Multilevel Models for Diary and Intensive Longitudinal Data

23.1 Process Questions and the Within-Person Regression

The defining move of intensive longitudinal analysis is to ask a question inside the person. Consider stress and negative affect. A between-person question asks whether people who report more stress also report more negative affect, a fact about how persons rank against one another. A within-person question asks whether a given person feels worse on the occasions when they are more stressed than is typical for them, a fact about how that person’s states move together over time. The two are logically distinct, they can even have opposite signs, and the whole apparatus of Chapter 13’s centering exists to keep them apart. Person-mean centering a momentary predictor, subtracting each person’s own average stress from each of their momentary reports, isolates the within-person deviation, and a regression on that deviation answers the within-person question directly. The person means, entered separately, carry the between-person information. This disaggregation, developed for panel data by Curran and Bauer (2011) and by Wang and Maxwell (2015), codified for centering by Enders and Tofighi (2007), Hoffman and Stawski (2009), and Hamaker and Muthén (2020), is the same doctrine as Chapter 13.4, now the daily bread of a chapter where every predictor is measured repeatedly within the person. The analysis of such data has a long methodological lineage, from the early treatment of daily-experience data by West and Hepworth (1991) to the experience-sampling handbook of Hektner, Schmidt, and Csikszentmihalyi (2007).

Figure 23.1 makes the within-person question visible. Each panel is one person’s own regression of momentary negative affect on their person-centered momentary stress, twelve people from the ema_stress data, a two-week experience-sampling study with six beeps a day in 120 people. The slopes differ, plainly. One person’s affect climbs steeply with their stress; another’s barely moves; a third’s, at the floor of the scale, is flat. That variation is not noise around a common slope to be averaged away. It is the phenomenon: people differ in how reactive their affect is to their own stress, and a model that only reported the average slope would discard the most interesting thing in the data. This is the substantive reading of a random slope, developed in the personality-process tradition of Bolger and Zuckerman (1995) and the daily-stress work of Bolger, DeLongis, Kessler, and Schilling (1989): the random slope’s variance is an individual-differences finding, and its explanation by person-level covariates is the next question.

Note. Each panel regresses one person’s momentary negative affect on their person-centered momentary stress, for twelve people from ema_stress. The slopes differ across people: some are steeply reactive, others nearly flat. That heterogeneity, not the average slope alone, is the object of study.

The multilevel model formalizes this. Writing \(\mathrm{na}_{it}\) for person \(i\)’s negative affect at occasion \(t\) and \(\mathrm{stress}^{wc}_{it}\) for their person-centered stress, the model \(\mathrm{na}_{it} = \beta_{0i} + \beta_{1i}\,\mathrm{stress}^{wc}_{it} + \gamma_2\,\overline{\mathrm{stress}}_i + e_{it}\), with \(\beta_{0i}\) and \(\beta_{1i}\) random across persons, estimates an average within-person reactivity in the fixed slope and its heterogeneity in the slope variance. Fitted to ema_stress, the average within-person stress reactivity is \(0.27\) on the affect scale, and the random-slope standard deviation is substantial, so people range from essentially unreactive to strongly reactive. Figure 23.2 shows the distribution of the person-specific reactivities and the two people at its extremes, one whose affect tracks their stress closely and one whose does not. The spread is the finding.

Note. (a) The distribution of person-specific stress reactivities (best linear unbiased predictors) from the random-slope model on ema_stress; the line marks the average. (b) The most and least reactive people: one person’s negative affect climbs with their momentary stress, the other’s is nearly flat. The variation across people is a substantive result, not statistical insurance.

