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RESEARCH ARTICLE |
Max Planck Institute for Human Development, Berlin, Germany.
Address correspondence to Paul B. Baltes, Max Planck Institute for Human Development, Lentzeallee 94, D-14195 Berlin, Germany; to Ulman Lindenberger, Saarland University, School of Psychology, Im Stadtwald, Building 1, D-66123 Saarbrücken, Germany; or to Tania Singer, Functional Imaging Laboratory, Wellcome Department of Imaging Neuroscience, 12 Queen Square, WC1N 3BG London, UK. E-mail: sekbaltes{at}mpib-berlin.mpg.de, lindenberger{at}mx.uni-saarland.de, or t.singer{at}fil.ion.ucl.ac.uk
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Abstract |
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 TOP Abstract Mortality-Associated Versus...Parent Sample Estimates for...MethodsResultsDiscussionReferences |
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In longitudinal investigations, one common threat to generalizability is sample selectivity, or nonrandom sample attrition. Sample attrition is called nonrandom when variables predicting attrition are related to variables of interest. Evidence from earlier investigations indicates that such selectivity occurs. For instance, individuals who participate in longitudinal studies for a longer period of time tend to be younger, healthier (McArdle, Hamagami, Elias, & Robbins, 1991), of higher intelligence (Baltes, Schaie, & Nardi, 1971), and of a higher social class (Powers & Bultena, 1972) than individuals who participate for shorter periods of time.
Sample selectivity compromises the generalizability of results, especially if substantive analyses are restricted to individuals observed at all measurement occasions (for a general treatment, see Baltes, Reese, & Nesselroade, 1988; Diggle, Liang, & Zeger, 1994). In the case of aging populations, generalizability can be impaired in two major ways. First, the average level of functional competence is overestimated if individuals with lower levels of functioning are less likely to be observed at all measurement occasions than individuals with higher levels of functioning. Second, the average amount of negative longitudinal change is underestimated if individuals with greater functional decline are less likely to be observed at all measurement occasions than individuals with less decline, stability, or gain (Siegler & Botwinick, 1979). In that case, the modal shape of observed longitudinal change is a biased representation of the full spectrum of longitudinal change trajectories at the population level, which, for instance, may include a greater proportion of precipitous decline patterns (e.g., Baltes & Labouvie, 1973, p. 174).
In longitudinal investigations of populations with mortality rates greater than zero, selectivity can originate from two different sources (cf. Baltes & Labouvie, 1973). First, individuals with higher mortality risks may differ on relevant attributes from individuals with lower mortality risks; henceforth, we refer to this kind of selectivity as mortality-associated. Second, individuals who are willing and able to participate in data collection may differ on relevant attributes from those who are unwilling or unable to do so but are also still alive; that kind of selectivity we designate as experimental. Mortality-associated selectivity gradually transforms the composition of the population under study and is best understood as a population process. Experimental selectivity captures the inability of the experimenter to assign an equal chance to all surviving individuals to continue participation in the study.
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Mortality-Associated Versus Experimental Selectivity: Introduction of Terminology |
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TOPAbstractMortality-Associated Versus...Parent Sample Estimates for...MethodsResultsDiscussionReferences |
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where Msurvivors is the mean level of performance among individuals who are alive at a given measurement occasion, Mparent sample is the mean level of performance of the original (full) sample, and SDparent sample is its standard deviation. The formula indicates the mean difference between the survivors and the parent sample in the standard deviation units of the parent sample. The same formula can also be used with dichotomous variables to quantify experimental selectivity effects on prevalence rates.
Mortality-associated selectivity can be further subdivided into an age-linked and an age-orthogonal component (which does not imply that the causal factors governing the two components have to be different). If the nature of selective mortality was entirely age linked, then individuals of the same age but different functional status would not differ in mortality risk. Under such conditions, mortality-associated selectivity would be zero after individual differences in age were controlled for. In contrast, any remaining association between mortality risk and a given variable would indicate that mortality also tends to be selective with respect to this variable among individuals of the same age. One possible way to examine this issue is to examine the magnitude of mortality-associated selectivity before versus after regressing the variables of interest on chronological age.
