Behavioural Science Section

Gerontology 2001;47:341–352

Understanding AgeingAn Evaluation of Research Designs for Assessing the Interdependence ofAgeing-Related Changes

Scott M. Hofera Martin J. Sliwinskib

a Department of Human Development and Family Studies, Pennsylvania State University, University Park, Pa.;b Department of Psychology, Syracuse University, Syracuse, N.Y., USA

Scott M. HoferDepartment of Human Development and Family Studies110 South Henderson Bldg., Pennsylvania State UniversityUniversity Park, PA 16802 (USA)E-Mail smh21@psu.edu

ABCFax + 41 61 306 12 34E-Mail karger@karger.chwww.karger.com

© 2001 S. Karger AG, Basel0304–324X/01/0476–0341$17.50/0

Accessible online at:www.karger.com/journals/ger

Key WordsAgeing W Methodology W Cognition

AbstractBackground: Cross-sectional studies of samples varyingwidely in age have found moderate to high levels ofshared age-related variance among measures of cogni-tive and physiological capabilities, leading researchersto posit common factors or common causal influencesfor diverse age-related phenomenon. Objective: The in-fluence of population average changes with age oncross-sectional estimates of association has not beenwidely appreciated in developmental and ageing re-search. Covariances among age-related variables incross-sectional studies are highly confounded in regardsto inferences about associations among rates of changewithin individuals since covariances can result from anumber of sources including average population age-related differences (fixed age effects) in addition to initialindividual differences and individual differences in ratesof ageing (random age effects). Analysis of narrow age-cohort samples may provide a superior analytical basisfor testing hypotheses regarding associations betweenrates of change in cross-sectional studies. Conclusions:

The use of age-heterogeneous cross-sectional designsfor evaluating interdependence of ageing-related pro-cesses is discouraged since associations will not neces-sarily reflect individual-level correlated rates of change.Typical cross-sectional studies do not provide sufficient

evidence for the interdependence of ageing-relatedchanges and should not serve as the basis for theoriesand hypotheses of ageing. Reanalyzing existing cross-sectional studies using a sequential narrow-age cohortapproach provides a useful alternative for evaluatingassociations between ageing-related changes. Longitu-dinal designs, however, provide a much stronger basisfor inference regarding associations between rates ofageing within individuals.

Copyright © 2001 S. Karger AG, Basel

Many age-related differences are evident when com-paring young and old individuals. For example, age-relat-ed differences are observed, on average, in memory andreasoning, immune function, forced expiratory volume,number of teeth, and muscle strength across people vary-ing in age. Investigations that aim to understand ‘ageing’often focus on associations between such time-dependentprocesses and emphasize particular questions. Are thecausal pathways of cognitive ageing few or many? Whatare the prior causes of these changes? Are the causes prox-imal or distal to the outcome? Are ageing-related changesdifferent across different people? To what degree are age-ing-related changes independent of one another? Thispaper is concerned with these questions and how differentstudy designs and analytic methods may lead to quite dif-ferent conclusions regarding the dimensionality andstructure of aging-related changes and of predictors andconcomitants of such changes.

342 Gerontology 2001;47:341–352 Hofer/Sliwinski

Most gerontological theories, certainly most cognitiveageing theories, are based on cross-sectional data of sam-ples varying broadly in age. A focus of many of these stud-ies is on the amount of shared age-related variance and onwhich variables account best for the individual differ-ences in other types of functioning. Indeed, analyses ofcross-sectional data have clearly demonstrated that thecovariance between chronological age and cognitive vari-ables (e.g., memory and speed) is rarely unique. This lackof unique age-related covariance among cognitive vari-ables has been interpreted as a demonstration that singleor common factor ageing theories are sufficient to accountfor cognitive age effects. Although the partitioning of vari-ance is a general paradigm in cognitive aging research,two hypotheses in particular, the general slowing hypothe-sis [1, 2] and the ‘common cause’ sensory-cognitive linkhypothesis [3, 4], have emphasized that a large proportionof age-related variance in varieties of cognitive function isaccounted for by measures of processing speed and senso-ry ability (e.g. visual and auditory acuity), respectively.Although few would consider aging to be a simple andgeneral process for all types of functioning, the evaluationof common and specific age associations among age-relat-ed processes are emphasized in research on cognitiveaging and usually involve cross-sectional research designsand variance partitioning approaches [5–9]. In general,however, large proportions of shared age-related varianceamong diverse variables are found in such studies al-though there have been reports of unique proportions ofage-related variance for particular variables [10–12].

