Longitudinal Data with Follow-up Truncated by Death: Match the Analysis Method to Research Aims

0.28 MB·English

by  Brenda F. Kurland

StatisticalScience
2009,Vol.24,No.2,211–222
DOI:10.1214/09-STS293
(cid:13)c InstituteofMathematicalStatistics,2009
Longitudinal Data with Follow-up
Truncated by Death: Match the Analysis
Method to Research Aims
Brenda F. Kurland, Laura L. Johnson, Brian L. Egleston and Paula H. Diehr
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Abstract. Diverse analysis approaches have been proposed to distin-

n guish data missing due to death from nonresponse, and to summarize
a
J trajectories of longitudinal data truncated by death. We demonstrate
5 how these analysis approaches arise from factorizations of the distri-
1 bution of longitudinal data and survival information. Models are illus-
trated using cognitive functioning data for older adults. For uncondi-
]
E tional models,deathsdonotoccur,deathsareindependentofthelongi-
M tudinalresponse,ortheunconditionallongitudinalresponseisaveraged
over the survival distribution. Unconditional models, such as random
.
t
a effects models fit to unbalanced data, may implicitly impute data be-
t yond the time of death. Fully conditional models stratify the longitu-
s
[ dinal response trajectory by time of death. Fully conditional models

1 are effective for describing individual trajectories, in terms of either
v aging (age, or years from baseline) or dying (years from death). Causal
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models (principal stratification) as currently applied are fully condi-
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tional models, since group differences at one timepoint are described
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2 for a cohort that will survive past a later timepoint. Partly conditional
1. models summarize the longitudinal response in the dynamic cohort of
0 survivors.Partlyconditionalmodelsareserialcross-sectionalsnapshots
0
of the response, reflecting the average response in survivors at a given
1
timepoint rather than individual trajectories. Joint models of survival
:
v
and longitudinal responsedescribetheevolving health status of the en-
i
X tire cohort. Researchers using longitudinal data should consider which
r method of accommodating deaths is consistent with research aims, and
a
use analysis methods accordingly.
Key words and phrases: Censoring, generalized estimating equations,
longitudinal data, missing data, quality of life, random effects models,
truncation by death.
Brenda F. Kurland is Ph.D. Assistant Member, Clinical [email protected]. Brian L. Egleston is Ph.D.
Research Division, Fred Hutchinson Cancer Research Associate Member, Biostatistics Facility, Fox Chase
Center, 1100 Fairview Ave. N. D5-360, P.O. Box Cancer Center, 333 Cottman Ave., Philadelphia,
19024, Seattle, Washington 98109-1024, USA e-mail: Pennsylvania 19111-2497, USA e-mail:
[email protected]. Laura L. Johnson is Ph.D. [email protected]. Paula H. Diehr is Ph.D.
Statistician, Office of Clinical and Regulatory Affairs, Professor, Departments of Biostatistics and Health
National Center for Complementary and Alternative Services, Box 357232 University of Washington, Seattle,
Medicine, National Institutes of Health, 6707 Washington 98195-7232, USA e-mail:
Democracy Blvd. Suite 401, MSC 5475 Bethesda, [email protected].
Maryland 20892-5475, USA, e-mail:
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2 KURLAND,JOHNSON,EGLESTON ANDDIEHR
1. INTRODUCTION data reflect assumptions about survival. The real
dataillustratehowstandardanalysistechniquessuch
Research studies often collect information at mul-
asrandomeffectsmodelsandgeneralized estimating
tiple timepoints. For example, the Cardiovascular
equations(GEE)maybeappliedtoaddressdifferent
Health Study(CHS),anobservationalstudyof5888
research aims involving longitudinal data truncated
older adults, conducted annual assessments of car-
by death. Bias, estimation and efficiency are impor-
diovascular functioning and other health measures
tantissues fordataanalysis.However, wefocusonly
for up to 18 years. With these longitudinal mea-
onourprimaryinterest,theinterpretation ofregres-
surements, CHS data have been used to study dis-
sion model estimands. Although each model is used
easecourse(Kaplan et al.,2005),healthintheyears
to fit a slope and expected response values, these
leading up to a diagnosis (Diehr et al., 2001b) and
estimands for the longitudinal response are “apples
the natural history of aging (Burke et al., 2001;
and oranges,” not directly comparable dueto differ-
Diehr et al., 2002).
ent factorizations of longitudinal response and sur-
Missing data can be an impediment to interpret-
vival.
ing longitudinal data. Suppose a cohort of 200 sub-
jects report self-rated health at age 70 years, and
2. BACKGROUND: FOLLOW-UP CENSORED
only 150 of these subjects are located for follow-up.
