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Psychology and Aging
2015, Vol. 30, No. 3, 573–588
© 2015 American Psychological Association
0882-7974/15/$12.00 http://dx.doi.org/10.1037/a0039493
Openness as a Buffer Against Cognitive Decline: The Openness-FluidCrystallized-Intelligence (OFCI) Model Applied to Late Adulthood
Matthias Ziegler and Anja Cengia
Patrick Mussel
Humboldt-Universität zu Berlin
Julius-Maximilians-University Würzburg
Denis Gerstorf
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Humboldt-Universität zu Berlin and German Institute for Economic Research (DIW) Berlin, Germany
Explaining cognitive decline in late adulthood is a major research area. Models using personality traits
as possible influential variables are rare. This study tested assumptions based on an adapted version of
the Openness-Fluid-Crystallized-Intelligence (OFCI) model. The OFCI model adapted to late adulthood
predicts that openness is related to the decline in fluid reasoning (Gf) through environmental enrichment.
Gf should be related to the development of comprehension knowledge (Gc; investment theory). It was
also assumed that Gf predicts changes in openness as suggested by the environmental success hypothesis.
Finally, the OFCI model proposes that openness has an indirect influence on the decline in Gc through
its effect on Gf (mediation hypothesis). Using data from the Berlin Aging Study (N ⫽ 516, 70 –103 years
at T1), these predictions were tested using latent change score and latent growth curve models with
indicators of each trait. The current findings and prior research support environmental enrichment and
success, investment theory, and partially the mediation hypotheses. Based on a summary of all findings,
the OFCI model for late adulthood is suggested.
Keywords: environmental enrichment and environmental success, investment theory, cognitive aging,
differential preservation, preserved differentiation
ated operationalization (McArdle, Ferrer-Caja, Hamagami, & Woodcock, 2002). In this article, cognitive abilities are therefore defined in
accordance with the Carroll-Horn-Cattell (CHC) model of intelligence as proposed by McGrew (2009). According to McGrew, who
synthesized the models proposed by Carroll, Cattell, and Horn, intelligence can be understood as a hierarchical construct with general
mental ability on top (Stratum III) and up to 16 lower-order, broader
abilities on Stratum II. Among these broader cognitive abilities are
fluid reasoning (Gf) and comprehension knowledge (Gc). Gf and Gc
refer to Cattell’s notion of fluid and crystallized intelligence, respectively. McGrew summarizes that Gf is the ability to solve novel and
complex problems that cannot be solved automatically. Gc “is primarily a store of verbal or language-based declarative (knowing what)
and procedural (knowing how) knowledge acquired through the investment of other abilities during formal and informal educational and
general life experiences” (p. 5). McGrew’s version of the CHC model
also includes around 80 Stratum I abilities. For example, he names
general sequential (deductive) reasoning, induction, quantitative reasoning, Piagetian reasoning, and speed of reasoning as lower-order
components of Gf. Of all the different constructs within the CHC
model, Gf and Gc have especially been shown to be predictors of
relevant life outcomes, such as academic or work success, as well as
longevity (Corley, Gow, Starr, & Deary, 2012; Ghisletta, McArdle, &
Lindenberger, 2006; Kuncel, Ones, & Sackett, 2010).
Aging influences physical as well as psychological parameters,
creating a complex picture of intraindividual development and
interindividual differences in this development. Specifically, the
decline of cognitive functions and its antecedents and consequences have attracted research interest. This article will add to
this body of literature by focusing on changes in indicators of fluid
and crystallized intelligence and their relation to personality in late
adulthood. More specifically, the role of openness to experience as
a buffer slowing down cognitive decline will be explored based on
the Openness-Fluid-Crystallized-Intelligence (OFCI) model
(Ziegler, Danay, Heene, Asendorpf, & Bühner, 2012).
Cognitive Ability and Its Development
Maintaining cognitive abilities in late adulthood is an important
factor of successful and healthy aging (Baltes, Lindenberger, &
Staudinger, 2006; Gottfredson & Deary, 2004; Rowe & Kahn, 1997).
