Essential Statistical Methods for Medical Statistics

355 Pages·2010·2.809 MB·English

by  J Philip Miller

Essential Statistical Methods
for Medical Statistics
A derivative of Handbook of Statistics:
Epidemiology and Medical
Statistics, Vol. 27
Edited by
C.R. Rao
Center for MultivariateAnalysis
Departmentof Statistics, ThePennsylvania StateUniversity
University Park,PA, USA
J.P. Miller
Division ofBiostatistics
School of Medicine, Washington University inSt. Louis
St. Louis,MO, USA
D.C. Rao
Division ofBiostatistics
School of Medicine, Washington University inSt. Louis
St. Louis,MO, USA
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Contributors
Bang, Heejung, Division of Biostatistics and Epidemiology, Department of Public
Health weill Medical college of Cornell University, New York, NY 10021;
e-mail: [email protected] (Ch. 9).
Bentler,PeterM.,UniversityofCalifornia,LosAngeles,UCLAPsych-Measurement,
Box 951563, 4627 FH, Los Angeles, CA 90095-1563; e-mail: [email protected]
(Ch.7).
Edwards, Lloyd J., Department of Biostatistics, CB 7420, University of North
Carolina,ChapelHill,NC27599-7420;e-mail:[email protected](Ch.5).
Edwards, Michael C., The Ohio State University, 1827 Neil Ave, Columbus, OH
43210; e-mail: [email protected] (Ch. 6).
Glidden, David V., Division of Biostatistics, Department of Epidemiology and
Biostatistics, University of California, San Francisco, 185 Berry Street, 5700,
San Francisco, CA 94143; e-mail: [email protected] (Ch. 3).
Greene,WilliamH.,NewYorkUniversity,HenryKaufmanMgmtCtr,44West4
Street, 7-78, New York City, NY 10012, NYU Mail Code: 0251; e-mail:
[email protected] (Ch. 4).
Gurka, Matthew J., Division of Biostatistics and Epidemiology, Department of
Public Health Sciences, School of Medicine, University of Virginia, Char-
lottesville, VA 22908-0717; e-mail: [email protected] (Ch. 5).
Hagan, Joseph L., Biostatistics Program, School of Public Health, Louisiana
State University Health Sciences Center, 2021 Lakeshore Dr., Suite 210, New
Orleans, LA 70122; e-mail: [email protected] (Ch. 2).
Hayashi, Kentaro, Department of Psychology, University of Hawaii at Manoa,
2430CampusDrive,Honolulu,HI96822;e-mail:[email protected](Ch.7).
Herna´ndez,Bernardo,DirecciondeSaludReproductivaCentrodeInvestigacionen
Salud Poblacional, Instituto Nacional de Salud Publica, Cuernavaca, Morelos,
Mexico 62508; e-mail: [email protected] (Ch. 10).
Hilbe,JosephM.,SchoolofSocialandFamilyDynamics,ArizonaStateUniversity,
Tempe, AZ 85232; e-mail: [email protected] (Ch. 4).
Horton, Nicholas J., Department of Mathematics and Statistics, Smith College,
Burton Hall 314, Northampton, MA 01063; e-mail: [email protected]
(Ch. 10).
ix
x Contributors
Laird, Nan M., Department of Biostatistics, Harvard School of Public
Health, Bldg 2, 4th FL, 655 Huntington Ave, Boston, MA 02115; e-mail:
[email protected] (Ch. 10).
Lecoutre, Bruno, ERIS, Laboratoire de Mathe´matiques Raphael Salem, UMR
6085 C.N.R.S. and Universite´ de Rouen, Avenue de l’Universite´, BP 12, 76801
Saint-Etienne-du-Rouvray (France); e-mail: [email protected]
(Ch. 11).
Litman,HeatherJ.,NewEnglandResearchInstitutes,9GalenStreet,Watertown,
MA 02472; e-mail: [email protected] (Ch. 10).
Looney, Stephen W., Department of Biostatistics, Medical College of Georgia,
112015thStreet,AE-3020,Augusta,GA30912-4900;e-mail:[email protected]
(Ch. 2).
Mazumdar, Madhu, Division of Biostatistics and Epidemiology, Department of
Public Health Weill Medical College of Cornell University, 411 East 69th
Street, New York, NY 10021; e-mail: [email protected] (Ch. 9).
McCulloch,CharlesE.,DivisionofBiostatistics,DepartmentofEpidemiologyand
Biostatistics, University of California, San Francisco, 185 Berry Street, 5700,
San Francisco, CA 94143; e-mail: [email protected] (Ch. 3).