BY NONRESPONSE (DROPOUT)
If the average self-rated health at age 75 is higher
thantheaverage at70,theincreasecouldreflectim- Abriefreviewofmodelsforlongitudinaldatawith
provement in individuals’ health, attrition of sicker monotone dropout provides a foundation for dis-
participants or death of sicker participants. cussing analysis of longitudinal data truncated by
Many analysis methodsfor longitudinal datawith death. Two common, widely applied analysis meth-
dropoutandnonresponsehave beenproposedtoad- ods for longitudinal data are random effects mod-
dress different research aims, study designs, missing els (Laird and Ware,1982)and generalized estimat-
data patterns and estimation strategies (Little and ing equations (GEE) (Liang and Zeger, 1986). By
Rubin, 1987; Rubin, 1987; Robins, Rotnitzky and modeling a structure for the correlation between
Zhao, 1995). Comparatively little work in statisti- subjects’ longitudinal responses, a correctly speci-
cal methodology has addressed data missing when fied random effects model will yield consistent, un-
deaths occur during the period of follow-up, and re- biased estimates of regression parameters by max-
centwork hasfocusedmostly on principalstratifica- imum likelihood estimation, even with unbalanced
tion (Frangakis and Rubin, 2002; Frangakis et al., data (Laird, 1988). For example, if sicker partici-
2007). Kurland and Heagerty (2005) characterized pants drop out, their trajectory of decline in self-
targets of inference for longitudinal data truncated ratedhealthiscontinuedimplicitlybyawell-specified
bydeathbyfactorizationsofthejointdistributionof random effects model. If trends for dropouts can be
survivalandlongitudinalresponse,f(S,Y).Thefac- inferred from observed data and parameters for lon-
torization was introduced primarily to provide con- gitudinal response and dropout are distinct (miss-
text for a single approach, the partly conditional ing at random, such as when scores decline before
mean model. In this article, we explore several tar- dropout), the missingness is ignorable and the over-
gets of inference in detail, and give guidance on ap- all rate of change may be analyzed as if no one has
propriate analysis techniques for common scientific dropped out. If the decline in health that leads to
questions arising for longitudinal data truncated by dropout starts after the last recorded measurement,
death. We present six modeling options, illustrated then dropout is nonignorable, and random effects
using both hypothetical and actual CHS data. The modelsarenotaneasysolution.Untestableassump-
hypothetical data without measurement error illus- tions must be made about nonignorable dropout
trate clearly how modeling choices for longitudinal processestomodellongitudinaltrends(Laird,1988).
GEE can accommodate data missing at random if
estimating equations are weighted by the inverse
This is an electronic reprint of the original article
probabilityofdropout(Robins, Rotnitzky and Zhao,
published by the Institute of Mathematical Statistics in
Statistical Science, 2009, Vol. 24, No. 2, 211–222. This 1995). Giving additional weight to observed data
reprint differs from the original in pagination and for people who were likely to drop out is similar to
typographic detail. implicit or explicit imputation of unobserved data.
LONGITUDINALDATATRUNCATEDBY DEATH 3
In fact, under some conditions, weighted GEE and Table 1
Longitudinal 3MSE scores for 4 hypothetical CHS
imputation will give the same results (Paik, 1997).
participants (X = deceased)
Missing at random is often a reasonable assump-
tion, especially when longitudinal observations are
Age
closelyspacedrelativetomechanismsactingonboth
Participant 70 71 72 73 74 75
dropout and response. For example, preclinical cog-
nitive changes could likely be detected by annual A (“normal”) 90 90 90 90 90 90
assessments before a CHS participant becomes im- B (“mild cognitive impairment”) 84 82 80 78 76 74
C (“terminal decline”) 84 80 76 X X X
paired by dementia in a way that would lead to
D (“terminal decline”) 65 50 35 X X X
nonresponse. However, analysis of longitudinal data
with MAR dropout still requires accurate modeling
oftheregressionmodel(fixedeffects)andeithercor-
scores (5174 of 31093) were missing due to partici-
relation (for random effects models) or dropout (for
pant nonresponse, but most participants with miss-
weighted GEE) (Kurland and Heagerty, 2004).
ingdatahadonlyoneortwoscoresmissing(n=948
participants). Someparticipants haddroppedoutof
3. DATA EXAMPLES AND NOTATION
the study, and were missing 7 or more scores each
3.1 Cardiovascular Health Study (CHS) (n=134). Nonresponse was accommodated by sin-
gleimputationusingtheMarkovChainMonteCarlo
The Cardiovascular Health Study (CHS) was a
method of PROC MI in the SAS/STAT software,
population-based prospective longitudinal study of
version 9.1 (SAS Institute, Inc., Cary, NC). Impu-
5888 adults aged 65 years and older at baseline
tationwasstratifiedbytimeofdeath(fordecedents)
(Fried et al., 1991). Cognitive functioning was as-
and recruitment group (for nondecedents), and was
sessed annually for up to 10 years by the Modi-
modeled based on observed 3MSE values, baseline
fied Mini-Mental State Examination (3MSE, scored
age, gender and recruitment group.
