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Structural Modeling of Dynamic Changes in Memory and Brain Structure Using Longitudinal Data From the Normative Aging Study

¹ Department of Psychology, University of Virginia, Charlottesville.
Departments of ² Psychiatry
³ Neurology, Massachusetts General Hospital, Boston.
⁴ Department of Radiology, Harvard Medical School, Boston, Massachusetts.
⁵ Department of Veterans Affairs, Boston University School of Public Health, Massachusetts.

Address correspondence to John J. McArdle, Department of Psychology, PO Box 400400, University of Virginia, Charlottesville, VA 22906-4400. E-mail: jjm{at}virginia.edu

	Abstract

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This is an application of new longitudinal structural equation modeling techniques to time-dependent associations of memory and brain structure measurements. There were 225 participants aged 30–80 years at baseline who were measured again after a 7-year interval on both the lateral ventricular size and Wechsler memory score. Multiple regression analyses show nonlinear associations with age but no relationships among longitudinal changes. Mixed-effects latent growth curve analyses and analyses based on latent difference scores indicate that longitudinal changes in both variables are reasonably well described by an exponential or dual change model. Bivariate dynamic structural equation modeling analyses indicate age-lagged changes operate in a coupled-over-time fashion, with the brain measure (lateral ventricular size) as a leading indicator in time of memory (Wechsler memory score) declines.

There are several well-known data-collection strategies for making scientific inferences about human brain functions and brain structure (e.g., La Rue, 1992). The first is a classical method based on the postmortem examination of brain structures of persons and the relation of these observations to premorbid behaviors. A second classical method is the collection of behavioral data following brain stimulation or insults, either surgical or accidental (e.g., lesions, tumors, and ablations). A more recent and less invasive methodology is based on the collection of data on brain structures and activity by use of some kind of imaging technique (e.g., computed tomography, magnetic resonance imaging, positron emission tomography, and functional MRI) together with the collection of data on concurrent behaviors under well-defined experimental conditions (e.g., Raz, Gunning-Dixon, Acker, Head, & Dupuis, 1998). A recent innovation in data-collection methodology has been short-term and long-term repeated measurements of imaging–behavior relations (see MacInnes, Paull, & Quaife, 1989; Mitra & Pesaran, 1999; Shaywitz et al., 2002).

The data examined here come from a subset of data from the Normative Aging Study (NAS; Albert, 1995). This is a study of 225 healthy persons aged 30–80 years who were examined at two points in time, separated by approximately 7 years. A selected set of data obtained from each individual subject is presented here in Figure 1. Computed tomography lateral ventricle size scores (LVS) are plotted in Figure 1A and Wechsler memory score data (scaled WMS) are plotted in Figure 1B. Each solid line in the plot is used to connect two scores obtained for the same person at the two occasions of measurement. A circle is used to represent a score obtained at a single age for a person not retested at follow-up. These data are similar to those of many studies of aging that have large differences in initial ages and a first follow-up after a few years.

View larger version (24K): [in this window] [in a new window]   Figure 1. Age-based graphic display of longitudinal data: A, lateral ventricle size; B, Wechsler memory score

There are many statistical models for the analysis of repeated-measures longitudinal data (e.g., Collins & Sayer, 2001; Diggle, Liang, & Zeger, 1994). The classical mixed-effect latent growth models allow us to deal with large amounts of incomplete data as well as direct age-based growth interpretations (McArdle & Anderson, 1990; McArdle, Ferrer-Caja, Hamagami, & Woodcock, 2002; Verbeke, Molenberghs, Krickeberg, & Fienberg, 2000). Here, we reformulate these models in terms of structural equation modeling (SEM) methods with latent difference scores (McArdle, 2001; McArdle & Hamagami, 2003; McArdle & Nesselroade, 1994). A bivariate difference score form of this model permits us to test dynamic hypotheses by using standard statistical theory and SEM software. In our final analysis we simultaneously model the longitudinal changes in memory performance, longitudinal changes in brain structure, their interrelationship over time lags, and the impact of other variables on these time-lagged associations.

All longitudinal analyses require the choice of the basis or timing for analysis. In research in which the participants are all measured at the same initial ages, or after a specific incident (e.g., recovery time from surgery), the time passed between tests (e.g., t = time lag) is often used as the basis of the trajectory (e.g., McArdle & Woodcock, 1997). In this study, as with many others, the participants were not all measured at the same initial ages, so "time of measurement" is not equivalent to "age of measurement." However, in situations in which we are interested in chronological age change, and there is no common starting point of specific interest (t = 0), it seems most natural to use a timing of observation based on the observed or chronological age at the occasion of measurement (i.e., t = age). As it turns out, this age-based model does not always yield the same results as a time-lag model, even when age is included as a covariate (e.g., Sliwinski & Buschke, 1999; cf. McArdle et al., 2002).