Prentice, Ross L., Division of Public Health Sciences, Fred Hutchinson Cancer
Research Center, 1100 Fairview Avenue North, Seattle, WA 98109-1024;
e-mail: [email protected] (Ch. 1).
Shiboski, Stephen C., Division of Biostatistics, Department of Epidemiology and
Biostatistics, University of California, San Francisco, 185 Berry Street, 5700,
San Francisco, CA 94143; e-mail: [email protected] (Ch. 3).
Vittinghoff, Eric, Division of Biostatistics, Department of Epidemiology and Bio-
statistics, University of California, San Francisco, 185 Berry Street, 5700, San
Francisco, CA 94143; e-mail: [email protected] (Ch. 3).
Woods, Carol M., Psychology Department, Washington University in St. Louis,
CampusBox1125,St.Louis,MO63130;e-mail:[email protected](Ch.6).
Xiong, Chengjie, Division of Biostatistics, Campus Box 8067, Washington
University School of Medicine, 660 S. Euclid Ave, St. Louis, MO 63110-
1093; e-mail: [email protected] (Ch. 8).
Yu, Kai, Biostatistics Branch, Division of Cancer Epidemiology and Genetics
National Cancer Institute Executive Plaza South, Room 8050 Bethesda,
MD 20892; e-mail: [email protected] (Ch. 8).
Yuan, Ke-Hai, University of Notre Dame, 105 Haggar Hall, Notre Dame, IN
46556; e-mail: [email protected] (Ch. 7).
Zhu, Kejun, School of Management, China University of Geosciences, Wuhan,
PRC 4330074; e-mail: [email protected] (Ch. 8).
HandbookofStatistics,Vol.27 1
ISSN: 0169-7161
r2007ElsevierB.V.Allrightsreserved
DOI: 10.1016/S0169-7161(07)27001-4
Statistical Methods and Challenges in Epidemiology
and Biomedical Research
Ross L. Prentice
Abstract
This chapter provides an introduction to the role, and use, of statistics in
epidemiology and in biomedical research. The presentation focuses on the
assessmentandunderstandingofhealth-relatedassociationsinastudycohort.
The principal context considered is estimation of the risk of health events in
relation to individual study subject characteristics, exposures, or treatments,
genericallyreferredtoas‘covariates’.Descriptivemodelsthatfocusonrelative
andabsoluterisksinrelationtoprecedingcovariatehistorieswillbedescribed,
alongwithpotentialsourcesofbiasinestimationandtesting.Therole,design,
and conduct of randomized controlled trials will also be described in this
preventionresearchcontext,aswellasintherapeuticresearch.Someaspectsof
the sources and initial evaluation of ideas and concepts for preventive and
therapeuticinterventionswillbediscussed.Thisleadsnaturallytoadiscussion
of the role and potential of biomarkers in biomedical research, for such pur-
poses as exposure assessment, early diseasediagnosis, or forthe evaluation of
preventive or therapeutic interventions. Recently available biomarkers, includ-
ing high-dimensional genomic and proteomic markers, have potential to add
much knowledge about disease processes and to add specificity to intervention
development and evaluation. These data sources are attended by many inter-
esting statistical design and analysis challenges. A brief discussion of ongoing
analytic andexplanatory analysesin theWomen’s Health Initiative concludes
the presentation.
1. Introduction
The topic of this chapter is too broad to allow an in-depth coverage of its many
important aspects. The goal, rather, will be to provide an introduction to some
specifictopics,manyofwhichwillbecoveredinlaterchapters,whileattempting
to provide a unifying framework to motivate statistical issues that arise in
1
2 R.L.Prentice
biomedical research, and to motivate some of the models and methods used to
address these issues.
Much of epidemiology, and biomedical research more generally, involves fol-
lowing a set of study ‘subjects’, often referred to as the study cohort. Much
valuable basic biological research involves the study of lower life forms. Such
studiesareoftenattendedbysubstantialhomogeneityamongstudysubjects,and
relatively short life spans. Here, instead, the presentation will focus on a cohort
of humans, in spite of the attendant greater heterogeneity and statistical chal-
lenges. For research purposes the individuals in a cohort are of interest through
their ability to yield health-related information pertinent to a larger population.
Such a larger population may, for example, include persons residing in the
geographic areas from which cohort members are drawn, who meet certain
eligibility and exclusionary criteria. The ability to infer health-related informa-
tion about the larger population involves assumptions about the representative-
ness of the cohort for the ‘target’ population. This typically requires a careful
characterizationofthecohortsothatthegeneralizabilityofstudyfindingscanbe
defined. The target population is often somewhat conceptual, and is usually
taken to be practically infinite in size. The major long-term goal of biomedical
researchistodecreasetheburdenofprematurediseasemorbidityandmortality,
andtoextendtheperiodoftimethatmembersoftargetpopulationslivewithout
major health-related restrictions.