Despite the positive influence the g factor of intelligence has on many
life outcomes, studying cognitive change requires a more differenti-
This article was published Online First July 6, 2015.
Matthias Ziegler and Anja Cengia, Psychological Institute, HumboldtUniversität zu Berlin; Patrick Mussel, Department of Psychology, JuliusMaximilians-University Würzburg; Denis Gerstorf, Psychological Institute, Humboldt-Universität zu Berlin and German Institute for Economic
Research (DIW) Berlin, Germany.
Correspondence concerning this article should be addressed to Matthias
Ziegler, Unter den Linden 6, Psychological Institute, Humboldt-Universität
zu Berlin, 10099 Berlin, Germany. E-mail: zieglema@hu-berlin.de
Development of Gf and Gc
McArdle et al. (2002) used latent change score models (LCSMs)
to analyze data from an intelligence test norming sample. The
sample was tested twice, with a time gap between less than 1 year
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574
ZIEGLER, CENGIA, MUSSEL, AND GERSTORF
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and up to 10 years. Using these data, the authors described the
development of Gf and Gc, identifying several marked differences.
For example, whereas Gf peaked at the age of 22.8, Gc peaked
around 35.6 years. Moreover, Gc had the slowest growth and
change rates. The latter point is especially important because it
underscores the necessity to include a longer time period to be able
to identify changes in Gc. Other research, focusing on individuals
over age 60, showed that many Gc abilities are indeed showing
average decline rates (Finkel, Reynolds, McArdle, & Pedersen,
2007; Singer, Verhaeghen, Ghisletta, Lindenberger, & Baltes,
2003; Whalley, Deary, Appleton, & Starr, 2004).
Developmental Relationship Between Gf and Gc
An important question is whether changes in Gf and Gc are
related. Cattell’s investment theory (Cattell, 1987) explains how
the investment of fluid ability leads to an increase in Gc. Ziegler
et al. (2012) found empirical support for the investment theory in
a longitudinal data set spanning a period of 6 years from 17 years
of age until 23. It is also reasonable to assume that changes in Gc
are still related to changes in Gf, even in late adulthood. In fact,
Ghisletta and Lindenberger (2003), using data from the Berlin
Aging Study (BASE), could show that changes in a Gc indicator
are dominated by changes in a Gf indicator. More empirical
support for the investment theory in late adulthood was provided
by McArdle, Hamagami, Meredith, and Bradway (2000). However, opposed to young age, the effect of Gf on the development of
Gc in late adulthood is more like a buffering. Thus, persons higher
in Gf have a slower decline in Gc.
The decline in Gf has mainly been explained by changes in the
underlying components. According to Finkel et al. (2007), the
leading indicator of changes in Gf are changes in processing speed.
To test this, Finkel et al. used a digit symbol test, as well as a figure
identification test. Thus, models trying to test mechanisms associated with changes in Gf should focus on processing speed as a
predictor of the decline. Previously, it was stated that the CHC
model as described by McGrew defines speed of reasoning as a
lower-order component of Gf. As Krumm et al. (2009) could show,
the latent general speed variable explaining interindividual differences in tests like the digit symbol test also explains the speed
component in reasoning tests. Thus, the current study used a digit
letter (DL) test, which represents this speed component as one
aspect of Gf.
It has been widely discussed that a variety of different factors
shape cognitive trajectories of change, including the accumulation
of disease and disability (Anstey, 2012; MacDonald, DeCarlo, &
Dixon, 2011; Spiro & Brady, 2008), genetics (Deary et al., 2009),
smoking (Corley et al., 2012; Deary et al., 2003), and social
participation (Lövdén, Ghisletta, & Lindenberger, 2004). Among
the many concepts that have been regarded as important for
cognitive aging are also personality domains.
Personality and the Development of Cognitive Abilities
The Big Five (Goldberg, 1990) or the five-factor model (Costa
& McCrae, 1995) are accepted as broad frameworks describing
interindividual differences in personality descriptions. Both models include the broad personality domains of neuroticism, extraversion, openness/intellect, agreeableness, and conscientiousness.
