Mutualistic Coupling Between Vocabulary and Reasoning Supports Cognitive Development During Late Adolescence and Early Adulthood

Method

Modeling framework

To tease apart candidate mechanisms of development, we fitted a series of LCS models (Kievit et al., 2017; McArdle & Hamagami, 2001; McArdle et al., 2000). These models conceptualize differences between successive measurements as latent change factors. Crucially, this allowed us to directly model within-subjects changes as a function of structural parameters, which made these models more suitable for our purposes than latent growth curve models (McArdle & Hamagami, 2001). The basic equation of the LCS model specifies the score of person i on test Y at time t as a sum of the score at time t – 1 and a change, or difference, score as follows:

Y i , t = β t , t − 1 Y i , t − 1 + Δ Y i , t .

A key step in the LCS model specification is to set the regression weight β _t, t–1 to 1 (McArdle & Hamagami, 2001), which allowed us to rewrite the change scores as follows:

Δ Y i , t = Y i , t − Y i , t − 1 .

These change scores were then modeled as perfect indicators of a latent factor of change scores. In cases in which there was only one observed variable, or indicator, per construct, the LCS factor was construed as the difference between these indicators over time. In the absence of coupling, the intercept of the simple LCS model gives approximately identical results as a paired-samples t test when comparing differences across two measurement occasions, but it allows two additional parameters of considerable theoretical importance to be modeled: the variance in change scores (i.e., whether individuals change homogeneously or not over time) and the covariance between scores at t – 1 and change scores. We extended the basic univariate LCS model to a bivariate LCS model with abilities Y1 and Y2 (McArdle et al., 2002) by modeling the change scores on two domains Y1 and Y2 (Vocabulary and Matrix Reasoning scores) as the function of two processes: a self-feedback process (β) and a coupling process (γ), as follows:

Δ Y 1 i , t = β 1 Y 1 i , t − 1 + γ 1 Y 2 i , t − 1 .

The self-feedback parameter (β) is thought to reflect a combination of effects, including regression to the mean and a dampening effect induced by an end horizon for rapid development (i.e., individuals reaching their performance ceiling). The coupling parameter (γ) is of special importance for several developmental accounts. It captures whether the change in Y1 is determined by the t – 1 scores in Y2 (and vice versa) and thus indexes the degree to which change in one domain is affected by the level of a cognitive ability in some other domain, above and beyond the self-feedback parameter. These gammas are conceptually similar to the M matrix in the mutualism model, which captures the strengths of the interactions between abilities in a complex system (van der Maas et al., 2006, p. 845).

We fitted models for g-factor theory, investment theory, and mutualism. First, for the g-factor model (Fig. 1a), we conceptualized observed scores on Vocabulary and Matrix Reasoning as a function of an underlying g score for each time point.

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Fig. 1.

Illustrations of the (a) g-factor, (b) investment, and (c) mutualism models. In each model, age at Time 1 (T1) was entered as a covariate of Vocabulary (Voc; capturing crystallized abilities) and Matrix Reasoning (Mat; capturing fluid abilities) scores at T1. Both tests were taken from the second edition of the Wechsler Abbreviated Scale of Intelligence (Wechsler, 2011). Circles indicate latent variables, and rectangles indicate observed variables. Thick single-headed arrows indicate regressions. Double-headed arrows indicate variance and covariance. Key parameters are indicated by dashed arrows, and triangles denote intercepts. A “1” indicates which values were constrained to unity. Factor loadings in the g model were equality-constrained across measurement occasions (thin single-headed arrows). T2 = Time 2.

Second, investment theory implies that scores in fluid abilities (here indexed by Matrix Reasoning scores) should positively influence the degree of change in crystallized abilities (indexed by Vocabulary scores), such that individuals with greater fluid ability will, on average, improve more in crystallized abilities than peers with lower Matrix Reasoning scores at Time 1. This process was modeled by a single coupling parameter from Matrix Reasoning scores at Time 1 on the Vocabulary change factor at Time 2 (Fig. 1b). Finally, the mutualism model (Fig. 1c) predicts bivariate coupling between both cognitive abilities; specifically, higher starting points in vocabulary would lead to larger gains in matrix reasoning and vice versa. In all models, we added age as a covariate to account for differences in baseline scores but did not include age anywhere else in the model (i.e., we hypothesized that the dynamics of change were fully captured by the change dynamics proposed by each theory).

