Modeling Within-Person Variability: Mixed-Effects Location-Scale Models

16.1 Variability as Signal

The premise of the chapter is a gallery of people who are identical in the mean and different in everything that matters. Figure 16.1 shows four persons with the same average negative affect and sharply different within-person variability, from nearly flat to wildly swinging. In a clinical setting these are four different patients: the stable one and the highly labile one may share a diagnosis and a mean symptom level yet face different risks, respond to different treatments, and warrant different monitoring. The tradition of studying such differences runs through the work on affective instability in psychopathology (Ebner-Priemer et al., 2009; Jahng et al., 2008), the study of performance inconsistency as an early marker of cognitive decline (Hultsch et al., 2002), and the broader program of treating short-term intraindividual variability as a stable individual-differences construct (Wang et al., 2012). In every case the claim is the same: the variance is the phenomenon, and a method that models only the mean discards it.

Note. Four simulated persons with the same average negative affect and increasing within-person variability, from stable to highly labile. The mean cannot distinguish them; the within-person standard deviation can, and it is frequently the quantity of clinical interest. This figure extends the intraindividual-variability display of Chapter 7 with an explicit substantive reading.

Chapter 7 measured this variability with the intraindividual standard deviation, computed one person at a time, and warned that the index is fragile. The warning can now be made precise as a four-part deficiency audit. First, the two-stage standard deviation is estimated with error that depends on the number of occasions, so a person measured on twenty beeps has a far noisier estimate than one measured on fifty, and Figure 16.2 shows the resulting scatter. Second, there is no shrinkage: each person’s estimate stands alone, so the extreme values, which fall disproportionately on the persons with the fewest occasions, are taken at face value rather than pulled toward the group. Third, the index confounds level with variability whenever the two are related, and the right panel of Figure 16.2 shows exactly the positive coupling between mean and standard deviation that a bounded or floor-prone scale induces, so that a difference in variability may be an artifact of a difference in level. Fourth, when the index is then carried into a second analysis as a predictor, its estimation error becomes error-in-variables that biases the downstream coefficient, an instance of the plug-in fallacy that recurs whenever a noisy estimate is treated as a known quantity. A model-based approach must repair all four: it must borrow strength across persons, admit covariates at both levels, model level and variability jointly, and propagate the uncertainty in the person-specific variability into whatever uses it. Table 16.1 sets the two approaches side by side.

Note. Left: the raw within-person standard deviation against the number of beeps, showing that fewer occasions give noisier and more extreme estimates with no shrinkage. Right: the raw within-person standard deviation against the person mean, showing that level and variability are confounded when the outcome has a floor. Both deficiencies are repaired by a model that estimates person volatility with shrinkage and separates it from the mean.

16.2 The Model, Built in Three Steps

The mixed-effects location-scale model is best assembled in three steps, each a recognizable extension of the last. The first step is heterogeneous level-one variance. The ordinary mixed model assumes a single residual variance \(\sigma^2\) for everyone; the first relaxation lets that variance differ by group or by a covariate, the familiar territory of the residual-variance functions of Chapter 14. The second step makes the variance a regression. Rather than a separate variance per group, a log-linear scale submodel writes the log of the within-person variance as a linear function of predictors, \(\log \sigma^2_{it} = \alpha_0 + \alpha_1 x_{it}\), so that a covariate can raise or lower momentary variability. The log link is not incidental: it keeps the variance positive for any values of the coefficients, and it gives the coefficients a multiplicative reading, because a one-unit change in the predictor multiplies the variance by \(e^{\alpha_1}\). The third step adds a random effect to the scale submodel. Persons differ in their residual volatility beyond what the covariates explain, and a random scale effect \(\omega_i\) captures that difference, \(\log \sigma^2_{it} = \alpha_0 + \alpha_1 x_{it} + \omega_i\), so that each person has a personal baseline volatility just as the location submodel gives each person a personal baseline level.