A multitude of studies have shown that these traits and their facets
are associated with a variety of other individual difference characteristics (Roberts, Kuncel, Shiner, Caspi, & Goldberg, 2007). Of
importance to the current research are findings showing that these
personality indicators affect scholastic, academic, and job performance (Poropat, 2009, 2014; Ziegler et al., 2014; Ziegler, Danay,
Schölmerich, & Bühner, 2010), areas where cognitive abilities are
known to be powerful predictors (Kuncel, Hezlett, & Ones, 2004;
Schmidt & Hunter, 1998). Research focused on the overlap between the Big Five and cognitive abilities has shown that it is
important to differentiate between Gf and Gc. For example, Ackerman and Heggestad (1997) reported that especially openness and
neuroticism overlap with cognitive abilities. In their meta-analysis,
they could show that openness is positively correlated with intelligence, especially with Gc. Neuroticism, on the other hand, was
found to be negatively associated with intelligence. Soubelet and
Salthouse (2011) investigated whether the relationships between
personality and cognitive ability varied as a function of age. They
also singled out openness as the personality trait most robustly
associated with cognitive abilities, reporting substantial correlations with Gf and Gc across the ages 18 to 96 years. Again,
neuroticism was also substantially and negatively correlated with
Gc and Gf performance. However, these relationships were much
less pronounced and partly restricted to Gf performance of older
adults. Whereas the literature cited so far has only looked at the
overlap, Curtis, Windsor, and Soubelet (2015) review literature
focused on mechanisms explaining the influence of personality on
the development of cognitive abilities. For openness, the authors
discuss two possible mechanisms by which an affect on cognitive
change could be exerted. On the one hand, the differential preservation hypothesis (Salthouse, 2006) states that more active individuals, that is, more open individuals, show a slower cognitive
decline. On the other hand, the preserved differentiation hypothesis (Salthouse, 2006) assumes that the trajectory of cognitive
decline is not affected by openness. Curtis et al. conclude that
more longitudinal research is needed before one of the hypotheses
can be favored. The mechanisms related to the other personality
traits are described as less clear, potentially reversed in nature, and
with only little empirical support. For example, Curtis et al. point
out a slower cognitive decline associated with conscientiousness.
At the same time, the mechanism is discussed as potentially being
reversed; that is, older adults behaving more conscientious to
compensate for declining ability. Regarding neuroticism, Curtis et
al. conclude that test anxiety might play an important part in
explaining the negative impact on cognitive development. In addition, they point out that the mechanism might also be reversed in
nature; that is, a decline in cognitive ability leading to an increased
emotional instability.
Summing up, as is the case in most process models integrating
cognitive development and personality, it is openness that plays a
vital role (Ackerman, 1996; Ziegler, Danay, Heene, Asendorpf, &
Bühner, 2012). Ackerman (1996) in his seminal process, personality, intelligence, knowledge (PPIK) model differentiated between intelligence-as-process and intelligence-as-knowledge,
which can be regarded as Gf- and Gc-type abilities. The PPIK
model describes how the interplay between these abilities, specific
interests, and personality traits leads to differentiated knowledge
structures across the life span. Of importance for the current study
is the fact that Ackerman also included openness as a key variable
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OFCI IN LATE ADULTHOOD
within this process. He also suggested that typical intellectual
engagement is influential for cognitive development. However,
work by Mussel (2010) shows that such traits might be regarded as
facets of a broader openness/intellect domain (Mussel, 2013).
Openness is regarded as an investment trait, which is a term coined
by von Stumm and Ackerman (2013) to summarize the personality
traits that influence when, where, and how people invest time and
effort into learning. In their meta-analysis, they could show that on
average there is a correlation of .30 between investment traits and
cognitive ability. Openness and its facets were again shown to be
related with cognitive ability tests. As a consequence, and in line
with these findings and process models, the current study derived
its hypotheses from a model in which openness is a key element.
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ment occasions spanning 6 years, the model assumptions could be
confirmed. Ziegler et al. (2012) also stated that the influence of
openness on Gf and Gc should only occur within critical time
periods. This means that an individual needs the degrees of freedom to act out open behavior and seek new learning environments.
