age-related changes in the mean and covariance structure of.pdf

Intelligence 48 (2015) 1529

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IntelligenceAge-related changes in the mean and covariance structure offluid and crystallized intelligence in childhoodand adolescenceUlrich Schroeders a,, Stefan Schipolowski b, Oliver Wilhelm c

a Department of Educational Science, University of Bamberg, Germanyb Department of Psychology, Humboldt-Universitt zu Berlin, Germanyc Department of Psychology and Education, Ulm University, Ulm, Germanya r t i c l e i n f o During the preparation of this manuscript, Stefanfellow of the International Max Planck Research SchEvolutionary and Ontogenetic Dynamics (LIFE). Corresponding author at: Bamberg Graduate Sch

University of Bamberg, 96045 Bamberg, Germany.E-mail address: [email protected] (

http://dx.doi.org/10.1016/j.intell.2014.10.0060160-2896/ 2014 Elsevier Inc. All rights reserved.a b s t r a c tArticle history:Received 6 July 2014Received in revised form 21 September 2014Accepted 16 October 2014Available online 8 November 2014Evidence on age-related differentiation in the structure of cognitive abilities in childhoodand adolescence is still inconclusive. Previous studies often focused on the interrelations orthe g-saturation of broad ability constructs, neglecting abilities on lower strata. In contrast,we investigated differentiation in the internal structure of fluid intelligence/gf (with verbal,numeric, and figural reasoning) and crystallized intelligence/gc (with knowledge in thenatural sciences, humanities, and social studies). To better understand the developmentof reasoning and knowledge during secondary education, we analyzed data from 11,756students attending Grades 5 to 12. Changes in both the mean structure and the covariancestructure were estimated with locally-weighted structural equation models that allowhandling age as a continuous context variable. To substantiate a potential influence ofschool tracking (i.e., different learning environments), analyses were additionally conductedseparated by school track (academic vs. nonacademic). Mean changes in gf and gc wereapproximately linear in the total sample, with a steeper slope for the latter. There was littleindication of age-related differentiation for the different reasoning facets and knowledgedomains. The results suggest that the relatively homogeneous scholastic learningenvironment in secondary education prevents the development of more pronounced abilityor knowledge profiles.

2014 Elsevier Inc. All rights reserved.Keywords:Fluid intelligenceCrystallized intelligenceAge differentiationLocal structural equation models1. Introduction

The structure and development of cognitive abilitieshave been a focus of intelligence research for over 100 years(e.g., Cudeck & MacCallum, 2007). Among the differentfactors discussed in many contemporary theories on thestructure of intelligence (Carroll, 1993; Horn & Noll, 1997;Schipolowski was aool The Life Course:

ool of Social Sciences,

U. Schroeders).McGrew, 2009), fluid intelligence (gf) and crystallizedintelligence (gc) are the most prominent ones. Gf reflectsindividual differences in decontextualized reasoning, thatis, the ability to arrive at understanding relations amongstimuli, comprehend implications, and draw inferences,while gc is defined as acculturation knowledge measuredwith tasks indicating breadth and depth of the knowledgeof the dominant culture (Horn & Noll, 1997, p. 69).

In the present study,we examine age-related changes in themean and covariance structure of fluid and crystallizedintelligence in order to better understand the development ofthese prominent abilities. Previous studies on differentiation ofcognitive abilities frequently used second-order factor models.

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16 U. Schroeders et al. / Intelligence 48 (2015) 1529Specifically, very broad ability constructs such as gf, gc, and gs(mental speed) are modeled as first-order factors below asecond-order g factor (e.g., Li et al., 2004; Tucker-Drob, 2009).In such models differentiation is expressed in terms of gsaturation and the magnitude of first-order factor loadings.However, these studies did not consider structural changes thatmay occur on lower strata of the ability hierarchy. In thepresent paper, we address this research desideratum byinvestigating age-related changes in the factor structure of gfand gc. We focus our examination to late childhood andadolescence because in these periods of time importantdecisions with respect to later academic or vocational trainingare made and the initially homogeneous learning environ-ments begin to diverge. Furthermore, Tucker-Drob and Briley(2014) showed in a comprehensive meta-analysis of longitu-dinal twin and adoption studies that the stability of cognitiveabilities approached asymptote in late childhood. From abehavior genetic perspective this may be due to the fact thatsources of interindividual differences in cognition (i.e., genetic,shared environmental, and nonshared environmental) reachedhigh levels of stability by early adulthood. Therefore, the periodof time before reaching adulthood seems of particular impor-tance in the present context.

1.1. Age-related changes in the mean structure of gf and gc

Studies on mean changes in gf and gc were alreadyconducted by Cattell, Horn, and colleagues in the context ofthe theory of fluid and crystallized intelligence (Cattell,1971; Horn & Cattell, 1967; Horn & Donaldson, 1980; Horn &Hofer, 1992). Summarizing these results, Horn (2008)emphasized the stability of gc with maintenance or im-provements through much of adulthood whereas gf reachesa peak in late adolescence or early adulthood followed by asteady decline. These results were also supported byfindings from developmental psychology over the life span.For instance, Baltes, Staudinger, and Lindenberger (1999)described a two-component model of life span intellectualdevelopment similar to the original distinction between gcand gf. They concluded that the crystallized cognitivepragmatics remain relatively stable until old age and startto decline only in very old age. In contrast, the age trajectoryof the fluid cognitive mechanics is marked by an early riseand decline. The characteristic mean changes with age havebeen conceptualized as the result of a combination ofbiological factors (e.g., maturation and aging of the brain)and cultural factors (Baltes, 1997).

Besides the characteristic differences in the meantrajectories between gf and gc, the changes in reasoningability or crystallized knowledge presumably also depend onthe specific measurement instrument. For instance, for gcthe assessment may focus on general knowledge typicallyacquired in school or on more subject-specific knowledgesuch as vocational knowledge. Because specialized knowl-edge is mainly acquired outside of school in early adulthood(e.g., vocational training, Ackerman, 2008), the learningcurve substantially increases even beyondmiddle adulthood(Ackerman, 2000; Ackerman & Rolfhus, 1999). Furthermore,there is evidence that the age trajectory of gc stronglydepends on the content domain captured. Ackerman (2000)reported for a sample of 228 adults aged 21 to 62 on the onehand positive correlations between age and knowledge inthe social sciences, humanities, and civics. On the otherhand, he found significantly negative correlations betweenage and the physical sciences. Among the different knowl-edge domains, the physical sciences had the relativelylowest correlations with a composite of traditional verbalgc indicators, but were comparatively highly associated withgf. Gf itself was substantially negatively correlated with age.These results suggest that the higher the relation of a specificmeasure to gf, for example, a science knowledge test, themore pronounced the age-related decline in that measure. Inthe case of gf measures, the prototypicality of the measurecould relate to the magnitude of the decline with age.Reversely, one could assume that the higher the relation of aspecific measure to gc, the higher the positive age-relatedgains.