This interpretation regarding the high degree of com-monality among ageing processes relies on the complexassumption that shared age covariance at cross-section (ofa sample varying in age) reflects correlated ageing-relatedchange – that rates of ageing within individuals are associ-ated. This assumption is complex because there are atleast several ways in which ageing-related change can beassociated. First, ageing-related variables can be related interms of magnitudes and patterns of population averagechange. This type of association is indexed by fixed ageeffects, which indicates the average population changeover time. A second type of association refers to individu-al differences in rates of change within the population.This association is indexed by random age effects, ex-pressed as deviations in rate of change from the popula-tion average change, which determine whether individu-als who decline rapidly on one variable also tend todecline rapidly on another variable. Third, systematicchange can be correlated within individuals over time.This type of change reflects whether, for a given individu-

al, the amount of change in one variable over a given tem-poral interval correlates with the amount of change in asecond variable over that same interval. Stated simplyand assuming a strong association between speed andmemory, if a person experiences relatively little slowingbetween the ages of 70 and 75, but much slowing between75 and 80, then they should also experience little memoryloss during the first interval relative to the second [13].This type of change or fluctuation in performance can alsobe assessed at much shorter intervals to indicate theextent to which processes systematically covary, perhapsinfluenced by environmental or health-related events, andthus provide information regarding systems of variablesand associated common influences. While each type ofcorrelated change is relevant for ageing theories in gener-al, and cognitive ageing theory in particular, we describeserious limitations regarding the use of cross-sectionaldesigns for understanding correlated rates of change anddiscuss designs that provide a much stronger basis forunderstanding ageing.

The central point of this paper is that cross-sectionalanalyses of age-heterogeneous samples are more informa-tive about population-level mean trends than associationsbetween rates of ageing, and therefore provide a weakbasis for examining the interdependence of ageing-relatedchanges within individuals. We show that estimates ofcovariance in cross-sectional samples are confoundedwith covariance related to average age trends, which alonewill produce positive correlations between variables evenif the true correlation between rates of change across pro-cesses is zero or negative. This simple fact, that virtuallyany variable that is associated with chronological age –that exhibits age differences on average – will result in apositively biased association, poses a major problem fortheories and empirical findings based on cross-sectionaldata. Indeed, cross-sectional studies of this type areinclined to find large proportions of shared age-relatedvariance and that common factor models of diverse age-related variables fit the data quite well. Unfortunately,findings of commonalities or common factors may notreflect anything substantive about the causal dimension-ality of age effects, but rather, these common factor solu-tions are, at least in part, simply detecting the passage oftime and that older individuals perform, on average, dif-ferently than younger individuals. That a variable exhib-its a particular pattern and magnitude of change with ageis important to know, but we would not want to base theo-ry or hypotheses on the simple fact that a set of variableschange over time.

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In gerontological research, time is an intrinsic aspect ofthe processes under study and can be seen as having aneffect both at the population level (known as the fixedeffect or average trend) and at the individual level (devia-tions from this average trend). In cross-sectional studies,these two types of effects cannot be easily disentangledand so the major problem resides in deriving inferentialstatements regarding common and specific effects relatedto age or aging from information about population aver-age changes (age differences in cross-sectional studies).Indeed, this is the major issue emphasized here – how thepopulation average, and deviations about the populationaverage effects, are an aspect of both cross-sectional andlongitudinal studies.

The problem is one of aggregation and in develop-mental and aging research, the aggregation is based ontime or age. In cross-sectional models focusing on time-dependent variables, age is a between individual effect. Inthe sections below, we demonstrate how covariancesresult, at least in part, from the aggregation of average age-based differences and therefore, results in a mixture ofbetween and within-individual effects. The problem asso-ciated with aggregation of this type is well known in par-ticular areas of science and statistics, though the cautionsin regard to developmental or aging phenomena have notinfluenced research practices [14–18]. That correlationsmay result from mean differences in individuals of differ-ent ages is closely related to the ecological fallacy [19, 20],Simpson’s paradox [21–24], and Lord’s paradox [25–27],where the essential problem results from aggregating orpooling data across groups. In 1903, Yule [24] referred tothis problem as an ‘illusory association’, the result of animproper mixture of distinct groups where differences inthe variables to be correlated were both associated withgroup status (e.g. different proportions or means of theattributes observed across sex groups). Yule describedsimilar problems in relation to sampling over time, re-ferred to as the time-correlation problem [28, 29]. Othercritical developments of these ideas for gerontologistsincludes differential selection of a population over timewhich result in estimates of population dynamics thatmay not reflect individual characteristics or patterns ofchange [30] and the establishment of measurement invar-iance across groups and time [31]. We should also notethat there are more general concerns regarding the utilityof variance partitioning for understanding the relativeimportance of variables [9, 32, 33] although these issueswill not be dealt with here.

The second major point is that an alternative cross-sec-tional design – the analysis of narrow-age cohorts (i.e.

samples of individuals of the same or nearly the samechronological age) – can inform about the second type ofcorrelated change in that the covariance between twovariables in the population at a given age is a function ofboth the covariances between initial level and covariancebetween rates of aging (random age effects). The narrow-age cohort design has been used to evaluate whetherincreasing covariance between age-related processes isobserved as a function of correlated rates of ageing [14,15] and has formed the basis for several gerontologicalstudies [34]. As to the third type of correlated change,there is no cross-sectional design that can inform aboutcorrelated change that occurs within individuals. Only thedirect observation of change in individuals within longi-tudinal studies can provide the necessary data to inferassociation at the intraindividual level [35].