Results

Raw scores and descriptive statistics for the Matrix Reasoning and Vocabulary subtests are shown in Table 1, and the association between age and score on each test is shown in Figure 2. Before fitting the models shown in Figure 1, we fitted two univariate LCS models to Vocabulary and Matrix Reasoning scores in order to quantify change within each domain. Both models fitted the data well: Matrix Reasoning: χ²(1) = 3.098, p = .078; RMSEA = .052, 90% confidence interval (CI) = [0.000, 0.119]; CFI = 0.995; SRMR = 0.014; Yuan-Bentler scaling factor = 0.904; Vocabulary: χ²(1) = 0.197, p = .657; RMSEA = 0.00, 90% CI = [0.000, 0.070]; CFI = 1.0; SRMR = 0.003; Yuan-Bentler scaling factor = 1.057. Both models showed evidence for change over time (unstandardized change-score intercepts ² —Matrix Reasoning: 10.252, SE = 0.747, z = 13.723; Vocabulary: 8.367, SE = 1.146, z = 7.301). Further, both models showed evidence for negative feedback: Higher scores at Time 1 were associated with less improvement at Time 2, a pattern compatible with regression to the mean and developmental-ceiling effects (Matrix Reasoning: −0.333, SE = 0.025, z = −13.29; Vocabulary: −0.134, SE = 0.19, z = −7.05). Finally, both models revealed significant evidence for individual differences in change scores (variance of Matrix Reasoning change scores = 2.89, SE = 0.23, z = 12.70; variance of Vocabulary change scores = 10.23, SE = 0.70, z = 14.54).

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Table 1.

Raw Scores and Descriptive Statistics for Matrix Reasoning and Vocabulary Scores

		Score
Task	N	Mean	Minimum	Maximum	SD	Skewness	Excess kurtosis
Matrix Reasoning Time 1	784	28.99	14	35	3.21			−0.88	1.47
Matrix Reasoning Time 2	563	29.61	17	35	2.89	−0.82	0.77
Vocabulary Time 1	784	58.44	27	78	7.81	−0.24	0.06
Vocabulary Time 2	563	59.06	23	77	7.51	−0.40	0.40

Note: The Matrix Reasoning and Vocabulary subtests were taken from the second edition of the Wechsler Abbreviated Scale of Intelligence (Wechsler, 2011).

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Fig. 2.

Scatterplots showing the association between age and score on the Matrix Reasoning subtest (top) and Vocabulary subtest (bottom) of the second edition of the Wechsler Abbreviated Scale of Intelligence (Wechsler, 2011). Lines connect the rescaled scores of those individuals who completed the test at both waves.

Having shown, as expected, a growth in scores in both domains, we next fitted all three models (g factor, investment, and mutualism) to determine which provided the best account of longitudinal development in these two cognitive domains across the two measurement occasions. To ensure comparability of factor scores across Time 1 and Time 2 for the g-factor model, we tested for longitudinal measurement invariance (Widaman, Ferrer, & Conger, 2010). We found that imposing weak invariance across time points (factor loadings) led to negligible decrease in model fit (ΔCFI = 0.004; Cheung & Rensvold, 2002). Imposing strong invariance (equality of both factor loadings and thresholds) also led to acceptable decrease in model fit (ΔCFI = 0.011). This suggests that longitudinal measurement invariance is tenable, and we interpreted changes in factor scores accordingly. Next, we fitted the investment and mutualism models, which differed only in the presence or absence of a Vocabulary-to-Matrix-Reasoning coupling parameter.

In Table 2, we report the fit statistics for each of the three competing models. This comparison suggests that the mutualism model fitted the data best, showing excellent model fit on all indices. The two alternative models (investment and g factor) showed comparable model fit between each other, and any difference was marginal according to conventional guidelines. As the mutualism model was also the most complex model, we plotted information criteria (AIC and BIC) for each of the three models to explicitly weigh parsimony, as shown in Figure 3a. This comparison showed a superior fit on both indices for the mutualism model. Finally, we computed Akaike weights. These are shown in Figure 3b, which illustrates that the mutualism model has by far the highest normalized probability (> 99.99%) of being the best model given our data. Compared with the other two models, the mutualism model was 2.45 × 10⁷ times more likely to be the best model. As the investment model was nested within the mutualism model, we compared the two with a chi-square test, which again showed that the mutualism model outperformed the investment model, Δχ²(1) = 22.829, p < .001.

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Table 2.