The principal focus of epidemiologic research is understanding the determi-
nants of disease risk among healthy persons, with a particular interest in mod-
ifiableriskfactors,suchasdietaryorphysicalactivitypatterns,orenvironmental
exposures.Thereisalonghistoryofepidemiologicmethods development,much
of which is highly statistical, whose aim is to enhance the likelihood that asso-
ciations between study subject characteristics or exposures and disease risk are
causal, thereby providing reliable concepts for disease prevention.
Theavailabilityofdiseasescreeningprogramsorservices,andthehealthcare-
seekingbehaviorofcohortmembers,havepotentialtoaffectthetimingofdisease
diagnosis.Earlydiseasedetectionmayallowthediseasecoursetobeinterruptedor
alteredinamannerthatisbeneficialtotheindividual.Diseasescreeningresearch
hasitsownsetofmethodologicchallenges,andiscurrentlythetargetofintensive
effortstodiscoverandvalidateearlydetection‘biomarkers’.
Muchbiomedicalresearchisdirectedtothestudyofcohortsofpersonhaving
a defined disease diagnosis, with emphasis on the characterization of prognosis
and, especially, on the development of treatments that can eradicate the disease
or can facilitate disease management, while avoiding undue adverse effects.
The ultimate products of biomedical research are interventions, biomarkers,
or treatments that can be used to prevent, diagnose, or treat disease. Addition-
ally, the knowledge of the biology of various life forms and the methodologic
knowledge that underlies the requisite research agenda, constitutes important
and durable contributions from biomedical research. These developments are
necessarily highly interdisciplinary, and involve a wide spectrum of disciplines.
Participating scientists may include, for example, molecular geneticists studying
biological processes in yeast; technologists developing ways to assess a person’s
genome or proteome in a rapid and reliable fashion; population scientists
Statisticalmethodsandchallenges 3
studying disease-occurrence patterns in large human cohorts; and expert panels
and government regulators synthesizing research developments and providing
recommendations and regulations for consumption by the general population.
Statisticians and other quantitative scientists have important roles to fulfill
throughout this research spectrum. Issues of study design, quality control, data
analysis, and reporting are important in each biomedical research sector, and
resolving methodologic challenges is crucial to progress in some areas. The
biomedical research enterprise includes natural tensions, for example, basic ver-
sus applied research; in-depth mechanistic research versus testing of current
concepts;andindependentversuscollaborativeresearch.Statisticianscanhavea
unifying role across related cultural research norms, through the opportunity to
bring ideas andmotivationsfrom onecomponentofthisresearchcommunityto
another in a non-threatening manner, while simultaneously applying critical
statistical thinking and methods to the research at hand.
2. Characterizing the study cohort
A general regression notation can be used to represent a set of exposures and
characteristics to be ascertained in a cohort under study. Let z(u)0 ¼{z (u),
1
z (u),y} be a set of numerically coded variables that describe an individual’s
2
exposuresandcharacteristicsat‘time’u,where,tobespecific,ucanbedefinedas
timefromselectionintothecohort,andaprime(0)denotesvectortranspose.Let
Z(t)¼{z(u), uot} denote the history of each covariate at times less than t. The
‘baseline’ covariate history Z(0) may include information that pertains to time
periods prior to selection into the cohort.
Denote by l{t, Z(t)} the occurrence rate for a health event of interest in the
targeted population at cohort follow-up time t, among persons having a pre-
ceding covariate history Z(t). A typical cohort study goal is to assess the rela-
tionship between aspects of Z(t) and the corresponding disease rate l{t; Z(t)}.
Doingsoinvolvesrecordingovertimethepertinentcovariatehistoriesandhealth
event histories for cohort members, whether the cohort is comprised of healthy
individuals as in an epidemiologic cohort study or disease prevention trial, or
persons having a defined disease in a therapeutic context. The notation Z(t) is
intended to encompass evolving, time-varying covariates, but also to include
more restrictive specifications in which, for example, only baseline covariate
information is included.
A cohort available for study will typically have features that distinguish it
from the target population to which study results may be applied. For example,
an epidemiologic cohort study may enroll persons who are expected to continue
livinginthesamegeographicareaforsomeyears,orwhoareexpectedtobeable
andwillingtoparticipateinresearchprojectactivities.Atherapeuticcohortmay
have characteristics that depend on institutional referral patterns and clinical
investigator experience and expertise. Hence, absolute health event (hereafter
‘disease’) occurrence rates may be less pertinent and transferable to the target
population, than are relative rates that contrast disease rates among persons
receiving different treatments or having different exposures.
