A Biometric Latent Curve Analysis of Memory Decline in Older Men of the NAS-NRC Twin Registry

Abstract

Previous research has shown cognitive abilities to have different biometric patterns of age-changes. We examined the variation in episodic memory (word recall task) for over 6,000 twin pairs who were initially aged 59–75, and were subsequently re-assessed up to three more times over 12 years. In cross-sectional analyses, variation in the number of words recalled independent of age was explained largely by non-shared influences (65–72%), with clear additive genetic influences (12–32%), and marginal shared family influences (1–18%). The longitudinal phenotypic analysis of the word recall task showed systematic linear declines over age, but several nonlinear models with more dramatic changes at later ages, improved the overall fit. A two-part spline model for the longitudinal twin data with an optimal turning point at age 74 led to: (a) a separation of non-shared environmental influences and transient measurement error (~50%); (b) strong additive genetic components of this latent curve (~44% at age 60) with increases (over 50%) up to age 74, but with no additional genetic variation after age 74; (c) the smaller influences of shared family environment (~15% at age 74) were constant over all ages; (d) non-shared effects play an important role over most of the life-span but diminish after age 74.

Appendices

Technical appendix

Mixed-effect biometric variance component models

Our analytic approach follows McArdle (1986, 2006) and McArdle and Prescott (2005) who showed how simultaneous estimation of the biometric parameters can be programmed using current mixed effects model programs (e.g., SAS PROC MIXED). The benefits of writing the model using structural equation techniques include maximum-likelihood estimation (MLE) and standard indices of goodness-of-fit (i.e., χ², df, ε_a). This analytical approach is based on the common features of path analysis models (PAM) and variance component models (VCM), and mixed-effects models can yield identical information as well as biometric inferences. This approach can easily include measured covariates, observed variable interactions, and multiple relatives within each family, and the programming logic is relatively standard. We briefly highlight the key elements of this approach here.

We denote the observed score (Y _n ) for any person (n = 1–N) but include different families (f = 1–F) and different persons (P = 1 or 2) within the family. We then write a structural equation model for any pair of persons as

$$ \begin{array}{*{20}c} {Y_{f,1} = \mu + A_{f,1} + S_{f} + I_{f,1} = \mu + \sigma_{a} \,a_{f1} + \sigma_{s} \,s_{f} + \sigma_{i} \,i_{f1} , \quad{\text{ and}}} \\ {Y_{f,2} = \mu + A_{f,2} + S_{f} + I_{f,2} = \mu + \sigma_{a} \,a_{f2} + \sigma_{s} \,s_{f} + \sigma_{i} i_{f2} ,\quad {\text{ with}}} \\ {E\{ a_{1} ,a_{2} \} = \rho_{a} .} \\ \end{array} $$

(1)

In this expression we represent the mean (μ) independent sources of the deviations that are additive genetic (A), non-genetic but shared by family members (S), and non-genetic factors independent across individuals (I) (note: we recognize the terms A, C, and E are more typical labels for the same biometric scores). These deviations can also be written with unobserved scores scaled to unit variance (E{a, a} = E{s, s} = E{i, i} = 1) so the coefficients (σ _j ) represent the standard deviation of each component, and the genetic correlation is assigned based on the genetic relationship of the pair (e.g., for MZ pairs, ρ _a  = 1, whereas for DZ pairs, ρ _a  = ½). In the simplest version of biometric theory we assume these components are all uncorrelated. More complex versions of this model include components to represent non-additivity (e.g., dominance deviations), examine correlations among these components (e.g., |ρ{A, I}| > 0), and consider interactions among components (e.g., A by I).

The standard model for a pair of relatives is often drawn as a latent variable path diagram that is familiar to behaviorial genetics (BG) researchers. As shown in detail by McArdle and Prescott (2005), some useful features emerge when we restrict our expression of any BG model so “all paths are fixed values for each person.” We can rewrite Eq. 1 for a pair of persons within the same family as

$$ \begin{array}{*{20}c} {Y_{f,1} = \mu + A_{f1} + S_{f} + I_{f1} ,{\text{with }}A_{f1} = w_{\text{ac}} AC_{f} + w_{\text{au}} AU_{f1} ,\quad{\text{and}}} \\ {Y_{f,2} = \mu + A_{f2} + S_{f} + I_{f2 } ,{\text{with }}A_{f2} = w_{\text{ac}} AC_{f} + w_{\text{au}} AU_{f2} ,} \\ \end{array} $$

(2)

where the additive genetic deviation is separated into two deviation scores: (a) AC _f is common for members of the same family, and (b) AU _fp is unique to the person. In this form the weights (W) are fixed at values which indicate the proportion of the additive genetic deviation shared between relatives. This general variance components model is drawn following Eq. 2 as a nested or higher-order latent variable or reduced form path diagram (see Fig. 1 in McArdle and Prescott 2005).