These dynamic SEM methods presented here can be useful for several reasons. First, these SEM methods avoid some statistical and psychometric problems with observed rates of change. Second, these methods formalize with age-based trajectories from incomplete or unbalanced longitudinal data. Third, these SEMs formalize questions about lead-lag dynamic relations among brain structure and behaviors. We use basic algebraic equations for longitudinal scores, and path diagrams describe the main features of these. We also attempt to show ways these new models can be effective in the common situation of mixed cross-sectional and longitudinal data (e.g., as in Figure 1).

	METHODS

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Participants
A total of 225 people participated in this study. At the first occasion of measurement, the participants ranged in age from 30 to 80 years, with an average of 40 individuals per decade (i.e., 30–39, 40–49, 50–59, 60–69, and 70–80). Approximately half the participants were men (n = 109) and half were women (n = 116), and all were measured as one part of a larger research project of the Massachusetts General Hospital Aging Study (Albert, 1995). The male participants were members of the NAS of the Boston VA Outpatient Clinic. These participants spanned a fairly broad socioeconomic range, including both white- and blue-collar workers. Their education ranged from 8 to 19 years, with a mean of 14.4 years over all ages. The female participants were recruited from the community (the NAS population included men only), but they met the same selection criteria, age, and educational level as the NAS male participants.

A follow-up assessment was conducted about 7 years after the initial measurements (with only a few months variation), and all individuals who agreed to participate were included, regardless of health status. Of the 225 individuals examined at baseline, 14 were deceased before their projected evaluation date. Of the remaining 211 individuals, 197 were reevaluated, representing 93% of those eligible. Of these 197 persons, 168 (85%) had reevaluations of all study procedures of interest. Of the remaining 29 participants, 21 were reevaluated with neuropsychological testing only and 8 completed two of the three study procedures (one of which was not examined here). Of the remaining 7% not evaluated, 9 persons were unwilling or unable to participate and 5 persons could not be located. Although the sample sizes differ for specific relationships, all available data will be used in the analyses to follow.

In order to be selected for the study, participants had to meet strict health criteria. Thus, at baseline, this sample did not represent "average" individuals of a particular age, but rather optimally healthy persons across the age range. The selection criteria excluded individuals with hypertension, diabetes, coronary artery disease, lung disease, kidney disease, learning disabilities, severe head trauma, psychiatric illness (including alcoholism), and neurological disease (including epilepsy). Information pertaining to these disorders was ascertained from medical history and standard laboratory tests (i.e., blood tests, urinalysis, electrocardiograms, and electroencephalograms). At follow-up, of the 197 participants evaluated, 124 (63%) had no evidence of major clinical disease. Of the remaining 73 individuals, 55 (28%) had a medical disorder (e.g., hypertension, coronary artery disease, or pulmonary disease), 16 (8%) had a neurological disorder (e.g., Alzheimer's disease or Parkinson's disease), and 2 (1%) had developed a psychiatric illness during the follow-up period.

Measurements
We assessed the brain structure by using one portion of the computerized axial tomography (CT) information gathered. We selected five CT slices for evaluation and analyzed them to determine the area of nine regions of interest (ROI). We normalized each ROI to the head size of the individual, so each value was expressed as a percentage of each individual's head size (Matsumae et al., 1996; Sandor et al., 1992). In this first set of analyses, we selected the LVS as a broad indicator of brain area. Because the lateral ventricles are present in both hemispheres of the brain, we used the mean of the measurements from the left and right hemispheres in the present analyses. Other more specific brain areas (e.g., temporal horn size) are likely to be a more appropriate reflection of the brain and memory dynamics (e.g., hippocampal formation; see Matsumae et al., 1996; Sandor et al., 1992; Zola-Morgan, Squire, & Ramus, 1994). We selected the global LVS variable for use here as a starting point for this dynamic modeling and to provide a general baseline for future models.

Participants were administered a large battery of neuropsychological tests chosen to span all major aspects of cognitive function, including assessments of sustained attention, language, memory, visuospatial ability, executive function, and broad intellectual abilities (see Albert et al., 1987). In the present study we focus on only the assessment of memory as measured by the original WMS (Wechsler, 1945; also see La Rue, 1992). This is a widely used standardized test of memory that includes an assessment of both verbal (story) and nonverbal (figural) explicit memory, and a brief evaluation of orientation and attention.

Given our focus on age-related changes, we analyzed raw WMS scores (scaled here as a proportion of their maximum value, 143). To simplify some of the subsequent SEM analyses, we extrapolated the two scores for each person to the nearest 5-year grouping to reflect a starting point at one of the 5-year ages (30, 35, 40, etc.) and change over a 5-year gap (30–34, 35–39, etc.). For example, in subsequent regression analysis, this implies that all intercept parameters will reflect Age = 30, and all one-unit changes will represent change over 5 years. Other choices of scaling are possible (1- or 7-year gaps), but we studied these alternatives and they do not alter the subsequent statistical interpretations based on the standard significance test levels ( =.05).

