Table Of ContentAnalysis of Repeated Measures
and Longitudinal Data in Health
Services Research
Juned Siddique, Donald Hedeker, and Robert D. Gibbons
Abstract Contents
Thischapterreviewsstatisticalmethodsforthe Introduction.......................................... 1
analysis of longitudinal data that are com- IssuesInherentinLongitudinalData................. 2
monly found in health services research. The HistoricalBackground................................ 3
chapterbeginsbydiscussingissuesinherentin StatisticalModelsfortheAnalysisofLongitudinal
longitudinaldataandprovideshistoricalback- andRepeatedMeasuresData................... 4
Mixed-EffectsRegressionModels.................... 5
ground on early methods that were used to
MatrixFormulation................................... 8
analyze data of this type. Next, mixed-effects
CovariancePatternModels........................... 9
regression models (MRMs) and covariance- CalculatingEffectSizes............................... 10
pattern models (CPMs) for longitudinal data
IllustrativeExample:TheWECareStudy......... 11
are introduced with a focus on linear models Mixed-EffectsRegressionModelsforContinuous
fornormallydistributedoutcomes.Asanillus- DataUsingtheWECareStudy.................... 12
CurvilinearGrowthModel............................ 13
trationoftheuseofthesemethodsinpractice,
CovariancePatternModels........................... 18
MRMsandCPMsareappliedtodatafromthe
EffectofTreatmentGrouponChange................ 19
WomenEnteringCare(WECare)study,alon-
ExtensionsandAlternatives........................ 21
gitudinal depression treatment study. Finally,
AnalysisofLongitudinalDatawithMissing
extensionsandalternativestothesemodelsare Values.............................................. 21
briefly described. Key phrases: mixed-effects GeneralizedEstimatingEquationModels............ 22
models; random-effects models; covariance- ModelsforCategoricalOutcomes.................... 23
GrowthMixtureModels.............................. 24
patternmodels;effectsizes.
Discussion............................................. 25
References............................................ 26
J.Siddique(*)
DepartmentofPreventiveMedicine,Northwestern
UniversityFeinbergSchoolofMedicine,Chicago,IL,
Introduction
USA
e-mail:[email protected]
Inhealthservicesresearch,atypicalstudydesign
D.Hedeker
DepartmentofPublicHealthSciences,Universityof is the longitudinal clinical trial in which patients
Chicago,Chicago,IL,USA arerandomlyassignedtodifferenttreatmentsand
e-mail:[email protected]
repeatedlyevaluatedoverthecourseofthestudy.
R.D.Gibbons Since the pioneering work of Laird and Ware
DepartmentsofMedicineandPublicHealthSciences,
(1982),statisticalmethodsfortheanalysisoflon-
UniversityofChicago,Chicago,IL,USA
gitudinal data have advanced dramatically. Prior
e-mail:[email protected]
#SpringerScience+BusinessMediaLLC2017 1
B.Sobolev,C.Gatsonis(eds.),MethodsinHealthServicesResearch,HealthServicesResearch,
DOI10.1007/978-1-4939-6704-9_1-1
2 J.Siddiqueetal.
to this time, a standard approach to analysis of (MRMs) and covariance-pattern models (CPMs)
longitudinal data principally involved using the aredescribedindetailandappliedtotheWECare
longitudinal data to impute end-points (e.g., last study.Attheendofthechapter,alternativestoand
observation carried forward) and then to simply extensions of linear MRMs are briefly discussed
discardthevaluableintermediatetime-pointdata, andconcludingremarksareprovided.
favoring the simplicity of analyses of change
scores from baseline to study completion (or the
IssuesInherentinLongitudinalData
last available measurement treated as if it was
what would have been obtained had it been the
Whilelongitudinalstudiesprovidefarmoreinfor-
end of the study), in some cases adjusting for
mation than their cross-sectional counterparts,
baseline severity as well. Laird and Ware (1982)
theyarenotwithoutcomplexities.Thefollowing
showed that mixed-effects regression models
sections review some of the major issues associ-
couldbeusedtoperformamorecompleteanaly-
atedwithlongitudinaldataanalysis.
sis of all of the available longitudinal data under
much more general assumptions regarding the
Heterogeneity
missing data (i.e., missing at random). The net
Particularlyinhealthservicesresearch,individual
resultwasamorepowerfulsetofstatisticaltools
differencesarethenormratherthantheexception.