Young adulthood with all the opportunities for development
clearly is such a time period (Figure 1 summarizes these ideas).
Moreover, Ziegler et al. also included interests (see Ackerman’s
[1996] PPIK model) in their model, which are seen as moderators
for the development of Gc. Late adulthood offers such degrees of
freedom to some degree as well because no job obligations are
keeping an individual from acting out open behavior. However,
health issues might have a negative influence counterbalancing
this.
Openness and Its Development
A person with a high degree of openness to experience has an
inner need to learn, is curious, imaginative, and willing to deal
with new problems, topics, persons, and much more. A metaanalysis by Roberts, Walton, and Viechtbauer (2006) shows that
openness to experience increases statistically significantly between
the ages 18 –22 and decreases between 60 and 70. In late adulthood, there were no significant decreases or increases, which
means that openness is staying relatively stable after 70. Despite
being relatively stable itself, openness might be related to the
development of other traits, especially cognitive abilities as described earlier (Curtis et al., 2015). One model that suggests such
relations is the OFCI model.
The OFCI Model
The OFCI model, proposed by Ziegler, Danay, Heene, Asendorpf, and Bühner (2012), is a developmental model integrating
openness, Gf, and Gc. The developmental aspects proposed in the
OFCI model combine several findings from research focused on
changes in personality and cognitive ability. The OFCI model
includes the investment theory (mentioned previously; Cattell,
1987), which states that Gf positively affects the development of
Gc. Another part of the OFCI model is the environmental enrichment hypothesis (as stated by Raine, Reynolds, Venables, & Mednick, 2002). This hypothesis assumes that openness to experience
has a positive longitudinal influence on Gf because individuals
higher on openness are more likely to encounter new learning
opportunities, which positively affects the development of Gf. A
further assumption of the OFCI model is the environmental success hypothesis. According to this hypothesis, Gf positively affects
the development of openness to experience. Individuals scoring
high on Gf are believed to have a higher probability of successfully
managing new problems, increasing the likelihood that these individuals will continue seeking new situations, thereby increasing
their openness to experience. Another hypothesis within the OFCI
model is the mediation hypothesis. It is assumed that openness to
experience also influences the development of Gc via the already
described effect on the development of Gf. In other words, in order
to learn, it does not suffice to experience new and stimulating
situations. It is also necessary that the new information be actively
processed using higher cognitive abilities. These model assumptions were successfully tested with data from young adults by
Ziegler et al. (2012). Using a LCSM and data from two measure-
The OFCI Model Applied to Late Adulthood
The present study focused on late adulthood, which means the
hypotheses of the OFCI model and the known developmental
aspects of Gc, Gf, and openness to experience, which were mentioned previously, have to be adapted to the developmental
changes occurring during this developmental stage. The major
difference is that Gf and Gc are not increasing in late adulthood, as
was the case in young adults, but are declining. Thus, it is assumed
(a) that openness to experience is associated with less steep declines in Gf (environmental enrichment); (b) Gf is associated with
the development of openness to experience (environmental success); (c) openness to experience is associated with flatter declines
in Gc indirectly, with changes in Gf as a mediator (mediation
hypothesis); and (d) Gf should slow down the decline in Gc
(investment theory).
Environmental Enrichment Hypothesis
The reason for Hypothesis (a) is that higher levels of openness
should still ensure a cognitively demanding and more enriched
environment, which can act as a training ground for Gf (environmental enrichment hypothesis). This would be in line with what
Salthouse (2006) called differential preservation hypothesis. Prior
research exists that training does indeed positively influence cognitive aging (Noack, Lövdén, Schmiedek, & Lindenberger, 2013).
Moreover, Graham and Lachman (2012) could show that older
adults who retained their level of openness had higher reasoning
scores after 8 years than did those individuals with a decline in
openness.