The computational approach suggested by McArdle and Prescott (2005) was based on this re-parameterization of the usual path model so the paths (coefficients) are all fixed weights defined by the application. These weights are typically scaled so the sum of squares is unity (w ² _ac  + w ² _au  = 1) so they do not impact the variance terms. Since the two individuals in any MZ pair are assumed to share the same genotypes and the same common family environment, we simplify the model for an MZ pair by fixing w _ac  = 1 and w _au  = 0 (in Eq. 2). In contrast, the assumption of no assortative mating implies that members of DZ pairs share ½ of their genotypes (due to segregation of alleles), and this implication can be represented as fixed weights of w _ac  = sqrt(½) and w _au  = sqrt(½). In order to estimate genetic dominance we add two new latent scores, D _fp for each person, decomposed into common DC _f and unique DU _fp , with both assigned equal variance (i.e., σ ² _d ). The analysis of MZ–DZ twins would require corresponding weights of 1 and 0 in MZ pairs for DC and weights of sqrt(½) for both unique DU _fp .

Approximating longitudinal results from cross-sections

As another cross-sectional view of the age-related longitudinal changes, the same kind of cross-sectional analysis was repeated for many different groups with “similar ages.” We created subgroups by centering on a specific age, and then considered individuals and pairs who were within 2 years of that age. Using sampling weights based on a normal smoother (W = [.04, .19, .50, .19, .04]) we could assure that all statistical information was based on a reasonable sample size. This exploratory analysis is conceptually simple, and we simply invoked the SAS Macro language and placed a DO-Loop around the SAS PROC NLMIXED code used above without the age covariate. The overall result of this sequential set of cross-sectional biometric analysis showed substantial variations in the additive variance within each age, not very much shared family variance within each age, and large non-shared variance (plus measurement error) within each age. Unfortunately, these sequential cross-sectional genetic estimates did not show distinct pattern over ages.

Latent curves as mixed effects models

The path analysis and variance components models have been extended for use in a longitudinal curve model for twin data (see McArdle 1986, 2006). In these analyses we are interested in the trajectory-over-time as well as the nested structure of the correlations among relatives, so we present the model needed to estimate all of the parameters of the model—both longitudinal and biometric. One generic form of a “latent curve model” based on a trajectory over time for observed variables is written as

$$ \begin{array}{*{20}c} {Y[t]_{n} = g0_{n} + u[t]_{n} ,} \\ {Y[t]_{n} = g0_{n} + B[t]g1_{n} + u[t]_{n} ,} \\ {Y[t]_{n} = g0_{n} + B1[t] g1_{n} + B2[t] g1_{n} + u[t]_{n} ,} \\ \end{array} $$

(3)

where for any individual (n = 1 to N) we write the score at any time in terms of unobserved latent scores. In the first model, there are two unobserved latent variables: The g0 are constant over time and termed the initial level scores while u[t] are unobserved and independent unique scores for measurements within each occasion. In some models there unique scores are assumed to be random errors, but in other models they also contain either systematic specific tests or systematic state variation as well (e.g., McArdle and Woodcock 1997). The second model adds one more unobserved latent variables—the g1 are slope scores, related to a set of basis coefficients Β[t] representing a function of the timing of the observations. In some cases Β[t] = [0, 1, 2, 3] represents the retesting wave, but in the alternative used here we write Β[t] = f{Age[t]} to represent the age at the time of the observation (see Fig. 2b in McArdle et al. 1998). The third model allows two slope scores (g1 and g2) and includes two sets of basis coefficients (B1[t] and B2[t]). The use of the third model is indicated in models using quadratic latent curves or multi-parts.

The typical application of this latent growth model further presumes that the initial level and slopes are assumed to be random variables with “fixed” means (μ _i ) but with unobserved deviations (di) that have “random” variances (σ ² _i ) and covariances (σ _jk ). The typical path diagram gives a standard representation of this kind of latent growth model. Model parameters representing “fixed” or “group” coefficients are drawn as one headed arrows while “random” or “individual” features are drawn as two-headed arrows. As usual, the unique error terms are assumed to be normally distributed with mean zero and constant variance (σ ²_u ) and are presumably uncorrelated with all other components. It is also possible to rewrite the standard model adding a new latent score c01, which is interpreted as a common factor of level and slope, with uncorrelated unique scores (u0, u1). Now by simple substitution, we can rewrite the reduced form of the basic growth model as

$$ \begin{array}{*{20}c} {Y[t]_{n} = (c01_{n} + u0_{n} ) + B[t](c01_{n} + u1_{n} ) + u[t]_{n} } \\ { = u0_{n} + B[t]u1_{n} + c01_{n} + B[t]c01_{n} + u[t]_{n} } \\ { = u0_{n} + B[t]u1_{n} + (1 + B[t])c01_{n} + u[t]_{n} ,} \\ \end{array} $$