from 0 to 100) (Teng and Chui, 1987). Our primary
Accommodation of deaths in CHS data also will
goalintheCHSanalysisistodescribethetrajectory
be described using simplified hypothetical data. Ta-
of cognitive functioning (3MSE) over time, and to
estimate 3MSE scores at different ages. Gender ef- ble 1 shows 3MSE data for 4 hypothetical partici-
fects areexploredin each analysis toaddabetween- pants with a baseline age of 70. Participant A, rep-
personvariableofinteresttothelongitudinal(within- resenting normal cognitive functioning, has a 3MSE
person changes) model, and to explore the effects of 90 points at all assessments. Participant B’s lin-
of differential survival on different regression ap- ear decline from 84 to 74 points over 5 years reflects
proaches for longitudinal and survival data. We ex- a possible trajectory of mild cognitive impairment
amine the 3814 participants aged 70 years and older (whichcouldbeinterpretedaspreclinicalAlzheimer’s
at baseline. In this cohort, 1356 participants (36%) disease).Participants CandDbothdeclinebetween
died during follow-up: 44% (744/1703) of men and baseline and age 72, and die before age 73.
29% (612/2111) of women. We will examine how
3.2 Notation
different analysis methods each yield 3MSE trajec-
tories (rates of change) and fitted values (expected VectorYi representsthelongitudinalresponse(i.e.,
cognitive status at specific ages), but address differ- cognitive functioningor quality of life), measured at
ent research aims. multiple timepoints for participant i. The dimen-
Although models may be constructed to accom- sion of Yi may differ for individuals (values of i),
modate both deaths and nonresponse due to death. For example, in Table 1 (hypotheti-
(Kurland and Heagerty, 2005), we imputed data cal CHS data), responses for participant A, YA, are
missing due to nonresponse for simplicity of presen- the vector (90,90,90,90,90,90), and for participant
tation.Datawerenotconsideredmissingiffollow-up C, YC, are (84,80,76). Scalar variable Si represents
was truncated by death, or censored by the end of survival time for participant i, such as age at death
the study period. (A later cohort to boost minority or weeks from baseline until death: in Table 1, SC
recruitment received 6 annual assessments instead is 73 years. The dimension of Y is determined by
i
of10.)Mostparticipants(n=2061,54%)completed the value of S , but is not in principle affected by
i
allscheduled3MSEassessments.About17%of3MSE data missing due to nonresponse. In practice, the
4 KURLAND,JOHNSON,EGLESTON ANDDIEHR
Table 2
Summary of statistical models for longitudinal response and survival (time of death)
Statistical model Sampleresearch setting Primary analysis Comments
method
a. Unconditional f(Yi)
Describe Yi (longitudinal re- Rate of local recurrence follow- Mixed effects / ran- May implicitly impute
sponse) in setting where sur- ing ablation of liver tumors dom effects / latent data beyonddeath
vival (Si) is unrelated to Yi, variable regression
or when death does not result
in missing data
b. Fully conditional: pattern-mixture f(Yi|Si=s)
Describe Yi separately for Longitudinal change in physi- Mixed effects / ran- Describes individual tra-
groups defined by survival calfunctioningfollowing stroke, dom effects regression jectories, but uses future
time separatelyfor<1,1–5,and>5- stratified by survival survival information to
year survivors time predict earlier responses
c. Fully conditional: principal stratification f(Yi(z)|Si(0)>s,Si(1)>s)
Describe average causal treat- Average QOL difference for Weighted averaging of Weights are unidentifi-
ment effect for stratum that toxic treatment with greater estimated outcomes able and must be ex-
would survive past time s re- survival, versus less toxic treat- from models such ploredthroughstrongas-
gardless of treatment ment with lesser survival, in as generalized linear sumptions and/or sensi-
subjects expected to live 6+ models tivity analysis
months on either treatment
d. Fully conditional: terminal decline f(Yi|Si=s)
Describe Yi counting back- Terminal decline studies Mixed effects / ran- Time scale is retrospec-
ward from time of death dom effects regression tive
e. Partly conditional: regression conditioning on being alive f(Yi|Si>t)
DescribeYiinthedynamicco- Average physical functioning in GEE with indepen- Describes longitudinal
hortofsurvivorsateachtime- survivors at 6 months and 5 dence working correla- trend of dynamic cohort,
point years after stroke tion not individuals
f. Joint model f(Yi|Si)