The full model couples the two submodels and, crucially, allows their random effects to correlate. The location submodel is \(y_{it} = \gamma_{0} + \gamma_{1} x_{it} + u_{i} + \varepsilon_{it}\) with \(\varepsilon_{it} \sim N(0, \sigma^2_{it})\), the scale submodel is \(\log \sigma^2_{it} = \alpha_0 + \alpha_1 x_{it} + \omega_i\), and the two person effects are drawn together, \((u_i, \omega_i)' \sim N(\mathbf{0}, \mathbf{\Sigma})\), with a correlation \(\rho\) between them. Figure 16.3 draws the architecture. The correlation \(\rho\) is the parameter with the substantive punch: it asks whether people who are higher on average are also more variable, whether, for instance, patients with higher mean negative affect are also more volatile in it, a question that no two-stage analysis can pose because it never estimates the two quantities in a common model. Table 16.2 is a glossary of the parameters and how each is read.

Note. The location submodel places a person random intercept \(u_i\) on the mean; the scale submodel places a person random intercept \(\omega_i\) on the log within-person variance; a within-person predictor can enter either submodel. The two random effects are drawn from a joint distribution with a correlation \(\rho\), the location-scale correlation that links a person’s level to a person’s volatility.

The model is harder to estimate than an ordinary mixed model because the likelihood integrates over two correlated random effects, one of which sits inside a variance, and the integral has no closed form. Two routes are available. Maximum likelihood by adaptive quadrature is implemented in Hedeker’s dedicated programs (Hedeker et al., 2008; Hedeker & Nordgren, 2013), which pioneered the model for ecological momentary assessment data. Bayesian estimation by Markov chain Monte Carlo, the route this book adopts, generalizes more gracefully to random scale slopes, to more than two levels, and to non-Gaussian location submodels (Rast et al., 2012; Williams et al., 2019). The worked analysis below is fitted by maximum likelihood through two-dimensional Gauss-Hermite quadrature, a transparent implementation that recovers the same estimand, while the recommended production tool is the Bayesian one, whose machinery Chapter 17 develops. This is the deliberate handoff: the location-scale model motivates the move to Bayes, and the next chapter delivers it.

16.3 Fitting the Model

The worked example asks whether momentary stress predicts the volatility of negative affect beyond its level, on the simulated diary dataset ema_lability, in which 120 persons are measured on unequal numbers of beeps and the data-generating process contains a genuine random scale effect and a positive location-scale correlation. Within-person stress is entered in both submodels: in the location submodel to allow stress to raise the level of negative affect, and in the scale submodel to allow it to raise the volatility. The location effect of stress is recovered at \(0.53\), close to its true value, but the finding of interest is in the scale submodel. The scale coefficient for stress is \(0.41\) with a standard error of \(0.03\), which on the multiplicative scale means that a one-unit increase in a person’s momentary stress multiplies the within-person variance of their negative affect by \(e^{0.41} = 1.50\). Figure 16.4 draws this as a predicted within-person standard deviation rising with stress, with an uncertainty band. Stress does not merely lift the average of negative affect; it destabilizes it, and the scale submodel is what makes that second, distinct claim estimable.

Note. Predicted within-person standard deviation of negative affect as a function of within-person stress, from the scale submodel, with a ninety-percent uncertainty band. A one-unit rise in momentary stress multiplies the residual variance by about \(1.5\). The scale submodel estimates this destabilizing effect separately from the effect of stress on the mean level.

The model repairs the two-stage index by shrinking the person-specific volatility estimates. Figure 16.5 compares each person’s raw log within-person standard deviation to the model’s empirical-Bayes estimate, plotted against the number of beeps. The raw estimates are extreme, especially for the persons with the fewest occasions, and the model pulls them toward the group in proportion to their unreliability, exactly the shrinkage of Chapter 13 now applied to a variance. That the shrunken estimates are better, not merely smoother, is checkable here because the truth is known: the model’s estimates of person volatility correlate \(0.90\) with the true random scale effects, against \(0.81\) for the raw indices. The location-scale correlation, the substantive target, is estimated at \(0.40\) with a standard error of \(0.10\), and Figure 16.6 shows the joint distribution of the person effects behind it: persons with a higher mean negative affect do tend to be more volatile in it. The wide standard error is itself a lesson. The location-scale correlation is estimated with far less precision than a mean effect, because it is a second-order quantity that depends on the spread of person volatilities, so studies aiming to estimate it need more persons and more occasions than studies aiming only at means, a power fact that connects to Chapter 4 and that honest reporting must acknowledge.

Note. Each person’s raw log within-person standard deviation (grey) and the model’s empirical-Bayes estimate (blue), joined by a segment, against the number of beeps. The model shrinks the noisy low-occasion estimates toward the group mean, and because the data-generating truth is known, the shrunken estimates can be shown to recover the true person volatility better than the raw indices (correlation \(0.90\) versus \(0.81\)).