for analysis of longitudinal data that led to more
The overall mean response in a sample drawn
powerfulstatisticalhypothesistests,moreprecise
from a population provides little information
estimatesofratesofchange(anddifferentialrates
regardingtheexperienceoftheindividual.Incon-
of change between experimental and control
trast to cross-sectional studies in which it is rea-
groups),andmoregeneralassumptionsregarding
sonable to assume that there are independent
missingdata,forexample,becauseofstudydrop-
random fluctuations at each measurement occa-
out. This early work has led to considerable
sion,whenthesamesubjectsarerepeatedlymea-
related advances in statistical methodology for
sured over time, their responses are correlated
the analysis of longitudinal data (see Hedeker
overtime,andtheirestimatedtrendlineorcurve
and Gibbons 2006; Fitzmaurice et al. 2012;
canbeexpectedtodeviatesystematicallyfromthe
Diggle et al. 2002; Goldstein 2011; Longford
overallmean trend line. Forexample, behavioral
1993; Raudenbush and Bryk 2002; Singer and
and/orbiologicalsubject-levelcharacteristicscan
Willett2003;VerbekeandMolenberghs2000for
increasethelikelihoodofafavorableresponsetoa
several excellent reviews of this growing
particular experimental intervention (e.g., a new
literature).
pharmacologictreatmentfordepression),leading
Thefollowingsectionsprovideageneralover-
subjectswiththosecharacteristicstohaveatrend
viewofrecentadvancesinstatisticalmethodsfor
with higher slope (i.e., rate of change) than the
the analysis of longitudinal data. The primary
overallaveragerateofchangeforthesampleasa
focus is on linear models for continuous data.
whole.Inmanycases,thesepersonalcharacteris-
Their application is illustrated using data from
ticsmaybeunobservable,leadingtounexplained
theWomenEnteringCare(WECare)study,alon-
heterogeneity in the population. Modeling this
gitudinal depression treatment study of low
unobserved heterogeneity in terms of variance
income minority women with depression. In
components that describe subject-level effects is
order to motivate the use of these advanced
one way to accommodate the correlation of the
methods,thefirstsectiondiscussesissuesinherent
repeated responses over time and to better
in longitudinal data and some of the history of
describe individual differences in the statistical
earlier methods for the analysis of longitudinal
characterizationoftheobserveddata.Thesevari-
data.Next,linearmixed-effectsregressionmodels
ancecomponentsareoftentermed“random-effects,”
AnalysisofRepeatedMeasuresandLongitudinalDatainHealthServicesResearch 3
leading to terms like random-effects or mixed- they have received full treatment benefit, an
effectsregressionmodels. end-point analysis might miss the fact that some
ofthesesubjectsmayhavehadarelapsehadthey
MissingData remained on treatment. Many other objections
Perhapsthemostimportantissuewhenanalyzing have been raised about these two simple
data from longitudinal studies is the presence of approaches of handing missing data, which have
missingdata.Statedquitesimply,notallsubjects led to more statistically reasoned approaches for
remain in the study for the entire length of the the analysis of longitudinal data with missing
study.Reasonsfordiscontinuingthestudymaybe observations.
differentially related to the treatment. For exam-
ple,somesubjectsmaydevelopsideeffectstoan IrregularlySpacedMeasurement
otherwise effective treatment and must discon- Occasions
tinue the study. Alternatively, some subjects It is not at all uncommon in real longitudinal
might achieve the full benefit of the study early studies either in the context of designed experi-
on and discontinue the study because they feel ments or naturalistic cohorts, for individuals to
that their continued participation will provide no vary both in the number of repeated measure-
added benefit. The treatment of missing data in ments they contribute and even in the time at
longitudinalstudiesisitselfavastliterature,with which the measurements are obtained. This may
major contributions by Laird (1988), Little beduetodrop-outorsimplyduetodifferentsub-
(1995), Rubin (1976), and Little and Rubin jects having different schedules of availability.
(2002) to name a few. The basic issue is that Whilethiscanbequiteproblematicfortraditional
eveninarandomizedandwell-controlledclinical analysisofvariancebasedapproaches(leadingto
trial,thesubjectswhowereinitiallyenrolledinthe highly unbalanced designs which can produce
study and randomized to the various treatment biased parameter estimates and tests of hypothe-
conditionsmaybequitedifferentfromthosesub- ses), more modern statistical approaches to the
jectsthatareavailableforanalysisattheendofthe analysis of longitudinal data are all but immune
trial. If subjects “drop out” because they already tothe“unbalancedness”thatisproducedbyhav-
have derived full benefit from an effective treat- ing different times of measurement for different
ment, an analysis that only considers those sub- subjects. Indeed, this is one of the most useful
jectswhocompletedthetrialmayfailtoshowthat features of the regression approach to this prob-
thetreatmentwasbeneficialrelativetothecontrol lem,namelytheabilitytousealloftheavailable
condition.Thistypeofanalysisisoftentermeda data from each subject, regardless of when the
“completer”analysis.Toavoidthistypeofobvi- datawerespecificallyobtained.