Once reactivity is recognized as varying, the question becomes what explains it, and the answer is a cross-level interaction: a person-level covariate that moderates the within-person slope. In ema_stress, neuroticism plays this role. Adding the interaction of person-centered stress with neuroticism, \(\beta_{1i} = \gamma_{10} + \gamma_{11}\,\mathrm{neuroticism}_i + u_{1i}\), estimates \(\gamma_{11}=0.13\) (\(SE=0.01\)): more neurotic people are more reactive to their own stress. Figure 23.3 probes the interaction by plotting the predicted within-person slope across the range of neuroticism with its confidence band, the person-specific reactivities scattered around it. The cross-level interaction has explained part of the slope variance, turning a described heterogeneity into a modeled one. The grammar of these specifications, from question to formula, is collected in Table 23.1.

Note. The predicted within-person stress-reactivity slope rises with neuroticism (line with 95% confidence band); points are the person-specific reactivities. The cross-level interaction explains part of the heterogeneity in reactivity that Figure 23.2 displayed.

23.2 Structure: How Many Levels Does the Design Create?

An intensive design nests observations in a hierarchy, and the analyst must decide how many levels to model. Beeps are nested in persons, giving the two-level structure used above; but beeps are also nested in days, and days in persons, giving a three-level structure; and because the same time-of-day slots recur across days, beeps are cross-classified by person and by slot. Figure 23.4 draws the three. The decision is not merely technical, because a day-level random effect represents something real, a person’s day being better or worse as a whole, above the momentary fluctuations within it, and ignoring it when it is present understates the clustering and can distort inference. The evidence for how many levels are needed is the same intraclass-correlation logic of Chapter 7, applied now to the extra day level.

Note. Beeps (blue circles) may be modeled as nested in persons only (a); as nested in days within persons (b), which adds a day-level random effect for whole-day highs and lows; or as cross-classified by person and by recurring time-of-day slot (c). The intraclass correlations at each level, estimated from a null model, guide the choice.

On ema_stress, a null three-level model partitions the variance of negative affect into a person share, a day-within-person share, and a beep-level share. The person intraclass correlation is \(0.42\), so a large part of the variance is stable between-person difference, the reason centering matters. The additional day-level share is \(0.06\), modest but nonzero: days do cluster the beeps within them, and a three-level model captures that a person’s Tuesday can be worse than their Wednesday as a whole. Whether to carry the third level is a judgment balancing this evidence against parsimony and convergence, and Table 23.2 lays out the decision. The related question of whether a predictor lives at the moment level or the day level, a momentary stress report versus a day’s aggregated stress, is a construction choice of the kind Chapter 5 raised: aggregating momentary reports to a daily mean creates a day-level predictor with its own between-day and within-day parts, and the aggregation should be deliberate, not accidental.

23.3 Time Within the Day and Week: Cycles

Affect is not flat across a day. It has a rhythm, rising or falling with the hours, and that diurnal cycle is either a nuisance to be removed or a phenomenon to be studied, but either way it must be modeled or it will contaminate everything else. The standard tool is trigonometric regression: a sine-cosine pair at the daily period, \(\beta_c\cos(2\pi\,h/24) + \beta_s\sin(2\pi\,h/24)\) where \(h\) is the hour, which represents a smooth cycle with two coefficients. The pair is more interpretable than it looks, because it reconstructs to an amplitude and a phase: the amplitude \(\sqrt{\beta_c^2+\beta_s^2}\) is the size of the swing from the daily mean to the peak, and the phase \(\arctan(\beta_s/\beta_c)\) locates the peak in the day. The foundations box gives the algebra and the delta-method confidence interval. On ema_stress, the fitted cosine and sine coefficients are \(0.23\) and \(0.12\), reconstructing to an amplitude of \(0.26\) (\(SE=0.01\)), with negative affect lowest around midday and rising across the afternoon toward an evening peak. Figure 23.5 draws the fitted rhythm and its amplitude, and a gallery of person-specific rhythms.

Note. (a) The population diurnal rhythm from a sine-cosine pair, amplitude \(0.26\): negative affect dips at midday and rises toward evening. (b) Person-varying rhythms from random sine and cosine coefficients; people differ in the size and shape of their daily swing. Amplitude and phase are reconstructed from the two trigonometric coefficients.