Describe both Yi and Si for Percent of stroke patients who Logistic regression, Continuous longitudinal
example,“probabilityofbeing are alive and can perform self- GEE (binary out- outcomesmayneedtobe
alive and healthy” care 6 monthsafter stroke come); specialized categorized for analysis
multiple response
methods
analysis in Section 4 uses imputation of data miss- the length of Y and is not fixed, regression models
i
ing due to nonresponse to ensure that the observed of longitudinal data truncated by death must ex-
responsevectorisoftheproperdimension.Thejoint plicitly or implicitly model survival as well. A single
distribution f(Yi,Si) describes the probability that regression model could be built for the joint dis-
Yi takes on a vector of specific values, and that par- tribution f(Yi,Si), or for factorizations based on
ticipant i dies at a specific time. We assume that thedefinitionsofjointandconditionaldistributions:
(Yi,Si) are independent and identically distributed f(Yi|Si)f(Si) or f(Si|Yi)f(Yi). We will character-
over i. ize models for Y as unconditional, fully conditional
i
or partly conditional based on how, or whether, the
4. STATISTICAL MODELS FOR
longitudinalresponsemodelconditions on S .These
i
LONGITUDINAL RESPONSE AND SURVIVAL
models are defined in more detail below, and sum-
Regression models for longitudinal data describe marized in Table 2. Each model is applied to the
the relationship between predictors and the longitu- hypothetical (Table 3) and actual (Figure 1) CHS
dinal response, Y . Because survival S determines data. Models are fit to data for all participants, but
i i
LONGITUDINALDATATRUNCATEDBY DEATH 5
Table 3
Hypothetical CHS data (Table 1) estimated age 75 3MSE score and 3MSE slope, using models that accommodate deaths in
different ways. Even a predicted value for the response, such as “3MSE at age 75,” represents a different estimand for each
model
Sample research question(s) 3MSEat age 75 Linear3MSEslope(annualchangein3MSE)
a. Unconditional f(Yi)
What is the expected 3MSE at age 75, in an 90+74+64+(−10) =54.5 0+(−2)+(−4)+(−15) =−5.25 points peryear
4 4
immortal cohort? points
b. Fully conditional: survivors f(Yi|Si>75)
What is the expected 3MSE at age 70 or at 90+74 =82 points 0+(−2) =−1.0 points peryear
2 2
age 75, for people who live to be at least 75?
Fully conditional: decedents f(Yi|Si≤75)
Whatistheexpected3MSEatage70forpeo- (bothdeceasedatage75) −4+(−15) =−9.5 points per year
2
ple who die at age 71–75?
c. Fully conditional: principal stratification f(Yi(z)|Si(0)>9,Si(1)>9)
Whatisthecausaleffectofgenderonexpected (causal effect not estimated for hypothetical data)
3MSE scores at the fourth or ninth survey
among those who would live to theninth sur-
vey regardless of gender?
d. Fully conditional: terminal decline f(Yi|Si=s)
What is the expected 3MSE two years before (not estimated directly) −9.5 points per year (same as row b
death? decedents—changing time scale does not
makeadifferencewith only one stratum of
decedents)
e. Partly conditional: regression conditioning on being alive f(Yi|Si>t)
Whatistheexpected3MSEatage70forpeo- 76.2 + 0.92∗5 = 80.8 0.92 pointsper year
ple wholive tobeat least 70, or at age 75 for points
people who live tobe at least 75?
f. Joint model f(Yi,Si)
What is the probability of being alive and 1 aliveand3MSE≥80at on average 1 lose health/life each year
4 10
healthy at age 75 for people who were alive age 75 (since 3 healthy at age 70, and 1 healthy
4 4
at age 70? at age 75)
Figure 1 shows fitted 3MSE values for a baseline distribution f(Y ) reflects averaging f(Y |S ) over
i i i
age of 70 years. Clearly the estimators (functions of the survival function f(S ), as demonstrated below.
i
sample data) are different for the 6 methods shown Unconditional regression models cognitive function-
for finding 3MSE fitted values and slopes. However, ing at all timepoints as if nobody died, in an “im-
we emphasize that the estimands are also different: mortalcohort”(Dufouil, Brayne and Clayton,2004).
the linear slope in an unconditional model is not The unconditional average 3MSE at age 75 years in
the same as the linear slope in a fully conditional
the CHS hypothetical data is as follows:
model. We assumeeach estimator is unbiased for its
estimand, and focus on interpretation of the esti- E(Y|age=75)
mands.
=E(Y|age=75,S >75)·P(S>75)
4.1 f(Yi) Unconditional
+E(Y|age=75,S ≤75)·P(S ≤75)
An unconditional model, f(Y ), is appropriate if
i 90+74 2 2
deathsdonotoccur,areindependentoftheresponse = · +?· .