Environmental Success Hypothesis
Adapted to late adulthood, this hypothesis assumes that Gf
influences the development of openness to experience. One prior
study partially supports this hypothesis in late adulthood. Von
Stumm and Deary (2013) used data from the Lothian Birth Cohort
(1936) to examine the relation between intellect, a facet of openness to experience, and verbal fluency, an indicator of Gc, at ages
70 and 73. A cross-lagged analysis was used. To eliminate differences in prior cognitive abilities, the measures were adjusted for
general IQ at ages 11 and 70 years. It was shown that IQ at age 11
was associated with Gc and intellect at age 70. Of further importance for the current study is the finding that Gc at age 70 was
ZIEGLER, CENGIA, MUSSEL, AND GERSTORF
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576
Figure 1. The original Openness-Fluid-Crystallized-Intelligence (OFCI) model for young age by Ziegler et al.
(2012). Auto-regressions are not displayed.
significantly associated with intellect at age 73. Thus, cognitive
ability affected the development of openness. A critical issue is
that no measure of fluid intelligence was included in the actual
developmental part of the model. Therefore, the present study will
revisit the issue of the relationship between changes in openness
and changes in cognitive ability in late adulthood but test hypotheses derived from an adapted OFCI model.
such large studies, the number of tests that can be applied is
limited. Thus, each of the constructs mentioned here is operationalized with one test only. Those tests were selected to achieve a
maximum of assessments. Moreover, tests were selected that are in
line with the theoretical assumptions stated previously. Therefore,
the following analyses should be seen as a first test of the OFCI
model for late adulthood.
Mediation Hypothesis
The mediation hypothesis adapted to late adulthood states that
openness to experience is associated with flatter declines in Gc
indirectly, with changes in Gf as a mediator. As before, the idea
here is that experiencing new and stimulating environments does
not necessarily affect knowledge unless the new information is
elaborated on.
Investment Theory
Adapting Cattell’s original idea to late adulthood leads to the
assumption that Gf should slow down the decline in Gc. The study
by Ghisletta and Lindenberger (2003) already used the same data
set used here to test just this hypothesis showing that Gf dominates
changes in Gc.
A comprehensive test of all the assumptions derived from the
adapted OFCI model would ideally be based on a multiwave
assessment of a large sample of older adults applying multiple
measures of Gf and Gc to accommodate research highlighting the
substantial breadth of both constructs. Moreover, openness should
ideally be operationalized with a facet measure to differentiate
between its different components (Mussel, 2013). The data used
here have the advantage of providing multiple waves of assessment within a large sample of older adults. As is often the case in
Method
The present study uses longitudinal data of the BASE that was
collected in six nested subsamples over 13 years. All in all, 6 data
collection periods took place: 1990 –1993 (T1; N ⫽ 516), 1993–
1994 (T2; n ⫽ 361), 1995–1996 (T3; n ⫽ 244), 1997–1998 (T4;
n ⫽ 164), 2000 (T5; n ⫽ 88), and 2004 –2005 (T6; n ⫽ 48). T2
was started an average of 1.95 years (SD ⫽ 0.71), T3 3.76 years
(SD ⫽ 0.66), T4 5.53 years (SD ⫽ 0.79), T5 8.94 years (SD ⫽
0.84), and T6 13.00 years (SD ⫽ 0.87) after T1. For detailed
descriptions of the longitudinal design and samples, assessed variables and procedures see Baltes and Mayer (1999) and Gerstorf,
Ram, Lindenberger, and Smith (2013).
Sample and Procedure
The baseline sample of BASE (mean age ⫽ 84.92 years, SD ⫽
8.66, range ⫽ 70 –103 years) was stratified by age and gender,
which means that each age group (70 –74, 75–79, 80 – 84, 85– 89,
90 –94, and 95⫹ years) consisted of 43 women and 43 men. A total
of 1,908 individuals from the Berlin city register were contacted,
of which 516 participants completed a 14-session assessment
protocol.