In emphasizing confounds related to population aver-age rates of change in age-heterogeneous cross-sectionalsamples [16, 17], we do not at all dismiss or undervalueother, better-known measurement and analysis issues thatare pertinent to developmental and aging research designs(e.g. no direct inference of within-individual change,selection and mortality, age-period-cohort confounds, dis-parate time spans) [36–38]. The issue of attrition, due tomortality or other causes, is a major problem in bothcross-sectional and longitudinal designs and this has ledto new developments in statistical methods to obtain bet-ter inferences of change under reasonable assumptionsregarding the attrition process [39, 40]. While attrition isusually regarded as a problem for longitudinal studies, itis arguably a greater problem for cross-sectional designsbecause the selection process is unobserved – individualsof different ages are not random samples of the popula-tion of individuals of younger ages [30, 41] – and there isno opportunity to acquire information regarding thecauses or magnitude of differential selection over timewithin such single occasion studies. Information regard-ing the selection process (i.e. attrition, mortality) may,however, be observed in longitudinal studies and becauseit is observed, it must be dealt with. In cross-sectionalstudies, however, attrition is generally ignored since noinformation is available and this is also an issue for longi-tudinal studies that begin with an age-heterogeneous sam-ple. Retest effects are a major source of bias in derivinginferences regarding longitudinal change within individu-als and a recurrent problem. Cohort effects may also leadto alterations in means and covariances, particularly inage-heterogeneous samples, and lead to generalizabilityissues when relying on narrow age-cohort samples. Whilewe do not describe further the differential impact of these

344 Gerontology 2001;47:341–352 Hofer/Sliwinski

Fig. 1. Demonstration of how average population-level age-relateddifferences (fixed effects) will influence the association between twoage-dependent variables.

and other sources of bias in cross-sectional and longitudi-nal studies, these influences clearly impact the designs wedescribe below. Our emphasis is on the examination of anunderappreciated source of covariance in cross-sectionalstudies – the influence of average population age differ-ences or mean trends.

In the following sections, we describe in detail theproblem of estimating associations between processes us-ing typical cross-sectional designs, and, in particular, theconfound in the covariance related to average populationtrends. We examine the alternative narrow age-cohortapproach to cross-sectional designs and demonstrate howinformation regarding correlated random age effects canbe recovered, and, therefore, why we may observe the pro-cess of dedifferentiation across samples differing in age.Finally, we describe briefly the implications for experi-mental ageing research and the utility of longitudinaldesigns that permit a stronger basis for inference regard-ing the association between ageing-related change.

Inference from Age-HeterogeneousCross-Sectional Designs

In addition to other reasons for differences betweencross-sectional and longitudinal results focusing on devel-opmental and ageing processes, associational analysis ofindividuals varying broadly in age emphasizes popula-tion-level mean change in addition to individual variationabout the population mean. Analyses of correlated changein longitudinal studies are based on detrended individualdifferences in change (expressed as deviations about thepopulation trend) and so may result in a less-biased esti-mate of association between change in processes becausethe fixed effects are not a component of the covariance.Indeed, age-heterogeneous cross-sectional analysis mayprovide little useful information regarding the interde-pendence of within-person changes than is known interms of mean differences or correlation with chronologi-cal age.

Association Due to Population Average TrendsResults from cross-sectional analysis provide very

weak evidence for associated ‘rates of change’ since mod-erate-to-high associations will arise even when the ageing-related changes are completely independent within indi-viduals. A simple example demonstrating how associa-tions may arise from mean trends is provided in figure 1a,which displays two variables (X and Y) that exhibit meandifferences across age groups. To simplify the diagram,only two groups are shown but a continuum of age groupscould be present. Note that the association between X andY within age groups is assumed to be zero and that there isno association between rates of change within the popula-tion. All that is present is population average changeacross chronological age. Under such circumstances, fig-ure 1b shows that the association between variables X andY may be quite high and result exclusively from meandifferences across age groups. This was described by Yulein 1903 [24, p 134]; ‘if two separate records, for each ofwhich the correlation is zero, be pooled together, a spu-rious correlation will necessarily be created unless themean of one of the variables, at least, be the same in thetwo cases’.

Hofer and Flaherty [16] show analytically how associa-tions between time-dependent variables are at least partlyand may be entirely due to mean trends in cross-sectionaldata and provide a demonstration of this using simulateddata. A distinction between fixed age and random ageeffects, terminology common to longitudinal modeling, todenote population average change and systematic indi-

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vidual-level rate of change expressed as deviations fromthe population average is shown. Assume a sample ofindividuals followed continuously over their lifespan suchthat each individual has a complete distribution of scoreson two processes, X and Y, as a function of their initiallevel and rate of change over time. Parameters sub-scripted by i refer to the random effects (deviations frompopulation means). An individual’s score at any particu-lar age (t) is a function of the population average interceptor level (Lx, Ly), deviation from population average inter-cept (Lxi, Lyi), population average rate of change or slope(Sx, Sy), deviation from population average rate of change(Sxi, Syi), and other sources of systematic and randomvariance (exi, eyi) which are assumed to be normally dis-tributed and independent.

xit = Lx + Lxi + Sxt + Sxit + exi (1)yit = Ly + Lyi + Syt + Syit + eyi

In the typical cross-sectional study, a sample of onemeasurement at time t from each individual is obtainedso that a broad range of ages is represented. In age-hetero-geneous cross-sectional samples, the individual scorescontain information on both the fixed and random effectsof the individual trajectories. This can be seen by insert-ing the age of each individual in the above equations giv-ing the individual scores at time t.