2 4 4
process, or do not result in truncation (if the re-
sponse has a well-defined value following death). If For analysis methods for which “missing at ran-
these stipulations are not met, the unconditional dom” nonresponse mechanisms are ignorable—such
6 KURLAND,JOHNSON,EGLESTON ANDDIEHR
Fig. 1. Fitted Modified Mini-Mental State Examination (3MSE) trajectories for CHSparticipants aged 70 years at baseline,
using different models to summarize longitudinal response (3MSE) and survival. Fitted values over time are shown as solid
lines for males, and as dashed lines for females.
as for likelihood-based modelsfitto unbalanced lon- wouldbewithreal data,sinceestimation of fixedef-
gitudinal data—the value of ? is imputed implic- fects in random effects models will be influenced by
itly because of the structure imposed by correla-
regressiontothemeanandothershrinkage(Robinson,
tion between each subject’s longitudinal observa-
1991).ExtendingtheTable1trajectories forPartic-
tions (Laird, 1988). We convey this implicit impu-
ipants C and D linearly, the “incomplete” response
tation by projecting the hypothetical 3MSE data
of deceased participants based on individual slopes. vectors (84,80,76) and (65,50,35) are imputed to
This extrapolation is more extreme than estimates (84,80,76,72,68,64) and(65,50,35,20,5,−10).Par-
LONGITUDINALDATATRUNCATEDBY DEATH 7
ticipantD’simputedresponseatage75(−10points) and 77 points at age 79. The unconditional model
is inappropriate, outside the range of the 3MSE. suggestsratherstrongage-associateddeclinesincog-
An unconditional model uses both observed and nitivefunctioning,buttheestimandisthetrajectory
implicitly imputed data to estimate linear 3MSE foranimmortalcohortinwhichtruncationbydeath
slope and 3MSE at age 75 (Table 3, row a). Com- does not occur.
pleting the estimate of 3MSE at age 75, In rare cases, implicit imputation beyond death
may be reasonable. When evaluating local recur-
90+74 2 64+(−10) 2
E(Y|age=75)= · + · rence following ablation of liver tumors, some liv-
2 4 2 4
ers may “die” due to transplant. Tumor marker lev-
=54.5.
els (Y ) and transplant candidacy (S ) are related.
i i
The age 75 fitted 3MSE (54.5 points) and linear However,sincethetransplantratewillneverbe100%,
3MSE slope (5.25 point decline per year) both yield an unconditional model and implicit imputation be-
lower estimates of cognitive functioning at age 75 yondtransplantmaybevalid.Theresearchquestion
than are observed for participants alive at age 75, addressed is, “What would the tumor marker levels
because 3MSE values are imputed beyond death. be if no transplants had occurred, but the markers
The unconditional model of 3MSE scores by age hadcontinuedalongtheexactpaththatledtotrans-
in the CHS data (Figure 1a) is a random effects lin- plant?”Whilethisquestionisrelevanttolivertrans-
ear regression, separating the effects of age and ag- plants, the CHS examples show that unconditional
ing (Neuhaus and Kalbfleisch, 1998) in a quadratic models are generally inappropriate for longitudinal
model with random intercept and first-order poly- data with considerable imbalance due to death.
nomial:
4.2 f(Yi|Si =s) Fully Conditional
3MSEij =β0+b0i+β1·malei+β2·age0i Sometimesonlysubjectswhosurvivetotheendof
2
+(β3+b1i)·yearij +β4·yearij the study are included in analysis, or decedents are
analyzed separately from nondecedents. An anal-
2
+β5·malei·yearij +β6·malei·yearij ogyinthemissingdataliteratureispattern-mixture
+ε , models (Little, 1995; Fitzmaurice and Laird, 2000),
ij
whichstratifybythetimeofdropout.Pattern-mixture
where age0 is the baseline age of participant i (in
i models may be fit using the same methods as un-
years),andyear isthestudyyear(yearssincebase-
ij conditional models (random effects regression, etc.)
line age) for participant i at timepoint j. male is a
i but are made fully conditional by fitting separate
dichotomousvariable(withvalue0iftheparticipant
regressionmodelstostratadefinedbytimeofdeath.
is female and value 1 if the participant is male), and
Generally a categorical variable defined by
random intercept, slope and error (b0i, b1i, and εij) survival time is used as a main effect (and inter-
are normally distributed with mean 0. Likelihood-
action term) in regression models,so that longitudi-
basedmethodssuchasrandomeffectsregressionwill
nal trajectories are fit for groups defined by time of
fit an unconditional model, since they treat any im-
death (Ribaudo, Thompson and Allen-Mersh,2000;
balance in the data as “missing at random.” Inter-
Pauler, McCoy and Moinpour,2003).Anadvantage
actions between sex and linear and quadratic terms
of this approach is accurate representation of in-
are included as potential predictors of interest. Fig-
dividuals’ scores over time. Principal stratification
ure 1a shows the fitted “average” trajectories for
(Frangakis and Rubin,2002;Rubin,2006)examines
males and females (thick lines, b0i =b1i =εij =0), causaleffectsbyusingcounterfactualsurvivaltocre-
and several fitted trajectories for individuals (thin
ate strata. Another stratification based on time of
lines, selected b0i and b1i). The random intercepts death examines the “dying process” for decedents
andslopesallowawiderangeofindividualfittedtra-
only, with years until death as the timescale for ex-
jectories. However, time trends and other covariate
aminingterminaldecline(Siegler,1975;Diehr et al.,
effects are generally interpreted based on the mean
2002; Wilson et al., 2003).