ous bias, investigators often resort to an analysis
inwhichthelastavailablemeasurementiscarried
forwardtotheendofthestudyasifthesubjecthad HistoricalBackground
actuallycompletedthestudy.Thistypeofanaly-
sis, often termed an “end-point” analysis, intro- Existingmethodsfortheanalysisoflongitudinal
duces its own set of problems in that (a) all data are an outgrowth of two earlier approaches
subjects are treated equally regardless of the forrepeatedmeasuresdata.Thefirstapproach,the
actualintensityoftheirtreatmentoverthecourse so-called repeated measures ANOVAwas essen-
of the study, and (b) the actual responses that tiallyarandominterceptmodelthatassumedthat
would have been observed at the end of the subjectscouldonlydeviatefromtheoverallmean
study, if the subject had remained in the study responsepatternbyaconstantthatwasequivalent
untilitsconclusion,mayinfact,bequitedifferent over time. A more reasonable view is that the
thantheresponsemadeatthetimeofdiscontinu- subject-specific deviation is both in terms of the
ation. Returning to the example of the study in baseline response (i.e., intercept) and in terms of
which subjects discontinue when they feel that the rate of change over time (i.e., slope or set of
4 J.Siddiqueetal.
trend parameters). This more general structure measuredondifferentoccasions.Finally,software
couldnotbeaccommodatedbytherepeatedmea- for the MANOVA approach often makes it diffi-
sures ANOVA. The random intercept model culttoincludetime-varyingcovariates,whichare
assumptionleadstoacompound-symmetricvari- oftenessentialtomodelingdynamicrelationships
ance-covariancematrixfortherepeatedmeasure- betweenpredictorsandoutcomes.TheMANOVA
ments in which the variances and covariances of approachhasbeenextendedintoasetofmethods
therepeatedmeasurementsareconstantovertime. referred to as CPMs which also estimate the
In general, it is common to find that variances parameters of the repeated measures variance-
increase over time and covariances decrease as covariancematrix,butwithinaregressionframe-
time-points become more separated in time. work. Additionally, CPMs allow for incomplete
Finally,basedontheuseofleast-squaresestima- data across time, and thus include subjects with
tion,therepeatedmeasuresANOVAbreaksdown incomplete data in the analysis. These methods
forunbalanceddesigns,suchasthoseinwhichthe arediscussedinthenextsection.
sample size decreases over time due to subject
discontinuation. Based on these limitations, the
repeated measures ANOVA and related StatisticalModels for theAnalysis
approaches are mostly no longer used for the of Longitudinaland Repeated
analysis of longitudinal data. Mixed-effects Measures Data
regression models, which are described in the
next section, build upon the repeated measures Inanattempttoprovideamoregeneraltreatment
ANOVA framework by allowing more than just of longitudinal data, with more realistic assump-
theintercepttermtovarybyindividualinorderto tionsregardingthelongitudinalresponseprocess
bettercapturebetween-subjectvariability.Inaddi- andassociatedmissingdatamechanisms,statisti-
tion, mixed-effects regression models use all cal researchers have developed a wide variety of
available data so that not all subjects need to be more rigorous approaches to the analysis of lon-
measuredatthesametimepoints. gitudinal data. Among these, the most widely
The second early approach for repeated mea- used include mixed-effects regression models
sures data was multivariate growth curve – or (Laird and Ware 1982), and generalized estimat-
MANOVA – models (Potthoff and Roy 1964; ing equation (GEE) models (Zeger and Liang
Bock 1975). The primary advantage of the 1986). Variations of these models have been
MANOVA approach versus the ANOVA developed for both discrete and continuous out-
approachisthattheMANOVAassumesageneral comes and for a variety of missing data mecha-
formforthecorrelationofrepeatedmeasurements nisms. The primary distinction between the two
over time, whereas the ANOVA assumes the general approaches is that mixed-effects models
much more restrictive compound-symmetric are “full-likelihood” methods and GEE models
form. The disadvantage of the MANOVA model are “partial-likelihood” methods. The advantage
is that it requires complete data. Subjects with ofstatisticalmodelsbasedonpartial-likelihoodis
incomplete data are removed from the analysis, that(a)theyarecomputationallyeasierthanfull-
leading to potential bias. In addition, both likelihoodmethods,and(b)theygeneralizequite
