An Application of a Mixed-Effects Location ScaleModel for Analysis of Ecological Momentary

Assessment (EMA) Data

Don Hedeker, Robin Mermelstein, & Hakan DemirtasUniversity of Illinois at Chicago

hedeker@uic.edu

Biometrics, in pressSupported by National Cancer Institute grant 5PO1 CA98262 (Mermelstein, PI)

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Ecological Momentary Assessment (EMA) dataaka experience sampling and diary methods

• Subjects provide frequent reports on events and experiences oftheir daily lives (e.g., 30-40 responses per subject collected overthe course of a week or so)

– electronic diaries: palm pilots or personal digital assistants(PDAs)

• Capture particulars of experience in a way not possible with moretraditional designse.g., allow investigation of phenomena as they happen over time

• Reports could be time-based, following a fixed-schedule, randomlytriggered, event-triggered

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Data are rich and offer many modeling possibilities!

• person-level and occasion-level determinants of occasion-levelresponses ⇒ potential influence of context and/or environmente.g., subject response might vary when alone vs with others

• allows examination of why subjects differ in variability ratherthan just mean level

– between-subjects variancee.g., subject heterogeneity could vary by gender or age

– within-subjects variancee.g., subject degree of stability could vary by gender or age

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Ecological Momentary Assessment (EMA) Study ofAdolescent Smokers (Mermelstein)

• 461 adolescents (9th and 10th graders); former and currentsmoking experimenters, and regular smokers

• Carry PDA for a week, answer questions when prompted

average = 30 answered prompts (range = 7 to 71)

• ∑Ni ni = 14, 105 total number of observations

Outcomes: positive and negative affect

Interest: characterizing determinants of affect level, as well as BSand WS affect heterogeneity

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Mixed-effects regression model for measurement y ofsubject i (i = 1, 2, . . . , N) on occasion j (j = 1, 2, . . . , ni)

yij = x′ijβ + υi + εij

xij = p × 1 vector of regressors (including a column of ones)

β = p × 1 vector of regression coefficients

υi ∼ N(0, σ2υ) BS variance

εij ∼ N(0, σ2ε) WS variance

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Log-linear models for variances

BS variance σ2υi

= exp(u′iα) or log(σ2

υi) = u′

iα

WS variance σ2εij = exp(w′

ijτ ) or log(σ2εij) = w′

ijτ

• ui and wij include covariates (and 1)

• subscripts i and j on variances indicate that these changedepending on covariates ui and wij (and their coefficients)(number of parameters does not vary with i or j)

• exp function ensures a positive multiplicative factor, and soresulting variances are positive

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WS variance varies across subjects

σ2εij = exp(w′

ijτ + ωi) where ωi ∼ N(0, σ2ω)

log(σ2εij) = w′

ijτ + ωi

• ωi are log-normal subject-specific perturbations of WS variance

• ωi are “scale” random effects - how does a subject differ in termsof the variation in their data

• υi are “location” random effects - how does a subject differ interms of the mean of their data

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Literature on Random Scale Effects

SummaryCleveland, Denby, & Liu (2002), Bell Labs technical report

FrequentistChinchilli, Esinhart, & Miller (1995), BiometricsJames, Venables, Dry, & Wiskich (1994), BiometrikaLin, Raz, & Harlow (1997), Biometrics

BayesianLeonard (1975), TechnometricsMyles, Price, Hunter, Day, & Duffy (2003), Public Health Nutrition

• Most use (square root) inverse gamma for random scaledistribution (for computational reasons)

• Do not allow random location and scale effects to be correlated

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Location random effects for two subjects

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Location and scale random effects for two subjects

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Model allows covariates to influence

• mean: level of solid line

• BS variance: dispersion of dotted lines

• WS variance: dispersion of points

additional random subject effects on: mean and WS variance

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Standardize the random effects via the Cholesky factorization

υiωi

=

συi 0

συω/συi

√√√√σ2ω − σ2

υω/σ2υi

θ1iθ2i

=

s1i 0s2i s3i

θ1iθ2i

The model is now, with θ1i, θ2i, eij all N(0, 1)

yij = x′ijβ + συiθ1i + σεijeij

BS std dev συi = exp

1

2u′

iα

WS std dev σεij = exp

1

2

[

exp(w′ijτ + s2iθ1i + s3iθ2i)

]

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• E(yij) = x′ijβ

• V (yij) = exp(u′iα) + exp

w′

ijτ + 12σ

2ω

BS variance WS variance

• C(yij, yij′) = σ2υi

= exp(

u′iα

)

for j 6= j′

• rij =exp(u′

iα)exp(u′

iα)+ exp(

w′ijτ+1

2σ2ω

)

⇒ ICC varies as a function of BS covariates (α), WS covariates (τ ),and variance of random scale effects (σ2

ω)

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Estimation

Model for the ni × 1 vector of responses, yi, of subject i

yi = Xiβ + 1is1iθ1i + exp

1

2[W iτ + 1is2iθ1i + 1is3iθ2i]

ei

The marginal density of yi in the population

h(yi) =∫

θ f(yi | θi) g(θ) dθ

The marginal log-likelihood from the sample of N subjects

log L =N∑

ilog h(yi)

⇒ can be solved using SAS PROC NLMIXED

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PROC NLMIXED GCONV=1e-12;