1.2. Age-related changes in the covariance structure of gf and gc

The research reviewed so far focused on themean structure,that is, on the development of the average ability level in thepopulation. However, possible changes in the covariancestructure of cognitive abilities are of particular importance forseveral reasons: The first reason concerns the invariance of ameasurement instrument which holds if the indicators of ameasure are invariant with respect to an external variable(e.g., gender or age; Grimm & Widaman, 2012). Measurementinvariance is often tested with multi-group confirmatory factoranalysis (MGCFA; Vandenberg & Lance, 2000). Given thatstatements concerning the mean structure of latent variablesand, thus, the interpretation of mean differences on a constructlevel are only feasible if strong measurement invariance holds,the inspection of possible changes in the covariance structureis an essential prerequisite for such statements. Second,assumptions about differentiationdedifferentiation processesare highly relevant for the development of theories oncognitive abilities. For example, in the domain of crystallizedabilities Carroll (1993, p. 145) advocated an age-relateddifferentiation of language skills, according to which languageskills are limited to understanding oral input and to rudimen-tary speaking skills in early childhood. But through informaland formal education they are hypothesized to becomeincreasingly more complex and diverse. The question ofdevelopment is inherently linked to questions of the general-izability of findings to different phases of the life span.

Research on age-related differentiation and dedifferentiationof cognitive abilities (e.g., Baltes, Cornelius, Spiro, Nesselroade,& Willis, 1980; Baltes et al., 1999) dates back to the earlyobservation that in childhood intellectual abilities might becorrelated more highly than in adulthood (Garrett, 1946).Accordingly, the factor structure of cognitive abilities might beless differentiated in childhood than it is in adolescence. Thisassumption is usually studied by investigating a) the number offactors required to account for individual differences in cognitiveabilities or b) the factor intercorrelations of a fixed set ofpresumed cognitive abilities. With maturation the factor struc-ture is hypothesized to differentiate and to be relatively stable inadulthood until old age when dedifferentiation takes place. Thelower complexity of the structure of cognitive abilities both atthe beginning and at the end of life spanmay be explainedwith astronger influence of neurobiological constraints on intellectual

Fig. 1. Theweight function for the focal age point 15.1 years. Note. Dashed linesindicate the age points at which an observation will get a weight of 0.5.

17U. Schroeders et al. / Intelligence 48 (2015) 1529functioning in these stages of ontogenesis (i.e., brain maturationand aging, including dementiawith a high prevalence in very oldage; see Baltes et al., 1999).

The age differentiationdedifferentiation hypothesis hasreceived mixed evidence. For example, Li et al. (2004)administered a broad battery of 15 cognitive tests includingmemory, reasoning, and verbal knowledge tasks, to 291participants aged 6 to 89 years. They considered the numberof dominant principal components (i.e., with eigenvalues N1)as an estimate for structural complexity in different age groupsand reported fewer dimensions at both ends of the life spancompared to middle-aged individuals. For a young age group,that is, children aged 3 to 7 years, Tideman and Gustafsson(2004) found empirical support for the notion that cognitiveabilities differentiate with increasing age.

However, there is also contradicting evidence. Gignac(2014) tested the strength of the g factor expressed by omegahierarchical across a wide age range (2.5 to 90 years) andobserved a small increase in its magnitude from 2.5 to10.0 years. This could be interpreted as dedifferentiation inearly childhood, even though this finding is possibly a statisticalartifact due to differences in test compilation across age groups(see Gignac, 2014). For this early age, also a reanalysis of theGerman standardization sample of the SON-R 2 1/2-7(Tellegen, Laros, & Petermann, 2007), a nonverbal intelligencetest battery for children between the ages of 2.5 and 7 years,showed very little change in the relationship between areasoning and a performance factor in early childhood (Hlr,Wilhelm, & Robitzsch, 2011). The age differentiationdediffer-entiation hypothesis was also rejected for other phases of life,that is, for early childhood to adulthood (Bickley, Keith, &Wolfle, 1995; Juan-Espinosa, Garca, Colom, & Abad, 2000) andearly maturity to senescence (Escorial, Juan-Espinosa, Garca,Rebollo, & Colom, 2003). Considering the entire life span,Tucker-Drob (2009) provided little evidence for age differen-tiation by means of nonlinear factor analyses of the standard-ization data of the WJ-III (Woodcock-Johnson III; Woodcock,McGrew, & Mather, 2001) which included measures of sevenbroad ability factors (among others gf and gc) and individualsaged 4 to 101 years. In contradiction to the assumptions of theage differentiationdedifferentiation hypothesis, the analyseseven hinted at the reverse effect pattern, that is, dedifferenti-ation in school age.

In contrast to most studies in the research literature ondifferentiation that investigated broad cognitive abilities,Tucker-Drob and Salthouse (2008) also considered differenti-ation of performance in different subtests for gf, gc, and otherability factors. In a sample of adults (aged 24 to 91 years) theloadings of the indicators on the factors changed only slightlywith age, providing evidence for a high stability of the factorstructure on lower-order strata.

1.3. Methodological aspects of testing for differentiation

Testing for differentiation of cognitive abilities poses anumber ofmethodological and statistical challenges. This is notlimited to the investigation of age differentiation, but also holdstrue to a greater extent for testing ability differentiation(e.g., Molenaar, Dolan, & Verhelst, 2010; Molenaar, Dolan,Wicherts, & van der Maas, 2010; Murray, Dixon, & Johnson,2013). In developmental research, samples are often dividedinto subsamples using arbitrary cutoff values to subsequentlyanalyze and compare these subsamples using MGCFA. Whilethis methodmay be appropriate or at least plausible for genderor migration status, most context variables such as age arecontinuous in nature. In these cases, defining artificial groupsleads to a loss of information and can produce spurious results.More precisely, results may differ depending on how onechooses the arbitrary cutoff values because near the thresholds,observations are more similar across groups than withingroups. In the research literature on differentiation of cognitiveabilities, different approaches have been proposed to avoidarbitrary groups. Specifically, Latent Moderated StructuralEquation Models (Klein & Moosbrugger, 2000) have been usedto introduce additional age terms to the standard model inorder to establish whether the linear relationship between twovariables is moderated by age (for an application see Tucker-Drob, 2009).