The primary reason for the decrease in sample size over time
was mortality, typically accounting for more than 75% of the
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OFCI IN LATE ADULTHOOD
variance in attrition; (voluntary) drop out often occurred because
of poor health. At T6, 17% (n ⫽ 89) of the initial 516 participants
were still alive and willing to participate. Baltes and Mayer (2001)
concluded with regard to possible attrition effects: “. . . selectivity
analyses did not provide any strong evidence in favor of a distortion of variances or covariances as a consequence of sample
attrition” (p. 331). Moreover, the longitudinal selectivity of the
BASE sample using an effect size metric was examined. The
analyses indicated the degree to which individuals who survived
and participated longitudinally differed from the 516 sample in
BASE at T1 (see Lindenberger, Singer, & Baltes, 2002). For
example, based on the 164 participants who provided data for four
or more occasions (on the DL test), it was found that better
performance on the DL at T1 (0.62 SD units, with SD referring to
that of the 516 sample), on the categories (CA) test at T1 (0.46
SD), more openness (0.31 SD), younger age (⫺0.74 SD), higher
socioeconomic status (SES; 0.19 SD), and fewer medical diagnoses (⫺0.31 SD) were all associated with subsequently lower mortality and higher participation rates among survivors. The covariates included in our models (e.g., age, health, SES) represent
attrition-informative variables and therefore helped to accommodate longitudinal selectivity for the variables under scrutiny here.
At the same time, we acknowledge that participants who provided the most change (i.e., longitudinal) information represent
a positively selected subset of the initial sample. Therefore,
additional analyses were conducted using only three measurement points (see Statistical Procedures section). Table 1contains descriptive statistics for the three construct indicators for
each measurement occasion.
Research assistants and medical personnel conducted all tests in
individual face-to-face sessions. Nearly all sessions took place in
the participant’s place of residence, except for the sessions that
involved geriatric medicine. The average time per session was 90
min and was split into shorter units if necessary.
Table 1
Age at Assessment and Descriptive Statistics for Data Entered
Into the Change Score Models
Test and occasion
Openness
T1
T3
T4
T5
T6
Digit letter test (Gf)
T1
T2
T3
T4
T5
T6
Categories test (Gc)
T1
T2
T3
T4
T5
T6
Note.
n
Age
M
SD
516
206
132
83
46
84.92
83.62
83.78
85.80
89.42
50
50.54
49.96
50.28
49.18
10
9.39
9.33
8.51
8.92
476
324
228
151
80
40
85.26
84.34
84.01
83.76
85.53
88.81
50
51.01
52.23
53.72
54.58
53.01
10
10.95
9.54
9.74
8.25
8.30
515
359
244
164
88
47
84.93
85.21
84.34
84.07
85.87
89.44
50
52.31
54.17
55.92
54.66
56.52
10
10.59
9.89
11.03
10.26
10.44
Gf ⫽ fluid reasoning; Gc ⫽ comprehension knowledge.
577
Measures
Cognitive measures.
DL task. As indicator of Gf, the digit letter (DL) task was
used. This task resembles the digit symbol substitution test of the
Wechsler Adult Intelligence Scale (Wechsler, 1981). As explained
previously, the assessed perceptual speed is a part of Gf (Krumm
et al., 2009; McGrew, 2009). A template with nine digit-letter
pairings was visible for the entire testing period of 3 min. Participants were shown a total of 21 sheets, each sheet containing six
digits. The subjects had to name the corresponding letter to all six
digits as fast as possible before the next sheet was presented
(Cronbach’s alpha ⫽ .96; see Lindenberger, Mayr, & Kliegl, 1993;
Lövdén et al., 2004).
CA. The task CA was used as an indicator of Gc. Here the
participants had to name as many different animals as possible
within 90 s. Nonvalid responses were morphological variants (e.g.,
“horses” after “horse”), repetitions and wrong category (e.g.,
“rose”). Therefore, all analyses were based on the number of
correct responses (Lindenberger et al., 1993). Intercoder reliability
for this measure was high (r ⫽ .99; see Lindenberger et al., 1993;
Lövden et al., 2004). Similar tests tapping fluency have been used
as indicators of Gc before (von Stumm & Deary, 2013).
