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CANSMOGMAKEUSUNHAPPY?…
4.Methods
11
A3-stagemethodologicalapproachisapplied,basedonquantitativetech-
niques.First,theexploratorycorrelationalanalysisisperformedtoidentifythe
relationsbetweenvariousSWBmeasuresandtheirhypothesisedpredictors.
Secondly,alog-linearanalysisisappliedtoexploretheinterrelationsbetween
perceivedAP,objectiveAP,andselectedSWBmeasures.Finally,a2-stagees-
timationisconducted,totakeaccountofthepotentiallyconfoundingeffectof
controlvariablesandtotestforexogeneityoftheperceivedAP.Allstatistical
procedureswereperformedwiththeRsoftware,usingthefollowingpackages:
MASS,PerformanceAnalytics,ordinal,olsrr.
Thelog-linearanalysisisastatisticaltechniqueusedtoexaminerelationships
between>2ordinalvariables(LiuandAgresti2005).Nodistinctionismade
betweenresponseandexplanatorycomponents,allowingforacomprehensive
explorationoftheinterrelationsamongpotentiallyendogenousvariables.Log-
linearmodelsdescribehowtheexpectedcellcountsinacontingencytablevary
inresponsetovariablesincluded.Tosolvetheestimation,aleastcomplexmodel
issoughtthatwouldaccountforthelargestpartofvarianceinobservedfrequen-
ciesofHappy/LS_tr,AQ_todandPM_ord.Thelog-linearanalysisexaminesif
thereisaninteractioneffectbetweenvariables,e.g.iftheinteractionbetween
perceivedandobjectiveAPleadstoasignificantvariationoftheinvestigatedre-
lationship.With3variables,thereare9possiblelog-linearmodelstofitthedata,
startingfromasimple3-factoradditivemodelandmovingtowardsthesaturated
model(Rodriguez2007).However,only4ofthemareconsideredhere,given
theirrelevanceforthisstudy(Equations1-4):
(1)Completeindependence
log
()
µ
ijk
=+
λ
λ
W
i
+
λ
P
j
+
λ
O
k
(2)Jointindependence
log
()
µ
ijk
=+
λ
λ
W
i
+
λ
P
j
+
λ
O
k
+
λ
PO
jk
(3)Conditionalindependence
log
()
µ
ijk
=+
λ
λ
W
i
+
λ
P
j
+
λ
O
k
+
λ
WP
ij
+
λ
PO
jk
(4)Homogenousassociation
log
()
µ
ijk
=+
λ
λ
W
i
+
λ
P
j
+
λ
O
k
+
λ
WP
ij
+
λ
PO
jk
+
λ
WO
ik
where:
W=SWB(Happy/LS_tr),P=perceivedAP(AQ_tod),O=objectiveAP(PM_
ord)
μ
ijkdenotesthemeanprobabilitythatanobservationfallsinthecell(i,j,k)ofthe
contingencytable
Tosatisfytheassumptionofanormaldistributionofobservedfrequencies
requiredforthelog-linearanalysis,asetofdatatransformationsisperformed.
Thevariablesincludedintheestimationarerecodedwithabridged3-pointscales
(seeTable4)tomergethecellsofthecontingencytablewithalimitednumber
ofobservations.
Finally,a2-stageestimationisemployedtocomprehensivelyinvestigatethe
relationshipofperceivedandobjectiveAPaspredictorsofSWB.Tothisend,
the2SCML(Two-StageConditionalMaximumLikelihood)estimationmethod