Summary Statistics
A brief description of summary statistics for both occasions of data appears in Table 1. This includes a list of the univariate statistics for both variables LVS and WMS at each occasion of measurement (e.g., LVS[1] and LVS[2]), for age at each occasion of measurement (i.e., Age[1] and Age[2]), as well as gender and educational level. We calculated the correlations of these measures for all available pairs of scores, and these are presented in Table 2. The WMS was completed at both occasions by most subjects, but this was not the case for the LVS measures. The CT scan for the LVS was not completed for all subjects at the second occasion of measurement. This incomplete CT scan occurred for a variety of reasons unrelated to a persons' health. For this reason, each element in this matrix includes correlations based on different sample sizes (i.e., all possible pairs of scores), and these coefficients are not analyzed directly here.

Alternative SEM Analyses
We use four related types of statistical models to examine the cross-domain and over-time relationships among neuroanatomical (brain structure) and neuropsychological (memory test) measurements. Each analysis uses a slightly different variation of standard computer software (and computer scripts are available on our Web site).

Type 1: Multiple regression of observed rates-of-change scores
In this approach we use standard forms of multiple regression analysis based on two separate equations, which are written as<--?1-->;

In this model, (a) we use the parameters of first equation (ß₀, ß₁) to describe the expected value of the initial measurement (or average score); (b) we use the parameters of the second equation (₀, ₁) to describe the expected rate-of-change score (Y_n/t_n) calculated from the two occasions; and (c) we index individual residuals by the other two terms (e and d).

Although this is not a model for a "pretest–posttest" design, this regression approach has similarities to the traditional multivariate analysis of variance (MANOVA) calculation of between-persons and within-persons effects, and it has been used in many studies of aging (e.g., Giambra et al., 1995; MacInnes et al., 1989; Sullivan, Marsh, Mathalon, Lim, & Pfefferbaum, 1995). Many researchers have pointed out limitations in using observed rates of change as outcome variables (but see Allison, 1990; Hamagami & McArdle, 2001).

Type 2: Mixed-effects growth models
These classical models are also known as multilevel, random coefficients, or latent growth models (McArdle & Hamagami, 1996, 2001; McArdle & Nesselroade, 2002; Verbeke et al., 2000). These statistical procedures are used to fit an implied trajectory over time directly to the observed scores based on various kinds of mathematical forms of growth. One common form of a growth model is based on a trajectory over time, written as

In this model, (a) the y₀ (in lower case) are unobserved or latent scores representing an individual's initial level; (b) the B[t] represents a function of the timing of the observations, such as B[t] = 0 or 1 or B[t] = Age[t]; (c) the y_s, termed the slopes, are unobserved latent scores (in lower case) for the individual change over time; and (d) the e[t] are unobserved but independent errors of measurements at each time.

The path diagram of Figure 2 is a standard representation of this kind latent growth model. We draw the observed variables as squares and the unobserved variables as circles, and we include the required constant as a triangle. We draw model parameters representing "fixed" or "group" coefficients as one-headed arrows whereas we draw "random" or "individual" features as two-headed arrows. In this case, we often assume the initial level and slopes to have to be random variables with "fixed" means (µ₀, µ_s) but "random" variances (²₀, ²_s) and correlations (_0s). (We draw the standard deviations, _j, in the picture to permit the direct representation of the covariances as scaled correlations.) We assume the error terms to be normally distributed with mean zero and variance (²_e), and they are presumably uncorrelated with all other components.

View larger version (19K): [in this window] [in a new window]   Figure 2. Path diagram of a latent growth curve model for repeated measures data

In this model (a) the corresponding latent true scores y[t] and y[t – 1] are separated from time-constant errors of measurement e[t] and e[t – 1]; (b) the additional latent variable {y[t]} is directly interpreted as a latent difference score; (c) the model of change is explicitly written in linear form; and (d) the summation (_i=1,t) or accumulation of the latent changes {y[t]} up to time t is defined by the model for change (for more complete development, see McArdle, 2001).

The structural path diagram in Figure 3 illustrates how we can directly represent this kind of latent change score model by using standard longitudinal SEM. We draw this set of equations by using (a) unit-valued regression weights among variables by fixed nonzero constraints, (b) a constant time lag by using additional latent variables as placeholders, (c) each latent change score as a focal outcome variable, and (d) a repetition (by equality constraints) of the additive () and proportional (ß) structural coefficients. Following the previous mixed-effects model concepts, we assume the unobserved initial level component (y₀) has a mean and variance (i.e., µ₀ and ²₀), while the error of measurement has mean zero, constant variance (²_e > 0) and is uncorrelated with every other component. As in Figure 3, the constant change component (y_s) has a non-zero mean (i.e., µ_s, the average of the latent change scores), a non-zero variance (i.e., ²_s, the variability of the latent change scores), and a nonzero correlation with the latent initial levels (i.e., _0s). This linear difference model creates expectations that permit the mixed-effects model to have a nonlinear trajectory, see B[t] in Equation 2, although the corresponding accumulation of differences remains linear. The numerical values of the parameters ( and ß) can be used to form many different kinds of individual and group trajectories over age.

