Time and Age as Longitudinal Timescales: Multiple Useful Models are Illuminating

AN AGE-OLD CONTROVERSY: WHICH TIMESCALE TO CHOOSE

The timescale debate on using “time” versus “age” in longitudinal analyses goes back decades,^1–5 alongside Age–Period–Cohort considerations.^6–8 While attention is often given to which timescale might be preferred, considering them as complementary can be beneficial. The work by Hayes-Larson and colleagues in this issue,⁹ is a tour de force of exploring timescale options in multi-cohort analyses. The authors examined relationships between memory decline and two primary predictors, on each of 10 studies, across 15 “time” and 12 “age” model formulations (540 different sets of estimates). The authors conclude that “timescale choice is not necessarily a ‘one or the other’ decision.” We agree with their take-home-message that standard practice should entail examining both timescales, with efforts to understand why they diverge when they do.

A TIMELESS TRADEOFF: BIAS VERSUS PRECISION/VARIANCE

The well-known “bias–variance tradeoff” describes how, as a model’s complexity increases, bias is expected to decrease, while variance is expected to increase. Figure 1 shows the common U-shaped complexity curve. Optimal model complexity balances bias–variance components. Hayes-Larson et al.⁹ offered empirical evidence on how estimates of APOE ε4 genotype (APOE4) and diabetes effects on cognitive decline might differ by timescale specification. Relating timescale choices to model complexity and bias–variance tradeoff decisions can further illuminate selection implications. Our aims with this commentary are to (1) help clarify relationships between “time” and “age” axes in longitudinal models, and (2) connect research goals, timescale choices, and bias–variance tradeoffs.

FIGURE 1.

The Bias–variance complexity curve. Bias, variance, and noise all contribute to overall error. Suppose Y=f(X)+ϵ with ϵ∼N(0,σ²); expected squared prediction error at a point x is: $E\left[{(Y-\widehat{f}\left(x\right))}^2\right]={(E\left[\widehat{f}\left(x\right)\right]-f\left(x\right))}^2+E\left[{(\widehat{f}\left(x\right)-E\left[\widehat{f}\left(x\right)\right])}^2\right]+{\sigma }^2$ i.e. Bias² + Variance + Irreducible error. As parameters are added, models generally become more flexible and fit the data better, reducing bias. However, more flexible models tend to have more uncertainty in their parameters and higher variance.

IN GOD WE TRUST, ALL OTHERS BRING DATA

Three cohorts described in Hayes-Larson et al.⁹ (Atherosclerosis Risk in Communities Study,¹⁰ Baltimore Longitudinal Study of Aging,¹¹ and Mayo Clinic Study of Aging¹²), generously provided data access for illustration; all were approved by their institutional review boards. Briefly, we followed Hayes-Larson and colleagues using Memory z-scores as the outcome, APOE4 positivity as an “exposure,” and sex and education as adjustors. With access to individual data, our pooled cohort included 80,956 observations on 21,913 participants, baseline ages 50–92 with up to 44 years of follow-up. See eAppendix1; https://links.lww.com/EDE/C240 for individual study contributions, study characteristics (eTable1; https://links.lww.com/EDE/C240), and other details (eFigure1; https://links.lww.com/EDE/C240).

TIME TO CONNECT ACROSS THE AGES

Hayes-Larson et al.⁹ specified the difference between cognitive decline trajectories (changes in memory z-scores) for a defined binary “exposure” as the estimand of interest. Figure 2 compares three linear trajectory exploratory plots representing potential estimates for “any APOE4 positivity” effects using different timescales. We show estimates using the pooled cohort and highlight the contrasts each model is based on.

FIGURE 2.

Three common timescales. Linear trajectory estimates by APOE4 status and their differences for three estimation approaches using the pooled data. Each panel shows data, slopes, slope estimates (per 10 years), standard errors (SE), and examples of information used across participants when estimating the slopes. Dashed lines represent between-person age comparisons and solid lines represent within-person aging. A, “Age₀,” shows baseline memory z-scores versus baseline age, regression slopes for all APOE4± participants, and scores from six fictitious example participants who are 20 years different in age, 3 for each group. B, “Time,” shows repeated-measure scores versus years spent on the study, regression slopes from mixed models (see eAppendices), and the observed trajectories of four participants from the six in (A) who each aged 20 years in the study; the two participants enrolled at age 90 did not contribute longitudinal information. Note that within-person slopes in (B) appear much shallower over time than the between-person slopes in A. C, “Age,” shows repeated-measure scores versus observed age, mixed model regression slopes, and scores for the same 6 participants. Figure 2 shows visually and numerically that using Age as the timescale axis induces a bias–variance type tradeoff by combining information from both within-person and between-person contrasts, leading to (in this simple example) a more precise (21% smaller SE) estimate of the trajectory difference between APOE4 groups that is still similar to the “time” axis estimate. While we specify linear relationships here to clarify relations between the timescale axes; adding smoothing curves (splines, fractional polynomials, other) to these plots can help diagnose nonlinearities for each axis (see eFigure 2).