Fit Statistics for Each of the Three Models

Model	χ2	df	RMSEA	CFI	SRMR
g factor	28.990	3	0.105 [0.076, 0.138]	0.982	0.031
Investment	26.477	3	0.100 [0.068, 0.136]	0.984	0.040
Mutualism	0.328	2	0.000 [0.000, 0.040]	1.00	0.002

Note: For root-mean-square errors of approximation (RMSEAs), 90% confidence intervals are given in brackets. CFI = comparative fit index; SRMR = standardized root-mean-square residual.

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Fig. 3.

Akaike’s information criterion and Bayesian information criterion (a) and normalized probabilities using Akaike weights (b), for each of the three models.

Having established the superior fit of the mutualism model, we next investigated its estimated parameters in more detail (see Fig. 4; Table S1 in the Supplemental Material available online contains all parameter estimates and 95% confidence intervals). As expected, Matrix Reasoning and Vocabulary scores at Time 1 were positively correlated, and age at first testing predicted scores on both tasks at Time 1. In addition to significant latent change intercepts (i.e., increasing scores), variance of change scores led to a substantial drop in model fit when fixed to 0—Matrix Reasoning: Δχ²(1) = 82.43, p < .001; Vocabulary: Δχ²(1) = 11.352, p < .001, which suggests that there were considerable individual differences in change between Time 1 and Time 2. Crucially, as predicted by the mutualism model, both coupling parameters were positive: Individuals who started out with a higher Matrix Reasoning score improved more on Vocabulary and vice versa. The coupling effect from Time 1 Vocabulary scores on gains in Matrix Reasoning scores was of typical size (r = .203, r² = 4.1%) for individual differences analyses, and the fully standardized estimate of Matrix Reasoning on Vocabulary gains was in the small to typical range (r = .155, r² = 2.4%; Gignac & Szodorai, 2016). Together, the self-feedback and coupling parameters accounted for 31.3% of the individual differences in Matrix Reasoning score changes and for 11.3% of the individual differences in Vocabulary score changes, which illustrates the considerable importance of longitudinal kinematics in cognitive development. Even in the presence of the bivariate coupling parameters, the residual change scores were still positively correlated. This is compatible with (although not direct evidence for) the idea of additional unmeasured cognitive abilities driving change in both vocabulary and matrix-reasoning ability. Further control analyses suggested that the mutualism model could be equality constrained across sexes without a notable drop in model fit, Δχ²(18) = 18.32, p = .44.

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Fig. 4.

Estimated parameters for the mutualism model. Values in Roman are standardized parameter estimates, and values in italics are unstandardized parameter estimates (with standard errors in parentheses). See Figure 1 for an explanation of the notational system used. Further results are given in Table S1 in the Supplemental Material. Mat = Matrix Reasoning; Voc = Vocabulary; T1 = Time 1; T2 = Time 2.

Using Equation 3 and the estimated parameters of the full mutualism model (Fig. 4), we next visualized the expected change between Time 1 and Time 2. To do this, we created a vector field plot (e.g., McArdle et al., 2000, p. 69) in which each arrow represents a (hypothetical) bivariate score at Time 1 (base of each arrow) and model-implied expected score at Time 2 (end of arrow) across a range of possible scores. Figure 5 shows the vector field plot and highlights regions where the mutualistic effects are easiest to see.

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Fig. 5.

Vector field plot for the mutualism model showing model-implied changes between Time 1 and Time 2. The dots represent the Time 1 Matrix Reasoning and Vocabulary scores of a randomly selected subset of individuals, and each arrow represents a model-implied change between Time 1 (base of arrow) and Time 2 (head of arrow). The horizontal shaded rectangle illustrates the positive effect of higher Vocabulary scores on expected change in Matrix Reasoning scores. The vertical shaded rectangle illustrates that there was a negligible expected Vocabulary improvement for low Matrix Reasoning ability (arrows below 24 on the y-axis) but considerable expected vocabulary improvement for individuals with high Matrix Reasoning starting scores (arrows above 28 on the y-axis). The dashed ellipse shows the 90% confidence interval for the raw data.

Although analytic work (van der Maas et al., 2006) has demonstrated that a g factor may arise through mutualism even in the complete absence of individual differences at the beginning of development, we think it most likely that g-factor and mutualistic processes operate in tandem. For example, it may be that children show (smaller or larger) consistent individual differences from very early ages (e.g., Gignac, 2014), which are then amplified by developmental processes, such as mutualism. This is in line with previous suggestions of gene-environment interactions whereby initial differences lead to a “reciprocal feedback loop between the phenotype and the environment” that amplifies initial differences (Beam & Turkheimer, 2013, p. 7; see also Briley & Tucker-Drob, 2013; Dickens & Flynn, 2001), a phenomenon observed even in genetically identical mice (Freund et al., 2013). Such models can also reconcile the high heritability of higher cognitive abilities (Briley & Tucker-Drob, 2013) with considerable environmental impacts and may partially reconcile more puzzling facts about heritability and the cultural load of cognitive tasks (Kan, Wicherts, Dolan, & van der Maas, 2013).