MANOVA and ANOVA models focus on com- easilytoawidevarietyofoutcomemeasureswith
parisonofgroup means andprovideno informa- quite different distributional forms. The price of
tion regarding subject-specific growth curves. this flexibility, however, is that partial likelihood
Finally, both ANOVA and MANOVA models methodsaremorerestrictiveintheirassumptions
require that the time-points are fixed across sub- regarding missing data than their full-likelihood
jects (either evenly or unevenly spaced) and are counterparts.Inaddition,full-likelihoodmethods
treatedasaclassificationvariableintheANOVA provideestimates ofperson-specificeffects(e.g.,
or MANOVA model. This precludes analysis of person-specifictrendlines)thatarequiteusefulin
unbalanceddesignsinwhichdifferentsubjectsare understanding inter-individual variability in the
AnalysisofRepeatedMeasuresandLongitudinalDatainHealthServicesResearch 5
longitudinal response process and in predicting time can be treated as a continuous variable in
futureresponsesforagivensubjectorsetofsub- MRMs, subjects do not have to be measured at
jectsfromaparticularsubgroup(e.g.,acounty,a thesametime-points.Thisisusefulforanalysisof
hospital, or a community). In the following sec- longitudinalstudieswherefollow-uptimesarenot
tions attention is focused on full-likelihood uniform across all subjects. Both time-invariant
methods,andpartial-likelihoodmethodsareonly and time-varying covariates can be easily
brieflydiscussedinsection“GeneralizedEstimat- included in the model. Thus, changes in the out-
ingEquationModels.” come variable may be due to both stable charac-
teristics of the subject (e.g., theirgender or race)
as well as characteristics that change across time
Mixed-EffectsRegressionModels (e.g., life-events). Finally, whereas traditional
approaches estimate average change (across
Mixed-effects regression models (MRMs) are time) in a population, MRMs can also estimate
now widely used for the analysis of longitudinal change foreachsubject.Theseestimatesofindi-
data. Variants of MRMs have been developed vidualchangeacrosstimecanbeparticularlyuse-
underavarietyofnames:random-effectsmodels ful in longitudinal studies where a proportion of
(Laird and Ware 1982), variance component subjects exhibit change that deviates from the
models(Dempsteretal.1981),multilevelmodels averagetrend.
(Goldstein1986),two-stagemodels(Bock1989), Tohelpfixideas,considerthefollowingsimple
random coefficient models (de Leeuw and Kreft linearregressionmodelforthemeasurementyof
1986),mixedmodels(Longford1987;Wolfinger individuali(i=1,2,...,Nsubjects)onoccasion
1993), empirical Bayes models (Hui and Berger j(j=1,2,...n occasions):
i
1983; Strenio et al. 1983), hierarchical linear
(cid:1) (cid:3)
models (Raudenbush and Bryk 1986), and ran- y ¼β þβ t þβ t (cid:2)Trt þe : (1)
ij 0 1 ij 2 ij i ij
domregressionmodels(Bock1983a,b;Gibbons
etal.1988).Abasiccharacteristicofthesemodels Ignoring subscripts, this model represents the
is the inclusion of random subject effects into regressionoftheoutcomevariableyontheinde-
regression models in order to account for the pendentvariabletime(denotedt).Thesubscripts
influence of subjects on their repeated observa- keep track of the particulars of the data, namely
tions.Theserandomsubjecteffectsthusdescribe whoseobservationitis(subscripti)andwhenthe
each person’s trend across time, and explain the observationwasmade(thesubscriptj).Theinde-
correlational structure of the longitudinal data. pendent variable t gives a value to the level of
Additionally, they indicate the degree of time, and may represent time in weeks, months,
between-subjectvariationthatexistsinthepopu- etc.Sinceyandtcarrybothiandjsubscripts,both
lationofsubjects. the outcome variable and the time variable are
There are several features that make MRMs allowed to vary by individuals and occasions.
especially useful in longitudinal research. First, ThevariableTrt isabinaryvariablethatindicates
i
subjects are not assumed to be measured the thetreatmentassignedtoindividuali.WhenTrtis
samenumberoftimes,thus,subjectswithincom- dummycodedasa1or0,with1indicatingmem-
pletedataacrosstimeareincludedintheanalysis. bership in the treatment group, the regression
The ability to include subjects with incomplete coefficient β is the mean of y when t = 0, β is
0 1
data is an important advantage relative to proce- theslope orrateofchangefor thecontrolgroup,
dures that require complete data across time and β is the difference in slopes between the
2
because (a) by including all data, the analysis treatmentandcontrolgroups.