Longitudinal studies are able to separate cross-sectional, between-person age effects from within-person aging effects. In Figure 2A, the “Age₀” (baseline age) axis uses only between-person information to estimate APOE4 associations with cross-sectional age, comparing different people who enrolled at different ages. Figure 2B, the “time” axis, uses only the within-person information on changes, contrasting for example, changes from e4- Jim and Gary, against changes from e4+ Todd and Josh, each of whom has aged 15–20 years while participating in the study. Although the average within-person slopes in Figure 2B are much shallower than the between-person slopes in Figure 2A for both APOE4 negative (−0.38 vs. −0.62) and positive (−0.53 vs. −0.72) individuals, the estimated APOE4 positivity effect (difference in slopes) is larger (−0.15 vs. −0.09 additional standard deviation [SD] decline per decade). Figure 2C, the “current age” axis, pools information from both within-person (Jim’s aging over 20 years) and between-person (Josh who is 70 vs. Elmer who is 90) contrasts, imposing an assumption that they have the same magnitude (shown below). While this can increase precision (e.g., $\frac{0.11}{0.14}\approx 21\mathrm{\%}$ smaller standard errors [SE] using “age” vs. “time”), it does so by potentially increasing biases. Luckily, we can examine these assumptions and implications.

YOU KNOW WHAT I “MEAN.” COMMON MODELS FOR AGE AND TIME

Throughout, we simplify notation to ease exposition and remove details to eAppendices; https://links.lww.com/EDE/C240. Here, we focus on parameters in the mean function E(Y), similar to the regression models used by Hayes-Larson et al.⁹ We use the common decomposition of “current age” into: age = (age₀ + time), where “age₀” is a person’s age at an index such as their first assessment (“baseline age”), again representing between-person (cross-sectional) age differences, and “time” is the time since their index measure, capturing within-person longitudinal changes. Other time-invariant ages can be used for the index instead of baseline age; the multilevel modeling literature often uses each person’s mean age as the index.

Age-time connections in longitudinal analyses are commonly introduced as follows:

1. Consider a simple, “current age” model, specifying a linear trajectory:

$${\displaystyle \begin{array}{l}{\displaystyle E\left(Y\right)={\beta }_0^a+{\beta }_1^a\mathrm{a}\mathrm{g}\mathrm{e}}\end{array}}$$ (1a)

$${\displaystyle \begin{array}{l}{\displaystyle ={\beta }_0^a+{\beta }_1^a(\mathrm{a}\mathrm{g}{\mathrm{e}}_0+\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e})}\end{array}}$$

$${\displaystyle \begin{array}{l}{\displaystyle ={\beta }_0^a+{\beta }_1^a\mathrm{a}\mathrm{g}{\mathrm{e}}_0+{\beta }_1^a\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}\end{array}}$$ (1b)

This shows that using current age in the model alone (Figure 2C, equation 1a) forces the longitudinal, within-person aging association for time, ${\beta }_1^a$, to be identical to the between-person, cross-sectional baseline age₀ coefficient (also ${\beta }_1^a$). This formalizes how “current age” models use both between-person and within-person information to estimate trajectories, as visualized in Figure 1. If between- and within-person associations are indeed similar and both information sources are pooled, estimation variance may decrease. Further, as in Figure 2C, sometimes biases can cancel each other when comparing trajectories for etiologic questions, (more on this below). However, if the between- and within-person associations diverge, ${\beta }_1^a$ is a biased estimate of either. Thankfully the magnitude of this bias can generally be checked:

• 2.Consider a model using two terms to define age; “age₀+time”:

$${\displaystyle \begin{array}{l}{\displaystyle E\left(Y\right)={\beta }_0^t+{\beta }_1^t\mathrm{a}\mathrm{g}{\mathrm{e}}_0+{\beta }_2^t\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}\end{array}}$$ (2)