In the present paper, we estimated Local Structural EquationModels (LSEMs; Hildebrandt,Wilhelm, & Robitzsch, 2009)whichare traditional SEMs with weighted observations. The weight isdefined by the proximity of an observation to a specific value ona context variable such as age at a certain focal age point. Withincreasing distance of the observation from the focal point,the weight decreases according to a normal distribution. Thisfunctional relationship is exemplarily depicted in Fig. 1 with afocal age point of 15.8 years. The core idea is that observationsnear the focal point provide more information for the corre-sponding SEM than themore distant observations (e.g., studentsaged13 years). If the age of the student is equal to the focal point,theweight is set to 1; if the difference is 10 months, theweight isabout .50, and so on. For each focal value of the context variable, aseparate SEM is estimated resulting in a series of models thatprovide gradients ofmodel parameters (e.g.,means, correlations,factor variances, and model fit indices). In the terminology ofLSEMs, the weights of aMGCFA are set to 1 in one age group andto 0 for all other age groups. The more the weights of the LSEMresemble such a dichotomousweighting, the closer the results ofboth methods will match. Hildebrandt et al. (2009) pointed outthat bothMGCFA and latentmoderated SEMs can be understoodas approximations of LSEMs.

18 U. Schroeders et al. / Intelligence 48 (2015) 1529Compared to MGCFA, LSEMs have some advantageousfeatures for analyzing differentiation. First, LSEMs make theartificial categorization of a continuous context variable such asage obsolete and avoid the associated loss of information. Inmany instances, this method would describe the underlyingrelationshipmore appropriately. Even in seemingly dichotomouscases (e.g., migration status) the question arises whether therelevant attribute is not in fact continuously scaled (e.g., masteryof test language). Second, LSEMs impose lower requirementsregarding sample size because for each focal point all cases areconsidered in the computation with varying weights. Thus, lowdensity or even small gaps in the sample distribution can besmoothed with adjacent observations. The higher flexibilityregarding sampling and the effective usage of observationsmakethis method particularly attractive for developmental researchacross the life span. These advantages also apply to latentmoderated SEM. Third and in contrast to nonlinear methods,LSEMs produce nonparametric age curves. Thus, the fluctuationof parameters (e.g., factor loadings) can be analyzed as a functionof a context variable. In the MGCFA approach parameters of twoor more groups are often considered without establishing theirfunctional relationship (e.g., negative linear vs. logarithmictrend) or the onset of change.1 Gradients of model parametersas derived from LSEMs can provide valid information for theorybuilding.

1.4. Research questions

To summarize, the evidence for age differentiation ofcognitive abilities is still inconclusive and often relying onstatistical methods that are not adequately dealing with thecontinuous context variable. But most importantly, researchefforts in this field have concentrated on broad ability constructswithout considering structural changes on lower strata of theability hierarchy. That is, the internal structure of broad factorssuch as gf and gc is considered stable over age. As a consequenceof examining differentiation with second-order factor models,our knowledge about changes in the factor structure of cognitiveabilities across the life span is still limited (Baltes et al., 1999) andoften restricted to broad abilities (i.e., stratum II factors). In thepresent study, we provide an in-depth analysis of age-relatedchanges in the mean structure and the factor structure of fluidand crystallized intelligence in late childhood and adolescence.

To study the internal structure of gf we assessed fluidintelligence with reasoning scales covering verbal, numeric,and figural content materials, because the distinction betweenthese content domains has been shown to be the mostimportant distinction under a common gf factor (Wilhelm,2004). In line with the description of gc as acculturationknowledge in the extended gfgc model (Horn & Noll, 1997,p. 69) we assessed gc with declarative knowledge items. Theknowledge tests covered factual knowledge in a broad varietyof 16 content domains from the natural sciences, the human-ities, and social studies.

In our examination, we first describe the average meanchanges over time for the gc factors (sciences knowledge,humanities knowledge, and social studies knowledge) and for1 Please note, however, that by imposing linear andquadratic age constraintsacross groups, it is also possible to examine age differences in the factorstructure by means of MGCFA (see Tucker-Drob & Salthouse, 2008).the reasoning factors (verbal, numeric, and figural reasoning).With regard to the age trajectories for gf and gc outlined above,we expected that the slope of the age gradients is higher for gcthan for gf. Furthermore, wewant to shed light on the questionwhether the knowledge gain is stronger for specific domains inchildhood and adolescence. In accordance with the findings ofAckerman (2000) for adults, one could assume that the higherthe relation of a specific measure to gc, the higher the positiveage-related gains. Measures depending more strongly on gfsuch as items tapping thenatural sciencesmay be subject to theearly decline in gf abilities which might result in a lesspronounced increase. On the other hand, such knowledgeitems were presumably more strongly anchored in the schoolcurriculumwhichmight cause a steeper knowledge gain in thesciences. With respect to the internal structure of gf, we wouldpredict the mean change to be especially small for the figuralaspect since it is often conceptualized as prototypical forreasoning (Wilhelm, 2004). On the other hand, the meanchange is presumably stronger for the verbal domain due to thehigher relations to gc.

Second, we investigate age-related changes in theinternal structure of gf and gc as well as the relationshipbetween both ability factors. The assumption is that overallgc is more prone to age-related differentiation than gf, sincegc is mainly the result of educational training and theopportunities for learning become more diverse in thecourse of educational training with the stratification of theschool system. Secondary education, usually starting withGrade 5, is a particularly important stage of formal trainingin Germany because it marks the beginning of a trackedschool system. Accordingly, students in different tracks areexposed to different learning environments (Baumert,Stanat, & Watermann, 2006; Becker, Ldtke, Trautwein,Kller, & Baumert, 2012). From Grade 9 on, most studentsalso get the opportunity to specialize through courseselection which should expand the knowledge in one areaat the expense of another (e.g., by dedicating more studytime to the sciences than to the humanities). Furthermore,differences in the exposure to out-of-school knowledge dueto different home environments and increasingly differentleisure time preferences (e.g., reading at home; Rolfhus &Ackerman, 1999) may contribute to the differentiation of gc.There is some evidence for complex knowledge structures inyoung and middle-aged adults (Ackerman, 2000; Rolfhus &Ackerman, 1999). Whether a differentiation of knowledgestructures is already present in secondary education has notyet been studied systematically. Gf has been shown alsoto be sensitive to differences in the individual learningenvironment (e.g., Becker et al., 2012; Ceci, 1991), eventhough to a lesser degree than gc. In order to substantiate apotential influence of school tracking (i.e., different learn-ing environments) on differentiation, results will also bereported for academic vs. nonacademic-track school types.

2. Method

2.1. Design and participants

Analyses were based on the German standardizationsample of the Berlin Test of Fluid and Crystallized Intelligence(BEFKI; see e.g., Wilhelm, Schroeders, & Schipolowski, 2014). A

19U. Schroeders et al. / Intelligence 48 (2015) 1529multiple-matrix booklet design (Gonzalez & Rutkowski, 2010)was used to keep the individual workload for the studentswithin manageable limits. That is, not all students wereadministered all items for all constructs. In total, 72 differentbooklets were used. Half of the booklets contained items onboth gf and gc, the other half either gf (24) or gc (12). Aftercompleting the test items all students were given asociodemographic questionnaire where they indicated theirage, among other demographics. Depending on the grade level,the complete test session including covariates and thequestionnaire took between 90 and 180 min.