4 R.L.Prentice
The hazard ratio regression model of Cox (1972) captures this relative risk
notion,withoutimposingfurtherrestrictionsoncorrespondingabsoluterates.It
can be written
lft;ZðtÞg¼l ðtÞexpfxðtÞ0bg, (1)
0
wherex(t)0 ¼{x (t),y,x (t)}isamodeledregressionp-vectorformedfromZ(t)
1 p
andproduct(interaction)termswitht,b0 ¼(b ,y,b )isacorrespondinghazard
1 p
ratio, or relative risk, parameter to be estimated, and l ((cid:2)) is an unrestricted
0
‘baseline’ hazard function corresponding to x(t)(cid:3)0. For example, x(t)(cid:3)x may
1
beanindicatorvariableforactiveversusplacebotreatmentinapreventiontrial,
or an indicator for test versus the standard treatment in a therapeutic trial, in
which case eb1 is the ratio of hazard rates for the test versus the control group,
and there may be special interest in testing b1¼0 (eb1 ¼1). Such a constant
hazard ratio model can be relaxed, for example, to x(t)¼{x , x log t} in which
1 1
case the ‘treatment’ hazard ratio function becomes eb1tb2; which varies in a
smooth manner with ‘follow-up time’ t. Alternatively, one may define x(t) to
include a quantitative summary of a study subject’s prior exposure to an envi-
ronmental or lifestyle factor in an epidemiologic context.
LetTbethetimetooccurrenceofadiseaseunderstudyinacohort.Typically
some,andperhapsmost,ofcohortmemberswillnothaveexperiencedthedisease
at the time of data analysis. Such a cohort member yields a ‘censored disease
eventtime’thatisknowntoexceedthefollow-uptimefortheindividual.LetYbe
a process that takes value Y(t)¼1 if a subject is ‘at risk’ (i.e., without prior
censoring or disease occurrence) for a disease event at follow-up time t, and
Y(t)¼0 otherwise. Then a basic independent censoring assumption requires
lft;ZðtÞ; YðuÞ¼1; uotg¼lft;ZðtÞg,
sothatthesetofindividualsunderactivefollow-upisassumedtohaveadisease
ratethatisrepresentativeforthecohortgivenZ(t),ateachfollow-uptimet.The
hazard ratio parameter b in (1) is readily estimated by maximizing the so-called
partial likelihood function (Cox, 1975)
2 3
Yk 6 expfxðtÞ0bg 7
LðbÞ¼ 4 P i i 5, (2)
expfxðtÞ0bg
i¼1 l i
l(cid:2)RðtiÞ
where t ,y,t are the distinct disease occurrence times in the cohort and R(t)
1 k
denotes the set of cohort members at risk (having Y(t)¼1) at follow-up time t.
Standardlikelihoodproceduresapplyto(2)fortestingandestimationonb,and
convenieRntsemiparametricestimatorsofthecumulativebaselinehazardfunction
O ðtÞ¼ tl ðuÞduarealsoavailable(e.g.,AndersenandGill,1982)therebyalso
0 0 0
providing absolute disease rate estimators.
The score test @logLðbÞ=@b for b¼0 is referred to as the logrank test in the
special case in which x(t)(cid:3)x is comprised of indicator variables for p of p+1
Statisticalmethodsandchallenges 5
groups, for which disease rates are to be compared. A simple, but practically
useful refinement of (1) replaces the baseline hazard rate l (t) by l (t) thereby
0 0s
allowingthebaselineratestodifferarbitrarilyamongstratas¼1,2,ythatmay
be time-dependent. This refinement allows a more flexible modeling of disease
ratesonstratificationfactors,formedfrom{Z(t),t},thanwouldconvenientlybe
possible through hazard ratio regression modeling. The partial likelihood func-
tionunderastratifiedCoxmodelissimplytheproductofterms(2)formedfrom
the stratum-specific disease occurrence and covariate data. Other modifications
areneededtoaccommodatetieddiseaseoccurrencetimes,andformorecomplex
disease occurrence time data as may arise with specialized censoring schemes or
withrecurrentorcorrelateddiseaseoccurrencetimes.TheCoxregressionmethod
has been arguably the major statistical advance relative to epidemiology and
biomedical research of the past 50 years. Detailed accounts of its characteristics
and extensions have been given various places (e.g., Andersen et al., 1993;
Kalbfleisch and Prentice, 2002).