Whether the cycle is nuisance or phenomenon depends on the question. If the interest is in stress reactivity and the diurnal rhythm is merely a source of variance that would otherwise inflate the residual and confound a predictor that also varies with time of day, the cycle is partialled out and forgotten. If the interest is in the rhythm itself, in chronotype, in whether some people’s affect peaks in the morning and others’ at night, the person-varying amplitude and phase are the estimands. A weekly cycle deserves the same care: negative affect can run lower on weekends, a weekday-weekend contrast or a day-of-week factor captures it, and Liu and West (2016) warn that an unmodeled weekly cycle is a common oversight in daily-diary data. In ema_stress the weekend effect on negative affect is \(-0.06\). Table 23.3 is the menu of cycle specifications.

23.4 Trends: The Fluctuation-Versus-Change Fork

Over two weeks a person’s affect may drift, and that trend forces a decision that turns entirely on the estimand. If the question is about fluctuation, about how a person’s affect moves around their own level on the occasions when they are stressed, then a trend is a contaminant: it inflates the within-person variance and, worse, it can leak into the within-person effect if the predictor also trends. If the question is about change, about how the person’s level itself evolves, then the trend is the signal and must be modeled, often in a growth-and-process hybrid that carries both a trajectory and within-person fluctuations around it. The wording of this fork is coordinated deliberately with the time-varying-covariate treatment of Chapter 14.4 and the dynamic models of Chapter 25, because it is one doctrine appearing at three sites.

The leakage is worth seeing. Figure 23.6 simulates a within-person predictor that trends over time and an outcome that depends on the predictor’s fluctuation and carries its own trend. When the predictor is person-mean centered in the usual way, the centered deviation still contains the trend, and the trend in the outcome correlates with it, so the estimated within-person effect is inflated, \(0.38\) against a true value of \(0.30\). Person-specific detrending, regressing out each person’s linear trend before forming the within-person deviation, recovers the truth at \(0.30\). The lesson is that person-mean centering assumes stationarity, and when a trend is present the centering must be preceded by detrending, or the trend must be entered as a covariate, or the within-person effect will absorb it. Table 23.4 states the fork.

Note. When a within-person predictor trends over time, naive person-mean centering lets the trend leak into the within-person effect, inflating it above its true value of \(0.30\). Person-specific detrending before centering recovers the truth. Bars are means over simulated samples; whiskers are the 2.5th-to-97.5th percentiles.

23.5 Yesterday Matters: Lags and Serial Dependence

Consecutive moments are not independent. A person feeling bad now tends to feel bad shortly after, and this serial dependence at level one must be handled, but how to handle it is a genuine fork with two branches that answer different questions and carry different biases. The first branch treats the dependence as a property of the residuals: the within-person errors are autocorrelated, an AR(1) structure in which \(e_{it}=\rho\,e_{i,t-1}+\text{noise}\), and modeling it corrects the standard errors of the within-person effects without changing their meaning. The within-person reactivity remains the estimand; the AR(1) simply acknowledges that the process noise is not white. On ema_stress, an AR(1) residual structure fitted by generalized least squares estimates the autocorrelation at \(0.30\), and Figure 23.8 shows that the residual autocorrelation, clearly present in a model that ignores it, is whitened once the AR(1) is included. The second branch puts the prior state into the mean model as a lagged outcome, regressing current affect on its own previous value, which changes the estimand: the coefficients are now conditional on the prior state, and the lagged-outcome coefficient is itself a substantive quantity, the carryover or inertia of affect. Figure 23.7 diagrams the two, and Table 23.5 states when each is preferred.

Note. (a) An AR(1) residual structure treats the within-person errors as autocorrelated (\(\rho\)); the within-person reactivity is unchanged and its standard errors are corrected. (b) A lagged-outcome specification puts the prior affect into the mean model (\(\phi\)); the carryover becomes the estimand and all effects become conditional on the prior state. The two answer different questions.

