadianactivitypatternswould
bemissed.FLMavoidsmaskingbyextendingthelinear
regressionmodeltotheanalysisofsmoothfunctions(i.
e.circadianactivitypatterns),anddifferencessuchas
describedinthisexamplebecomeapparent.
Theconceptualchangegoingfromclassicallinear
regressiontoFLMisthatthemodelregressioncoeffi-
cients,(e.g.
b
0
,
b
1
),anderrortermarefunctions.To
illustratetheuseofFLMsforanalyzingactigraphydata,
foursubjectsfromourdatabasewiththehighestapnea
scoresandfoursubjectswiththelowestapneascores
wereselected.apneaisameasureofapnea-hypopnea
indexusedroutinelyinsleepmedicine,andmeasures
theseverityofsleepapneawithhighvaluesindicating
moreseveredisease.InFigure2,thecircadianactivity
patternsfittedbyFourierexpansionforeachofthe8
subjectsareshowninseparateplotswithtimerecorded
ontheXaxis,andactivitylevelontheYaxis.Thetop4
plotsshowthehighapneasubjects(severesleepapnea)
andthebottom4plotsshowthelowapneasubjects
(mildornosleepapnea).Visuallythereisalargediffer-
encebetweenthecircadianpatternsinthehighandlow
apneasubjects.
Usingthissubsetofsubjects,functionalsmoothing
andlinearmodelingisillustratedinthissection.Inthe
followingsectionthemethodsareappliedtothefull
dataset.

Page4of10

Totestwhetherhighandlowapneapatientshavedif-
ferentactivitylevels,standardapproacheswouldreduce
eachsubject

sdatatoanaverageactivitylevel,anda
classicalstatisticalmethodsuchaslinearregression
wouldtestifthesevaluesarethesameordifferent.For
example,alinearregressionmodeltotestifthereare
differencesinaverageactivitybetweenthehighapnea
(averageactivity=78,76,80and76)andlowapnea
(averageactivity=370,397,482and421)groupsis
definedas
Activity
k
=
β
0
+
β
1
×
AHI
+
ε
k
.
(7)
wherek=1,2,...,8arethesubjectsinFigure2,apneais
thegroupmembershipindicatorwithapnea=1forlow
apneasubjects,apnea=-1forhighapneasubjects,and
ε
k
istheerrorterm.Theresultingmodelfittothisdata
isActivity
k
=247.9+169.9×apnea,P<0.001,andR
2
=0.97.Theestimatedmeanactivityinthe4lowapnea
subjectsis247.9+169.9=417.8,andinthe4high
apneasubjectsis247.9-169.9=78.Thisstatisticalana-
lysisconfirmstheclinicalbeliefthatapneaimpacts
activity,andconfirmswhatisseeninFigure2.However,
itdoesnottelluswhenduringthedayactivitylevelsare
different.
Figure3illustrateshowfunctionallinearmodelingis
appliedtoactigraphydatatotestfordifferences
betweenthetwoapneagroups,andshowwhereduring
thedaythosedifferencesoccur.Plot(a)showsthe8
individualcircadianactivitypatternswithblueandred
lineforhighandlowapneagroups,respectively.The
overallmeancircadianactivitypatternisthesolid

Figure2
Smoothedactivityof8subjectsfittedbyFourierexpansionandshowninseparateplotswithtimerecordedontheXaxis,
andactivitylevelontheYaxis
.Thetop4plotsshowthehighapneasubjectsandthebottom4plotsshowthelowapneasubjects.