hasincreasedstatisticalpower,and(b)complete- In linear regression models, the errors e are
ij
caseanalysismaysufferfrombiasestotheextent assumed to be normally and independently dis-
thatsubjectswithcompletedataarenotrepresen- tributed in the population with zero mean and
tativeofthelargerpopulationofsubjects.Because
6 J.Siddiqueetal.
commonvarianceσ2.Thisindependenceassump- longitudinaldata,andsotheυ termswilldeviate
0i
tion makes the typical general linear regression from0.Sinceindividualsinasamplearetypically
modelunreasonableforlongitudinaldata.Thisis thoughttoberepresentativeofalargerpopulation
because the outcomes y are observed repeatedly of individuals, the individual-specific effects υ
0i
fromthesameindividuals,andsoitismuchmore are treated as random effects. That is, the υ are
0i
reasonable to assume that errors within an indi- consideredtoberepresentativeofadistributionof
vidual are correlated to some degree. Further- individual effects in the population. The most
more, the above model posits that the change common form for this population distribution is
across time is the same for all individuals since thenormaldistributionwithmean0andvariance
the model parameters (β , the intercept or initial σ2.Inaddition,themodelassumesthattheerrors
0 υ
level,andβ ,thelinearchangeacrosstime)donot of measurement (e ) are conditionally indepen-
1 ij
vary by individuals except in terms of treatment dent, which implies that the errors of measure-
assignment.Forbothofthesereasons,itisuseful ment are independent conditional on the random
to add individual-specific effects into the model individual-specific effects υ . Since the errors
0i
that will account for the data dependency and now have the influence due to individuals
describedifferentialtime-trendsfordifferentindi- removed from them, this conditional indepen-
viduals. This is precisely what MRMs do. The dence assumption is much more reasonable than
essential point is that MRMs therefore can be theordinaryindependenceassumptionassociated
viewed as augmented linear regression models. with the linear regression model in Eq. 1. The
Notealsothathereandelsewhereinthischapter, random intercept model is depicted graphically
a main effect for treatment is not included in the intheleftpanelofFig.1.
model. That is, it is assumed that there is no As can be seen, individuals deviate from the
difference in the expected outcomes between regression of y on t in a parallel manner in this
treatmentgroupsatbaseline.Thisisareasonable model (since there is only one subject effect υ )
0i
assumption in a clinical trial where participants (for simplicity, it is assumed the treatment effect
are randomized prior to receiving treatment. β =0).Inthisfigurethesolidlinerepresentsthe
2
Alternatively, in an observational study where population average trend, which is based on
treatment(orexposure)isnotrandomized,itusu- β andβ .Alsodepictedaretenindividualtrends,
0 1
ally makes sense to include a main effect for both below and above the population (average)
treatmenttoaccountfordifferencesbetweentreat- trend. For a given sample there are N such lines,
mentgroupsatbaseline. one for each individual. The variance term σ2
υ
representsthespreadoftheselines.If σ2 isnear-
υ
RandomInterceptModel zero, then the individual lines would not deviate
Asimpleextensionofthelinearregressionmodel much from the population trend and individuals
describedinEq.1istherandominterceptmodel, donotexhibitmuchheterogeneityintheirchange
which allows each subject to deviate from the across time. Alternatively, as individuals differ
overall mean response by a person-specific con- from the population trend, the lines move away
stantthatappliesequallyovertime: fromthepopulationtrendlineandσ2increases.In
υ
(cid:1) (cid:3) thiscase,thereismoreindividualheterogeneityin
yij ¼β0þβ1tijþβ2 tij(cid:2)Trti þυ0iþeij (2) time-trends.
where υ represents the influence of individual RandomInterceptandTrendModel
0i
i on his/her repeated observations. Notice that if Forlongitudinaldata,therandominterceptmodel
individuals have no influence on their repeated is often too simplistic for a number of reasons.