Note. The autocorrelation of the level-1 residuals from a random-intercept model on ema_stress (orange) shows the AR(1) signature: a lag-1 correlation near \(0.28\) decaying over lags. Modeling an AR(1) residual structure whitens them (blue). This diagnostic belongs in every intensive-longitudinal report.

The lagged-outcome branch carries a hazard that must be stated plainly. Regressing current affect on its prior value while also including a person random intercept produces a biased carryover coefficient when the number of occasions per person is small, because the random intercept is correlated with the lagged outcome and cannot be estimated well from few observations. Figure 23.9 quantifies it by simulation: with a true carryover of \(0.40\), the random-intercept estimate is inflated to \(0.62\) at five occasions and only settles onto the truth as the occasions grow into the dozens. This is the multilevel cousin of the Nickell (1981) bias met in Chapter 21, where the fixed-effects version of the same estimator was biased in the opposite, downward direction; neither is trustworthy at small \(T\), and McNeish and Stapleton (2016) survey the small-sample corrections. The honest conclusion is that a lagged-outcome model needs many occasions per person, and that when the dynamics are the real interest the dynamic structural equation models of Chapter 25 estimate them properly. Within the range where a lagged-outcome model is defensible, it earns its keep on genuinely dynamic questions: does today’s conflict predict tomorrow’s mood controlling for today’s mood, the cross-day carryover contrasts that are the signature analyses of the Bolger and Laurenceau (2013) and Laurenceau, Barrett, and Pietromonaco (1998) tradition. Figure 23.10 shows the same-day and lagged effects side by side on ema_stress: momentary stress moves affect strongly in the moment, the carryover of affect is substantial, and the lagged stress effect, conditional on the prior state, is small and even slightly negative, a reminder that lagged and contemporaneous effects are different quantities.

Note. With a true carryover coefficient of \(0.40\), the random-intercept lagged-outcome estimate is inflated when occasions per person are few and settles onto the truth only as the number grows. The fixed-effects version of the same estimator is biased downward (Chapter 21). Below Chapter 25’s tools, both mislead at small \(T\).

Note. On ema_stress: momentary stress moves negative affect strongly within the occasion; prior affect carries over substantially; and the lagged stress effect, conditional on the prior state, is small and slightly negative. Separating same-day from lagged effects is a signature intensive-longitudinal analysis.

23.6 Within-Person Mediation and Moderation

A process question often has a mechanism: stress raises rumination, and rumination raises negative affect, a chain in which all three variables are measured repeatedly within the person, a 1-1-1 mediation. The multilevel apparatus estimates it, but with a subtlety that is the single most common error in the area. The \(a\)-path from stress to rumination and the \(b\)-path from rumination to affect both vary across persons, and when two random slopes are multiplied their expected product is not the product of their means but the product plus their covariance: the average within-person indirect effect is \(a\,b + \mathrm{cov}(a_i,b_i)\). Omitting the covariance term, as a naive two-model approach does, biases the indirect effect. The remedy of Bauer, Preacher, and Gil (2006) is a single stacked multilevel model that estimates both equations jointly and allows the \(a\)- and \(b\)-path random effects to covary, from which the covariance term is read directly. Figure 23.11 draws the 1-1-1 model with its random paths.

Note. All three variables are measured repeatedly within the person. The \(a\)-path (stress to rumination) and \(b\)-path (rumination to affect) vary randomly across persons, so the average within-person indirect effect is \(a\,b + \mathrm{cov}(a_i,b_i)\), the product of the average paths plus their covariance. The direct path \(c'\) is the reactivity net of the mediated route.