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2011,
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Page5of10

expansiontotheactigraphydataforeachsubjectk)
Activity
k
(
t
),themeancircadianactivitypatternoverall
subjects
b
0
(
t
),thefunctionalcoefficientindicatinghow
themeancircadianactivitypatternschangesforlow
apneasubjects(apnea=1,
b
0
(
t
)+
b
1
(
t
)),orforhigh
apneasubjects(apnea=-1,
b
0
(
t
)-
b
1
(
t
)),and
ε
k
(
t
)isthe
functionalerrorterm.Inotherwords,thelowapnea
groupispredictedtohaveameancircadianactivitypat-
ternfoundbyaddingthetwofunctions
b
0
(
t
)+
b
1
(
t
),
andthehighapneagroupispredictedtohaveamean
circadianactivitypatternfoundbysubtractingthetwo
functions
b
0
(
t
)-
b
1
(
t
).InFigure3A
b
0
(
t
)isthethick
blacklinerepresentingtheoverallmean,
b
0
(
t
)+
b
1
(
t
)is
thethickredlineforthemeanofthelowapneagroup,
and
b
0
(
t
)-
b
1
(
t
)isthethickbluelineforthemeanof
thehighapneagroup.
Equation8canbeformulatedasamatrixanalysispro-
blemasdescribedaboveusingaNx2designmatrix
Z
withrowsindicatingsubjectsandcolumnsindicating
Figure3
FLMresultfor8subjects
.Plot(a)showsthe8individual
themeanfunction(column1)andeffectsontheactivity
circadianactivitypatternswithblueandredlineforhighandlow
duetoapnealevel
g
(column2).Instandardmatrix
apneagroups,respectively.Theoverallmeancircadianactivity
notationeachrowisavectorof1

sand-1

sindicatingif
patternisthesolidblacklineandthemeancircadianactivity
patternsforthehighandlowapneagroupsarethickblueandred
thesubjectbelongstohighapnea(1,-1)andlowapnea
line,respectively.Plot(b)showsF-testresulttheredsolidcurve
(1,1).Thetwofunctionallinearcoefficientsarerepre-
representstheobservedstatisticF(t)ateachtimepoint,theblue
sentedinmatrixnotationasa

functionalvector

dashedanddottedlinescorrespondtoaglobalandpoint-wisetest

ofsignificanceatsignificantlevel
a
=0.05,respectively.
β
0
(
t
)
β
(
t
)=
β
1
(
t
)
blacklineandthemeancircadianactivitypatternsandthesmoothedfunctionaldatarepresentedin
separatelyforthehighandlowapneagroupsarethematrixformby
thickblueandredline,respectively.Plot(a)showsa

clearseparationofthemeancircadianactivitypatterns
Activity
1
(
t
)
forthetwoapneagroupsandidentifieswhenduring
⎢⎢
Activity
2
(
t
)
Act
(
t
)=
daytimethosecurvesdiffer.Inaddition,circadian
⎢⎣
.
activitybehaviorsbecomeapparentwiththisanalysis.
Activity
N
(
t
)
Forexample,themaximumactivityinthehighapnea
group(thickblueline)occursinthemorningwithawhereeachrowrepresentsasubject

sfittedactivity
steadydeclineinactivitytheremainderoftheday,values.Finally,thefunctionalerrormatrixisdefinedas
’comparedtolowapneagroup(thickredline),the
ε
(
t
)=(
ε
1
(
t
),
ε
2
(
t
),...,
ε
N
(
t
)).Equation8inmatrixnotation
maximumactivityoccursatabout3PMandisstablebecomes,
fromabout9AMtonoonandfromabout6PMto9
PM.
Act
(
t
)=
Z
β
(
t
)+
ε
(
t
).
(9)
Asinthelinearregressionmodeldescribedabove,weThecoefficients
b
(
t
)areestimatedbyminimizinga
areinterestedinestimatingregressioncoefficientsthatleastsquaresestimate
willproducethegroup-specificmeancircadianactivity
patterns,andtestifthesemeancircadianactivitypat-
N

2ternsaredifferentacrossgroups.Thismodel,forapnea,
LMSSE
(
β
)=
Act
k
(
t
)