Experimental selectivity
Experimental selectivity, in the narrow sense adopted in this article, refers to an association between phenomena of interest and the likelihood of participating in the study among individuals who are still alive. With respect to mean differences, the magnitude of experimental selectivity can be computed as follows:
where Mselect is the mean level of performance among individuals who were measured at a given occasion, Msurvivors is the mean level of performance among individuals still alive at this occasion, and SDparent sample is the standard deviation for the original sample. As was true for mortality-associated selectivity, experimental selectivity can be subdivided into an age-linked and an age-orthogonal part. The standard deviation of the parent sample, rather than that of the survivors, is used to norm mortality-associated and experimental components to the same metric. As a consequence, total selectivity can be defined as the sum of mortality-associated and experimental components (see following text).
Total selectivity
Total selectivity designates the extent to which individuals measured at a given occasion differ from their parent sample, for both mortality-associated and experimental reasons. If selection is transitive (cf. Aitkin, 1934), that is, if the survivors are a subsample of the parent sample and if the select sample is a subsample of the survivors, then total selectivity is equal to the sum of its mortality-associated and experimental components:
Alternatively, total selectivity can be computed directly as the normed difference between the select sample and the parent sample:
A numerical example
The following example refers to the mean age at the first measurement occasion (T1) for three nested samples. These three samples also form the empirical basis of the analyses reported in this article. First, the T1 parent sample refers to all individuals who participated in the Intensive Protocol of the first measurement occasion of the Berlin Aging Study (BASE; N = 516, mean age at T1 = 84.9 years, SD = 8.7; see Appendix, Note 1). Second, the T3 survivors refer to all individuals who were still alive at the time when the Intensive Protocol of the third measurement occasion took place, that is, about 3.7 years after T1 (N = 313, mean age at T1 = 81.5 years, SD = 7.6). Finally, the T3 sample refers to all individuals who actually participated in the Intensive Protocol of the third measurement occasion (N = 206, mean age at T1 = 79.8 years, SD = 6.9). Note that the T3 survivors are a subset of the T1 parent sample, and the T3 sample is subset of the T3 survivors. Thus, the relationship among the three samples is nested, or transitive.
For age at T1, mortality-associated selectivity of the T3 sample is computed according to Equation 1:
At T1, then, individuals still alive at T3 were 0.39 standard deviation units younger than the T1 parent sample. Experimental selectivity is computed according to Equation 2:
Total selectivity can be computed directly with Equation 4:
Equation 7 indicates that the T3 sample was 0.59 standard deviation units younger than the T1 parent sample. As is easily seen, -0.39 + -0.20 = -0.59; that is, total selectivity is the sum of mortality-associated and experimental selectivity, as proposed in Equation 3. This is so (a) because the mean difference in Equation 7 is the sum of the mean differences in Equations 5 and 6, and (b) because the mean differences of all three equations are normed to the same reference sample (i.e., the parent sample), which is common practice (cf. Hedges & Olkin, 1985).
The preceding analysis reveals that total selectivity for chronological age amounts to -0.59 standard deviation units, a medium-sized effect (cf. Cohen, 1977). About 66% of this total amount (i.e., [-0.39/-0.59] x 100) is due to mortality, indicating that older individuals in the T1 parent sample were less likely to be still alive at T3 than younger individuals. Conversely, 34% of total selectivity is experimental in nature and refers to individuals still alive at T3.
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Parent Sample Estimates for Variables Assessed at T3 |
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Aitkin (1934) and Lawley (1943) showed that the PLSF can be applied repeatedly. In our case, a three-step procedure was followed. First, variables available at T1 were used to arrive at parent sample estimates for variables assessed at T2. Second, parent sample parameters for T1 and T2 (i.e., the observed values from T1, and the parent sample estimates from T2) served as selection variables and were used to arrive at parent sample estimates for variables available at the T3 Intake Assessment. Third, parent sample parameters for T1, T2, and the T3 Intake Assessment were taken to arrive at parent sample estimates for variables available at the T3 Intensive Protocol.