View larger version (20K): [in this window] [in a new window]   Figure 3. Path model of a univariate latent difference dynamic structural equation system

In this model, (a) both variables are written as latent difference scores (y[t], x[t]); (b) each dual change score model is represented by their own parameters ( and ß); (c) one coupling parameter (_yx) represents the time-dependent effect of latent x[t] on y[t]; and (d) another coupling parameter (_xy) represents the time-dependent effect of latent y[t] on x[t].

The key features of this model are drawn in Figure 4 and include the used of fixed unit values, to define y[t] and x[t], and equality constraints, for the , ß, and parameters. These latent difference score models can lead to more complex nonlinear trajectory equations (e.g., nonhomogeneous equations), but we can describe these simply by writing the respective bases in terms of the linear accumulation of first differences. This bivariate model was developed as an extension of difference equations for the latent variables (McArdle, 2001; McArdle & Hamagami, 2001, 2003; McArdle et al., 2001). Hypotheses can be formed about (a) parallel growth, (b) the covariance among latent components, (c) proportional growth, and (d) the dynamic coupling over time. That is, in addition to restrictions on the dynamic parameters ( = 0, = 1, and ß = 0), we can evaluate models where one or more of the coupling parameters is restricted (i.e., _yx = 0 and _xy = 0). To the degree that at least one of these coupling parameters is large and repeatable, we can say we have estimated a coupled dynamic system with "leading indicators" in the presence of growth. To the degree that these parameters are zero, we can say we have estimated an "uncoupled system." A vector field display is used to interpret the direction and the strength of dynamics (Boker & McArdle, 1995).

View larger version (34K): [in this window] [in a new window]   Figure 4. Expected growth curves based on the exponential and dual change score model for Y, lateral ventricle size and X, Wechsler memory scores

In a similar way, the latent difference score SEM analyses (Type 3 and 4) are based on fitting structural models to the raw score information for each person on each variable at each time (e.g., McArdle et al., 2002). A few available computer programs can be used to estimate the parameters of all latent difference SEMs described here. Here we used the Mx computer program (Neale, Boker, Xie, & Maes, 1999), and it deals with incomplete data patterns by using an "unbalanced pedigree" approach from behavior genetics (McArdle & Hamagami, 2003). We assess the goodness of fit of each model presented here by using classical statistical principles about the model likelihood (f_MLE).

	RESULTS

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We fit the four types of longitudinal models to the LVS and WMS scores including all points and lines of data in Figure 1. To preserve space, we present only the results based on the latent difference score models (Type 3, Table 3) and the bivariate difference score models (Type 4, Table 4) in detail.

A second set of four equations used the observed rate scores (LVS/t and WVS/t) as outcomes. For Model (1d), including a constant only, we obtained a significant intercepts for change in the LVS (₀ =.05), but no significant change for the WMS (₀ =.00). For Model (1e), the addition of age as a predictor of the change scores adds only a small amount to the LVS (₀ =.04 ₁ =.002, with R² <.01), and very small effects for the WMS (₀ =.003, ₁= –.007, with R² =.06). For Model (1f), the addition of the first time score as a predictor of the change scores adds a substantial amount to both the LVS (₀ =.08, ₁ =.001, ₂ = –.003, with R² =.16) and the WMS (₀ =.34 ₁ = –.008, ₂ = –.003, with R² =.22), and the negative sign shows both scores have limits on the changes. For Model (1g), the final equation adds a cross-variable prior time predictor of the change scores, that is, LVS 1 on WMS 2, and in each case the cross-variable does not add in any significant effects or predictions (for both LVS and WMS, ₃ = 0).

From these multiple regression results, we conclude that the trajectories over time for each variable are not linear, have self-imposed limits (i.e., are slowing), and are moving in different directions (up for LVS and down for WMS); the observed changes in the two scores are not predictable from each other. In addition, there are only small cross-variable correlations of the initial levels (–.04), the rates of change (.07), and the residuals (.06).

Type 2: Age-Based Mixed-Model Results
We fit a set of four alternative age-based linear and nonlinear mixed models to the data by using the observed ages. These results turned out to be almost identical to the latent difference score models (Type 3) described in the paragraphs that follow. This is important, because some information could have been lost when the raw data were organized into 5-year age segments. However, because the basic latent growth results are identical, we present details on only the newer approach.

Type 3: Age-Based Latent Difference Score Results
The results of four alternative age-based latent difference scores models (Type 3) are presented in Table 3. The numerical results for four models listed include the model parameters and overall goodness-of-fit comparisons. We fit a no-change model (3a) with only three parameters, and the results using this SEM approach are comparable with the mixed-effects approach. The LVS results include the three parameters that are expected to be constant over age: (a) a constant mean (µ₀ =.32); (b) a constant level variance (₀ =.14), and (c) a constant error variance (_e =.03). The ratio of the two variance components leads to a high estimate of the intraclass correlation (² =.96), and this simply implies people are different from one another. This model has a likelihood (f_MLE = –389) that represents a notable statistical improvement (with ² = 94 on df = 1) over the entirely random (i.e., zero correlation) case. The initial results for the WMS variable includes three parameters (µ₀ =.90, ₀ =.08, and _e =.06), with a strong implied intraclass correlation (² =.64) and a corresponding model likelihood (f_MLE = –833), which is also a clear statistical improvement (with ² = 113 on df = 1) over the entirely random case.