Z
k
β
(
t
)
dt
(10)
isdefinedas[12]:
k
=1
thActivity(
t
)=
β
0
(
t
)+
β
1
(
t
)
×
AHI
+
where
Z
k
isthe
k
rowofthedesignmatrix
Z
.
k(8)Afterweestimate
β
ˆ
(
t
)
for
b
(
t
)infunctionlinear
ε
k
(
t
),
k
=1,2,...
N
regression,wealsowanttomeasuretheaccuracyofour
wherethe(t)notationindicatesfunctionsoverthecir-estimationresult.Wecalculatethepoint-wise95%con-
cadianperiodforactivity(fittedbytheFourierfidencelimitsfortheseeffectsusingresidualsfromthe

⎤⎥⎥⎥⎦
Wang
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2011,
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model.Thisformulationisthesameasthestandardlin-
earmodelexceptthatinsteadofnumericcoefficientswe
arenowestimatingfunctionalcoefficientsdefinedover
the24hourcircadianperiod.Astatisticaltestofthe
nullhypothesisthatthecircadianactivitypatternsare
thesameinbothgroupsisgivenbythefunction[12]:
Var
[(
Z
β
ˆ
)
k
(
t
)]
F(t)=1
N

2
(11)
(
Act
k
(
t
)

(
Z
β
ˆ
)
k
(
t
))
Nk1=where
Z
isthedesignmatrixand
β
ˆ
isavectorofthe
estimatedregressioncoefficientfunctions.
Becauseofthenatureoffunctionalstatistics,itisdiffi-
culttoattempttoderiveatheoreticalnulldistribution
foranygiventeststatistic.Instead,weappliedanon-
parametricpermutationtestmethodology.Ifthereisno
relationshipbetweenactivitypatternandapnealevels,it
shouldmakenodifferenceifwerandomlyrearrangethe
apneagroupassignment.Theadvantageofthisisthat
wenolongerneedtorelyondistributionalassumptions
whilethedisadvantageisthatwecannottestforthesig-
nificanceofanindividualcovariateamongmany.The
p
valueofthetestcanthenbecalculatedbycountingthe
proportionofpermutation
F
valuesthatarelargerthan
the
F
statisticsfortheobservedpairing.Hereweused
twodifferentwaystocountingtheproportion:global
testandpoint-wisetest.Globaltestprovidesasingle
numberwhichistheproportionofmaximized
F
values
fromeachpermutation.Point-wisetestprovidesacurve
whichistheproportionofallpermutation
F
valuesat
eachtimepoint.
Plot(b)inFigure3providesadisplayforthestatisti-
calsignificancetestforthedifferencesincircadianactiv-
itypatternscontinuouslyovertime.Thebluedashed
anddottedlinescorrespondtoaglobalandpoint-wise
testofsignificanceatsignificantlevel
a
=0.05,respec-
tively,andtheredsolidcurverepresentstheobserved
statisticF(t)ateachtimepoint.WhenF(t)isabovethe
bluedashedordottedline,itisconcludedthetwo
apneagroupshavesignificantlydifferentmeancircadian
activitypatternsatthosetimepoints.Theglobalcritical
value(bluedashedline)ispreferredsincethisrepresents
amoreconservativetest.Forthesedata,thetwoapnea
groupsarestatisticallydifferentinactivityfromapproxi-
mately7AM-9PM.
Thestatisticalandcomputationaldetailsforfitting
FLMmodelsarewelldescribedelsewhereandareout-
sidethescopeofthispaper.Thereaderinterestedin
thesedetailsarereferredtoRamsayandSilverman[12].
Thisillustrationwasmeantasanintroductiontothe
methodologyonly,andnotanindicatorofaclinical
conclusion.Inthefollowingsection,thesemethodsare

Page6of10

appliedtotheentire395subjectdataset,andshowhow
apneaandBMIclinicallyimpactscircadianactivity
patterns.
3.Results
3.1DemographicInformation
Table1showsbasicdemographicinformationandsam-
plecharacteristics.Baselinecovariateshavebeencol-
lectedfrom395participants(196females),ageranging
from18to80yearsold.Theaverageapneascoreis22.1
(standarddeviation=28.1)andaverageBMIis34.7
(standarddeviation=8.9).Clinically,BMI>30isused
toseparatesubjectsintoobeseandnon-obesecate-
gories.However,subjectsinourdatabasewererecruited
fromasleepcenterandhadhigherBMIthanfoundin
thegeneralpopulation,sowecannotgeneralizeour
conclusionsoftheimpactofBMIoncircadianactivity
totheentirepopulation.