This model is often presented as examining within-person aging effects over time, ${\beta }_2^t$, while “adjusting for baseline age differences,” ${\beta }_1^t$. Comparing equation (2) versus (1b) reveals that support for the simple “current age” timescale in (1), which assumes $({\beta }_2^t={\beta }_1^t)={\beta }_1^a$, can be examined using the linear contrast (${\beta }_2^t-{\beta }_1^t$) after estimating (2). If (${\beta }_2^t-{\beta }_1^t$) is negligible, the “current age” model and its implicit equality assumption are supported.

When (${\beta }_2^t\ne {\beta }_1^t$), choosing the best model requires understanding why the coefficients diverge. The time-coefficient (${\beta }_2^t$) may be distorted compared with typical within-person aging by (a) test-retest/learning/practice effects (common with cognitive assessments), (b) study protocol changes, or (c) differential survivorship or attrition after enrollment. Participants can “learn” the brief cognitive tests used in cohort studies and perform better upon successive encounters compared with the first. Similarly, cognitive aging studies often alter their assessment protocols over time, for example, remote assessments were necessary during the COVID pandemic. If remote assessments routinely understate an older adult’s cognition compared with in-person assessments, time coefficients, including pandemic data will appear distorted. Additionally, survivorship/attrition differences in longitudinal cohort studies are also common; participants with cognitive measurements over long study times have inherently escaped potential death, dementia, and other missing not at random biasing events.

In contrast, the age₀-coefficient (${\beta }_1^t$) may deviate from the rate of aging due to between-person confounding phenomena such as (a) cohorts born in different periods averaging different cognitive scores for reasons unrelated to aging (b) rates of aging differing across birth cohorts or (c) differential survivorship before enrollment. For example, different life experiences for people born in different eras, (e.g., education quality in 1950 versus 1980, pre–post personal computing, etc.), could distort cross-sectional age representations of aging. Similarly, cumulative life experiences that affect rates of aging (1960s Vietnam war, 1970s disco, 1980s stock market crash, etc.) may bias the age₀-coefficient away from the rate of change individuals in any given generation experience. Survivorship bias may occur because a new enrollee who is 90 years old may have selection influences making them intrinsically different than a new enrollee who is 50 years old. Whenever the within-person “time” (${\beta }_2^t$) and between-person “age₀” (${\beta }_1^t$) coefficients diverge substantially, additional care should be taken, as neither may be the obvious best estimate for changes in cognitive aging.

Additional variables may be included in the models above without affecting generality, (with care taken in interpretations when using interactions with age, age₀, or time terms). Using standard exposure (e4⁺=0/1) interaction terms to study APOE4 trajectory differences:

• 3.“age” timescale:

$${\displaystyle \begin{array}{l}{\displaystyle E\left(Y\right)={\beta }_0^a+{\beta }_1^a\mathrm{a}\mathrm{g}\mathrm{e}+({\beta }_2^a+{\beta }_3^a\mathrm{a}\mathrm{g}\mathrm{e})\cdot \mathrm{e}{4}^{+}}\end{array}}$$ (3a)

$${\displaystyle \begin{array}{l}{\displaystyle ={\beta }_0^a+{\beta }_1^a\mathrm{a}\mathrm{g}{\mathrm{e}}_0+{\beta }_1^a\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}+({\beta }_2^a+{\beta }_3^a\mathrm{a}\mathrm{g}{\mathrm{e}}_0+{\beta }_3^a\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e})\cdot \mathrm{e}{4}^{+}}\end{array}}$$ (3b)

• 4.“age₀+time” timescale:

$${\displaystyle \begin{array}{l}{\displaystyle E\left(Y\right)={\beta }_0^t+{\beta }_1^t\mathrm{a}\mathrm{g}{\mathrm{e}}_0+{\beta }_2^t\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}+({\beta }_3^t+{\beta }_4^t\mathrm{a}\mathrm{g}{\mathrm{e}}_0+{\beta }_5^t\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e})\cdot \mathrm{e}{4}^{+}}\end{array}}$$ (4)