We next fit the constant change score model (3b), and this is the same as a linear age mixed-effects model. The LVS results include three time-constant parameters (µ₀ =.15, ₀ =.12, and _e =.06) plus three time-dependent slope parameters indicating small but positive systematic increases in means over every 5 years of age (µ_s =.006) and increases in variances and covariances over every 5 years of age (_s =.004, _0s = –.66). The linear model likelihood (f_MLE = –505) is a statistical improvement over the entirely random model (with ² = 208, on df = 4) and, more critically, a statistical improvement over the previous no-change level model (with ² = 114, on df = 3). The WMS results for the linear age model show small negative changes over age (µ_s = –.01, _s =.002, _0s = –.28), with a correspondingly small statistical improvement in fit (with ² = 13, on df = 3).

The next model included only the proportional changes (3c) where ß is estimated but = 0. The proportional coefficients for LVS (ß =.02) and for WMS (ß = –.01) generate expected curves with exponential shapes based on different orientations. This not a typical mixed-effects model, but this simple alternative did offer substantial improvement in fit for LVS (² = 109 on df = 1) and minor improvement for WMS (² = 5 on df = 1).

Finally, we fit a dual change model (3d), with the mean of the slopes allowed to be free (µ_s) and = 1 fixed (for identification purposes). This resulted in MLE for the effects for LVS and WMS as (a) limiting effects (ß =.03 and.05), (b) initial level means (µ₀ =.18 and.92 at Age = 30), and (c) linear slope means (µ_s = –.001 and –.05) for each 5-year period (after Age = 30). The goodness of fit of the dual change score (DCS) model can be compared with every other nested alternative, and these comparisons (listed in Table 3) show the best fit was achieved for both LVS and WMS when this model was used.

These equations lead to an expected trajectory over time for each variable, and this is plotted in Figure 5 (with 95% confidence boundaries). It is now clear that these parameters generate expected curves moving in different directions. The expected LVS scores over age are seen to start rather low (.18) at age 30, but they increase as an exponential curve over age with a negligible constant decrease (–.001) and with large inertial relationship (.03) every 5 years. This WMS equation starts high (at.92), but it shows a large constant decrease (–.05) plus a similar sized inertial correction (.05) every 5 years.

View larger version (16K): [in this window] [in a new window]   Figure 5. Path model of a bivariate latent difference dynamic structural equation system

We fit three additional models to examine whether one or more of the coupling parameters () were different than zero. In the first alternative model (second set of columns in Table 4), the parameter representing the effect of LVS[t] on WMS[t] was fixed to zero (_x = 0), and this leads to a notable loss of fit (² = 20 on df = 1). The second alternative assumed no effect from WMS[t] on LVS[t] (_y = 0), and this is a much smaller loss of fit (² = 3 on df = 1). Finally, we fit the last model listed where no coupling was allowed (_x = 0 and _y = 0), and this leads to a notable loss of fit (² = 23 on df = 2) so this coupling over time is needed.

The use of age-based incomplete data vectors adds complexity to the calculation of the confidence boundaries of the parameter estimates (e.g., SE_MLE), especially with small samples. To better identify these problems, we supplemented this technique with confidence interval estimates based on well-known bootstrap estimates. The confidence intervals from interval estimation and bootstrapping show that the results presented herein are quite robust. For example, the slope variance is small but has a range (5–95%) that does not include zero either the bootstrap or confidence interval approach is used. In addition, the confidence interval around the ² statistic shows that the DCS is the most reasonable model in all cases. Another potential problem is the unstable estimate of the correlation of the levels and slopes (_0s .9), but this estimate is likely to be a result of the wide cross-sectional age spread and does not seem to alter the stability of the other parameters (see Hamagami & McArdle, 2001).

We fit four additional models to examine a common growth factor model proportionality hypothesis (following McArdle & Woodcock, 1997; McArdle et al., 2002). In this case, the factor model has two indicators at each time, LVS[t] and WMS[t], and we combined it with the previous univariate dual change model (Equation 3). The basic model required only nine parameters in common factor loadings (_y = 1, _x =.35) and common factor dynamic parameters (_z = 1, ß_z =.14, µ_sz = – 0.13, with no ) and achieved convergence (f_MLE = –1133). The comparison of alternative models within this common factor yielded large constant changes. However, the fit of this common factor DCS was much worse than the bivariate DCS model (² = 262 on df = 11), and we take this as strong evidence that separate process models are needed for LVS and WMS.