Equation 3 shows the covariance between two time-dependent processes, X and Y, which is the result of sub-stituting equation 1 into the formula for a covarianceshown as equation 2,

Cov (X,Y ) = E[(X – Ìx)(Y – Ìy)] (2)

and

Cov (X,Y ) =E{[(Lx + Lxi + Sxti + Sxiti + exi) – E(Lx + Lxi + Sxti + Sxiti + exi)][(Ly + Lyi + Syti + Syiti + eyi) – E(Ly + Lyi + Syti + Syiti + eyi)]}, (3)

where ti denotes an individual i’s age at time t.Equation 4 shows the expansion of the covariance of

the fixed and random effects for Level and Slope aftersubstituting the mean of the X and Y processes and omit-ting terms with expectations of zero (which includes termsinvolving error). Given that this is a cross-sectional sam-pling of individuals varying in age, a single measurementassociated with a particular age location is sampled froman individual’s trajectory. We use the notation ti to repre-sent individual i’s age at time t. Equation 4 demonstratesthat the covariation between two variables from a cross-sectional sampling scheme is a function of the fixedeffects indicating population average change in additionto covariance related to initial individual differences

(covariance between intercepts), covariation betweenrates of change (random effects), and covariance betweenintercepts and rates of change.

Cov (X,Y ) = E(LxiLyi + LxiSyti + LxiSyiti + SxtiLyi + SxtiSyti +SxtiSyiti – SxtiSyt- + SxitiLyi + SxitiSyti + SxitiSyiti – SxitiSyt- –Sxt-Syti – Sxt

-Syiti + Sxt-Syt

- (4)

To use an extreme example, suppose we have two pro-cesses that exhibit systematic average change over timewithin a population. Assume, however, that these twoprocesses are completely independent and therefore haveno association with one another in terms of individual dif-ferences in either initial level or rate of change. This is thecondition shown in figure 1 where there is no associationbetween processes within particular age groups but whereboth processes exhibit mean differences across age. Withonly fixed effects present in the expected covariance,equation 4 simplifies to

Cov (X,Y ) = SxSyVar(ti). (5)

Equation 5 demonstrates that covariation will resultfrom simple linear trends in cross-sectional samples vary-ing in age. This covariation will be a product of thesquared age of the individual multiplied by the rates ofchange in processes X and Y (centered at the average ageof the sample). This presents a fundamental problemwhen evidence from cross-sectional designs is used toevaluate the interdependency of process. Associationsproduced by simple mean age trends in the populationshould not be taken as evidence for a common causal age-ing mechanism or common process change within indi-viduals over time. Cross-sectional designs based on broadage samples are the most confounded in terms of sourcesof covariation, in contrast to other designs, since associa-tions between time-dependent processes may arise fromtrends in mean level in addition to correlated rates ofchange and initial individual differences. We should notethat this expectation is based on a simple linear modeland that curvilinear change and the additional influenceson such study designs cited above will necessarily com-pound the complexity of the expected covariance in actualdata.

The problem of mean trends in producing spuriousassociations between uncorrelated processes has beenknown for nearly a century. For example, in regards tosampling across time it is important to detrend to evalu-ate associations between processes, otherwise mean-in-duced autocorrelations will arise. One of the problemswith cross-sectional sampling of age (time) is that withcoincident measurements obtained at an arbitrary occa-sion in time (i.e. age), we do not know the previous values

346 Gerontology 2001;47:341–352 Hofer/Sliwinski

of these measurements. Because we do not have a historyon each individual from such sampling, we cannot de-trend the individuals’ scores to evaluate patterns of covar-iation that are separate from the average population age-change trajectories, a problem further compounded byunknown sample selection over time. What is observed inthe cross-sectional covariance is the between-individualeffects – the covariances that arise from the mean levels offunctioning at each age represented in the sample.

Inference from Age-HomogeneousCross-Sectional Designs

An alternative analysis of cross-sectional data focuseson associations between variables in samples that do notvary significantly in chronological age. A less-utilizedcross-sectional design, the narrow age-cohort (NAC) sam-ple may be more suitable for deriving inferences regardingthe interdependence of ageing-related change. NAC sam-pling designs, particularly sequential comparison of nar-row age-cohorts (SNAC), provide stronger evidence fordisassociation of ageing effects than age-heterogeneouscross-sectional studies.

The NAC design is based on the idea that there are indi-vidual differences in rates of ageing and that, given suffi-cient time, the rank ordering of individuals at cross-section(e.g. age 75) will increasingly reflect the rank ordering ofindividual rates of ageing-related change. If ageing has acommon effect on different processes (i.e. the rates of age-ing are associated), moderate and increasing correlationsacross systems of variables should be observed.