Table1Demographicinformationandsample
characteristics
VariableN(%)Mean±std
(NTotal395)
Female
196(49.87%)
Race
African-American134(35.08%)
Caucasian237(62.04%)
PresentingSymptoms
Snoring279(70.63%)
Gasping93(23.54%)
Morningheadache67(16.96%)
RLSsymptoms26(6.58%)
PLMS3(0.76%)
Witnessedapneas146(36.96%)
Insomnia42(10.63%)
Excessivedaysleepiness91(23.04%)
Nonrestorativesleep9(2.28%)
Mallampatiscore
Class4145(41.55%)
Class3136(38.97%)
Class253(15.19%)
Class115(4.30%)
DiagnosisResult
OSA292(73.92%)
RLS5(1.27%)
Insomnia8(2.03%)
Hypersomnia20(5.06%)
BMI>30
241(60.86%)
BMI
34.66±8.88
(Median=34)
Age(years)
47.9±14.8
apnea
22.11±28.11
(Median=12.95)

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Figure4
SmoothedActivityforindividualsasblacksolid
curvesandoverallmeanasredcurves
.
3.2SmoothedFunctionalActigraphyData
RawactigraphydatawerereadintotheRstatisticalsoft-
wareforanalysisusingtheFDApackageandsoftware
writtenbyourgrouptoapplyFLMmethods.Twohun-
dredandeightyninepatientshaveactigraphydata.Each
patient

sdatafrommidnightMondaythroughmidnight
Fridaywereaveragedandfitbya9basisFourierexpan-
sionandtheircircadianactivitypatternsplottedinFig-
ure4.Themeancircadianactivitypatternacrossall
subjectsisshownbytheredline.Whilegeneralstruc-
tureisvisible(e.g.,loweractivityduringsleephours),
theoverlapofthesecurvesmakesclinicallymeaningful
interpretationdifficult.
3.3.FunctionalLinerModel(FLM)Results
WeapplyFLMtomeasuretheimpactofapneaand
BMIonsubjectcircadianactivitypatternsandtestthe
nullhypothesisthatcircadianactivitypatternsarethe
sameregardlessofapneaandBMIvalues.Thealterna-
tivehypothesisisthatapnea,BMI,and/ortheirinterac-
tioneffectactivitybehaviorinastatisticallysignificant
way.Inadditiontothetestsofhypotheses,FLMpro-
videsagraphicalviewofthesubgroupcircadianactivity
patternsthatcanaidinterpretationofbehavioral
differences.
Tofitthesemodels,eachsubjectiscategorized
accordingtotheirapneaandBMIvaluesby:

Page7of10

1ifAHI
<
MedianofAHI
AHI=-1ifAHI

MedianofAHI
BMI=1ifBMI
<
30
-1ifBMI

30.
Forthe295subjects,235subjectshaddataonapnea
andactigraphy,277subjectshadBMIandactigraphy,
and232subjectshadapnea,BMI,andactigraphy.The
followinganalysesarebasedonthesesubsets.
Wefitthefollowingthreefunctionallinearmodelsas
definedinTable2.Thefirsttwomodelsmeasurethe
univariateimpactofapnea(N=235)andBMI(N=
277)separately,andthethirdmodelmeasurestheir
multivariateimpact(N=232).Themodelsarepre-
sentedinthisordertogofromalesstomorecompli-
catedanalysis.ConvertingapneaandBMIintobinary
categorieswasdoneforsimplificationbutisnotneces-
saryforfunctionallinearmodeling,andcontinuous
apnea,BMI,orothercovariatescouldbeused.Atthe
end,weshowhowBMIcanbeanalyzedbyFLMasa
continuousvariable.
3.3.1ApneaMainEffectModels
Theimpactofapneaasamaineffectoncircadianactiv-
itypatternswastestedwithModel1,Table2.Thenull
hypothesisisthatthecircadianactigraphypatternsare
thesameinthetwoapneagroups.Ofthe235subjects
inthisanalysis,118haveapnealessthanthemedian
apnea=10.8,and117patientshaveapnealargerthan
orequalto10.8.
Figure5presentstheestimatedgroupmeanswith95%
confidencebandsinplot(a).Thelowapneagroupindi-
catinglessdiseaseseverity(redsolidline)hashigher
activityduringthedaycomparedtothehighapnea
group(bluesolidline).Theconfidencebandsaround
thetwogroupmeancurvesdonotoverlapduringthe
daysuggestingthevariabilityinthegroupcircadian
activitypatternsdonotcross.TheF-testintheplot(b)
indicateswhenthesecurvesarestatisticallydifferent
duringtheday.TheF-testresultshowsthatthetwo
apneagroupsaresignificantlydifferentfromabout7
AMto9PM.
3.3.2.BMImaineffect
Next,theimpactofBMIasamaineffectoncircadian
activitypatternswasmeasuredusingModel2,Table2.
Thenullhypothesisisthatthecircadianactivitypat-
ternsarethesameinnon-obese(BMI<30)andobese
(BMI>=30)groups.182patientsareclassifiedasobese

Table2ThreeFunctionalLinearModels
Model1apneaMainEffectOnlyActivity
k
(
t
)=
b
0
(
t
)+
b
AHI
(t)×AHI
k
+
ε
k
(
t
)
Model2BMIMainEffectOnlyActivity
k
(
t
)=
b
0
(
t
)+
b
BMI
(t)×BMI
k
+
ε
k
(
t
)
Model3apnea+BMI+interactionActivity
k
(
t
)=
b
0
(
t
)+
b
AHI
(t)×AHI
k
+
b
BMI
(t)×BMI
k
+
b
AHI
×
BMI
(t)×AHI
k
×BMI
k
+
ε
k
(
t
)

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Page8of10

Table3Samplesizeforapnea,BMImode
apneaLowapneaHighTotal
(<10.75)(>=10.75)
BMI>=30
6194155
BMI<30
552277
Total
116116232
differenceincircadianactivitypatternsacrossthese
groupswasnotobserved.
3.3.3ApneaandBMIeffect,withinteraction
Model3,Table2wasusedtomeasuretheimpactof
apnea,BMI,andtheapnea×BMIinteractiontermon
circadianactivitypatterns.Thenullhypothesisisthat
thecircadianactivitypatternsmeasuredarethesamein
thefourapnea×BMIgroupsversusthealternativethat
apneaand/orBMIand/ortheirinteractionimpactscir-
cadianactivitypatternsinastatisticallysignificantway.
Figure5
FLMresultforapneamaineffectmodel
.Plot(a)is
Table3showsthesamplesizesforeachgroup.
estimatedactivitypatternsfortwoapneagroupsand95%
Thisinteractionmodelhasfourfunctionalcoefficients
confidenceband.Plot(b)isF-testresultforthismodel.
β
ˆ
o
(
t
),
β
ˆ
AHI
(t),
β
ˆ
BMI
(t),
β
ˆ
AHI
×
BMI
(t)
whichincombina-
tiondefinethefourclinicalgroups(e.g.,lowBMIand
and95asnon-obese.Figure6presentsestimatedgrouplowapnea;lowBMIandhighapnea,etc).Thefoursub-
meanswith95%confidencebandandF-testresult.Thegroups

circadianactivitycanbeestimatedbyaddingor
highBMIgroup(bluesolidline)hashigheractivitydur-subtractingthefunctionalcoefficientsasshowninTable
ingnightandloweractivityduringdaytime,butactivity4.
patternsforthetwogroupsareonlysignificantlydiffer-Whenasubject

sapneaorBMIislow,thefunctional
entaround3AMand6PM.coefficientforthatfactorisaddedtothemeanactivity
Weemphasizethatthepopulationofparticipantsinpattern.Whenasubject