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Methods |
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TOPAbstractMortality-Associated Versus...Parent Sample Estimates for...MethodsResultsDiscussionReferences |
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BASE is an interdisciplinary study with a wide range and a very large number of variables from psychology, sociology, psychiatry, and internal medicine (Baltes & Mayer, 1999). Only a small fraction of variables could be included in the present analyses. Our main criterion for inclusion was construct centrality, that is, the degree to which a given variable is at the core of an important domain of functioning (cf. Little, Lindenberger, & Nesselroade, 1999). In addition, we decided to oversample variables from psychological domains such as intellectual and sensory functioning (see also Baltes & Lindenberger, 1997; Lindenberger & Baltes, 1997).
Measures
Twenty-three and 16 variables were considered at T1 and T3; they are listed in Tables 1 and 2, respectively (for detailed descriptions, see Baltes & Mayer, 1999). In addition, 11 variables from the second measurement occasion of the BASE (T2, N = 361, 1.7 years after T1), which involved a reduced measurement protocol, were used to enhance the estimation of selectivity effects in T3 variables with the PLSF. Specifically, the following variables measured at T2 were included: activities of daily living (ADLs); depression/depressivity, based on a checklist of depressive symptoms (0 = no depression, 1 = subthreshold depression, 2 = depression); hearing (same as T1 and T3); Digit Letter, a test of perceptual speed; self-rated mobility; the Short Mini-Mental State Examination (SMMSE)]; social network size; well-being (same as T1 and T3); and vision (same as T1 and T3).
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When projecting selectivity onto variables observed after selection using the PLSF, the following rule needs to be kept in mind: The more closely variables assessed at earlier measurement occasions (i.e., selection variables) are associated with variables at later occasions (i.e., dependent variables or variables after selection), the more confidence one can have in the estimates. Therefore, we examined the degree to which the linear combination of T1 and T2 variables was associated with each of the T3 variables. Analyses were restricted to the T3 sample (N = 206). On average, the proportion of predicted variance was quite high (mdn R2 = .63). The highest value (R2 = .89) was found for hearing, and the lowest for the clinical diagnosis of depression (R2 = .28). Thus, insofar as there was selectivity on variables assessed at T3, the predictive power of the system of selection variables was generally sufficiently large to detect it. Note that low predictability for a given variable may point either to an absence of related measures at earlier occasions or to an intrinsic lack of relative stability in that variable.
To partition selectivity into its mortality-associated and experimental components for variables measured after selection, we computed two independent series of PLSF analyses, one to obtain means and prevalence estimates of T3 variables for the T1 parent sample (N = 516) and the other to obtain means and prevalence estimates on T3 variables for the T3 survivors (N = 313). Using Equations 1 and 2, we then computed mortality-associated and experimental selectivity, respectively.
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Results |
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Mortality-Associated and Experimental Selectivity Components in Relation to Age
As can be seen by comparing the left and right panels of Figure 1, selectivity effects were considerably reduced when variables were regressed on age. Specifically, for the 48 variables considered, the average size of the mortality-associated component was reduced from 0.18 to 0.08, and the average size of the experimental component was reduced from 0.10 to 0.05 standard deviation units after partialing age. In an analysis of variance with age (without vs with statistical control for age) and component (mortality-associated vs experimental), we found that the Age x Component interaction was statistically reliable (MSE = 0.0004), F(1,47) = 103.93, p < .01, 
2 = 0.69. The interaction indicates that the mortality-associated selectivity component was more strongly related to age than the experimental component (see Figure 2). In addition to the interaction, the main effects of age (MSE = 0.003), F(1,47) = 108.10, p < .01, 2 = 0.70, and component (MSE = 0.004), F(1,47) = 28.66, p < .01, 2 = 0.38, were also reliable (see Appendix, Note 3).