Commonly, ${\beta }_3^a$ from (3a) and ${\beta }_5^t$ from (4) are taken as describing the differences in group trajectories of etiologic interest. (3b) shows the underlying within–between equality pooling assumptions of the “age” model, contrasted against the less restricted “age₀+time” model. Hayes-Larson and colleagues empirically evaluate whether ${\beta }_3^a$ ≈ ${\beta }_5^t$. We note that even when the between-versus-within-person effects differ in (4) if the differences are the same for both groups (e4+ vs e4−), then the differences will cancel out and the estimated trajectory contrasts across APOE4 positivity will indeed be equivalent in the two models (${\beta }_3^a={\beta }_5^t$; see eAppendix2; https://links.lww.com/EDE/C240). This is a key concept for evaluating bias–variance trade-off decisions for etiologic investigations. Sometimes (as in Figure 1) despite different trajectories using within-only (time) versus between-only (age₀) information, the forces driving these differences operate similarly across the groups of interest (e4+ vs e4−). When this occurs the “age” and “age₀+time” models give similar trajectory contrasts, and although group trajectories in the “age” model would be biased, potentially more precise trajectory comparisons would not be.

These formulations all make an implicit assumption that effects across different cohorts are similar, that is, trajectories are assumed to be the same for a cohort starting in 1980 compared with one starting in 2000, as discussed in Age–Period–Cohort literature.

WHAT ELSE COULD YOU MEAN? INTERACTIONS AND NONLINEARITIES

Hayes-Larson and colleagues consider both interactions of baseline age₀ with time/age axes and nonlinear mean trajectories. The main points above hold for mean functions like these as well; that is, one can expand “current age” into its two components of “age₀” and “time” to better understand within-versus-between-person contributions and related bias–variance tradeoffs. The mean trajectory formulations in Hayes-Larson et al.⁹ (even those using “current age”) can be thought of as inducing alternative assumptions about the mean outcome surface $E\left(Y\right)=f(age0,time)$ over the two-dimensional predictor space of age₀ and time; they consider generalized additive models (GAM)¹³ of the form:

$${\displaystyle \begin{array}{l}{\displaystyle E\left(Y\right)=f(age0,time)=f_1\left(\mathrm{a}\mathrm{g}{\mathrm{e}}_0\right)+f_2\left(\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\right)+f_3\left(\mathrm{a}\mathrm{g}{\mathrm{e}}_0\right)\cdot f_4\left(\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\right),}\end{array}}$$

where the trajectory functions f_k(·) are parameterized using linear and/or restricted cubic spline formulations. Figure 3 shows selected 3-dimensional surface fits of induced E(Y) = f(age₀, time) for common f(age) formulations, (eAppendix3; https://links.lww.com/EDE/C240 gives details and additional examples). The linear “age” model (equation 1a, Figure 3A) assumes that the outcome surface E(Y)=f(age₀, time) is a plane defined by two identical slopes (for age₀ and time) defining the plane. Specifying a linear interaction between age₀ and age (Figure 3B) is equivalent to assuming an interaction between “age₀” and within-person “time,” with certain coefficients restricted to be equal; this model also imposes an additional “hidden” assumption of a quadratic association with cross-sectional age₀ that is identical to the age₀-by-time interaction. Similarly, if quadratic(age) is specified, (Figure 3C), it induces equivalent quadratic(age₀) and quadratic (time) associations, alongside a linear age₀-by-time interaction term with a restricted coefficient. These restricted functions of f(age₀, time), induced by the “age” timescale, would often not be the initial choice; for example, if a quadratic cross-sectional age₀ association and simultaneous linear age₀-by-time interaction are of interest, the less restricted comparison model in 3B would be a natural candidate. Figure 3 and eAppendix3; https://links.lww.com/EDE/C240 give suggested corresponding nonrestricted E(Y) = f(age₀, time) models that can be compared with assumed E(Y) = f(age) models to examine degrees of potential bias induced by “age” timescale restricting assumptions.

FIGURE 3.

Selected E(Y)= f(age) models and corresponding induced three-dimensional surface fits for E(Y)=f(age₀, time). In general, every chosen “current age” model, (linear, quadratic, interactions, etc.), induces assumed forms on both axes of the (age₀, time) space, with additional parameter equivalences and potential interactions. Links for online rotatable versions are in eAppendix3.