(4)

where there are two sets of basis coefficients (Β[t] and 1 + Β[t]) but all the model components (c01, u0, u1, u[t]) are presumed to be uncorrelated. As it turns out, the orthogonal reduced form of the latent curve model written as Eq. 4 is especially useful for representing correlated latent variables as orthogonal variance components calculations (e.g., SAS MIXED).

Longitudinal biometric variance component models

The biometric decomposition of the latent components of level and slope used here follows the model of McArdle (1986). First, we assume the longitudinal growth representation of Eq. 3 is appropriate for all individuals. Next, using pairs of relatives, we consider the initial level (g0) to be decomposable into three new latent variables that are added within each person (A0 _n , S0 _f , E0 _n ) and where the genetic correlation is assigned for the group (e.g., ρ _a = 1 or ρ _a = ½). In this way, the biometric parameters (σ ² _0a , σ ² _0s , σ ² _0i ) indicate features of the latent initial level variance. We write the same model for the latent slopes (g1) across pairs and include a regression on the initial level latent variables, although we recognize this choice of the “slope as an outcome” is usually arbitrary. A path analysis diagram of this model is presented in other research reports (e.g., McArdle 1986, 2006)—of most importance here, this model can be fitted as a standard structural equation model with multiple groups.

The typical interpretation of latent curve models relies on a model with an arbitrary position for the intercept, and this creates the need to estimate a level and slope covariance (σ ₀₁ ). Unfortunately, this approach implies that the variance of the slope (σ ² ₁ ) is not equal to the variance of the change (i.e., σ ²_Δ  = σ ² ₁  + 2σ ₀₁ ). In order to clarify this separation, we fit this same model using mixed-effects techniques we can use the orthogonal variance components approach established in Eq. 4. We re-parameterize both common and unique parts of these components to specify the model as a set of orthogonal variance components, and we rewrite the model in terms of fixed weights and in the reduced form of

$$ \begin{array}{*{20}c} {Y[t]_{f,p} = u0_{f,p} + B[t] \cdot u1_{f,p} + (1 + B[t]) \cdot c01_{f,p} + u[t]_{p} ,} \\ {u0_{f,p} = \mu_{0} + Au0_{f,p} + Su0_{f} + Iu0_{p} \,Au0_{f,p} = w_{\text{ac}} \,ACu0_{f} + w_{\text{au}} \,AUu0_{p} ,} \\ {u1_{f,p} = \mu_{1} + Au1_{f,p} + Su1_{f} + Iu0_{p} \,Au1_{f,p} = w_{\text{ac}} \,ACu1_{f} + w_{\text{au}} \,AU1_{p} ,{\text{and}}} \\ {{\text{c}}01_{f,p} = Ac01_{f,p} + Sc01_{f} + Ic01_{p} \,Ac01_{f,p} = w_{\text{ac}} \,ACc01_{f} + w_{\text{au}} \,AUc01_{p} . } \\ \end{array} $$

(5)

In this model the additive genetic weights are defined based on the genetic resemblance of the pairs (e.g., W _mz  = [1, 0], W _dz  = [sqrt(½), sqrt(½)]), so we estimate two means, one error variance, and nine orthogonal variance components (three which represent covariance information). The higher-order version of the orthogonal model is portrayed in McArdle (2006) and we can now fit the biometric latent curve models to these data using any of several programs (SAS, Mplus, Mx) and obtain the same basic results (all programs used here are available from the first author).

Longitudinal biometric expectations

The appropriate calculation of proportions of variance can be a more complex problem when dealing with multiple time points (see McArdle 1986, 2006; Eaves et al. 1986; Hewitt et al. 1988; Loehlin et al. 1989). One issue that needs to be considered is the changing baseline of expected variance at each age, and another issue is the addition of the independent and common components of the slope combined to form the change variance. The way we have dealt with this problem in prior research is to compare the “changes in the expected statistics at specific ages” (see McArdle 1986, 2006; McArdle et al. 1998; McArdle and Hamagami 2003), and these are calculated for the final two models in Tables 8 and 10.

Examples of computer program scripts

Script 1: example of univariate biometric model

Script 2: example of univariate biometric linear longitudinal model

Script 3: example of univariate biometric non-linear longitudinal model