outcomes, then all of the υ terms would equal First, it is unlikely that the rate of change across
0i
0. However, it is more likely that subjects will time is the same for all individuals. It is more
have positive or negative influences on their likely that individuals differ in their time-trends;
AnalysisofRepeatedMeasuresandLongitudinalDatainHealthServicesResearch 7
Random intercept model Random intercept and slope model
6 6
2 2
5 5
2 2
ble 24 ble 24
a a
ari ari
V V
ent 23 ent 23
d d
n n
e e
p p
e e
D 2 D 2
2 2
1 1
2 2
Average Trend Average Trend
Individual Trends Individual Trends
0 0
2 2
0 1 2 3 4 0 1 2 3 4
Time Time
Fig. 1 Simulated longitudinal data based on a random the overall population (average) trend. The dashed lines
intercept model (left panel) and a random intercept and representindividualtrends
slopemodel(rightpanel).Thesolidboldlinerepresents
not everyone changes at the same rate. Further- groups (by specifically indicating how the slope
more,thecompoundsymmetryassumptionofthe for Trt coded 1 is different than the slope for Trt
random intercept model is usually untenable for coded0).Intermsoftherandomeffects,υ isthe
0i
mostlongitudinaldata.Ingeneral,measurements intercept deviation for subject i, and υ is the
1i
at points close in time tend to be more highly slopedeviationforsubjecti(relativetotheirtreat-
correlated than measurements further separated mentgroup).Asbefore,e isanindependenterror
ij
in time. Also, in many studies subjects are more term distributed normally with mean 0 and vari-
similar at baseline due to entry criteria, and anceσ2.Aswiththerandominterceptmodel,the
change at different rates across time. Thus, it is assumption regarding the independence of the
natural to expect that variability will increase errorsisoneofconditionalindependence,thatis,
overtime. they are independent conditional on υ and υ .
0i 1i
For these reasons, a more realistic MRM With two random individual-specific effects, the
allows both the intercept and time-trend to vary population distribution of intercept and slope
byindividuals: deviations is assumed to be bivariate normal
(cid:1) (cid:3) N (0, Συ), with the random-effects variance-
y ¼β þβ t þþβ t (cid:2)Trt þυ covariancematrixgivenby
ij 0 1 ij 2 ij i 0i
þυ t þe : (3) (cid:4) (cid:5)
1i ij ij X σ2 σ
¼ υ0 υ0υ1 : (4)
In this model, β is the overall population υ συ0υ1 σ2υ1
0
intercept, β is the overall population slope for
1 ThemodeldescribedinEq.3canbethoughtof
thegroupwithTrtcoded0,andβ indicateshow
2 as a personal trend or change model since it
the population slopes vary between treatment
8 J.Siddiqueetal.
representsthemeasurementsofyasafunctionof For example, in the random intercepts and
time,bothattheindividualυ andυ andpopu- slopes MRM just considered, for a participant in
0i 1i
lation β and β (plus β ) levels. The intercept thetreatmentgroup(Trt =1)thedatamatricesare
0 1 2 i
parameters indicate the starting point, and the writtenas
slope parameters indicate the degree of change
2 3 2 3
over time. The population intercept and slope yi1 1 ti1 ti1
parameters represent the overall (population) 66yi2 77 66 1 ti2 ti2 77
trend, while the individual parameters express yi ¼66(cid:3)(cid:3)(cid:3)77andXi ¼66(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) 77andZi
4(cid:3)(cid:3)(cid:3)5 4(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) 5
how subjects deviatefrom the populationtrends.
y 1 t t
The right panel of Fig. 1 represents this model 2 ini 3 ini ini
graphically. 1 t
i1
As can be seen, individuals deviate from the 6 1 t 7
6 i2 7
averagetrendbothintermsoftheirinterceptand ¼6(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3)7
6 7
intermsoftheirslope.Aswiththerandominter- 4(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3)5
cept model, the spread of the lines around the 1 t
in
i
average intercept is measured by σ2 in Eq. 4.