TURNING TIME INTO AGE: MARGINAL STANDARDIZATION

Three-dimensional surfaces can be arduous to interpret. Figure 4 shows surface slices at selected baseline ages using pooled data fits of the most flexible mean surfaces Hayes-Larson et al.⁹ consider: Restricted cubic spline (RCS) functions for age₀ interacted with those functions for “time” and/or “current age,” (adjusted for sex and education). Briefly, Figure 4A,B) show “time” axis estimates, for:

FIGURE 4.

Estimated trajectories for APOE4 groups (e4−, e4+) and related differences from mean models using restricted cubic splines (RCS) for baseline age₀, time since baseline and/or current age, and interactions of RCS’s for age₀ with RCS’s for time and/or current age. Akaike\Bayesian Information Criterion slightly favored the “time” model in panels C\D (AIC=179,027; BIC=179,369) over the “age” model in panels A\B (AIC =179,038; BIC=179,380), (lower is better). However, APOE4 difference estimates from the “age” model are nearly identical to the “time” model, and appear more precise when there is less information, e.g. the small group of 80–90-year-olds who entered early (age₀ = 50) and were observed for 30–40 years. See eAppendix4 for details.

$${\displaystyle \begin{array}{ll}& E\left(Y\right)=f(\mathrm{a}\mathrm{g}{\mathrm{e}}_0,\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e})=RC{S}_1\left(\mathrm{a}\mathrm{g}{\mathrm{e}}_0\right)+RC{S}_2\left(\mathbf{t}\mathbf{i}\mathbf{m}\mathbf{e}\right)+RC{S}_1\left(\mathrm{a}\mathrm{g}{\mathrm{e}}_0\right)\cdot RC{S}_2\left(\mathbf{t}\mathbf{i}\mathbf{m}\mathbf{e}\right)\end{array}}$$

while Figure 4C,D show “age” axis estimates, for:

$${\displaystyle \begin{array}{ll}& E\left(Y\right)=f(\mathrm{a}\mathrm{g}{\mathrm{e}}_0,\mathit{a}\mathit{g}\mathit{e})=RC{S}_1\left(\mathrm{a}\mathrm{g}{\mathrm{e}}_0\right)+RC{S}_2\left(\mathbf{a}\mathbf{g}\mathbf{e}\right)+RC{S}_1\left(\mathrm{a}\mathrm{g}{\mathrm{e}}_0\right)\cdot RC{S}_2\left(\mathbf{a}\mathbf{g}\mathbf{e}\right)\end{array}}$$

All Figure 4 estimated trajectories and 10-year differences between APOE4 groups are constructed using postestimation marginal standardization.^14–16 This approach allows projections of the estimates onto the “current age” dimension for both models. Similar to the models in Figure 3, the f(age₀, age) model induces specific forms for the RCS(age₀) & RCS(time) parameters, while E(Y) = f(age₀, time) does not. Even so, the model fits and APOE4 trajectory differences were nearly identical, potentially due to the similarity of between-person and within-person effects, or to the flexibility of restricted cubic spline interactions. Model fit statistics showed little difference, slightly favoring f(age₀, time) over f(age₀, age). However, while most APOE4 differences were very similar between the models in Figure 4 (eAppendix4; https://links.lww.com/EDE/C240, eTables 2 and 3; https://links.lww.com/EDE/C240), estimates for the few participants who entered the study frame in their 50s–60s and were observed into their 80–90s (i.e., over 30–40 years), are more precise in the “age” model than the “time” model. Simple exploratory data analyses (Figure 1, eFigure2; https://links.lww.com/EDE/C240), reveal that only a handful of participants contributing 30+ years of within-person information for the “time” model, whereas many 80–90 years old participants contributed between-person information. Although the age timescale pooling assumptions do not always decrease variance, (estimates in time models may be more precise than in age models), in many settings precision is gained by leveraging the cross-sectional information. In this case, the Akaike\Bayesian information criterion (AIC\BIC) slightly favors the “time” model, but given better precision, the “age” model may be more valuable. Again, considering “age” and “time” models together provides a useful understanding of why the estimates perform as they do and helps with bias–variance tradeoff choices.

MIXING IT UP: VARIANCE MODELS

Connections above for mean functions hold for both generalized estimating equation and linear mixed model estimation approaches. Hayes-Larson et al.⁹ use linear mixed models with random intercepts, -slopes and -splines for time and age. Here, we make three comments. First, when using random slopes/functions on age, centering can be important for interpretation (and sometimes convergence). If “current age” is used without centering, random intercepts correspond with an extrapolated age = 0, and their variance and covariance with the random slopes can be artificially inflated. Using noncentered random age slopes in the pooled data led to random intercept variances over 10 times greater compared with centering age at 65 and a nearly perfect inverse correlation between random intercepts and slopes. We centered both age and age₀ for fixed and random terms across models (detailed descriptions in eAppendix4, eTable4; https://links.lww.com/EDE/C240).