sapneaorBMIishigh,the
thisstudyhadahigheroverallBMIcomparedtothefunctionalcoefficientforthatfactorissubtractedfrom
generalpopulationwhichmayexplainwhytheexpectedthemeanactivitypattern.Theinteractioncoefficientis
addedwhenapneaandBMIareconcordant(high/high
orlow/low)andsubtractedwhenapneaandBMIare
discordant(low/high,high/low).Figure7showsthe
activitycurvesforeachofthefourgroupsdefined
accordingtotheirapnea/BMIstatus.TheF-testshowsa
significantdifferenceamongthesefourgroupactivity
patternsbetweenabout7AMto11AMand12:30PM
to8PM.

Table4Fourgroupcircadianactivityresult
apneaBMIGroupMean
LowLow
β
ˆ
o
(
t
)+
β
ˆ
AHI
(t)+
β
ˆ
BMI
(t)+
β
ˆ
AHI
×
BMI
(t)
LowHigh
β
ˆ
o
(
t
)+
β
ˆ
AHI
(t)

β
ˆ
BMI
(t)

β
ˆ
AHI
×
BMI
(t)
HighLow
β
ˆ
o
(
t
)

β
ˆ
AHI
(t)+
β
ˆ
BMI
(t)

β
ˆ
AHI
×
BMI
(t)
Figure6
FLMresultforBMImaineffectmodel
.Plot(a)isHighHigh
estimatedactivitypatternsfortwoBMIgroupsand95%confidence
ˆ
o
(
t
)

β
ˆ
AHI
(t)

β
ˆ
BMI
(t)+
β
ˆ
AHI
×
BMI
(t)
band.Plot(b)isF-testresultforthismodel.

β
Wang
etal
.
JournalofCircadianRhythms
2011,
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Figure7
FLMresultforapneaandBMImodel
.Plot(a)is
estimatedactivitypatternsforthefourgroupsand95%confidence
band.Plot(b)isF-testresultforthismodel.

Itisanestablishedstatisticalpracticeinalinear
regressionmodeltotestthemaineffectsoftwocovari-
atesandtheeffectoftheinteractionofthetwocovari-
ates.Weextendedthismethodtothefunctionallinear
model.Thecomparisonsofall4groupsinthissection
areactuallytheevaluationofthecombinationofthe
mainandinteractioneffectswhichshouldbeconsistent
witha2-wayANOVA.
3.3.4BMIasaContinuousVariable
Asnotedabove,BMIshowedlittleimpactoncircadian
activitypatternswhichdoesnotcorrespondtogeneral
clinicalbelief.Thisismostlikelyexplainedbythefact
thatoursubjectpopulationhashighBMIrelativetothe
generalpopulation,sothedistinctionbetweenobese
andnon-obesewaslesspronounced.Inthissection,we
fitafunctionallinearmodeltreatingBMIasacontinu-
ousvariable.BMIrangesfrom17to67inthisdataset.
Figure8presentsestimatedmeansandF-testresult.In
thisplot,eachcolorrepresentsoneBMIgroup.Thelar-
gestBMIgrouphashigheractivityduringnightand
loweractivityduringdaytime.BMIimpactissignificant
around1AMto4AMand4PMto8PM.Itisnoted
thatthesignificantlydifferenttimeperiodsarelonger
thanthoseobtainedfromcategorizedBMIeffectmodel.
4.Discussion
Traditionally,actigraphydataistransformedintosum-
marynumbers,suchastotalsleeptime,sleepefficiency,
wakeaftersleeponset,andothermeasurements.These
transformationsallowdataanalyststotesthypothesis

Page9of10

Figure8
FLMresultforBMImodeltreatingBMIascontinuous
.
Plot(a)isestimatedactivitypattersforBMIgroups.Plot(b)isF-test
resultforthismodel.