Describing Bivariate Dynamic Results
These results show notable and systematic coupling across the LVS[t] and WMS[t] variables. The age trajectories of the best fitting model (14) can be represented as

where we list the estimates of the fixed values of the intercepts and the random deviations (in parentheses). The estimated correlations among the latent components are given as

Thus the intercepts across variables are negatively correlated, the slopes across variables are negatively correlated, and these may be an additional important characteristic of the individual differences. However, on the basis of the respective latent change accumulations, we can write the resulting dynamic equations as

In this form, we highlight a dynamic statistical result that is important in substantive terms: An increase in the relative size of the lateral ventricle is a leading indicator of lower scores on memory, and the reverse is not apparent. The parameters just listed are specific to the time interval chosen (i.e., t = 5), so any calculation of explained variance requires a specific interval of age (McArdle & Woodcock, 1997). This also means these seemingly small differences can accumulate over longer periods of time so the LVS[t] is expected to account for an increasing proportion of variance in WMS[t] over age.

These mathematical results are also displayed in Figure 6 in the pictorial form of a vector field plot (see Boker & McArdle, 1995). This representation of the simultaneous difference equations places a directional vector (arrow) in the location {y[0], x[0]} of each combination of initial values for WMS and LVS. Each arrow then points in the expected direction of change (y, x) for any individual starting at the previous pair of values. This mathematical display shows the range of different changes that are expected from a particular set of difference parameters. One addition to this picture is our display of the location of the actual data points (and the 95% confidence ellipsoid) underneath this flow. From this perspective, the arrows that should be taken most seriously are those inside the data ellipse. We can also interpret the level–level correlation (_y0,x0 = –.23), which describes the initial location of the individuals in the vector field, and the slope–slope correlation (_ys,xs = –.65), which describes the location of the subsequent scores for individuals in the vector field. In any case, the resulting "flow" shows a dynamic process in which scores on LVS have a tendency to affect score changes on WMS, and the reverse effect is not apparent.

View larger version (54K): [in this window] [in a new window]   Figure 6. Vector field expectations from the bivariate latent difference score parameters (n = 225)

	DISCUSSION

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The univariate regression (Type 1) results exhibit a lack of association, especially for the change equations. We also note that the change equation parameters (e.g., ₀ =.13, ₁ =.001) differ from the expected change based on the Time 1 equations (e.g., ß₁ = –.06, ß₂ =.02l). These nonconverging results may be due to a true lack of association or a misspecification of the longitudinal equation as a result of the implicit choice of a time basis; they may be due to the error inherent in the calculation of the observed rates of change; or they may be due to the loss of power required by the use of only complete-case data.

The univariate age-based latent change results (Type 2 or Type 3) suggested that either an exponential or dual change model is a reasonably good expression of the changes over a 7-year interval in both memory test performance and brain structure among individuals across the adult age range. For both sets of measurements, the peak levels were found at about age 60, and most of the changes were found to occur after this age. This finding extends and clarifies previous cross-sectional reports of the nonlinear nature of age-related changes in memory and brain structure (e.g., Albert et al., 1987; Sullivan et al., 1995).

In our final analyses (Type 4), we expanded these techniques to study the dynamical relationships across variables—the sequence of relationships among the memory measures and the brain structure measures. As in the regression analyses, we can simply place one variable as a predictor of another, but now we can do this with a latent difference score as our focal outcome. The bivariate latent difference score results show that individual differences in these age-lagged changes operate in a coupled over time fashion, with the LVS acting as a leading indicator in time of WMS declines.

As in other longitudinal studies, the persons who dropped out were slightly lower in WMS performance at baseline than those who participated in the follow-up. We attempted to account for this nonrandom attrition by including all longitudinal and cross-sectional data in the models, but we recognize this is still a potential confound, especially if it is related to other key variables, such as age selectivity of cohort (e.g., McArdle & Anderson, 1990; Miyayzaki & Raudenbush, 2000). In order for an age-based model (B[t] = Age[t]) to be viable, we need to extend the untestable MAR assumptions to this age dimension. In general, although we do observe some small amount of nonrandom attrition, we have included all the longitudinal and cross-sectional data to provide the best estimate of the parameters of change as if everyone had continued to participate (Diggle et al., 1994; Little, 1995; McArdle & Hamagami, 1991).

These results are broadly consistent with prior research on neurophysiological impacts on neuropsychological measurements and behavior changes, but the current results represent a unique quantitative test of this cross-domain temporal hypothesis within a person over time. To some degree, these dynamic results confirm previous cross-sectional studies suggesting a relationship between declines in memory with age and alterations in brain structure (e.g., Jones et al., 1991; MacInnes et al., 1989; Matsumae et al., 1996; Raz et al., 1998; Sullivan et al., 1995). This bivariate result will be clarified by further multivariate analyses with more specific brain and memory measurements. It is possible that the leading indicator of memory decline is not specific to the isolated brain structures (i.e., not specific to the hippocampus; Sullivan et al., 1995), or it could mean that memory loss as measured by the global WMS scores (see Park, 2000) is more broadly an outcome of features associated with larger ventricular size. Group differences based on demographics, prior experiences, or current illness can be studied by using any of the same techniques. The dynamic results presented here may differ over these kinds of groupings of persons.