The second point is that expanding age = ” age₀ + time” shows that using random “age” slope models induces equivalence assumptions about variance contributions from between-person age₀ and within-person time terms, similar to the equivalence assumptions on mean structures shown earlier. As before:

$${\displaystyle \begin{array}{l}{\displaystyle E\left(Y|{\mathit{b}}_i^a\right)=X\beta +b_{0i}^a+b_{1i}^a\mathrm{a}\mathrm{g}{\mathrm{e}}_{ij}}\end{array}}$$ (5a)

$${\displaystyle \begin{array}{l}{\displaystyle =X\beta +b_{0i}^a+b_{1i}^a(\mathrm{a}\mathrm{g}{{\mathrm{e}}_0}_i+\mathrm{t}\mathrm{i}\mathrm{m}{\mathrm{e}}_{ij})}\end{array}}$$

$${\displaystyle \begin{array}{l}{\displaystyle =X\beta +b_{0i}^a+b_{1i}^a\mathrm{a}\mathrm{g}{{\mathrm{e}}_0}_i+b_{1i}^a\mathrm{t}\mathrm{i}\mathrm{m}{\mathrm{e}}_{ij}}\end{array}}$$ (5b)

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}:{\mathit{b}}_i^a=\left(\begin{array}{l}{b}_{0i}^a\\ b_{1i}^a\end{array}\right)\sim MVN(\left[\begin{array}{l}0\\ 0\end{array}\right],\left[\begin{array}{ll}{\tau }_{00}^a& {\displaystyle }\\ {\tau }_{01}^a& {\displaystyle {\tau }_{11}^a}\end{array}\right])}\end{array}}$$

So the random-slopes $b_{1i}^a\mathrm{a}\mathrm{g}{\mathrm{e}}_{ij}$ assumption induces both a $b_{1i}^a\mathrm{t}\mathrm{i}\mathrm{m}{\mathrm{e}}_{ij}$, and a $b_{1i}^a\mathrm{a}\mathrm{g}{{\mathrm{e}}_0}_i$ random-coefficients with identical variance terms. Although $\mathrm{a}\mathrm{g}\mathrm{e}{0}_i$ is constant within-person, the $b_{1i}^a\mathrm{a}\mathrm{g}{{\mathrm{e}}_0}_i$ random term can be thought of as introducing heteroscedastic variance across age₀. To allow both within-person slopes over time and differential overall variance across baseline age₀, one could examine a less restricted “age₀+time” model:

$${\displaystyle \begin{array}{l}{\displaystyle E\left(Y|{\mathit{b}}_i^t\right)=X\beta +b_{0i}^t+b_{1i}^t\mathrm{a}\mathrm{g}{{\mathrm{e}}_0}_i+b_{2i}^t\mathrm{t}\mathrm{i}\mathrm{m}{\mathrm{e}}_{ij}}\end{array}}$$ (6)

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}:{\mathit{b}}_i^t=\left(\begin{array}{l}{b}_{0i}^t\\ b_{1i}^t\\ b_{2i}^t\end{array}\right)\sim MVN(\left[\begin{array}{l}0\\ 0\\ 0\end{array}\right],\left[\begin{array}{l}{\tau }_{00}^t\\ {\displaystyle {\tau }_{01}^t}{\displaystyle {\tau }_{11}^t}\\ {\displaystyle {\tau }_{02}^t}{\displaystyle {\tau }_{12}^t}{\displaystyle {\tau }_{22}^t}\end{array}\right])}\end{array}}$$

where $b_{1i}^t$ and $b_{2i}^t$ and their variance terms are allowed to differ. This shows again that using “current age” for random slopes in (5a) induces additional equality pooling assumptions. As before, if the “age” model assumptions hold, ($b_{1i}^t\approx b_{2i}^t$, ${\tau }_{12}^t\approx 1$, and ${\tau }_{11}^t=var\left(b_{1i}^t\right)\approx var\left(b_{2i}^t\right)={\tau }_{22}^t$ in (6)), then pooling the information and using $b_{1i}^{a}ag{e}_{ij}$ in (5a) is supported and may provide more precision. The commonly used “time” random-slopes variance model:

$${\displaystyle \begin{array}{l}{\displaystyle E\left(Y|{\mathit{b}}_i\right)=X\beta +b_{0i}+b_{1i}\mathrm{t}\mathrm{i}\mathrm{m}{\mathrm{e}}_{ij}}\end{array}}$$ (7)

simply assumes there is no additional heteroscedastic variance in outcomes across baseline age₀ (${\tau }_{11}^t=0$) after accounting for the fixed effects, random intercepts and random-slopes over time. Variance assumptions in (5a) and (7) can be examined with the alternative variance model (6) and its (age₀,time) sources. The authors and others⁵ purport that random slopes in “age” versus “time” models cannot be made mathematically equivalent. This is true if you compare basic models (5a) and (7) above, ignoring the additional age₀ term in the variance model. Equation (6) above shows that you can make a random-slopes “time” model equivalent to a random-slopes “age” model by including the random “age₀” term in the “time” model with the identical random effect (5b); this can be fit using constraints in standard software packages. Note that while the models can be made mathematically equivalent, model convergence depends on both the model and data. Impossibilities of fitting a model offer evidence that it is inconsistent with the data or over-specified/overly complex. eTable5; https://links.lww.com/EDE/C240 gives results from the pooled data for all variance models (5)–(7). Although variance terms for random age₀ and random time slopes in model (6) show 95% confidence intervals that support the inclusion of both. Model (5a), random age slopes assuming ($b_{1i}^t=b_{2i}^t=b_{1i}^a$) shows better AIC/BIC fit statistics compared with (6) or (7), Model (5b) was fit using constraints ($b_{1i}^t=b_{2i}^t$) and has equivalent estimates, log-likelihood and AIC\BIC to (5a). Additional useful work related to both variance points 1 & 2 (centering and random-slopes equivalence) has appeared in the multilevel model literature for disaggregating “conflated” within-cluster and between-cluster fixed and random effects.¹⁷

Our third point on variance models is that we used semi-robust “sandwich” variance estimates^18,19 throughout and suggest others do so whenever possible. Semi-robust variance estimates help protect against mis-specifications when we have guessed a useful but incorrect variance model, and per the famous George Box quote, all models are wrong.

ALL MODELS ARE WRONG, SOME ARE USEFUL… AND MULTIPLE USEFUL MODELS ARE ILLUMINATING

The old joke goes: “That person uses statistics like a drunk uses a lamppost, more for support than illumination.” Examining multiple approaches to scientific queries can offer deeper insights than using a single model and simply stating a number for support (P value, AIC, BIC, etc.). We have attempted to illuminate technical aspects underlying the rich set of example timescale models examined by Hayes-Larson and colleagues. Many discussions regarding longitudinal timescale choice try to answer the question “which timescale should you use?” We take Box’s position that they are all wrong. However, recognizing how “age” and “time” are connected can help lead to using more useful models. Comparing timescales across multiple mean and variance structures helps to better understand within-person and between-person contributions to the estimates, even though they may cancel out when comparing groups. When contributions diverge, examining factors such as learning/practice effects,^20–22 survivorship/attrition/selection effects,^23–25 and Age–Period–Cohort effects,^26,27 can help illuminate optimal estimates and truths about the world.

ACKNOWLEDGMENTS

We thank the ARIC, BLSA, and MCSA studies for making their data available. We thank Gwen Windham for constructive editorial comments and Terry Therneau for invigorating discussions on timescales, centering and general modeling approaches.

ABOUT THE AUTHORS

MICHAEL GRISWOLD is Professor of Biostatistics and Data Science in the MIND Center at the University of Mississippi Medical Center and adjunct faculty in the Department of Biostatistics at the Johns Hopkins University. His work involves translational biostatistics, data harmonization, and longitudinal models for complex data with a substantive emphasis on healthy aging and Alzheimer’s Disease and Related Dementias.

MARIA GLYMOUR is Chair of the Department of Epidemiology at Boston University School of Public Health. Her work evaluates social, behavioral, and clinical factors that influence Alzheimer’s Disease and Related Dementias, with attention to methodologic challenges in dementia research.

Supplementary Material

ede-36-572-s001.pdf ^{(753.3KB, pdf)}