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Mortality-associated and experimental selectivity components were substantially correlated (r = .76, p < .01). The corresponding correlation was lower for variables regressed on age (r = .43, p < .01); for the difference between the two correlations, z = 2.52 (p < .01). Thus, a significant portion of the link between mortality-associated and experimental mortality was collinear with chronological age.
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Discussion |
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At the same time, however, there was significant heterogeneity in the magnitude of selectivity effects. Specifically, experimental selectivity effects for chronological age and for measures of intellectual functioning varied around 0.22, rather than 0.10, standard deviation units. At least with respect to age and intelligence, the nonrandom sampling of survivors introduced a nonnegligible degree of mean level bias in the T3 sample.
Age-associated selectivity: Mortality-associated versus experimental components.
A substantial portion of experimental selectivity was related to the fact that surviving individuals were less likely to participate when they were older. This association between experimental selectivity and age implies that both longitudinal and cross-sectional age relations observed in the T3 sample are likely to be less pronounced than they actually are in the aging population. With respect to intelligence, experimental selectivity remained substantial after controlling for age (T1: 0.26 vs 0.19; T3: 0.24 vs 0.20). Thus, surviving individuals of a given age were less likely to participate in the T3 Intensive Protocol when their levels of intellectual functioning were low.
The reduction in selectivity after controlling for chronological age was more pronounced for the mortality-associated component (0.18 vs 0.08) than for the experimental component (0.10 vs 0.05; see Figure 2). The relatively strong link between mortality-associated selectivity and chronological age arises because older individuals are less likely to survive and show lower levels of functioning.
Experimental Selectivity: A "Precursor" of Mortality-Associated Selectivity?
Both conceptually and statistically, mortality-associated and experimental components of selectivity are free to vary independently of each other (cf. Baltes et al., 1988). In the present heterogeneous sample of very old individuals, however, experimental and mortality-associated selectivity components shared 58% of their variance across the 48 variables considered. A plausible explanation for this finding is that some of the causes underlying experimental and mortality-associated selectivity are identical. For instance, it appears that both life expectancies and participation rates among survivors are lower among individuals with low levels of functioning in intelligence, ADLs, and sensorimotor performance. It seems likely that the average life expectancy of individuals who were still alive at T3 and did not participate in the T3 Intensive Protocol (N = 107) will be lower than the average life expectancy of individuals who were still alive at T3 but did participate in the T3 Intensive Protocol (N = 206). In this limited sense, experimental selectivity may be seen as a "precursor" of mortality-associated selectivity.
Validity of the PLSF
The PLSF were used to project selectivity into variables measured after selection. The similarity of effects sizes at T1 and T3 for identical variables (see middle and lower panels of Figure 1) lends credibility to this estimation method. Note that the PLSF can also be used to estimate selectivity effects for variables that were measured at a later measurement occasion for the first time, especially if these variables are well predicted by variables assessed at earlier occasions (cf. Lindenberger et al., 1999).
Selectivity of the T3 Sample With Respect to Variances and Correlations
So far, our findings suggest that the degree of selectivity of the T3 sample relative to the T1 parent sample was relatively small by conventional standards. However, this conclusion is warranted only with respect to average experimental selectivity effects on means and prevalence rates, and must not be generalized to other aspects of the data. A more comprehensive picture of selectivity effects also has to consider variability information (cf. Lindenberger et al., 1999). Therefore, we briefly address selectivity in the T3 sample regarding variances and correlations.
Overall, the T3 sample was less variable than the T1 parent sample on 26 of the 32 continuous variables listed in Tables 1 and 2 (z = 3.36, p < .01; sign test). Variability decrements tended to be more pronounced for variables that also showed larger selectivity effects with respect to means, such as chronological age, ADLs, and intelligence.
Variance restrictions can exert a strong influence on covariances and correlations (e.g., Nesselroade & Thompson, 1995). As an illustration, Table 3 displays simple correlations among age, vision, hearing, balance/gait, and intelligence at T1 for the T1 parent sample, the T3 survivors, and the T3 sample. Almost all interrelations were attenuated by mortality-associated selectivity (T1 parent sample vs T3 survivors) and by experimental selectivity (T3 survivors vs T3 sample). Table 3 also shows that the corresponding changes were close to zero for age-partialed correlations. Apparently, then, the observed reductions in correlations were collinear with reductions in the variance of age.