After excluding students that did not indicate their age(n = 212), the total sample for the following analysesconsisted of N = 11,756 students (50.6% female; nmiss. = 7)attending Grades 5 to 12. Students from all German federalstates (except for the smallest one) and all school types of theGerman general educational system were included in thesample. Overall, almost one third of the sample (31.5%)attended academic-track Gymnasium schools, 16.4% wereenrolled in intermediate-track Realschule schools, and 20.2%attended vocational-track Hauptschule schools. The rest ofthe sample (31.9%) was enrolled in mixed-track schools(e.g., Gesamtschule) or advanced vocational schools. In thefollowing, we refer to Gymnasium as academic-track schoolsand all remaining school types as nonacademic-track schools.Mean age was 14.81 years (SD= 2.29) and 95% of the samplewas between 10.6 and 18.7 years old (range 9.123.6). Testingtook place between April 2008 and June 2010 at different timepoints within the school year.

Despite the overall good coverage of the different schooltracks within the sample, there were some restrictionswith respect to the sample composition in Grade 8. Morespecifically, the BEFKI standardization sample was embed-ded in an educational assessment study that intentionallytargeted students who pursued the least demanding regularschool certificate. As a consequence, no students from theFig. 2.Number of observations for students attending academic-track schools and nonahere are restricted to the range of focal age points between 11.6 and 18.5, even thousubsequent LSEManalyses. Left panel: nacad.= 3705 comprised of all students enrolled ithe distribution can be ascribed mainly to the missingness in Grade 8. Right panel: nnwho worked on at least one of the intelligence tests.intermediate-track Realschule or academic-track Gymnasi-um were included in the sampling frame for Grade 8,resulting in a considerable gap in the age distribution foracademic-track students between 13.5 and 14.5 years (seeFig. 2, for the number of students in the academic group, leftside, and nonacademic group). The few remaining observa-tions in the gap represented students from adjacent grades(i.e., mostly students that either repeated or skipped one ormore grades). As pointed out above, through the weightingof observations in LSEMs, it was possiblewithin certainlimitsto compensate for gaps within the distribution of thecontext variable. Furthermore, the overrepresentation ofstudents attending Hauptschule and comparable tracksleading to the lowest school certificate also resulted in adistinct ability-restricted subsample for this particulargrade. These restrictions in the sampling procedure haveto be kept in mind when interpreting the following results.

2.2. Measures

To allow for efficient measurement, three age-adapted testforms were used to assess fluid and crystallized intelligence.Specifically, students in Grades 5 to 7 received the easiest testform, whereas the most difficult test was administered inGrades 11 and 12; scales of intermediate difficulty were used inGrades 8 to 10. The test forms were linked with anchor items,allowing placing students of all grades on a common metric.

The fluid intelligence part of the BEFKI covered the threecontent facets verbal, numeric, and figural. The content materialhas been shown to be the most prominent distinction under acommon gf factor (Wilhelm, 2004) which in turn is very closelyrelated or even identical to g (Kyllonen & Christal, 1990). Theverbal aspect of fluid intelligence (gfv) was measured with tasksfor relational reasoning; participants had to derive logically validconclusions based on a set of given premises. The numeric part(gfn) consisted of mathematical text problems; solving thesecademic-track schools at all focal age points.Note. The age distributions depictedgh adjacent observations outside this range were included in the weighting inn academic-trackwhoworked on at least one of the intelligence tests. The gap inonacad. = 8051 comprised of all students enrolled in nonacademic-track schools

20 U. Schroeders et al. / Intelligence 48 (2015) 1529items primarily required mathematical modeling, but only basicmathematical knowledge. The figural reasoning scale (gff) wascomposed of a sequence of geometric drawings that changedtheir shading and form according to certain rules. Participantshad to infer these rules in order to determine the missingdrawings. The individual student worked on 16 to 32 multiplechoice items per content facet. Considering the different testforms, a total of 192multiple choice itemswere used in the study(64 per facet). There were 24 anchor items linking Grades 57with Grades 810 and 24 items linking Grades 810with Grades1112. A total of 12 of these anchor items were identical in allthree test forms. An additional set of 12 items form a linkbetween Grades 57 and Grades 1112.

Crystallized intelligence was assessed with declarativeknowledge items covering 16 content domains from threebroad areas: natural sciences (physics, chemistry, biology,medicine, geography, technology), the humanities (art, litera-ture, music, religion, philosophy), and social studies (history,law, politics, economy, finance). Item development and thechoice of knowledge domains aimed at covering the breadthand depth of the knowledge of the dominant culture (Horn &Noll, 1997, p. 69), taking into account knowledge that iscommonly rewarded in society, considered important, and acultural goodin contrast to trivial, short-lived knowledge(such as soccer results or the bus schedule). Both curriculum-related and out-of-school knowledge was included. Theindividual student worked on 64 to 128 multiple choice itemsper knowledge area. Considering the different test forms, 240knowledge items were used (90 for knowledge in the naturalsciences, and 75 for the humanities and the social sciences,respectively). Adjacent test forms were linked with 32 anchoritems, that is, there were 32 items linking Grades 57 withGrades 810 and 32 items linkingGrades 810with Grades 1112. A total of 16 of these anchor itemswere part of all three testforms. An additional set of 16 items were presented both inGrades 57 and in Grades 1112.

2.3. Statistical analyses

In this paper, we implemented the recently introducedmethod of Local Structural Equation Models (LSEMs). As men-tioned above, in LSEMs weights are used to treat contextvariables as continuous variables instead of creating artificialcategories. The statistical procedures for defining the bandwidthof the kernel function, calculating the weights for every fo-cal point and rescaling of the weights have been describedelsewhere (Hildebrandt, Sommer, Herzmann, & Wilhelm, 2010;Hildebrandt,Wilhelm, Herzmann, & Sommer, 2013; Hildebrandtet al., 2009; Hlr et al., 2011). Note that weightedmodels couldnot be calculated for focal points at the very edges of thedistribution of the context variable because estimation for eachfocal point requires a sufficient number of cases belowand abovethat focal point (symmetrical weighting). Measurement errorwas taken into account by estimating the 95% confidence intervalfor all reported correlations.