Note. The joint distribution of the empirical-Bayes location random effects (person mean) and scale random effects (person log volatility). The positive slope is the estimated location-scale correlation, here about \(0.40\) with a standard error of \(0.10\). This correlation, inaccessible to any two-stage analysis, is typically the substantive target of a location-scale study.

16.4 Extensions and Relatives

The scale submodel accepts the same elaborations as any regression, and two are especially useful. A predictor of variability may be time itself, turning the model into a statement about how volatility changes over a study. Figure 16.7 shows a variance trajectory in which within-person volatility declines across the weeks of a therapy, recovered by a time-varying scale model: variability is here a treatment outcome in its own right, the stabilization of a symptom rather than the reduction of its level, and a therapy might succeed on the second even where it is slow on the first. The distinction between between-person and within-person variance is the second elaboration. The random scale effect models between-person differences in volatility, while the scale covariates model within-person, occasion-to-occasion changes in it, and Hedeker’s framework separates the two explicitly (Hedeker et al., 2012). The multilevel machinery for individual differences in within-person variation predates the fully coupled model and remains a useful special case (Hoffman, 2007).

Note. A time-varying scale model in which the within-person residual standard deviation declines across therapy weeks. The fitted decay (blue) tracks the true generating decay (dashed). When variability itself is the outcome, its trajectory over an intervention is the finding, distinct from any change in the mean level.

The location-scale model sits within a family of distributional models that let parameters other than the mean depend on covariates. Table 16.3 maps the neighborhood. Double generalized linear models put a submodel on the dispersion of a generalized linear model; generalized additive models for location, scale, and shape allow every parameter of a flexible distribution, including skewness and kurtosis, to be a smooth function of covariates (Rigby & Stasinopoulos, 2005); variance-function mixed models and the dispersion formulas of modern software fit fixed scale submodels quickly; and the variance-modeling tradition in multilevel education research addresses the same structure from another field (Leckie et al., 2014). All share the premise that the spread, not only the center, is worth a model.

A final extension concerns using the estimated variability downstream. When a person’s scale effect \(\omega_i\) is to serve as a predictor or an outcome in a further analysis, it must not be reduced to a point estimate and plugged in, because that discards its uncertainty and biases the second-stage result, the same plug-in fallacy that afflicts any use of a shrunken estimate as data. The correct practice carries the full posterior distribution of \(\omega_i\) forward, either by fitting the two stages jointly or by propagating draws, so that a person whose volatility is poorly estimated contributes proportionally less. This is one more reason the Bayesian route is natural here, since it produces the posterior draws that make honest propagation straightforward.

16.5 Validation, Pitfalls, and Reporting

A location-scale model must be checked where it makes its claims, on the variance structure, and the tool is a posterior predictive check aimed at variance features rather than at the mean. Figure 16.8 compares the observed distribution of per-person within-person standard deviations to the distribution replicated from the fitted model; the observed distribution falls comfortably within the replicated spread, evidence that the model reproduces the pattern of volatility it was built to explain. Checks of this kind, targeting the dispersion of the iSD across persons or the calibration of person-specific variances, are the location-scale analogue of the ordinary posterior predictive check and belong in any report. Table 16.4 is the reporting checklist, which extends the mixed-model checklist of Chapter 13 with the estimation-quality diagnostics that a Bayesian fit requires and that Chapter 17 explains in full.

Note. The observed distribution of per-person within-person standard deviations (red) against forty distributions replicated from the fitted location-scale model (grey). The observed distribution lies within the replicated spread, indicating that the model reproduces the between-person pattern of volatility. Posterior predictive checks for a location-scale model should target variance features, not the mean.

Three pitfalls recur. Interpreting a scale coefficient as a raw standard-deviation effect ignores the log link and understates the effect; scale coefficients are multiplicative on the variance. Reading low variability off a bounded scale can be an artifact, because a person near a floor or ceiling has little room to vary, so an apparent difference in volatility may be a difference in level in disguise, the same caution the relative-standard-deviation discussion of Chapter 7 raised. And using a point estimate of a person’s volatility as an outcome launders away its uncertainty and biases the downstream analysis. Each is avoidable with the discipline the model itself supplies.

