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[(Y−f^(x))2]=(E[f^(x)]−f(x))2+E[(f^(x)−E[f^(x)])2]+σ2 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., 0.110.14≈21% 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: Consider a simple, “current age” model, specifying a linear trajectory: E(Y)=β0a+β1aage =β0a+β1a(age0+time) =β0a+β1aage0+β1atime This shows that using current age in the model alone (Figure 2C, equation 1a) forces the longitudinal, within-person aging association for time, β1a, to be identical to the between-person, cross-sectional baseline age0 coefficient (also β1a). 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, β1a 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”: E(Y)=β0t+β1tage0+β2ttime This model is often presented as examining within-person aging effects over time, β2t, while “adjusting for baseline age differences,” β1t. Comparing equation (2) versus (1b) reveals that support for the simple “current age” timescale in (1), which assumes (β2t=β1t)=β1a, can be examined using the linear contrast (β2t−β1t) after estimating (2). If (β2t−β1t) is negligible, the “current age” model and its implicit equality assumption are supported. When (β2t≠β1t), choosing the best model requires understanding why the coefficients diverge. The time-coefficient (β2t) 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 (β1t) 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” (β2t) and between-person “age0” (β1t) 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: E(Y)=β0a+β1aage+(β2a+β3aage)⋅e4+ =β0a+β1aage0+β1atime+(β2a+β3aage0+β3atime)⋅e4+ 4. “age0+time” timescale: E(Y)=β0t+β1tage0+β2ttime+(β3t+β4tage0+β5ttime)⋅e4+ Commonly, β3a from (3a) and β5t 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 β3a ≈ β5t. 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 (β3a=β5t; 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(Y)=f(age0,time) over the two-dimensional predictor space of age0 and time; they consider generalized additive models (GAM)13 of the form: E(Y)=f(age0,time)=f1(age0)+f2(time)+f3(age0)⋅f4(time), where the trajectory functions fk(·) 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 can be to Figure shows surface at selected baseline ages using pooled data fits of the flexible mean Hayes-Larson et al.9 cubic spline functions for age0 with those functions for “time” and/or “current for sex and Briefly, Figure show “time” axis trajectories for APOE4 groups and related differences from mean models using restricted cubic for baseline age0, time since baseline and/or current age, and interactions of for age0 with for time and/or current the “time” model in over the “age” model in is However, APOE4 difference estimates from the “age” model are identical to the “time” model, and appear more precise when is less information, the group of who (age0 = and were observed for See for while Figure show “age” axis Figure estimated trajectories and differences between APOE4 groups are using This of the estimates the “current age” for both to the models in Figure the f(age0, age) model induces forms for the while E(Y) = f(age0, time) does the model fits and APOE4 trajectory differences were potentially due to the of between-person and within-person or to the of restricted cubic spline interactions. fit f(age0, time) over f(age0, However, while APOE4 differences were similar between the models in Figure 2 and https://links.lww.com/EDE/C240), estimates for the participants who the study in their and were observed into their over years), are more precise in the “age” model than the “time” model. exploratory data analyses (Figure https://links.lww.com/EDE/C240), that only a of participants years of within-person information for the “time” model, years old participants between-person information. 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