In the NAC and SNAC designs, the fixed effects areconstant within age group and do not contribute to thecross-products. Note also that since the age within NACgroups is constant, t is not subscripted and represents afixed age effect. Hofer and Flaherty [16] derive the co-variance of two time-dependent processes as a function ofthe random effects of initial level and rate of change. Co-variance terms containing fixed age effects are constant,means of the random effects terms are zero, and termsinvolving error all drop from the model leaving us withthe general case for covariance within a single age group,

Cov (X,Y ) = Cov (LxiLyi) + [t]Cov (LxiSyi) + [t]Cov (SxiLyi) +[t2]Cov (SxiSyi). (6)

The key concept motivating the NAC design is that therank order across individuals of the same age will becomemore and more informative regarding the associationsbetween aging-related rates of change and that average

population change (fixed age effects) will not enter intothe estimate of association. Equation 6 demonstrates howthe covariance of any single NAC, representing a singleslice of time, will be a function of covariance betweenLevel and Slopes ([t]Cov(LxiSyi), [t]Cov(SxiLyi)), initialLevel covariance (Cov(LxiLyi)) and, most importantly,covariance related to rates of change ([t2]Cov (SxiSyi)).Equation 7 shows that as time increases, the effect of thecovariance between the rates of change in the processesincreases quadratically,

t→Cov (X,Y ) =Cov (LxiLyi) + [t]Cov (LxiSyi) + [t]Cov (SxiLyi) + [t2]Cov (SxiSyi)

1→Cov (X,Y ) =Cov (LxiLyi) + [1]Cov (LxiSyi) + [1]Cov (SxiLyi) + [1]Cov (SxiSyi)

2→Cov (X,Y ) =Cov (LxiLyi) + [2]Cov (LxiSyi) + [2]Cov (SxiLyi) + [4]Cov (SxiSyi) (7)

3→Cov (X,Y ) =Cov (LxiLyi) + [3]Cov (LxiSyi) + [3]Cov (SxiLyi) + [9]Cov (SxiSyi)

4→Cov (X,Y ) =Cov (LxiLyi) + [4]Cov (LxiSyi) + [4]Cov (SxiLyi) + [16]Cov (SxiSyi)

The major feature of the NAC design is that, as timeelapses, the magnitude of the covariance becomes increas-ingly due to the covariance associated with rates of changerelative to the other sources of covariance. Therefore, inolder samples of individuals, more time will have trans-pired and this will increase the contribution to the covar-iance of NAC samples that reflects individual differences inrates of ageing. In the analysis of a single age-cohort sample,associations between variables may be due to ‘initial’ indi-vidual differences as well as to common rates of ageing. Wemake the assumption that intraindividual change due to‘ageing’ overwhelms any initial (early adulthood) individu-al differences in functioning. It would be only in the unusu-al case where the observed association among variables isattenuated because of the cancellation of initial individualdifferences with correlated rates of change (e.g. negativecorrelation of initial individual differences with positivecorrelation between rates of change) that no association inolder age-cohort groups would be observed. Sequential nar-row age-cohort (SNAC) samples permit evaluation of in-creases or decreases in covariation across age (or othergroup status variables), as shown in figure 2, and permitdistinction between initial individual differences and com-mon rates of ageing as the key sources of covariation be-tween processes. The key concept is that the rank ordering(e.g. NAC covariance) will indicate the interdependencebetween rates of ageing to increasing degrees in sampleswhere the developmental and ageing processes have in-fluenced the rates of change from the initial values.

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This SNAC approach is related to the analysis of differ-entiation and dedifferentiation in that changes in the co-variation among variables or factor structure are assessedacross sequential narrow age-cohort groups [42, 43]. Whatis described here is the statistical mechanism by whichcovariational evidence for dedifferentiation (and differ-entiation in childhood) might arise [16]. In NAC andSNAC designs, only the outcomes of time-dependent pro-cesses are observed and time is treated as the continuaover which a process unfolds. As shown above, the covar-iances among outcome variables carry information re-garding the correlated change over time within individu-als.

These methods might be usefully applied to under-standing the interdependence resulting from other formsof time-dependent processes, such as span to death as inthe terminal decline hypothesis, changes in health status,or changes in cognitive functioning during the preclinicalphase of dementia that occurs prior to clinical diagnosis.Additionally, changes in covariance structures can beevaluated across time within the same individuals usingdata from longitudinal studies. We must also be clear thatthe analytical expectations described above predict in-creasing or decreasing covariances in narrow age-cohortsamples if selection (i.e. mortality), cohort, and othereffects are negligible. Cross-sectional designs remain high-ly confounded, with age, cohort, and time of measure-ment remaining difficult or impossible to disentangle andthe influence of mortality and attrition expected to havedifferential effects on results for particular outcome vari-ables and across different age-cohort samples. Of course,these are all problems for the typical, age-heterogeneousdesign as well. Narrow age-cohort designs are, neverthe-less, less confounded than age-heterogeneous designs inthat mean trends do not contribute to the covarianceamong processes.