model (thick lines), which reflects both observed
dataanddataimplicitly imputedbeyonddeath.Ac- 4.2.1 Pattern-mixture. Computing linear slopes
cordingtothisaveragetrajectory,theexpected3MSE separately for decedents and survivors in the hy-
for both males and females is 86 points at age 75, pothetical CHS data (Table 3, row b) demonstrates
8 KURLAND,JOHNSON,EGLESTON ANDDIEHR
minimal decline in survivors [1 point per year] and years of study enrollment, and 0 otherwise.
terminal decline for decedents [row b(2), 9.5 points LetS (z)similarlydenotethepotentialsurvivaltime
i
per year]. However, since the time of death is not for person i with gender z, and S (z) determines
i
known in advance, these models could not be used the dimension of Y (z). We are interested in the
i
to predict an individual’s trajectory based on base- 3MSE scores among those in whom D (0)=0 and
i
line information. Di(1)=0. Let Xi represent a vector of potentially
In a pattern-mixture model fit to CHS data, confoundingcovariates;weonlyincludeageasacon-
quadratic time trends are stratified by year of death founding covariate. To identify potential outcomes
relative to baseline. Figure 1b shows fitted mean ateachtimepoint,wemaketheexplainablenonran-
3MSE trajectories for a baseline age of 70 years. dom survival assumption and use the estimator of
The pattern-mixture model demonstrates terminal Hayden, Pauler and Schoenfeld (2005). Explainable
decline in participants who died (fitted lines that nonrandomsurvivalassumesthatDi(z)⊥Di(1−z)|Xi
end before age 79), and reasonably stable cognitive and Di(z)⊥Yi(1−z)|Xi,Di(1−z)=0 where ⊥ rep-
functioning in participants who survived. The fit- resents conditional independence. This assumption
ted 3MSE at age 75 ranged from 75 points in males heuristicallystatesthatcounterfactualoutcomeswill
who died by age 76, to 91 points in males and fe- be conditionally independent of the observed out-
males who enrolled at age 70 and survived at least comes. We need such an assumptionsince we donot
to age 79. The fitted mean trajectories are closer to observe counterfactual outcomes for study
trajectoriesobservedforindividualsthantheuncon- participants; for example, we do not observe 3MSE
ditional model, but require conditioning on survival scores that women in the study would have had if
time, which is not known at baseline. they had been born men. We also make a pseudo-
randomization assumption that the potential out-
4.2.2 Principal stratification. Another fully con-
comes are independent of gender given X as de-
i
ditional approach is to estimate causal effects for
tailedinEgleston et al.(2007).Pseudo-randomization
selected principal strata defined by potential
basically states that the “assignment” of gender is
survival outcomes (Frangakis and Rubin, 2002;
analogous to a random coin toss given a set of con-
Hayden, Pauler and Schoenfeld,2005;Egleston et al.,
foundingcovariates. We appreciate that somemight
2007; Egleston, Scharfstein and MacKenzie, 2009). feel that gender is inappropriate for such a causal
Men are generally more likely to die than women analysis even as a didactic example, since gender
at any given age. Do women have lower incidence cannot be manipulated (Holland, 1986).
of mental and physical decline, or are they able to Figure 1c shows fitted 3MSE scores for partici-
survive with greater deficits than men? By examin- pants aged 70–75 at baseline who would survive 9
ing only a hypothetical group that would survive to yearsregardlessofgender.Menandwomenareboth
a certain timepoint regardless of gender, we can de- predicted to have stable scores on average for four
couple the association of gender and cognitive func- years after baseline, and to decline an average of
tion from the association of gender and survival. 3.7 points in men and 5.0 points in women over
We estimate the association of gender and cognitive the following five-year period. Trajectories for men
function only in the stratum of patients expected to and for women look similar to the survivor cohort
live nine years from age 70–75 regardless of gender. in the pattern-mixture model (Figure 1b). In other
The interpretation of these fully conditional models words, when survival is held comparable for both
differs from pattern-mixture models, because prin- genders, men may show an advantage in cognitive
cipal stratification models are conditioned on both functioning. This could reflect an ability for women
observed and counterfactual survival status. Gener- to survive when cognitive functioning is diminished.