One challenge of this new method is that the samplesused for the LSEMs are not independent (as in MGCFA), butoverlapping and therefore dependent. This prevents the use ofestablished indices to assess the significance of changes inmeasurement parameters. In order to allow for traditionalinference tests we applied a permutation test (Hildebrandtet al., 2009; Hlr et al., 2011). In this permutation test, LSEMswere run with 1000 datasets for statistical inference. In eachdata set age was randomly assigned to individuals, althoughthe distribution of age originated from the observed sample.This approach ensured that the ability data were completelyindependent of the context variable in the permuted data set.Thus, allowing testing whether changes of the gradients in thereal data set are connected to age. Through the randomassignment, the results of the permutation test were adjustedfor a general effect of age on ability. Therefore, the shape of theparameter estimates has to be compared between the observedand the permuted data sets rather than absolute values.

In order to estimate the ability of all participants on a singlescale regardless of the test form, we used Weighted LikelihoodEstimates (WLEs; Warm, 1989) derived from estimations withthe one-parameter logistic model. WLEs were standardized(M = 0; SD = 1) with all students in Grade 9 as referencegroup. IRT scaling is the recommended procedure to compen-sate for slight deficits in test compilation with regard to thedistribution of item difficulties (Tucker-Drob, 2009). Scalingwas conducted separately for each gf facet and for each of fouritem parcels per facet. For gc, the scaling was conductedseparately for each of the 16 content domains. Accordingly,12 gf and 16 gc indicatorswere used in the LSEMs. From the 420items aggregated in the 28 parcels only 2 items had aninformation-weighted item fit value (infit) above 1.1 andnonehad an infit above 1.2which is often used as a cutoff-valuefor good item fit (Bond & Fox, 2001).

In the total sample, skewness was not an issue for thecontext variable age (see statistical objection of Murray et al.,2013). R statistics (R Development Core Team, 2014) was usedfor data preparation, weighting, and plotting. WLEs wereestimated in ConQuest 2.0 (Wu, Adams, Wilson, & Haldane,2007). LSEMs were estimated with Mplus 7.11 (Muthn &Muthn, 19982014) taking into account the nested structureof the data, that is, students within classes, with the CLUSTERstatement.

3. Results

3.1. Changes in mean structure across age

The subject of the first research question was theaverage gain in gf and gc in late childhood and adolescence.Graphical representations of age-related changes in themean structure of the gf and gc facets in the total sampleare given in Fig. 3. The data points represent the means oflatent variables that were estimated with locally-weightedthree-dimensional measurement models in separate runsfor gf and gc. In order to freely estimate the means of thelatent variables, the effects-coding method was imple-mented. This method of identifying and scaling latentvariables fixes the mean of the loadings to 1 and the sum ofthe intercepts to 0 for each factor in each measurementmodel (Little, Slegers, & Card, 2006). As a consequence,factor loadings, variances, and latent means can beestimated and tested simultaneously. In contrast to otherscaling methods, effects-coding has the advantage that theestimated latent variances and means reflect the observedmetric of the indicators, optimally weighted by the degreeto which each indicator represents the underlying latent

2 Because in LSEM the effective n varies between the focal points of thecontext variable, model comparisons between the one and three-dimensionalmodels also depend on the sensitivity of the fit measure to sample size. Wechose the Comparative Fit Index (CFI) and the Root Mean Square Error ofApproximation (RMSEA) which are less dependent on sample size than otherindices.

Fig. 3. Change in the means of fluid and crystallized intelligence facets across age in the overall sample. Note. ngf = 10,652; ngc = 8673. Parameters are standardizedmeans of the latent variables in the locally-weighted SEMs. Shaded areas indicate the 95% confidence interval. gfv = verbal fluid intelligence, gfn = numeric fluidintelligence, gff = figural fluid intelligence, sci = science, crystallized intelligence, hum = humanities, crystallized intelligence, soc = social studies, crystallizedintelligence.

21U. Schroeders et al. / Intelligence 48 (2015) 1529construct. (Little et al., 2006, p. 63). As Tucker-Drob andSalthouse (2008, p. 455) pointed out, the variances can bedirectly interpreted as the average amount of variance ineach indicator accounted for by the factor. In the contextof differentiationdedifferentiation testing, decreasingfactor loadings and factor variances over age thereforeindicate differentiation at the indicator level. In Fig. 3 thegradients around the means mark the 95% confidenceinterval (mean 1.96 SE) for the means of the latentvariables.

For all variables therewas an approximately linear increase,whereas the difference between the youngest participants(aged 11.7) and the oldest participants (aged 18.8) wassubstantial for both gf and gc. Table 1 gives the mean changeper year for all subdimensions. The average age-related changeper annumwas higher for gc (d= 0.33) than for gf (d= 0.22).The slight decrease in the means around age 14 which is moreprominent for the gf curves was due to the ability restriction inthe subsample for Grade 8 (see Design and participants).Knowledge gains over timewere slightly higher for the naturalsciences and social studies (average d = 0.36) than for thehumanities (d = 0.27, see Table 1). Considering the shape ofthe trajectories for the different subdimensions, there was noindication of differential effects across secondary education.

Fig. 4 displays themeans for all latent variables separated byschool track. Age-related changes in the nonacademic trackwere essentially linear for the gf and gc subdimensions. Thetrajectories of the mean changes in the academic track weresomewhat s-shaped. Beyond age 16 there is no substantial gainin the reasoning or knowledge scale (Table 1). As in the totalsample, the slopes of the gradients were higher for gc.Interestingly, for gcwith the exception of the humanitiesthe mean differences between the academic-track vs. thenonacademic-track students increased with age whereas for gf,the difference between the school types decreased. Similar tothe total sample, the age trajectories were very similar for allsubdimensions of gf and gc.

3.2. Age-related differentiation

To check for structural differentiation in gf and gc, weestimated 1) a three-dimensional model for gf distinguishingbetween the verbal, numeric, and figural facets and 2) a modelfor gcwith three correlated factors capturing knowledge in thenatural sciences, the humanities, and social studies. Accordingto widely used cut-off values (Hu& Bentler, 1999), the fit of thethree-dimensional models in the total samplewas good (for gf:CFI .992, RMSEA .009; for gc: CFI .969, RMSEA .016).

Fig. 5 displays the correlations between latent variablesrepresenting the gf and gc facets with their corresponding 95%confidence intervals in the overall sample. The correlationsbetween the broad knowledge domains were exceedingly high( N .90). Despite the high correlations, the three-dimensionalmodel described the data slightly better than the one-dimensional model across the entire age range; the differencein model fit ranged between .002 (CFI) .005 in favor ofthe three-dimensional model.2 As has been pointed outpreviously (Molenaar, Dolan, & Verhelst, 2010; Molenaar,Dolan, Wicherts, & van der Maas, 2010), the factor correlationsof gf and gc represent only one potential source of differenti-ation. For the three-dimensional models considered in ouranalyses, other possible sources of differentiation were a) thevariances of the latent variables, b) the factor loadings of theindicators, and c) the residual variances of the indicators. The

Table 1Change in gf and gc facets per year expressed as effect sizes.