Fitted to ema_stress as a stacked model, the \(a\)-path is \(0.34\), the \(b\)-path is \(0.23\), and the direct path is \(0.19\); the fixed indirect effect \(a\,b\) is \(0.076\) with a Monte Carlo 95% confidence interval of \([0.062, 0.091]\), and the estimated \(a\)-\(b\) covariance adds a further \(0.006\), so the total average indirect effect is \(0.081\). The Monte Carlo interval, drawing the two paths from their joint sampling distribution and taking the percentiles of their product, is the appropriate uncertainty for an indirect effect, whose sampling distribution is not normal; the same logic as Chapter 21’s mediation, here inside persons. A Bayesian implementation, in the manner of Vuorre and Bolger (2018), handles the random covariance and the interval in one step and is often the cleanest route when the software is available. The multilevel structural equation framework of Preacher, Zyphur, and Zhang (2010) is the other principled option, separating the within- and between-person parts of the mediation explicitly. Within-person moderation, the interaction of two within-person predictors, follows the same centering discipline: both predictors are person-centered, and their product is formed from the centered versions, or the interaction will conflate within- and between-person effects.

23.7 The Complete Worked Study and Reporting

The chapter’s craft assembles into a single workflow, and ema_stress carries it end to end. The research questions are three: whether affect is reactive to momentary stress and how that reactivity varies and is explained (reactivity and its moderation), whether affect carries over and how stress operates across time (persistence and lagged effects), and whether rumination mediates the stress-affect link within persons (mechanism). The structure decision, from the intraclass correlations, is a three-level model with a modest day-level share. The cycles are modeled, a diurnal sine-cosine pair and a weekend contrast, as nuisance to be partialled here. Trends are checked and, being a fluctuation study, removed where present. Serial dependence is handled as AR(1) residuals, since reactivity, not carryover, is the primary estimand, with the residual autocorrelation diagnosed and whitened. The reactivity model returns an average within-person effect near \(0.27\) with substantial heterogeneity explained by neuroticism; the mediation model returns the indirect effect and its covariance term; the diagnostics confirm the residuals are whitened. Figure 23.12 is the study in one view: the indirect effect’s Monte Carlo distribution and the key effects against their generating truth, every one recovered near its true value.

Note. (a) The Monte Carlo distribution of the within-person indirect effect (stress to affect via rumination), with its 95% interval and the true \(a\,b\). (b) The study’s key effects, estimated (blue) against their generating truth (red crosses); every effect the design targeted is recovered near its true value. The complete script is shipped as ch23_analysis_V01.R.

The results section that reports such a study has obligations beyond a table of coefficients, and Table 23.6 extends the reporting checklist of Chapter 13 with the intensive-longitudinal specifics: the centering statement, saying exactly which predictors were person-centered and how the person means were entered; the lag construction, saying how lags were formed and whether they crossed the overnight gap; the cycle and trend decisions, saying what was modeled and what was partialled; the dependence handling, saying whether AR(1) residuals or a lagged outcome was used and why; and the software and versions, because these models are sensitive to estimator and optimizer. Gabriel and colleagues (2019) and the open handbook of Myin-Germeys and Kuppens (2021) collect the field’s emerging standards, and Nezlek (2011) and Hoffman (2015) give book-length treatments of the modeling. The pitfalls, practice, and software boxes gather the recurring hazards, the convergence economics, and the syntax.

Exercises

1. 23.1 Specification grammar. For five “on occasions when” questions, write the multilevel formula and the centering plan, then trade critiques with a peer using Table 23.1 as the key.
2. 23.2 Reactivity and moderation. On ema_stress, fit the random-slope reactivity model, add the neuroticism cross-level interaction, probe it, and write the results paragraph.
3. 23.3 Cycle lab. Recover the diurnal amplitude and phase from the sine-cosine pair, add random trigonometric slopes, and interpret two people’s fitted rhythms.
4. 23.4 Trend leak. Reproduce the leakage simulation of Figure 23.6, then write the decision memo explaining when to detrend and when to model the trend.
5. 23.5 The lag fork. Fit both an AR(1) residual model and a lagged-outcome model to ema_stress, and reconcile their estimands in words, naming what each coefficient means.
6. 23.6 Within-person mediation. Estimate the 1-1-1 mediation with a Monte Carlo interval, report the \(a\)-\(b\) covariance term, and state the random-effect caveats.

References