ally, untestable assumptions are necessary to iden- Cognitiveimpairmentmayoccuratthesameratein
tify effects within the principal strata. menandwomen,butmaybeassociatedwithgreater
Specification of principal stratification models re- mortality risk in men.
quires additional notation to describe potential out- One concern with this approach is that explain-
comes. For z={0 for women, 1 for men}, let D (z) able nonrandom survival is a very strong
i
be an indicator for potential death by a specified assumption. A number of sensitivity analysis ap-
time and Y (z) denote potential 3MSE scores. For proaches are available to investigate whether devia-
i
this analysis, D (z) = 1 if a person dies within 9 tions from this assumption could influence
i
LONGITUDINALDATATRUNCATEDBY DEATH 9
inferences (Hayden, Pauler and Schoenfeld, 2005; on the subjects being alive at time t . This con-
ij
Egleston et al., 2007; Egleston, Scharfstein and ditioning may seem trivial: after all, data are not
MacKenzie, 2009). Still, some investigators have collected posthumously. However, as is well docu-
advocated that the use of relatively strong mented for data missing due to nonresponse (Laird,
assumptions to uniquely identify principal strata ef- 1988), analysis methods that model the correlation
fects is justifiable (Joffe, Small and Hsu, 2007; structure of longitudinal data (such as mixed mod-
Elliott, Joffe and Chen, 2006). els) will implicitly impute responses, whether miss-
Theprincipalstratificationapproachisbettersuited ing due to dropout or death (Dufouil, Brayne and
to estimating causal effects for interventions than Clayton,2004;Kurland and Heagerty,2005).Partly
to describing the prognosis for individuals. Survival conditional regression models may be estimated by
status and nonidentifiable assumptions about coun- assuming independence among the longitudinal re-
terfactual survival are both part of the estimand for sponses. In that sense, they may be fit using lin-
causal effects within each principal stratum. ear regression or generalized linear models. How-
ever, generalized estimating equations (with inde-
4.2.3 Terminal decline. A third fully conditional
pendence working correlation) allow estimation of
CHS model examines terminal decline. Rather than
sandwich standard errors.
counting forward in years of age, the time scale for
Partly conditional “regression conditioning on be-
this analysis counts backward from death. The2458
ing alive” (Kurland and Heagerty, 2005) (RCA) de-
participants who are alive at the end of follow-up
scribes3MSEscoresatdifferentagesamongthesur-
(64%) are excluded, since their age at death is not
viving participants. For the hypothetical CHS data,
known.As inearlier models,arandomeffects model
RCA estimates of average 3MSE at age 75 and lin-
is fit with quadratic time and random intercept and
ear 3MSE slope are calculated using linear regres-
slope. However, the time scale is changed:
sion (Table 3, row e). Implicitimputation is avoided
3MSEij =β0+b0i+β1·malei+β2·age0i by treating observations from the same person as
+(β3+b1i)·yr fr deathij independent.RCA accurately shows that the preva-
lent cognitive functioning level is slightly higher at
2
+β4·yr fr deathij age 75 (82 points is the average 3MSE for survivors
A and B, estimated as 80.8 by imposing a single
+β5·malei·yr fr deathij
linear slope to all observed data), compared to age
2
+β6·malei·yr fr deathij +εij, 70 (80.8 point average for A–D, estimated as 76.2
by linear regression). The partly conditional slope
where yr fr death ranges from −1 (one year before
predicts that average 3MSE increases 0.92 points
the year of death) to −9 (9 years before), and other
per year, despite that no individuals have increas-
variables are as described in Section 4.1. Figure 1d
ing 3MSE scores. Partly conditional regression re-
showsfittedaverage terminaldeclinetrajectories for
flects 3MSE in the dynamic cohort of survivors, not
menandwomenwithbaselineage70.Theestimated
individual subjects change in cognitive functioning.
rate of decline is about 4.7 points per year, plus
ThepartlyconditionalRCAmodelavoidsimplicit
the effect of a negative quadratic coefficient (−0.3).
imputation of data for deceased subjects, and de-
Thefitted 3MSE score 6 years before death is about
scribeslongitudinal3MSEforthedynamiccohortof
87 points, close to the average baseline value (88
survivors (Dufouil, Brayne and Clayton, 2004;
points) for the 70-year-olds at baseline. For this co-
Kurland and Heagerty, 2005). The regression equa-
hort of decedents, averaging over yr fr death and
tion is similar to that for the unconditional and
male, the expected 3MSE at age 75 is 85 points,
pattern-mixturemodels,butdoes notincludearan-
and at age 79 is 82 points. The rate of terminal de-
dom intercept or slope:
clinereflectsthecombinedinfluenceofthemostdra-
matic declines and the more stable 3MSE patterns 3MSEij =β0+β1·malei+β2·age0i+β3·yearij
observed in the multiple pattern-mixture fitted tra-
2
jectories for decedents. +β4·yearij +β5·malei·yearij
2
4.3 f(Yi|Si >t) Partly Conditional +β6·malei·yearij +εij.