Ability construct Effect size

d(12/13) d(13/14) d(14/15) d(15/16) d(16/17) d(17/18) Avg. (d)

Total samplegf Verbal 0.12 0.05 0.38 0.33 0.32 0.33 0.25

Numeric 0.17 0.10 0.41 0.33 0.28 0.24 0.25Figural 0.07 0.07 0.16 0.24 0.28 0.31 0.16

gc Science 0.31 0.21 0.47 0.38 0.37 0.41 0.36Humanities 0.23 0.11 0.36 0.32 0.29 0.32 0.27Social studies 0.28 0.30 0.51 0.37 0.33 0.36 0.36

Academic subsamplegf Verbal 0.18 0.28 0.56 0.29 0.13 0.08 0.25

Numeric 0.21 0.27 0.48 0.25 0.11 0.03 0.23Figural 0.16 0.14 0.21 0.25 0.20 0.02 0.16


Nonacademic subsamplegf Verbal 0.13 0.20 0.37 0.29 0.30 0.33 0.27

Numeric 0.23 0.28 0.41 0.29 0.26 0.27 0.29Figural 0.05 0.02 0.15 0.17 0.23 0.33 0.16


Note. Effect sizes are given as standardized mean differences between consecutive years (e.g., d(12/13) indicates the mean difference between students aged 13 andstudents aged 12); higher values indicate better mean performance of older students. In the last column the average change per annum is given.

22 U. Schroeders et al. / Intelligence 48 (2015) 1529gradients of factor variances are suited to assess differentiationon the construct level, whereas the other parameters wouldreflect changes in the indicators (i.e., issues with the measure-ment instrument). An age-related decrease in factor varianceswould imply differentiation in the ability constructs (see alsoTucker-Drob & Salthouse, 2008). The changes in the gradientsof factor variances were small (see Fig. 6). Similar to the factorcorrelation gradients, there was a small decline in gf and gcFig. 4.Age-related change in themeans of gf and gc across age split by school track.Note. Tgray; ngf, acad. = 3590; ngc, acad. = 2750) versus all remaining school types (ngf, nonacad. =intelligence, gff= figural fluid intelligence, sci= science, crystallized intelligence, hum=variances above age 17 which indicated a small differentiationeffect. For all other potential sources of differentiation, the age-related parameter changes were small and unsystematic overtime. Changes of all measurement parameters are given inTables 1 and 2 of the online supplement.

The gradients of the permutation test ran parallel to thex-axis (see Fig. 5). Because the results of the permutation testwere trimmed for the main effect of age, not the differences inhe samplewas split into academic-track schools (data pointswith lighter shades o7062; ngc, nonacad. = 5923). gfv = verbal fluid intelligence, gfn = numeric fluid

humanities, crystallized intelligence, soc= social studies, crystallized intelligencef

.

Fig. 5. Age differentiation in terms of factor correlation gradients of gf and gc subdimensions.Note. ngf= 10,652; ngc= 8673. gfv= verbal fluid intelligence, gfn= numericfluid intelligence, gff = figural fluid intelligence, sci = science, crystallized intelligence, hum = humanities, crystallized intelligence, soc = social studies, crystallizedintelligence.

23U. Schroeders et al. / Intelligence 48 (2015) 1529the height of parameters but in the progression are decisive forexamining differentiation. Therewas no indication of structuraldifferentiation for any of the gc dimensions; all gradients in thereal data set corresponded to their permutated counterparts.For gf, however, therewas a small decrease. The first small dentfor the gf factor correlationsespecially those that included thefigural aspectat about age 14 might be attributed to theability restriction of the sample in Grade 8 (see Design andparticipants). This explanation was reaffirmed by splitting thesample into the academic track and nonacademic-track schooltypes (see Fig. 7). When considering the academic trackforwhich there was no ability restriction or overrepresentation ofstudents with low educational aspirationsthe gradients wereessentially linearly declining. The slight and constant decrease

Fig. 6. Age differentiation in terms of factor variance gradients of gf and gc subdimensions. Note. ngf = 10,652; ngc = 8673. gfv= verbal fluid intelligence, gfn= numericfluid intelligence, gff = figural fluid intelligence, sci = science, crystallized intelligence, hum = humanities, crystallized intelligence, soc = social studies, crystallizedintelligence.

24 U. Schroeders et al. / Intelligence 48 (2015) 1529that was observable in the academic track subsample, butnot in the nonacademic track subsample could be taken asevidence of an interaction between ability and age. In the totalsample, these relationships were less clearly visible due todifferent selection effects. Taking into consideration theconfidence intervals of the factor correlations, however, theobserved differentiation effects were small and of no practicalconsequences. It is interesting to note that small differentiationeffects seemed to occur when changes in the mean structureare comparatively weak, that is, in the academic sample and forFig. 7. Age differentiation as indicated by the gradients of factor correlations split bngc, nonacad. = 5923. The results for gf and gc are reported separately for academicpoints in lighter gray represent correlations between the gc-facets. gfv = verbintelligence, sci = science, crystallized intelligence, hum = humanities, crystallizgf. From a measurement perspective, one might have expectedthat large changes in the means provide optimal conditions forthe expression of differentiation.

One anonymous reviewer suggested that the fact thatchildren of different age groups received ability-tailored testswas problematic because if parameters showed age-relatedchanges over time, it remained unclearwhether thiswas due tothe fact that the children in the different grades completeddifferent items or due to differentiation. In order to examinethe influence of linking on the results, we replicated the LSEMsy school track. Note. ngf, acad. = 3590; ngc, acad. = 2750; ngf, nonacad. = 7062-track schools (left panel) and all remaining schools. The gradients of dataal fluid intelligence, gfn = numeric fluid intelligence, gff = figural fluided intelligence, soc = social studies, crystallized intelligence.;

25U. Schroeders et al. / Intelligence 48 (2015) 1529with a reduced item pool of 12 gf and 16 gc items that wereadministered to all subjects (i.e., anchor items present in all testforms). Even though the gradients were expectedly lessprecisely estimated for these anchor items, the gradients forboth the gf facets and the gc facets were similar to the originalanalyses, especially when taking into account the underlyingnumber of items (gf: 12 vs. 180; gc: 16 vs. 240) and thedifferentestimators (ML for the continuously scaled WLEs vs. WLSMVfor the dichotomously scored items). A comparison of thegradients obtained from the different scoring methods can befound in the online supplement (Fig. 1). A further argument forusing age-adjusted test formswith subsequent linking to locatethe participants' ability on a common scale is that it exertsconsiderable advantages in contrast to alternatives (e.g., sumscore of a fixed set of items). First, it is challenging to constructitems that are suited for both 12 and 18 year old students withrespect to item difficulty. Second, substantially more items andtesting time would be required for each individual. Third, incontrast to a sum score, IRT scaling can compensate specificitiesin test compilation. Tucker-Drob (2009) showed in a simula-tion study that sum scores are particularly problematic whenthere is a mismatch between item difficulties and thedistribution of person abilities and that IRT scaling can reducesuch biases.