υ0
The variance of the slopes around the average andthepopulationandindividualtrendparameter
trend is measured by σ2υ1 in Eq. 4. By allowing vectorsarewrittenas,
theindividualslopestovary,itisnowpossiblefor
2 3
individual trends to be positive even though the β (cid:4) (cid:5)
0 υ
overalltrendisnegative. Thetermσυ0υ1inEq.4 β¼4β15andυ0i ¼ υ0i
measures the association (covariance) between β 1i
2
therandominterceptandslope.Whenthisquan-
tityisnegative,individualswithlargerintercepts respectively. The distributional assumptions
(β +υ )willhavesteeperslopes(β +υ ). abouttherandomeffectsandresidualsare:
0 i0 1 i1
υ (cid:4)Nð0,Σ Þ
i υ
MatrixFormulation e (cid:4)Nð0,σ2I Þ:
i n
i
A more compact representation of the MRM is As a result, it can be shown that the expected
affordedusingmatricesandvectors.Thisformu- valueoftherepeatedmeasuresy is
i
lationhelpstosummarizestatisticalaspectsofthe
model.Forthis,theMRMfortheni(cid:2)1response EðyiÞ¼Xiβ (6)
vectoryforindividualicanbewrittenas:
andthevariance-covariancematrixofy isofthe
i
yi ¼ Xi β þ Zi vi þ ei (5) form:
ni(cid:2)1 ni(cid:2)p p(cid:2)1 ni(cid:2)r r(cid:2)1 ni(cid:2)1
VðyÞ¼ZΣ Z0þσ2I : (7)
with i = 1 ... N individuals and j = 1 ... n i i υ i ni
i
observations for individual i. Here, y is the
i Forexample,withr=2,n=3,and
n (cid:2)1dependentvariablevectorforindividuali,
i
X isthen (cid:2)pcovariatematrixforindividuali,β 2 3 2 3
i i 1 0 0 1 0
isthep(cid:2)1vectoroffixedregressionparameters, X41 1 15andZ ¼41 15
Z is the n (cid:2) r design matrix for the random i i
i i 1 2 2 1 2
effects,υ isther(cid:2)1vectorofrandomindividual
i
effects,and« isthen (cid:2)1residualvector.
i i
AnalysisofRepeatedMeasuresandLongitudinalDatainHealthServicesResearch 9
Theexpectedvalueofyis modelthecovariancestructurethroughtheuseof
CPMs.Thesemodelsareadirectoutgrowthofthe
2 3
β multivariategrowthcurvemodelsdescribedinthe
0
4 β þβ þβ 5 Historical Background section where the covari-
0 1 2
β0þ2β1þ2β2 ance structure of the repeated observations was
assumedtofollow ageneralformandallparam-
andthevariance-covariancematrixequalsσ2Inþ eters of the matrix were estimated. Rather than
i
estimating every parameter of the covariance
2 3
4 σ2υ0þσ2υ0συ0υ1 σ2υ0þσ2υ02þσυ0συ1υ0þυ1σ2υ1 σ2υ0þσ2υ03þσυ02υσ1υþ0υ12σ2υ15 mmaattrriixx,ofCtPhMersepaesastuemdoebstheervavtaiorinasnfcoel-lcoowvsaraiasnpcee-
σ2υ0þ2συ0υ1 σ2υ0þ3συ0υ1þ2σ2υ1 σ2υ0þ4συ0υ1þ4σ2υ1 cific structure. For example, the compound sym-
metry (CS) covariance model has only two
which allows the variances and covariances to parameters σ2 (variance) and ρ (correlation) and
changeacrosstime.Forexample,if συ0υ1 isposi- assumes that observations Yij have constant vari-
tive, then clearly the variance increases across
ance over time and the correlation between any
time. Diminishing variance across time is also
twoobservationsonthesamesubjectisthesame
possible if, for example, (cid:5)2συ0υ1 >σ2υ1. Other no matter how far apart those observations
patterns are possible depending on the values of
occurred.Avarietyofcovariancestructuresexist
thesevarianceandcovarianceparameters.
andareavailableinmostsoftwarepackages.See
Modelswithadditionalrandomeffectsarealso
Weiss(2005)fordetaileddescriptionsofanumber
possible, as are models that allow autocorrelated
ofdifferentcovariancematrices.
errors,thatis« (cid:4)N(0,σ2Ω).Here,Ωmight,for
i i Using the matrix notation in Eq. 5, a CPM
example, represent an autoregressive (AR) or
wouldbe
moving average (MA) process for the residuals.
Autocorrelated error regression models are com- y ¼Xβþe (8)
i i i
monineconometrics.Theirapplicationwithinan
MRM formulation is treated by Chi and Reinsel
Where instead of assuming the residuals are
(1989) and Hedeker (1989), and extensively independent, it is assumed « (cid:4) N (o, Ω). Some
i i
described in Verbeke and Molenberghs (2000). common choices for Ω include the previously
i
By including both random effects and auto-
mentioned compound symmetry where for three
correlated errors, a wide range of variance-
observationsonsubjectithecovariancematrixis
covariance structures for the repeated measures
0 1
ispossible. Thisflexibilityisinsharp contrastto 1 ρ ρ
the traditional ANOVA models which assume VðyÞ¼σ2@ρ 1 ρA
i
eitheracompoundsymmetrystructure(univariate ρ ρ 1
ANOVA) or a totally general structure
(MANOVA). Typically, compound symmetry is andtheparameterρisthecorrelationbetweenany
toorestrictiveandageneralstructureisnotparsi-
two observations on the same subject. An auto-
monious.MRMs,alternatively,providethesetwo
regressiveorAR(1)covariancestructurealsohas
andeverythinginbetween,andsoallowefficient
two parameters like the compound symmetry
modeling of the variance-covariance structure of
structurebuttakesonadifferentform,namely,
therepeatedmeasures.