Regression Partial of Chronological AgeFrom the perspective of the SNAC design, there is a

loss of information when chronological age is partialedfrom associations in cross-sectional samples varyingbroadly in age. It is also the case that partialling for chro-nological age variance only partially removes true agingeffects that result from covariance associated with thecumulative effects of correlated rates of change (equation6 above [16]). In other words, the use of regression to con-trol for variance associated with chronological age doesnot provide an estimate of the association that is indepen-dent of age but rather of the association at the average ageof the sample. After controlling for age-related variance,

Fig. 2. Utility of the narrow age-cohort (NAC) design for demon-strating increasing covariances between outcomes due to commonrates of ageing.

the covariance will still provide information about thevariation associated with true within-person change overtime, as in the NAC covariance expectations above, aswell as the static individual differences between the pro-cesses. Figure 3 shows that the regression partial of age-

348 Gerontology 2001;47:341–352 Hofer/Sliwinski

Fig. 3. Effect of regression partial of chronological age on associationbetween two age-dependent variables.

related variance can be conceptualized as averaging co-variance across SNAC samples – similar to a pooledbetween-group covariance with each age represented as agroup. In this example, there is no association between Xand Y in the young group and a moderate association in

the old group. The aggregation across age groups, shownat the bottom of figure 3, removes mean differences in Xand Y associated with chronological age and results in anaverage covariance between X and Y processes acrossSNAC samples, shown as the long dashed line. The age-partialed association may remain appreciable and due, atleast in part, to the average common ageing influences butalso to initial individual differences that are presentacross the age range of the sample. As we described ear-lier, the SNAC design is useful in that the change in covar-iation (increasing or decreasing interdependence) may beevaluated.

Inference from Longitudinal Designs

The study of ageing is the study of change. The study ofchange requires that the same individual be followed overtime and longitudinal designs are necessary for clearassessment of the interdependence of age-related pro-cesses [35]. In longitudinal analyses, the random effectscan be separately estimated from the fixed effects suchthat estimates of association between rates of change are‘detrended’ from the population average change. How-ever, longitudinal studies necessarily begin as cross-sec-tional studies, based on either age-heterogeneous or age-homogeneous sampling designs, and have similar limita-tions to those described above.

Age-Heterogeneous Longitudinal DesignsLongitudinal studies do not provide sufficiently strong

evidence for definitive statements about age-dependentcauses and associated outcomes. Observing that time-dependent trajectories (rates of change) are associated isnecessary but not sufficient to infer common causation.As described above, one reason that association in rates ofchange may be found is related to the fact that many lon-gitudinal studies begin as cross-sectional samples of indi-viduals varying broadly in age. In such studies, estimatesof association between rates of change may be evaluatedby fitting latent growth curves across occasions of mea-surement [44] and thus aggregating across individuals ofdifferent ages. Associations among time-dependent pro-cesses may arise due to different magnitudes of change,expressed in longitudinal analysis as deviations from thepopulation average age trend (i.e. random effects), andresult from age-defined periods of greater change (i.e. lateadulthood versus middle adulthood). For example, in alongitudinal study that initially was comprised of a sam-ple 50–80 years of age at the first occasion, estimates of

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individual trajectories will show that 75-year-old individ-uals will exhibit greater change over the course of thestudy than 50-year-old individuals in many measuredvariables. Since these rates of change are expressed asdeviations from population average change based on theentire age range of the sample, associations between ‘ratesof change’ in these processes may be misleading in thatthey may not reflect truly coupled processes or mutuallyinfluenced processes within individuals but describe theprobability for greater change or curvilinear change as afunction of chronological age or events related to the ageperiod of the individual. The caution raised here againfocuses on the interpretation of associations between ‘rateof change’ since it would be confounded in age-heteroge-neous studies with age-related population average change.This is definitely not to say that we regard the informationregarding average rates of change in the population asunimportant. Understanding developmental and agingphenomena requires focus on both population averageand individual differences about the average rate ofchange. We simply caution that the correlation amongslopes (rates of change) within longitudinal modelingapproaches may result from average age-dependent andnonlinear changes at the population level when such anal-yses are based on an aggregated sample composed of indi-viduals of different ages.

Nevertheless, results from longitudinal studies do pro-vide stronger evidence for coupled age processes thancross-sectional studies do. Longitudinal designs that beginwith a relatively homogeneous age sample at the firstoccasion may permit a clearer interpretation of the asso-ciations in much the same way as the cross-sectional age-homogeneous designs do, generalizations across cohortsnot withstanding.