We also specified a two-dimensional higher-order modelwith gf and gc as second-order factors and the six facets(i.e., verbal, numeric, and figural reasoning, and knowledge inthe natural sciences, the humanities, and social studies) as first-order factors to examine the relationship between gf and gc inmore detail. The correlation between gf and gcwas stable overage, both for the total sample (.74 .84) and thesubsamples (academic track: .61 .75; nonacademictrack: .73 .80; see Fig. 8). To summarize, with theexception of the above mentioned local decrease for gff, wefound no indication of age-related differentiation, neitherFig. 8. Correlation of gf and gc in a two-dimensional higher-order model overtime. Note. ntotal = 11,756; nacad. = 3705; nnonacad. = 8051.between gf and gcnorwithin these constructs. This observationalso holds for the subgroup-specific analyses.

4. Discussion

In the present study we investigated developmentalchanges in the mean and the covariance structure of fluid andcrystallized intelligence. In comparison to previous researchon age differentiationdedifferentiation of cognitive abilities(e.g., Li et al., 2004; Tucker-Drob, 2009), we did not limit ourinvestigation to the broad ability constructs, but particularlyconsidered changes in the internal structure of gf and gc.

4.1. Age-related changes in the mean structure of gf and gc facets

Considering the subdimensions of gf (i.e., verbal, numeric,and figural reasoning) and gc (i.e., knowledge in the sciences,humanities, and social studies), we found empirical support forthe hypothesis that crystallized abilities show stronger age-related gains. The steeper slope for gc-type abilities duringsecondary educationwas expected since gc is especially sensitiveto learning and (formal) education, arguably more so than fluidintelligence (Hunt, 2008), even though this is still subject todebate (see Gustafsson, 2008, for diverging results). In the agerange considered in this studyfrom 11.5 to 18 yearswe alsofound an approximately linear positive trend for gf-type abilitiesin the total sample.

No differential effects could be ascertained for the meangradients of the different subdimensions, neither for the gf facetsnor for the gc content domains. For example, knowledge gains inthe natural sciences were neither higher (as might be assumedbecause science knowledge is more strongly connected to theschool curriculum) nor lower (as might have been the case dueto the higher overlapwith gf) than gains for the other knowledgedomains. It is conceivable that the opposing effects for thescience domain neutralized one another. A slight deviation fromthe approximately linear trend couldbeobservedonly for gff thatis probably due to fidelity problems of the gff measure in lowergrades (i.e., a lack of discriminative power at the lower end of theability distribution). Furthermore, the small dent in the meangradients at age 14 can be attributed to the sampling effects inGrade 8 (see Design and participants).

The school track-specific investigation revealed lin-ear trends for both the gf and the gc subdimensions inthe nonacademic group whereas the mean trajectories ofstudents attending academic-track schools were s-shaped.That is, for students in the most demanding school typegains in gf and gc over time followed a continuous functionthat is flattened at the beginning and particularly at theend of secondary education when compared to the steepergradients in the middle age range. These different func-tional relationships across school track may be due toselection effects: Whereas the academic group is homoge-neous over the course of secondary education (i.e., moststudents enrolled in Gymnasium in Grade 5 remain in thisschool type), the composition of the nonacademic group ischanging. That is, especially after completing Grade 9 andagain after Grade 10 many students in nonacademic tracksleave the general educational system and as a result theremaining students are on average more capable and strivefor more advanced certificates.

26 U. Schroeders et al. / Intelligence 48 (2015) 1529In the same vein, the group of students in academic-trackschools represents an ability-selected subsample. We founddiminishing gains for both gf and gc at the end of adolescence.The mean gf trajectories were consistent with the literature onthe development of cognitive abilities assuming an early peak offluid abilities in early adulthood (e.g., Horn, 2008). Surprising-ly, this curvilinear trend was also present for the gains inknowledge. One explanation for this finding is that the schoolingsystemmay be less effective in conveying additional knowledgeto competent students at the end of upper secondary education,but further research is needed to investigate this possibility.Another explanation that cannot be ruled out completely is thatthe measures were not sensitive enough to adequately capturethe knowledge gains that occur at the end of secondaryeducation. However, the use of age-adapted test forms (seeMeasures) and the absence of ceiling effects in this (or any other)age group contradict this explanation.

With regard to mean differences between school tracks,Becker et al. (2012) examined the influence of a trackedschool system on decontextualized reasoning in a longitu-dinal data set from Grades 7 to 10 and reported increasingdifferences between the tracks. Constraining the findingsof our study to the same age range (1316 years), we couldreplicate these results. However, taking into accountthe extended age range investigated here, the differencebetween academic vs. nonacademic-track schools wassmaller at the end of secondary education (see Fig. 4). Thismay be largely due to the composition of the nonacademicgroup described earlier, since only the more competentstudents will continue their general school educationbeyond Grades 9 and 10, respectively (e.g., instead of takingup an apprenticeship), which reduces the performance gapto the academic-track group.

4.2. Age-related changes in the covariance structure of gf and gcfacets

Despite its importance for interpreting changes in themean structure and ensuring the validity of the measure, thecovariance structure is often neglected in research onthe development of cognitive abilities. For gf, changes ofthe internal factorial structure in the course of secondaryeducation may seem unlikely since decontextualized rea-soning is often characterized as being less sensitive toeducation (Ackerman, 2000). However, empirical evidencechallenging the strong assumption that changes in reasoningability are independent of the context (i.e., schooling) isaccumulating (Becker et al., 2012; Cahan & Cohen, 1989). Forgc, on the other hand, the structural invariance of theconstruct over age is questionable: As the dependency onlearning, education, and acculturation is at the core of thedefinition of crystallized intelligence (Cattell, 1943, 1971), itis plausible to assume that gc changes its very naturewith culture and age (Cattell, 1971, p. 128). Beier andAckerman's (2001) examination of knowledge about eventsfrom the 1930s to the 1990s is an example illustrating thevariability in declarative knowledge over time. Cattell(1971) argued that a structural differentiation of gc takesplace only after school (p. 121; italics in original) due tothe lack of a common, standardized educational treatment.However, individual differences in the learning environmentand increasing specialization are already present duringsecondary education and may cause an early differentiationof knowledge structures. Hitherto, this assumption hasnot yet been systematically tested by means of a broadknowledge assessment.