In the three models examined here, we included age as a linear covariate to account for individual differences due to age at Time 1 (we will describe alternative parametrizations of age in the Discussion). This reflects a hypothesis that age affects scores at Time 1 but that all aspects of development over time can be captured within the model. Allowing age to directly predict change scores did not improve model fit, Δχ²(2) = 0.33, p = .85, in line with this hypothesis. Notably, this does not necessarily imply that cognitive development occurs at the same rate across development. The decelerating improvement in late adolescence was captured by the negative self-feedback parameter in Matrix Reasoning and Vocabulary. A second analytic choice is to assume a linear effect of age on scores at Time 1. An age-squared term as predictor of scores at Time 1 could be fixed to 0 without a decrease in model fit, Δχ²(2) = 6.50, p = .039, which suggests that a linear term would suffice. Third, we included age as a predictor of the raw Vocabulary and Matrix Reasoning scores at Time 1 for the mutualism and investment models but allowed age to predict the g factor only in the g-factor model (under the assumption that this factor captures the “true” shared variance). Although this is in line with the conceptualization proposed here, we wanted to ensure that this analytic choice did not favor or disfavor the g model artificially. We therefore fitted two additional versions of the g-factor model by including age either as a covariate of only the observed scores at Time 1 (alternative A) or as covariates of both the observed scores and the g factor (alternative B). The mutualism model was preferred to all three conceptualizations of the g model—ΔBIC = 29.36 (original g-factor model), ΔBIC = 51.85 (alternative A); ΔBIC = 7.27 (alternative B). Together, these analyses suggest that characterizing age as a linear effect was sufficient within this sample, that differences in change scores were not affected by age beyond the indirect effect, and that the mutualism model provided a compelling account of dynamic processes during cognitive development.

Discussion

In a large (N = 784) developmental cohort of adolescents and young adults, we compared three competing accounts that could explain age-related changes in key cognitive abilities. We found that mutualism outperformed alternative accounts based on g-factor and investment theory. Specifically, we found evidence for bivariate coupling between Matrix Reasoning scores (as an index of fluid abilities) and Vocabulary scores (as an index of crystallized abilities); specifically, higher starting points in one cognitive domain were associated with greater developmental gains in the other domain. Our findings refine the understanding of cognitive development in several ways. They suggest that covariance between cognitive abilities is, at least in part, a consequence of a developmental process rather than of a single underlying causal entity g. Our data provide strong evidence that a model of intellectual development that omits coupling parameters is incomplete.

We can hypothesize several mechanisms to explain the coupling parameters, both direct and indirect. One direct pathway may be that a greater facility with vocabulary and verbal skills allows for swifter, more accurate decomposition of reasoning problems into constituent elements, as well as decreased working memory demands for maintenance of such elements, especially in younger adults. A more indirect pathway, in line with the gene-environment interactions mentioned previously, is that greater vocabulary may be an easily detectable marker of higher cognitive ability, which leads to real-world feedback effects in the form of more academically challenging classes or environments to support perceived ability in a manner that generalizes to other domains. A final, intriguing possibility is that traditionally fluid tasks such as Matrix Reasoning may in fact reflect a hybrid of purely fluid abilities (or learning potential) and more strategic, verbal components akin to crystallized abilities (Kühn & Lindenberger, 2016). This would explain both the life-span trajectories of fluid abilities and the considerable secular gains in fluid abilities in the 20th century (Flynn, 1987).