Implications for Research on Aging

This critique focuses on the problem of how time-dependent processes might best be studied and is meantto encourage further discussion and use of optimal de-signs for understanding individual variation and covaria-tion among developmental and ageing processes. Devel-opment and ageing occurs within individuals and al-though there are certainly commonalities to such changes,there is also evidence for many and varied influences onindividual rates of ageing [44]. Clearly, changes withinindividuals contribute to average or population-levelchanges in function. However, the processes which con-tribute to or are an outcome of development and ageing

may exhibit different magnitudes of effects within indi-viduals, occur at different ages or periods, or have qualita-tively different effects depending on subsequent or ante-cedent conditions. Research designs used to evaluate suchassociations among aging processes must be sensitive tothe within-person processes and permit the separation ofaging-related changes that occur on average from thosethat occur within the individual. In addition to longitudi-nal studies, experimental research designs and measure-ment intensive designs have demonstrated or potentialutility in this regard.

Experimental DesignsThe comparison of averages (fixed age effects) across

age groups is the focus of the experimental approach andso is not influenced by the criticisms raised here. Theexperimental approach focuses on making inferences re-garding the relative sensitivity of cognitive processes toageing. Such approaches are valid when attempting toidentify cognitive mechanisms that are fundamental toageing. However, experimental procedures may be quitelimited in finding broad or specific mechanisms that pro-duce population-level age effects.

Measurement Intensive Longitudinal DesignsLong-term ageing effects can be gradual and enduring,

decline abruptly, or fluctuate about a characteristic levelwith the potential for rebound to previous levels. In allthese conditions, systematic fluctuation around an indi-vidual’s characteristic trend will be confounded if singlemeasurements or widely spaced measurements are ob-tained. A more sensitive process-oriented approach is themeasurement intensive design, involving short-term mea-surement bursts, which permits estimates of covariationbetween different processes to be assessed over bothshort- and long-term intervals and permits stronger state-ments to be made regarding the effect of a commoninfluence or dynamic across functions [46]. Numeroustypes of designs may be used that would greatly enhancestatements regarding the interdependence of ageing pro-cesses and include measurement burst designs (permittingprecise estimates of level and state variation) and mea-surement burst designs with factor indicators (permittingdistinction between systematic state and random fluctua-tion). These designs are currently being implemented in anumber of studies focused on both physiological and cog-nitive outcomes.

350 Gerontology 2001;47:341–352 Hofer/Sliwinski

Summary and Conclusions

Theories and models of ageing should not be undulyinfluenced by the simple fact that age-related changeoccurs. This is the critical problem for theories andhypotheses that are based solely on cross-sectional studiesof samples varying broadly in age. In this paper we soughtto elaborate the analytical problems in analyzing associa-tions of ‘ageing-related’ changes and to clarify the assump-tions and interpretational basis for several commonlyused approaches. The following points formed the basis ofour discussion of optimal designs for understanding asso-ciations between ageing-related changes:

(1) Cross-sectional analysis of samples varying broadlyin age will result in positively-biased estimates of associa-tion between processes that change with time. This resultsfrom the combination of fixed and random effects con-tributing to an individual’s score.

(2) An alternative cross-sectional method based onexamination of narrow age-cohort samples provides a use-ful approach for understanding the structure of ageing-related change. This is one method for ‘detrending’ thepopulation-level effects within cross-sectional applica-tions.

(3) Experimental ageing designs comparing young andold individuals are not susceptible to these problemsbecause of their focus on population mean differencesacross age groups (fixed age effects).

(4) Longitudinal designs, particularly those involvingintensive intraindividual measurements, provide directinformation on within-individual change, variation, andcovariation of ageing-related processes. Longitudinal de-signs, of course, present other challenges such as sampleselection, attrition, and retest effects.

Ageing occurs within individuals and it is at this levelthat theory must be evaluated. Although comparing indi-viduals varying in age has long been considered a usefulproxy for understanding within-individual ageing, theconfounds associated with cross-sectional analysis seri-ously compromise any statement regarding associationsbetween age-related variables and their shared or com-mon influence. The concerns raised here in regards to theinfluence of mean trends are in addition to general con-cerns with the variance decomposition approach. In termsof model sufficiency, we must regard the typical cross-sec-tional design as confounded and potentially misleadingregarding the commonality of ageing-related influencesand outcomes. Cross-sectional designs will often lead tocommon factor explanations for influences that might beindependent in terms of both causes and effects. Alterna-

tive models will fit the data equally well and all could beimplausible. It is essential that methodological designsand analytical approaches be sensitive to causal multiplic-ity. Fundamentally, this is a question of the degree towhich group-level processes can inform us of individual-level processes and is related to problems of aggregationand ergodicity, the assumption that all individuals areidentical representations of one another. A major themeof the issues raised here is that understanding ageing – thedimensionality, structure, and causal pathways – willrequire multiple perspectives and designs sensitive towithin-individual patterns of change.

Another problem associated with an emphasis on pop-ulation-level effects is the relatively weak sensitivity tolow prevalence rates within individuals. Different indi-viduals, or subgroups of individuals, will exhibit differentpatterns and rates of change. For example, very stronginfluences such as preclinical dementia could account formost of the within-individual change in cognition prior todementia diagnosis, which will only weakly be demon-strated in analysis across individuals and only weakly berelated to chronological age [47]. In this way, it is possiblefor a variable to not exhibit large population meanchanges but still account for a large proportion of within-individual ageing. Causal multiplicity of particular out-comes, when there are numerous weak ageing effectswhich cumulate over time, are not well described by manystudy designs. As we have shown, a very large number ofcauses that vary across subsets of people may appear ascommon effects due to the time dependencies of the pro-cesses. Typical cross-sectional approaches are insensitiveto the multiplicity of ageing processes.