0 1
1 ρ ρ2
VðyÞ¼σ2@ ρ 1 ρ A:
i
CovariancePatternModels ρ2 ρ 1
An alternative to using random effects to model
correlatedmeasurementsovertimeistoexplicitly
10 J.Siddiqueetal.
(cid:6) (cid:7)
Thus,thefartheraparttwoobservationsarein E y jTrt ¼1,t ¼2 ¼β þ2β (10)
ij i ij 0 1
time, the lower the correlation between them
(assumingρ>0).Ingeneral,CPMsapplystruc-
turebyspecifyingaspecificrelationshipbetween sincethemeanoftherandomeffectsandvariance
termsare0.Thusthedifferencebetweenthetwo
repeated observations on the same subject and
groupsis2β .Thevarianceforbothgroupsattime
assuming constant (homogenous) variance over 2
2is
time(thoughthehomogeneityofvariancecanbe
relaxed). (cid:6) (cid:7)
Whenchoosingacovariancemodelforrepeated Var yijjtij ¼2 ¼Varðv0iÞþ22Varðv1iÞ
measures data, one wishes to choose the most
þ4Covðv ,v Þ
parsimonious model that fits the data well. This (cid:1) (cid:3)0i 1i
þVar e (11)
can bedonebyfirstmodelingthe meanofobser- ij
vations over time and then using likelihood ratio
¼σ2 þ4σ2 þ4σ þσ þσ2: (12)
testsaswellasmodelfitindicessuchastheBayes- υ0 υ1 υυ υ0υ1
ian Information Criteria (BIC) and the Akaike
Information Criteria (AIC) to select the model Inmatrixnotation,thisiswrittenas
thatbestfitsthecorrelationandvariancestructure (cid:6) (cid:7)
ofthedata.Moredetailsonmethodsforassessing Var y jt ¼2 ¼½1 2(cid:6)Σ ½1 2(cid:6)T þσ2:
ij ij υ
and comparing model fit of the variance-
covariance structure are described by Wolfinger
Thus,theeffectsizeoftheinterventionattime
(1993)andGradyandHelms(1995).
2is
2β
CalculatingEffectSizes Effect Size¼ 2 :
σ2 þ4σ2 þ4σ þσ2
υ0 υ0 υ0υ1
EffectSizesforMixed-EffectsModels
EffectSizesforCovariancePattern
Itisoftenofinteresttosummarizeresultsfroman
Models
intervention in terms of effect sizes. The effect
sizeofaninterventionisdefinedasthedifference Calculating effect sizes for a covariance pattern
model is slightly different than for the mixed-
inmeansbetweentheinterventionandthecontrol
effectmodelinEq.16because,althoughitisnot
(or its comparator) divided by the standard devi-
necessary take into account the variance of the
ationoftheoutcome.Assumearandomintercept
randomeffects,theerrortermsarenolongerinde-
andslopeMRMasinEq.16,thatis
pendent.Themodelis
(cid:1) (cid:3)
y ¼β þβ t þþβ t (cid:2)Trt þυ þυ t (cid:1) (cid:3)
ij 0 1 ij 2 ij i 0i 1i ij y ¼β þβ t þβ t (cid:2)Trt þe (13)
ij 0 1 ij 2 ij i ij
þe
ij
where « (cid:4) N (0, Ω). As in Eqs. 9 and 10 the
i i
To estimate the effect size of the treatment differenceinpredictedmeansbetweentreatments
effectattime2,beginbycalculatingthepredicted andcontrolsis2β .Thevarianceforbothgroups
2
meanforasubjectinthetreatmentgroupattime attime2issimply
2(Trt =1t =2):
i ij (cid:6) (cid:7)
(cid:6) (cid:7) Var y jt ¼2 ¼Varðe Þ (14)
ij ij i3
E y jTrt ¼1,t ¼2 ¼β þ2β þ2β (9)
ij i ij 0 1 2
¼σ2 (15)
33
and the predicted mean for a control subject at
time2is