usingsimpleclassicalstatisticalmethods.However,large
amountofinformationcanbelostandproblemsof
maskingcircadianpatternsmayarise.
Themeritoffunctionallinearmodelingreliesin
determiningwhenalongthe24-hourscalegroupsdiffer.
Resultsfromparametertestsinacosinorapproach
wouldprovideinformationastodifferencesinharmonic
contentbetweengroups.Anotheradvantageofthefunc-
tionallinearmodelingapproachisexemplifiedinFigure
8,whereBMIisusedasavariableinsteadofcomparing
groupswithhigherversuslowerBMIvalues.
Inthispaperwehavepresentedanovelapproachfor
analyzingthefullactigraphydatawhichwebelieve
avoidssignificantinformationlossandmaskingeffect.
Representingactigraphydataassmoothcontinuous
functions,andapplyingFunctionalLinearModeling
methodsallowedustodirectlycompareandtestdiffer-
encesofcircadianactivitypatternsacrossapneaand
BMIsubgroups.OtherFunctionalDataAnalysismeth-
odsusingprincipalcomponentsanalysis([15];Zeitzer,
etal.

PhenotypingapathyinindividualswithAlzhei-
mer

susingfunctionalprincipalcomponentanalysis

,
RevisedandResubmitted)foridentifyingsourcesof
variabilitywithincircadianactivitypatternsacrosssub-
groups,andmixedeffectmodels(Ding,etal.,

Func-
tionalLinearMixedEffectsModelforActigraphyData

,
InPreparation)forincorporatingadditionalsourcesof
withinsubjectvariabilityarecurrentlybeingdeveloped
inourlabandappliedtothistypeofdata.Functional
linearmixedmodelsarealsobeingdevelopedinourlab

Wang
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JournalofCircadianRhythms
2011,
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:11
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whichwillallowwithin-subjectvariabilitysuchasday-
to-dayorpre-treatmenttopost-treatmentdifferencesin
activitytobeanalyzed.
Acknowledgements
Weareparticularlygratefultotheeditorandreviewerswhohavegreatly
increasedourknowledgeofexistingworkincircadianrhythmdataanalysis.
ThisworkwassupportedbyR01HL092347

NewDataAnalysisMethodsfor
ActigraphyinSleepMedicine

(Shannon,PI),theWashingtonUniversity
Dept.ofMedicine

sBiostatisticsCenter(Shannon,Director),andtheDept.of
NeurologySleepCenter(Duntley,Director)
Authordetails
1
Dept.ofMedicine,WashingtonUniversitySchoolofMedicine,(660South
EuclidAvenue),St.Louis,(63110),USA.
2
St.LouisVAMedicalCenter,Research
Service,(501NorthGrandAve),St.Louis,(63103),USA.
3
Dept.ofNeurology,
WashingtonUniversitySchoolofMedicine,(212NKingshighway),St.Louis,
(63108),USA.
4
Dept.ofMathematics,WashingtonUniversity,(OneBrookings
Drive),St.Louis,(63130),USA.
Authors

contributions
JWandHXcarriedoutstatisticalanalysis,contributedtodevelopmentof
methodologyandwrotesectionsofthemanuscript.ALprovidedclinical
inputandoversight.EDdevelopedtheclinicaldatabase,contributedto
statisticalprogrammingandreviewedthemanuscript.JDdeveloped
theoreticalmathematicalbasisfortheanalysisandwrotesectionofthe
manuscript.JMandCTactedasclinicalcoordinators,enteredthedata,
wrotesectionsandcriticallyreviewedthemanuscript.TLprovided
programmingandmathematicalsupportandcriticallyreviewedthe
manuscript.SDisco-PIontheproject,oversawallclinicalaspectsofthe
project,providedclinicaltheoreticalperspectivesandwrotesectionsofthe
manuscript.WSwasthePIontheproject,developedstatistical
methodology,oversawtheworkofstatisticiansandprogrammers,wrote
sectionsofthemanuscriptandcriticallyreviewedallitscontents.Allauthors
havereadandapprovedthefinalmanuscript.
Competinginterests
Theauthorsdeclarethattheyhavenocompetinginterests.
Received:5August2011Accepted:13October2011
Published:13October2011
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