In our analyses of cross-sectional data with the recentlyintroduced method of LSEMs, we found little evidence for agedifferentiation in the internal structure of reasoning from latechildhood to the onset of adulthood. Even though in compar-ison to the results of the permutation test there was a slightdecline in the gradients of factor correlations, the effect wasrather small, which is consistent with Cattell's (1971; seealso Ackerman, 1996) assumptions. We also observed a highstability between the second-order factors, that is, thecorrelation between gf and gc (see Fig. 8). The small deviationsfrom stability of the gradients over time were likely due tosampling issues combined with effects of ability differentiation(i.e., stronger decline in the academic-track sample). Inaddition, as mentioned earlier the test assessing the figuralcontent facet of fluid intelligence seemed to be affected byfidelity problems at the lower end of the ability distribution;due to the sampling procedure these issues were especiallyprominent in Grade 8.

The lack of clear evidence for age-related differentiationfor both gf and gc was surprising given that the learningenvironments apparently become increasingly differentfor individuals in the course of secondary education. On theother hand, our findings coincide with previous resultsshowing a relatively high stability of interindividualdifferences in primarily knowledge-based achievementtests even during primary education (Weinert & Helmke,1998). The main reason for this could be that all partici-pants shared a learning environment thatdespite thementioned differenceswas comparatively homogeneous,as they all attended regular school classes. Moreover,secondary education in Germany provides only limitedopportunities for individualized courses or subject choices.Thus, differences in domain-specific interest, motivation,or intellectual investment traits such as typical intellectualengagement that are all assumed to steer differences inknowledge acquisition (von Stumm & Ackerman, 2012)may not be decisive within the homogeneous scholasticlearning environment. More precisely, these traits maydrive knowledge acquisition in general, but not (yet)specialization. More pervasive specialization begins onlyat the end of compulsory education (e.g., with importantcourse choices in Grade 11 and above in academic-trackschools or by starting vocational training) and in adulthood(e.g., by acquiring specialized occupational knowledge;Ackerman, 1996, 2000).

Recently, Kan, Kievit, Dolan, and van der Maas (2011)questioned the status of gc as a psychological capacity, that is,as a causal variable by arguing. The authors argued that a causalrelationship requires that latent variables can be distinguishedfrom their indicators. Through the reanalysis of a data setfrom theHuman Cognitive Abilities Project (McGrew, 2009) theydemonstrated that gc can be conceptualized as a purelystatistical entity. However, their line of argumentation relieson rather strong assumptions: First, the perspective that gf isthe decisive source of the correlations between crystallizedabilities is problematic since gc has been shown to also depend

27U. Schroeders et al. / Intelligence 48 (2015) 1529on additional sources such as investment traits (von Stumm &Ackerman, 2012) and interests (Su, Rounds, & Armstrong,2009). Kan et al. (2011, p. 296) also acknowledged the fact thatinvestment theory is broader in scope than the investmenthypothesis, nevertheless their conclusions were based on thisassumption. Second, we think that neither the presence northe absence of transfer of training across domains can beinterpreted as proof or counterproof of gc being a psychologicalconstruct. In this study, we conceptualize gc as behaviordomain scores (Markus & Borsboom, 2013), which do notnecessarily imply a causal interpretation (McDonald, 2003;Schipolowski, Wilhelm, & Schroeders, 2014).

4.3. Limitations and future research

A limitation of the present study is that our analyses werebased on cross-sectional data, rendering it impossible to makedefinite statements about the developmental perspective.While we assume that cohort effects were a minor problem,replication of the results with longitudinal data is pending.Second, although the sample can be considered representativefor students in German general-education schools and theoverall sample sizewas very large, the sample characteristics inGrade 8 deviated strongly from thepopulation. Third, it was notpossible to consider all types and tracks of vocational schoolsduring sampling, although more and more students aredropping out of general education after Grade 9.

Future research could extend the study inmultipleways: Todraw a more complete picture, the extension to a larger agerange in a longitudinal design would be desirable; moreprecisely, to begin assessment in preschool and to extend itinto adulthood when according to Cattell (1971, p. 121), the gcfactor may extend into Protean forms. There is empiricalevidence that the predictive power of reasoning and knowl-edge is not constant over the life span, but that in the course ofeducational training domain-specific knowledge becomesmore and more important for the prediction of scholastic(Baumert, Ldtke, Trautwein, & Brunner, 2009), academic(Kuncel, Hezlett, & Ones, 2001), and vocational success(Ackerman, 2000). The question as to what extent changes inthe factor structure are responsible for this finding is anobjective for future research. In our study with high schoolstudents, the learning environments were comparativelyhomogeneous. The diversification of learning opportunities inadulthood is presumably accompanied by a substantial differ-entiation of knowledge structures (see Ackerman & Rolfhus,1999; Rolfhus & Ackerman, 1999) that may lead to factorsbeing more predictive for vocational success. To gain furtherinsights into the underlying mechanisms of knowledgeacquisition, it would be interesting to supplement the assess-ment with relevant covariates such as domain-specific interestor investment traits such as typical intellectual engagement.

From a psychological assessment perspective an exten-sion to a much larger age range would require a very largeitem pool that covers the huge differences in cognitiveabilities from childhood to middle adulthood. This is asurmountable obstacle for the assessment of gf, butespecially challenging for gc since it would be necessaryto capture the breadth and depth of (specialized) knowl-edge in adulthood, including but not limited to occupa-tional knowledge. As Cattell (1971, p. 121) put it, such anapproach would amount to producing as many differenttests as there are occupations, etc.. Naturally, even themost comprehensive knowledge assessment would fallshort of this demand. We question if an assessment of gc inan extensive longitudinal design can be conducted withfixed item sets because the knowledge considered relevantfor gc is not only culture-specific, but also changing withtime; this problem is more relevant for some domains(e.g., technology) than for others. Nevertheless, we concurwith Ackerman (2000) that in spite of these hurdles, theassessment of specialized knowledge is a prerequisite forunderstanding intellectual development and changes inthe structure of intelligence across the life span.

Appendix A. Supplementary data

Supplementary data to this article can be found online athttp://dx.doi.org/10.1016/j.intell.2014.10.006.

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Age-related changes in the mean and covariance structure of fluid and crystallized intelligence in childhood and adolescence1. Introduction1.1. Age-related changes in the mean structure of gf and gc1.2. Age-related changes in the covariance structure of gf and gc1.3. Methodological aspects of testing for differentiation1.4. Research questions

2. Method2.1. Design and participants2.2. Measures2.3. Statistical analyses

3. Results3.1. Changes in mean structure across age3.2. Age-related differentiation

4. Discussion4.1. Age-related changes in the mean structure of gf and gc facets4.2. Age-related changes in the covariance structure of gf and gc facets4.3. Limitations and future research

Appendix A. Supplementary dataReferences

age-related changes in the mean and covariance structure of.pdf

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introductionthe structure

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