To our knowledge, this is the first report demonstrating the time lag longitudinal relationship between in vivo brain measurements and cognitive function across the adult age range. This study also represents an attempt to organize and fit a set of formal structural models based on these substantive problems. In order to examine these substantive issues in more detail, other measured and unmeasured constructs have to be included in the network (e.g., hippocampal size, speed of response). We also recognize that more complex dynamic models may be needed to more closely approximate these relationships (e.g., Mitra & Peseran, 1999). At the very least, these initial models of age-dependent dynamics do seem well matched to the problems of brain structure and function, and we hope this approach can be useful in future research studies.

	Acknowledgments

 
This research was supported by grants from the National Institute on Aging (Grant AG-007137 to J. McArdle and Grant P01-AG04953 to M. Albert). We acknowledge Dr. Tamas Sandor for the development of the software used to analyze the MRI data, and for assistance in applying them to the MRI images.

We thank our colleagues John L. Horn, John R. Nesselroade, and the journal editors for their helpful comments on earlier drafts of this paper. We also thank the staff and participants of the NAS for their valuable contributions to this research (see Albert, 1995).

Computer scripts for all analyses are available on our website (http://kiptron.psyc.virginia.edu) under the scripts for "Journal of Gerontology 2004."

	Footnotes

 
Decision Editor: Christopher Hertzog, PhD

Received for publication June 12, 2000. Accepted for publication May 27, 2004.

	References

TOP Abstract Methods Results Discussion References

 