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Methodological Conclusions
In this article, we proposed a simple set of formulae for the additive decomposition of selectivity effects into mortality-associated and experimental components. Partitioning total selectivity into these two components serves to separate selectivity as a population process (i.e., selectivity associated with mortality) from selectivity as a sampling problem (i.e., selectivity associated with nonparticipation among survivors). In addition, we advocated the use of the PLSF to project selectivity onto variables measured after selection, and the comparison of results before and after regressing the relevant variables on age to document the extent to which mortality-associated and experimental components of selectivity are age linked. In combination, the three methods yield central and useful information about the magnitude and nature of selectivity effects regarding means and prevalance rates in aging populations. Because of its standardized (effect-size) format, this information is easily compared across measurement occasions, variables, and studies.
At the same time, we recognize the limitations of the data-analytic strategy pursued in this article and would like to mention three of them. First, the PLSF assume linearity as well as homoscedasticity, and generate estimates for first and second-order moments only (i.e., means, prevalence rates, variances, covariances). Hence, the PLSF do not provide selectivity-adjusted estimates for missing data at the individual (i.e., raw data) level, and are not well suited to capture selectivity in patterns of change. Second, our approach identifies correlates for surviving up to a specific point in time (e.g., T3). This is appropriate whenever one would like to describe sample selectivity as a function of differential survival at exactly that time point, as was the case in the present article. For most other purposes, individual differences in survival should be described on a temporal continuum using hazard functions or related procedures (e.g., Ghisletta & Lindenberger, 2000; Maier & Smith, 1999). Third, our approach focused on selectivity at the level of individual variables. In a large multidisciplinary project such as BASE, this initial focus on individual variables is legitimate because researchers often want to know the degree of selectivity for their specific variable of interest. However, such variable-centered approaches to selectivity need to be complemented by methods that focus on latent factors as the unit of selection (Dolan & Molenaar, 1994) or on differences between types of individuals (Smith & Baltes, 1997).
Despite these limitations, the data and methodology presented in this article highlight the shortcomings of common longitudinal and cross-sectional methods for the study of age differences and age changes in old and very old age. Quite often, these methods are based on the notion of a unitary and immutable population spanning the entire age range under study. For instance, the identification of cross-sectional age gradients for intellectual abilities from age 70 to age 100+ presumes that individuals observed at age 100+ are drawn from the same population as individuals observed at age 70. Aside from cohort differences, this assumption is no longer tenable if survival is selective (cf. Baltes, Mayer, Helmchen, & Steinhagen-Thiessen, 1999, p. 44; Lindenberger et al., 1999, p. 78). To arrive at better representations of interindividual differences in intraindividual change under conditions of selective mortality, we need to develop and use methods that represent change and selection processes within the same data-analytic framework (e.g., Lindenberger & Ghisletta, in press; McArdle et al., 1991; Singer, Verhaeghen, Ghisletta, Lindenberger, & Baltes, 2002).
SPECIAL SECTION
Notes
1. Sample selectivity prior to the Intensive Protocol of the first measurement occasion has been reported elsewhere (Lindenberger et al., 1999).
2. The magnitude of selectivity effects did not vary reliably as a function of measurement occasion (i.e., T1, T2, T3), F(2,45) = 0.87, MSE = 0.016, p = .43. Therefore, it appeared justified to collapse the data across measurement occasions despite the fact that selectivity effects at T1 were computed directly, whereas selectivity effects at T2 and T3 were computed after application of the PLSF.
3. Note that our rationale for identifying the age-orthogonal component of mortality-associated and experimental selectivity effects was entirely descriptive. From a developmental perspective, the age-linked component of selectivity is at least as much in need of explanation as the age-orthogonal component.
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Acknowledgments |
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Received for publication July 31, 2001. Accepted for publication July 27, 2002.
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