Personality measures. Six items from the Neuroticism Extraversion Openness (NEO) Inventory (Costa & McCrae, 1985) were
selected to measure openness to experience. Those items represented the facets fantasy, ideas, feelings, aesthetics, and actions
thereby providing an operationalization of the core elements of
openness as defined in the NEO. Items were read aloud and
participants were asked to rate their agreement with the statements
on a 5-point Likert scale ranging from (1) “does not apply to me at
all” to (5) “applies very well to me”. Test–retest reliability for the
measure was .77 for a time period of approximately 8 weeks
(Freund & Smith, 1999). Further details about the measurement
properties of the personality trait items as assessed in the BASE
are reported in Smith and Baltes (1999). A table containing correlations between all measures used can be found in the Appendix.
Covariates
To control for the potential influence of health the extent of
comorbidities was measured as the number of physician-observed
diagnoses (determined in clinical examinations supported by additional blood and saliva laboratory assessments) of moderate to
severe chronic illnesses (according to the International Classification of Diseases, Ninth Revision). For details, see SteinhagenThiessen and Borchelt (1999). In addition, a composite score
containing (a) equivalent income, defined as the net household
income weighted by the number of people sharing the household;
(b) occupational prestige, based on a standard rating scale for
Germany; and (c) number of years of education was used as a
marker of SES. Controlling for SES ensures that the findings are
not driven by factors associated with this variable (Lang et al.,
2008). Finally, age was used as a covariate.
Data Preparation
All measures were standardized using a T norm (M ⫽ 50, SD ⫽
10), with the parent (T1) BASE sample (n ⫽ 516) as reference. For
ZIEGLER, CENGIA, MUSSEL, AND GERSTORF
578
the DL task, missing data not related to attrition added up to 8.6%
(122 of 1,421 data points over six assessments), which could be
explained by poor vision of these participants. For the CA task,
missing data amounted to 0.3% (4 of 1,421 data points over six
assessments) and for assessing openness to 7.3% (77 of 1,060 data
points over five assessments). A full information maximum likelihood (FIML) method was used in all structural analyses.
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Statistical Procedures
For describing longitudinal development, LCSMs were used. A
graphical representation is given in Figure 2. The analyses were
divided into several steps. Because of the complexity, we first
estimated a LCSM for each of the three constructs separately.
The diagram shows the LCSM for CA (Model 1a) and is mostly
equivalent to the single models for DL (Model 1b) and openness
(Model 1c). The error term eX is assumed to be equal for all
manifest variables and uncorrelated with all other components. For
every assessment (X1, X2, X3, X4, X5, X6), a corresponding
latent variable (x1, x2, x3, x4, x6, x8) was created. The variables x5
and x7 for CA are unmeasured “node” variables with a path to ⫻1
set to zero. These were added to represent occasions on which the
given variable was not assessed, so that an equal time interval of
approximately 2 years was assured across the models for DL, CA,
and openness.
The latent change scores ⌬x[t] represent the latent difference
score between x[t⫺1] and x[t]. The intercept Ix represents the
baseline and is estimated at population level like the slope Sx. The
slope has a constant loading of 1 on all change scores ⌬x[t],
thereby explaining linear change or constant growth. As an extension of typical latent growth curve models, an auto-proportion
parameter (␤x) is added. That parameter shows the effect of
variable x at time t⫺1 on subsequent change between times t⫺1
and t. For reasons of simplicity and model identification, ␤x is set
to be equal across time. If ␤x would be set to zero, the LCSM
would be equivalent to a linear latent growth curve model. Three
equations can be specified which describe how the model estimates the observed score X for each person n on a specific
measurement occasion t (McArdle, Hamagami, Meredith, & Bradway, 2000):
X关t兴n ⫽ x关t兴n ⫹ e关t兴n ,
⌬x关t兴n ⫽ x关t兴n ⫺ x关t ⫺ ⌬t兴n ,
⌬x关t兴n ⫽ ␣x*n ⫹ ␤x关t ⫺ ⌬t兴n .
The first equation describes that the observed score X is explained
by a time-specific latent variable x and a time-specific error term
e. The following two equations describe how the model specifies
change. The second equation specifies a latent change score ⌬x for
time t and each person n as the difference between the latent
v