• Albert, M. (1995). The Massachusetts General Hospital Aging Study (NIA Research Grant P01-AG04953). Bethesda, MD: National Institute on Aging.
• Albert, M. (1993). Neuropsychological and neurophysiological changes in healthy adult humans across the age range. Neurobiology of Aging, 14,623.[Medline]
• Albert, M., Duffy, F. H., Naeser M. (1987). Non-linear changes in cognition with age and their neurophysiological correlates. Canadian Journal of Psychology, 41,141-157.[Medline]
• Allison, P. D. (1990). Change scores as dependent variables in regression analysis. In C.C. Clogg (Ed.), Sociological methodology 1990 (pp. 93–114). San Francisco: Jossey-Bass.
• Arbuckle, J., Wotke, W. (1999). AMOS 4.0 user's guide. Chicago: SmallWaters Corp.
• Boker, S. M., McArdle, J. J. (1995). A statistical vector field analysis of longitudinal aging data. Experimental Aging Research, 21,77-93.[Medline]
• Cnaan, A., Laird, N. M., Slasor, P. (1997). Using the general linear mixed model to analyse unbalanced repeated measures and longitudinal data. Statistics in Medicine, 16,2349-2380.[Medline]
• Collins, L., Sayer, A.G., (Eds.). (2001). New methods for the analysis of change. Washington, DC: American Psychological Association.
• Cudeck, R., Browne, M. W. (1983). Cross-validation of covariance structures. Multivariate Behavioral Research, 18,147-167.
• Diggle, P. J., Liang, K.-Y., Zeger, S. L. (1994). Analysis of longitudinal data. New York: Oxford University Press.
• Giambra, L. M., Arenberg, D., Zonderman, A. B., Kawas, C., Costa, P. T. (1995). Adult life span changes in immediate visual memory and verbal intelligence. Psychology & Aging, 10,123-139.
• Hamagami, F., McArdle, J. J. (2001). Advanced studies of individual differences linear dynamic models for longitudinal data analysis. In G. Marcoulides & R. Schumacker (Eds.), New developments and techniques in structural equation modeling (pp. 203–246) Mahwah, NJ: Erlbaum.
• Heyting, A., Tolboom, J. T. B. M., Essers, J. G. A. (1992). Statistical handling of drop-outs in longitudinal clinical trials. Statistics in Medicine, 11,2043-2061.[Medline]
• Jones, K., Albert, M., Aldwin, C., Duffy, F., Naeser, M., Hyde, M. (1991). Modeling age using cognitive, physiological and psychosocial variables. Experimental Aging Research, 17,227-242.[Medline]
• La Rue, A. (1992). Aging and neuropsychological assessment. New York: Plenum Press.
• Littell, R. C., Milliken, G. A., Stroup, W. W., Wolfinger, R. D. (1996). SAS system for mixed models. Cary, NC: SAS Institute.
• Little, R. T. A. (1995). Modeling the dropout mechanism in repeated-measures studies. Journal of the American Statistical Association, 90,1112-1121.
• MacInnes, W. D., Paull, D., Quaife, M. (1989). Longitudinal changes in regional cerebral blood flow in a normal elderly group. Archives of Clinical Neuropsychology, 4,217-226.
• Matsumae, M., Kikinis, R., Morocz, I., Lorenzo, A., Sandor, T., Albert, M., et al. (1996). Age-related changes in intracranial compartment volumes in normal adults assessed by MRI. Journal of Neurosurgery, 84,972-981.[Medline]
• McArdle, J. J. (2001). A latent difference score approach to longitudinal dynamic structural analyses. In R. Cudeck, S. du Toit, & D. Sorbom (Eds.), Structural equation modeling: Present and future. Lincolnwood, IL: Scientific Software International (pp. 342–380).
• McArdle, J. J., Anderson, E. (1990). Latent variable growth models for research on aging. In J. E. Birren & K. W. Schaie (Eds.), The handbook of the psychology of aging (pp. 21–43). New York: Plenum Press.
• McArdle, J. J., Ferrer-Caja, E., Hamagami, F., Woodcock, R. W. (2002). Comparative longitudinal multilevel structural analyses of the growth and decline of multiple intellectual abilities over the life-span. Developmental Psychology, 38,115-142.[Medline]
• McArdle, J. J., Hamagami, F. (1991). Modeling incomplete longitudinal and cross-sectional data using latent growth structural models. Experimental Aging Research, 18,145-166.
• McArdle, J. J., Hamagami, F. (1996). Multilevel models from a multiple group structural equation perspective. In G. Marcoulides & R. Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques (pp. 89–124). Hillsdale, NJ: Erlbaum.
• McArdle, J. J., Hamagami, F. (2001). Linear dynamic analyses of incomplete longitudinal data. In L. Collins & A. Sayer (Eds.). Methods for the analysis of change (pp. 137–176). Washington, DC: American Psychological Association.
• McArdle, J. J., Hamagami, F. (2003). Structural equation models for evaluating dynamic concepts within longitudinal twin analyses. Behavior Genetics, 33,137-159.[Medline]
• McArdle, J. J., Hamagami, F., Meredith, W., Bradway, K. P. (2001). Modeling the dynamic hypotheses of Gf-Gc theory using longitudinal life-span data. Learning and Individual Differences, 12,53-79.
• McArdle, J. J., Nesselroade, J. R. (1994). Structuring data to study development and change. In S. H. Cohen & H. W. Reese (Eds.), Life-span developmental psychology: Methodological innovations (pp. 223–268). Hillsdale, NJ: Erlbaum.
• McArdle, J. J., Nesselroade, J. R. (2002). Growth curve analyses in contemporary psychological research. In J. Schinka & W. Velicer (Editors), Comprehensive handbook of psychology, volume two: Research methods in psychology (pp. 447–480). New York: Pergamon Press.
• McArdle, J. J., Woodcock, R. W. (1997). Expanding test-retest designs to include developmental time-lag components. Psychological Methods, 2,403-435.
• Meredith, W., Tisak, J. (1990). Latent curve analysis. Psychometrika, 55,107-122.
• Mitra, P. P., Pesaran, B. (1999). Analysis of brain imaging data. Biophysical Journal, 76,691-708.[Medline]
• Miyazaki, Y., Raudenbush, S. W. (2000). Tests for linkage of multiple cohorts in an accelerated longitudinal design. Psychological Methods, 5,24-63.
• Neale, M. C., Boker, S. M., Xie, G., Maes, H. H. (1999). Mx: Statistical modeling. Richmond, VA: Medical College of Virginia, Virginia Institute for Psychiatric and Behavioral Genetics.
• Park, D. C. (2000). The basic mechanisms accounting or age-related decline in cognitive function. In D. Park & N. Schwarz (Eds.), Cognitive aging: Primer (pp. 3–21). Philadelphia: Psychology Press.
• Peters, A., Rosene, D., Moss, M., Kemper, T., Abraham, C., Tigges, T., et al. (1996). Neurobiological bases of age-related cognitive decline in the rhesus monkey. Journal of Neuropathology and Experimental Neurology, 55,861-874.[Medline]
• Raz, N., Gunning-Dixon, F. M., Acker, J., Head, D., Dupuis, J. H. (1998). Neuroanatomical correlates of cognitive aging: Evidence from structural magnetic resonance imaging. Neuropsychology, 12,95-114.[Medline]
• Sandor, T., Jolesz, F., Tieman, J., Kikinis, R., Jones, K., Albert, M. (1992). Comparative analysis of CT and MRI scans in Alzheimer patients and controls. Archives of Neurology, 49,381-384.[Abstract/Free Full Text]
• Shaywitz, B., Shaywitz, S. E., Pugh, K. R., Mencl, W. E., Fulbright, R. K., Skudlarski, P., et al. (2002). Disruption of posterior brain systems for reading in children with developmental dyslexia. Biological Psychiatry, 52,101-110.[Medline]
• Sliwinski, M., Buschke, H. (1999). Cross-sectional and longitudinal relationships among age, cognition, and processing speed. Psychology & Aging, 14,18-33.
• Sullivan, E., Marsh, L., Mathalon, H., Lim, K., Pfefferbaum, A. (1995). Age-related decline in MRI volumes of the temporal lobe gray matter but not hippocampus. Neurobiological Aging, 16,591-606.
• Verbeke, G., Molenberghs, G., Krickeberg, K., Fienberg, S., (Eds.). (2000). Linear mixed models for longitudinal data. New York: Springer-Verlag.
• Wechsler, D. (1945). A standardized memory scale for clinical use. Journal of Psychology, 19,87-95.
• Zola-Morgan, S., Squire, L., Ramus, S. (1994). Severity of memory impairment in monkeys as a function of locus and extent of damage within the medial temporal lobe memory system. Hippocampus, 4,483-495.[Medline]

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