Our findings suggest a need for a shift away from a narrow focus on desirable cognitive end goals (e.g., adequate performance on abilities such as vocabulary or mathematics) and the incorporation of a simultaneous view across abilities that may have less intrinsic interest but are essential in their capacity to support successful development. For example, skills such as processing speed or working memory may be less important in isolation but may be coupled to other cognitive skills (Kail, 2007), which in turn may affect later life socioeconomic outcomes. In other words, to facilitate early detection and possibly even effective intervention, it may pay off to focus on abilities that have the strongest coupling strengths rather than solely on outcomes that are currently below some desirable threshold. For example, Quinn, Wagner, Petscher, and Lopez (2015) used dynamic models to show that vocabulary was a leading indicator of gains in reading comprehension but not vice versa. Such a finding offers insight into the causal pathways of children with reading difficulties, as well as informing appropriate interventions. Similarly, disruptions to typical development were reported by Ferrer, Shaywitz, Holahan, Marchione, and Shaywitz (2010), who observed that within a subgroup with dyslexia (or “persistently poor readers,” p. 94), the coupling between IQ and reading ability observed in typical groups was absent. This suggests not only a possible mechanism for developmental disorders, but also shows how multivariate longitudinal models can allow for early detection of developmental challenges that are likely to self-reinforce over time.

Although we compared various developmental models and quantified longitudinal coupling, our research has certain limitations. First and foremost, we focused on two cognitive subtests alone, which yielded a relatively simplistic g model. Although both are well validated, have highly similar reliabilities, and represent broad cognitive domains, it will be desirable in future studies to represent cognitive abilities by more than one indicator variable and to sample a wider range of cognitive abilities. Our sample was measured on two occasions, and undoubtedly, measurement on more occasions would allow a more precise decomposition of kinetics and kinematics, such as the modeling of lead-lag relations using bivariate dual-change-score models (e.g., Ghisletta & Lindenberger, 2003). Here, we showed that baseline scores are positively associated with cross-domain rates of change. With three or more waves, it is possible to use the change scores at time t to predict the change scores at time t + 1 (Grimm, An, McArdle, Zonderman, & Resnick, 2012). Moreover, if age is sampled at sufficient frequency, it is possible to examine latent changes as a function of age itself (∆Y_agei ) rather than as testing occasion (∆Y_ti ), which would obviate the need for covariates by binning individuals’ scores by age and estimating models using methods that account for missingness (e.g., Voelkle & Oud, 2017).

An additional challenge with repeated measures data is the improvement in test scores due to practice effects, which may inflate developmental gains or attenuate age-related decline (Rabbitt, Diggle, Smith, Holland, & Mc Innes, 2001; Salthouse & Tucker-Drob, 2008). Although, in our sample, practice effects may have led to greater increases in scores between Time 1 and Time 2, it is unlikely that these effects affected our conclusions regarding mutualism. First, such practice effects would lead to an increase in test scores that are a combination of true (developmental) gains and increases due to practice effects (although see Lövdén, Ghisletta, & Lindenberger, 2004, on the interpretation of practice effects). Notably, if one interprets the gains between Time 1 and Time 2 as a combination of “true” gains and practice effects, this would entail an underestimate of the mutualism effect (as the effect size reflects the prediction of the total gains rather than the non-practice-related gains). In principle, a sufficiently large number of time points spaced at unequal retest intervals would allow for a decomposition of retest effects, but both practical difficulties as well as the inherent collinearity of retest occasions with time intervals has proved methodologically challenging (Hoffman, Hofer, & Sliwinski, 2011).

Finally, we observed our effects in adolescents and young adults, which limited the generalizability of our observations to this developmental period alone. We hypothesize that the coupling effects we observed are likely to be stronger earlier in life and the self-feedback parameters weaker, as developmental change in higher cognitive abilities is most rapid during pre- and early adolescence. Considering these effects at the other end of the life span yields several intriguing questions. It is conceivable that mutualism occurs only during early development, with other processes and mechanisms taking over after initial peaks are reached. However, we suggest that studying later life decline from the perspective of mutualism might prove a promising avenue for future work. If dynamic coupling is crucial for maintenance of cognitive abilities in later life, this may explain why declines are often strongly correlated (see Ghisletta & Lindenberger, 2003; Tucker-Drob, 2011, for further exploration of this hypothesis). Using large longitudinal cohorts and similar tests across the entire life span will allow for the investigation of possible “regime changes” within the same cohort.

Future work should study multiwave, multidomain cognitive data using principled model-selection methods to better capture the underlying dynamics of cognitive development. Data of high temporal resolution would allow one to move beyond group-level dynamics of individual differences to the ultimate goal, namely that of estimating individual differences in intraindividual dynamics over time. The investigation of individual coupling parameters across domains and across the life span is likely to yield a wealth of information on cognitive development in health and disease. The recent convergence of novel modeling techniques, large-scale data-gathering ability via tools such as smartphones, and the integration of behavioral data sets with data from neural and genetic sources of evidence together promise to provide new insight into some of the most elusive, yet fundamental, questions in cognitive psychology.

Supplemental Material

Supplemental Material