For partly conditional models, the expected value Figure 1e shows expected 3MSE scores for partici-
of Y (response of subject i at time t ) conditions pants who entered the study at age 70, given that
ij ij
10 KURLAND,JOHNSON,EGLESTON ANDDIEHR
they were alive at the time 3MSE was measured. aroundthePAHestimate (Johnson,2002).Theper-
The difference in expected 3MSE at different ages cent alive and healthy is not always decreasing: in-
is smaller than for the unconditional or fully condi- dividuals may regain “health,” such as when a low
tional models. The expected 3MSE score of partic- 3MSE score was due to short-term side effects of
ipants who entered CHS at age 70 is 91 points for medication. Like the partly conditional model, the
surviving 75-year-olds, and 87 points for surviving joint model describes the cohort, rather than trends
79-year-olds. However, this does not imply an aver- for individuals.Unlike the partly conditional model,
age decline of 1 3MSE point per year, since the av- the entire cohort is described at each timepoint, not
erage for survivors is not the same as the trajectory only survivors. (The end of follow-up for the minor-
from following individuals. The partly conditional ity recruitment group could affect differences seen
model tracks the prevalent average 3MSE score in between ages 70–76 and 77–79. We avoid implicit
the survivors at each timepoint. imputation beyond 6 years by computing the em-
pirical PAH as a simple proportion.)
4.4 f(Yi,Si) Joint Model
For participants aged 70 years at baseline, the
A joint response encompassing both survival and probability of being alive and having 3MSE ≥ 80
the longitudinal response also may be of interest. at age 75 is 0.82 for females and 0.75 for males.
A patient facing a diagnosis may ask not only the At age 79, the PAH is 0.70 for females and 0.54
chance of 5-year survival but the chance of 5-year for males. Summing the area under the curve, the
survivalwithacceptablequalityoflife.Ajointmodel average years of healthy life (of 9 possible) is 7.6
of the probability of being alive and healthy for females, and 6.7 for males (Diehr et al., 1998).
(Diehr et al., 1995) characterizes the status of the The average gender difference appears to be greater
entirecohortwithrespecttoalongitudinalresponse for the joint model than for the other fitted mod-
anddeath.Arelatedapproachrescalesresponsemea- els in Figure 1. This reflects a survival advantage
sures to predict the probability of being alive and for females, which was not apparent in models that
healthy in a prescribed amount of time, such as focused on the 3MSE.
one year (Diehr et al., 2001a). Because the joint re-
4.5 Other Factorizations
sponse (alive and healthy) is defined at all time-
points for all individuals, longitudinal data are bal- A factorization of the joint distribution f(Y ,S )
i i
anced. Therefore, analysis methods (random not yet discussed is f(S |Y )f(Y ). This framework
i i i
effects,GEE,etc.)willnotbeaffectedbydifferential is especially applicable to predicting survival (S )
i
survival. Joint models also may assess usinginformationfromlongitudinalbiomarkers(Y )
i
treatment effects simultaneously for longitudinal re- (De Gruttola and Tu, 1994; Wulfsohn and Tsiatis,
sponse and survival (Gray and Brookmeyer, 2000;
1997;Ye, Lin and Taylor,2008).Applyingthismodel
Ratcliffe, Guo and Ten Have,2004),orintegratemor-
to hypothetical CHS data, we could conclude that
bidity and mortality in utility measures such as
33% (1/3) of participants with declining 3MSE sur-
quality-adjusted life years.
vive to age 75, while 100% (1/1) with stable 3MSE
Defining 3MSE scores ≥ 80 points as healthy, the
survive to age 75. This class of models would be
probability of being alive and healthy at age 75 in
categorized as unconditional in the framework dis-
the hypothetical CHS data is 1/4 (Table 3, row f),
cussed here, but is not considered in detail because
which reflects adecline in the cohort, since 3/4 were
survival, not longitudinal data, is the primary re-
alive and healthy at baseline (see Table 1). Assum-
sponse of interest.
ing a linear trend, the decline from 3/4 healthy to
1/4 healthy over 5 years reflects a rate of decline of
5. DISCUSSION
1/10 of the cohort losing health or life each year.
Figure 1f shows the proportion of CHS partici- Through analysis of hypothetical and actual data
pants who are alive and healthy at ages 70–79. The sets, we have shown that choice of analysis has a
percent alive and healthy (PAH) shown is a pro- greatinfluenceoninterpretationoflongitudinaldata
portion reflecting the status of study participants truncated by death. No single approach is appro-
enrolled at age 70. Modeling of PAH also is pos- priate in all situations, so the analysis should be
sible, as well as constructing confidence intervals chosen to address the aims of a research project.