The Cox model provides a powerful and convenient descriptive tool for
assessing relative associations with disease incidence.Thereare other descriptive
models, such as accelerated failure time models
(cid:2)Z (cid:3)
t
lft;ZðtÞg¼l exðuÞ0bdu exðtÞ0b,
0
0
for which the regression parameter may have a more useful interpretation in
somecontexts.Thismodeltendstoberatherdifficulttoapply,however,though
workable implementations have been developed, with efficiency properties
dependent on the choice of model for l ((cid:2)) that is used to generate estimating
0
functions (e.g., Jin et al., 2003).
In some settings mechanistic or biologically-based disease occurrence rate
models are available (e.g., Moolgavkar and Knudson, 1981). The parameters in
such models may characterize aspects of the disease process, or provide specific
targets for treatments or interventions that allow them to valuably complement
descriptive modeling approaches. Biologically based models with this type of
potential also tend to be more challenging to apply, but the payoff may some-
times justify the effort. Of course, it is useful to be able to examine a cohort
dataset from more than a single modeling framework, to assure robustness of
principal findings, and to garner maximal information.
The statistical issues in study design, conduct, and analysis differ somewhat
between the epidemiologic, early detection, and therapeutic contexts, according
to differences in disease outcome rates and outcome ascertainment issues, and
according to covariate definition and measurement issues. However, there are
also some important commonalities; for example, issues of multiple hypothesis
testing,especiallyinrelationtohigh-dimensionalcovariatedataandstudymon-
itoring procedures, arise in each context. We will proceed by describing some of
the context-specific statistical issues first, and subsequently include a discussion
of shared statistical issues.
6 R.L.Prentice
3. Observational study methods and challenges
3.1. Epidemiologic risk factor identification
3.1.1. Sampling strategies
Cohort studies provide a mainstay epidemiologic approach to the identification
of disease risk factors. A single cohort study has potential to examine the
associationsbetweenmultipleexposures,behaviorsorcharacteristicsandtherisk
of various diseases, and has potential to examine both short- and long-term
associations. A distinguishing feature of the epidemiologic cohort study is the
typical low incidence rates for the diseases under study. Even such prominent
chronicdiseasesascoronaryheartdiseaseorlungcancertypicallyoccuratarate
of1%orlessperyearamongostensiblyhealthypersons.Itfollowsthatepidemio-
logiccohortsmayneedtobequitelarge,oftenintherangeoftensofthousandsto
more than 100,000, depending on the age distribution and on the frequencies of
‘exposures’ of interest in the cohort, to provide precise estimates on association
parametersofinterestinapracticaltimeframe.Well-characterizedcohortstendto
befollowedforsubstantialperiodsoftime,astheirvaluetypicallyincreasesasmore
diseaseeventsaccrue,andmarginalcostsforadditionalyearsoffollow-uptendto
diminish.
The rare disease aspect of epidemiologic cohort studies opens the way to
various design and analysis simplifications. For example, the partial likelihood-
based estimating function for b from (2) can be written
8 9
P
Xk >>< xlðtiÞWilðbÞ>>=
@logLðbÞ=@b0 ¼ xðtÞ(cid:4)l(cid:2)RðtPiÞ , (3)
i¼1>>: i i WilðbÞ >>;
l(cid:2)RðtiÞ
where W ðbÞ¼expfxðtÞ0bg; which contrasts the modeled regression vector for
il l i
the individual developing disease at time t (the case), to a suitably weighted
i
average of the regression vectors, x(t), for cohort members at risk at t
l i i
(the controls). Most of the variance in this comparison derives from the ‘case’
regressionvector,andthesummationsoverthe‘riskset’att canbereplacedbya
i
summationoverafewrandomlyselectedcontrolsfromthisrisksetwithlittleloss
of estimating efficiency. This ‘nested case–control’ (Liddell et al., 1977; Prentice
and Breslow, 1978) approach to estimation is attractive if the determination of
some components of x(t) is expensive. Often only one, or possibly two or three,
controls will be ‘time-matched’ to the corresponding case. Depending somewhat
onthecovariatedistributionandhazardratiomagnitude,theefficiencyreduction
for a nested case–control versus a full-cohortanalysis is typically modest if, say,
fiveormorecontrolsareselectedpercase.Withlargecohorts,itisoftenpossible
to additionally match on other factors (e.g., baseline, age, cohort enrollment
date,gender)tofurtherstandardizethecaseversuscontrolcomparison.Another
within-cohort sampling strategy selects a random subcohort, or a stratified ran-
domsubcohort,foruseasthecomparisongroupforthecaseateacht,insteadof
i
the entire risk set R(t) in (3). If some care is taken to ensure that the subcohort
i
is well aligned with the case group, there will be little to choose between this