Nevertheless, the robustness of cross-sectional findingsdoes require explanation. Why are processing speed andsensory acuity the ‘best’ predictors of age-related variancein varieties of cognitive functioning? Are variables thatcorrelate most highly with chronological age (i.e. exhibitthe greatest magnitude of age differences) the best predic-tors of age-related variance? There are numerous possibil-ities regarding why some behavioral and biological indica-tors of ageing are better predictors than others in cross-sectional studies and these reasons may include relativepopulation mean change in the context of other predictorsin the model, reliability, and the ratio of systematic inter-individual to intraindividual variability. However, it maybe that the range of differences in these predictive associa-tions is relatively narrow and the analytical explanationslie within a delimited range. In our view, attempts at suchexplanation are difficult since these associations are high-ly related to population mean trends. For example, an

Interdependence of Ageing-RelatedChanges

Gerontology 2001;47:341–352 351

explanation based on the relative ranking of correlationswith chronological age is uninformative since the relativemean changes with age in a population may not be impor-tant for understanding common causal paths and out-comes of ageing-related change within individuals. Theo-retical interpretations that focus on the relative degree ofcommon and unique effects and that rely on empiricalevidence from cross-sectional studies must be carefullyconsidered.

The alternative design based on narrow age-cohortsamples provides a useful approach to evaluating thestructure of age-related changes. This approach treats co-variances between individuals as the product of growthand decline, research topics known as differentiation anddedifferentiation. We have described analytically the ef-fect of dedifferentiation in relation to ageing as the out-come of correlated individual differences in rates and pat-terns of change. In NAC population samples from threeNordic countries of individuals aged 75 years, Hofer et al.[14, 15] find no systematic evidence for the common-cause hypothesis. Few associations between sensory acui-ty, balance, and cognitive variables were observed andthese could more parsimoniously be explained by generalhealth and socioeconomic status (e.g. number of teeth wasone of the strongest predictors of cognitive functioning)and peripheral sensory changes (cognitive tests requiringsimilar sensory functioning were more highly correlated).Using statistical simulations to provide examples of theanalytical demonstration of mean-induced covariances,Hofer and Flaherty [16] show that the associations ob-served at any cross-section will increase with subsequentage samples and closely approximate the population val-ues of the covariance between rates of change under thesimulation conditions. This simulation demonstratedthat the time-sampling frame is an important consider-ation regarding the strength of reorganization of individu-al differences based on associated rates of change andwith different initial conditions. A strength of the NAC/SNAC approach is that it permits reanalysis of existingcross-sectional data and provides an alternative to thetypical cross-sectional analyses of association that areconfounded by population mean trends.

In summary, theories and hypotheses of ageing-relatedchange in cognitive functioning have been based almostentirely on cross-sectional research designs. There aregood reasons for this, though most of which are based oneconomy, of time and money. We would argue that thesetradeoffs may not be cost-effective since such studies pro-vide little basis for evaluating the interdependence be-tween processes that change with age. Indeed, only weak

evidence for correlated aging processes or outcomes canbe derived from cross-sectional studies including sharedage effects or that a common ageing factor is sufficient toaccount for all or most of the individual differences incognitive change. Indeed, analyses focused on shared ageeffects or the magnitude of common to specific factorvariances will reflect mean trends in the population andcannot permit strong inferences in regards to associationsbetween rates of change and common influences on thesechanges.

This conclusion does not imply that a common factortheory is necessarily unlikely, only that support for such atheory cannot be obtained from analysis of age-relatedcovariance in samples varying broadly in age. At present,we consider the case for common factor ageing theories asweak, given that the preponderance of supporting evi-dence comes from cross-sectional studies. Because of themethodological problems described here, theories, hy-potheses, and empirical generalizations of ageing phe-nomenon that are based solely on cross-sectional samplesvarying broadly in age must be considered weak withoutalternative evidence. Clearly, cross-sectional designs willremain useful for describing important variables thatexhibit age-related differences and for the development ofmeasurement instruments for use across broad age spansor particular subgroups. Alternative cross-sectional de-signs (e.g. NAC, SNAC) are useful, but optimally, longitu-dinal designs will be used to evaluate associations be-tween rates of ageing. This will lead to increased under-standing of the structure of ageing, in terms of sequencesand patterns of changes to the ageing body, brain, andbehavior.

Acknowledgements

The concepts presented here were initially reported at the 1998Gerontological Society of America Conference, Philadelphia, Pa.,and subsequently at the 2000 Cognitive Ageing Conference, Atlanta,Ga. We thank Fumiaki Hamagami, John McArdle, Jennie Noll,Andrea Piccinin, and two anonymous reviewers for their helpfulcomments on previous versions of the manuscript.

352 Gerontology 2001;47:341–352 Hofer/Sliwinski

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