16.6 Fitting Location-Scale Models in R

The distributional-model syntax of brms states the two submodels in one formula, the first for the mean and a second, prefixed sigma, for the log standard deviation, each with its own fixed and random effects.

library(brms)
# Location: na ~ stress + (1 | person); Scale: log sigma ~ stress + (1 | person)
f <- bf(na    ~ stress_wc + (1 | p | person),
        sigma ~ stress_wc + (1 | p | person))   # shared "p": correlates the REs
m <- brm(f, data = ema, chains = 4, cores = 4)   # priors + MCMC; see Chapter 17
summary(m)                                        # scale effects are on log-SD

The shared grouping tag | p | is what correlates the location and scale random effects, yielding the location-scale correlation. A fixed-only scale model, without the random scale effect, is available more quickly through nlme residual-variance functions or the glmmTMB dispersion formula, useful as a stepping stone or when the random scale is not needed.

library(nlme)   # fixed log-linear scale: residual SD ~ exp(theta * stress)
m_fixed <- lme(na ~ stress_wc, random = ~1 | person, data = ema,
               weights = varExp(form = ~ stress_wc))
# glmmTMB alternative: glmmTMB(na ~ stress_wc + (1|person),
#                              dispformula = ~ stress_wc, data = ema)

The complete analysis, including the two-stage deficiency demonstration, the full location-scale model fitted by two-dimensional Gauss-Hermite maximum likelihood with empirical-Bayes person effects, the scale-effect and location-scale displays, the variance trajectory, and the variance-targeted posterior predictive check, is the shipped script ch16_analysis_V01.R, with figures drawn by ch16_figures_V01.R and the dataset generated by gen_ema_lability_V01.R. The script fits the model by maximum likelihood for transparency and portability; brms fits the same model by Markov chain Monte Carlo in a single call and is the recommended tool for applied work, especially when the scale submodel carries its own random slopes.

16.7 Common Misconceptions

Several beliefs about within-person variability mislead. The first is that a well-estimated intraindividual standard deviation is as good as a model; it is not, because even with many occasions it lacks shrinkage, cannot carry covariates, and confounds level with variability whenever they are coupled. The second is that the scale submodel is a heteroscedasticity correction; here it is the hypothesis, and the within-person variance is the outcome of interest, not a nuisance to be swept away. The third is that variance findings are inherently less replicable; the truth is that they demand more of the design, more persons and more occasions, and their apparent fragility is usually a power problem that the honest planning of Chapter 4 addresses. The fourth is that a five-wave panel can support within-person variance modeling; it cannot, because a handful of occasions per person yields no usable estimate of that person’s volatility, and the method belongs to intensive designs.

Where to Go Next

This chapter deliberately ends at the threshold of Bayesian estimation. The location-scale model is fit here by maximum likelihood for transparency, but its natural engine is Markov chain Monte Carlo, which handles the random scale slopes, the multilevel extensions, and the posterior propagation that maximum likelihood strains to provide. Chapter 17 supplies that machinery in full, and a reader who wants the recipe before the theory can fit the models of this chapter in brms now and return to Chapter 17 for the understanding. The within-person variance reappears throughout the intensive-longitudinal part of the book: Chapter 25 models heterogeneity in the innovation variance of a dynamic model, the moving cousin of the random scale effect, and Chapters 33 and 35 deploy affective instability as a clinical and personality construct. The lesson that the spread deserves a model, not only the center, recurs wherever a process is measured densely enough to reveal how much it swings.

Exercises

1. 16.1 Deficiency demonstration. Simulate persons with unequal numbers of occasions and a common true volatility, and show that the ranking of persons by raw intraindividual standard deviation is unstable while the location-scale estimates, shrunken, are not.
2. 16.2 Fit the scale submodel. On a diary outcome, fit a location-scale model with a within-person predictor in both submodels, and write the scale-submodel results paragraph, reporting the effect on the multiplicative variance scale.
3. 16.3 A scale random slope. Extend the model to let the effect of the predictor on volatility vary across persons, assess convergence honestly, and report whether the data support the added complexity.
4. 16.4 Variance-targeted checks. Propose and implement two posterior predictive checks aimed at the variance structure, and interpret what each would reveal about a misspecified scale submodel.
5. 16.5 Location, scale, or both. For three research questions, decide whether each concerns the level, the variability, or their relationship, and specify the submodels that would answer it.

References