PARMS b0=.25 b1=-.5 b2=.3 lnv1=1 v2=.05 c12=0

alp1=0 lnve=1 tau1=0 tau2=0;

z = b0 + b1*x1 + b2*x2 + u1;

v1 = EXP(lnv1 + x2*alp1);

ve = EXP(lnve + x1*tau1 + x2*tau2 + u2);

MODEL y ∼ NORMAL(z,ve);

RANDOM u1 u2 ∼ NORMAL([0,0], [v1,c12,v2]) SUBJECT=id;

RUN;

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Simulation Study1000 datasets of size N=200 and ni between 10 and 30covariates x1ij and x2i as standard normals

true value estimate bias coverageIntercept β0 0.0 0.00071 0.00071 0.950x1ij effect β1 0.1 0.09928 -0.00073 0.945x2i effect β2 0.2 0.20078 0.00078 0.952WS variance (ln) τ0 0.2 0.19949 -0.00051 0.953BS variance (ln) α0 0.2 0.18820 -0.01180 0.938

Intercept β0 0.0 -0.00066 -0.00066 0.950x1ij effect β1 0.1 0.09802 -0.00198 0.954x2i effect β2 0.2 0.20147 0.00147 0.964WS variance (ln) τ0 0.2 0.19698 -0.00303 0.972BS variance (ln) α0 0.2 0.19008 -0.00992 0.911BS var of WS var σ2

ω 0.0 0.00462 0.00462 0.986covariance συω 0.0 0.00117 0.00117 0.966

latter model converges 562 of 1000LR rejection rate = .0445 of true model in favor of latter (2 df, 1-tailed .05)

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true value estimate bias coverageIntercept β0 0.00 -0.00005 -0.00005 0.948x1ij effect β1 0.10 0.09914 -0.00086 0.953x2i effect β2 0.20 0.20032 0.00032 0.944WS variance (ln) τ0 0.20 0.19826 -0.00175 0.951WS x1ij effect τ1 0.10 0.09915 -0.00085 0.944WS x2i effect τ2 0.20 0.19892 -0.00108 0.941BS variance (ln) α0 0.20 0.17807 -0.02193 0.939BS x2i effect α1 0.20 0.19385 -0.00615 0.954BS var of WS var σ2

ω 0.25 0.22853 -0.02148 0.859covariance συω 0.20 0.19853 -0.00147 0.946

Intercept β0 0.00 0.00066 0.00066 0.948x1ij effect β1 0.10 0.09944 -0.00056 0.946x2i effect β2 0.20 0.20082 0.00082 0.947WS variance (ln) τ0 0.20 0.34314 0.14314 0.018BS variance (ln) α0 0.20 0.20649 0.00649 0.937

first model converged 987 of 1000, average model deviance = 12960latter model converged all 1000, average model deviance = 13298

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Ecological Momentary Assessment (EMA) Study ofAdolescent Smokers (Mermelstein)

• 461 adolescents (9th and 10th graders); former and currentsmoking experimenters, and regular smokers

• Carry PDA for a week, answer questions when prompted

average = 30 answered prompts (range = 7 to 71)

• ∑Ni ni = 14, 105 total number of observations

Outcomes: positive and negative affect

Interest: characterizing determinants of affect level, as well as BSand WS affect heterogeneity

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Dependent Variables

• Positive Affect mood scale (mean=6.797 and sd=1.935)

– Before signal: I felt Happy

– Before signal: I felt Relaxed

– Before signal: I felt Cheerful

– Before signal: I felt Confident

– Before signal: I felt Accepted by Others

• Negative Affect mood scale (mean=3.455 and sd=2.253)

– Before signal: I felt Sad

– Before signal: I felt Stressed

– Before signal: I felt Angry

– Before signal: I felt Frustrated

– Before signal: I felt Irritable

⇒ items rated on 1 (not al all) to 10 (very much) scale

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Positive and Negative Affect - ML ests and std errs

Positive Affect Negative Affectparameter estimate se estimate se

β0 6.779 .058 3.482 .071

WS variance τ0 .622 .036 .741 .047

BS var of location α0 (ln var of υi) .367 .069 .793 .069

BS variance of scale σ2ω .518 .039 .963 .069

covariance συ ω -.386 .048 .765 .080

BS variance = exp(α0) 1.443 2.210

WS variance = exp(τ0 + .5σ2ω) 2.413 3.396

ICC .374 .394

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Correlation of Empirical Bayes PA and NA estimates

• mean estimates (υi) are correlated at -.573⇒ subjects higher on PA are lower on NA

• scale estimates (ωi) are correlated at .641⇒ subjects consistent on PA are also consistent on NA, or⇒ subjects inconsistent on PA are also inconsistent on NA

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Observed vs model-based subject means of PA

yi = observed mean of responses for subject i

yi = model-based mean for subject i ( = β0 + υi)

mean std devsample N yi yi yi yi corr slope

overall 461 6.78 6.78 1.24 1.18 .999 .95

ni ≤ 15 21 6.63 6.63 1.04 .90 .994 .86

ni ≥ 43 17 6.50 6.49 1.39 1.35 .999 .97

erratic 13 5.63 5.66 .92 .74 .998 .80

consistent 16 8.36 8.36 1.11 1.10 .999 .99

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Observed vs model-based subject sds of PA

Syi = observed std dev of responses for subject i

σyi = model-based std dev for subject i(

=√

exp[τ0 + ωi])

mean std devsample N Syi σyi Syi σyi corr slope

overall 461 1.44 1.40 .51 .44 .995 .86

ni ≤ 15 21 1.49 1.41 .62 .43 .992 .69

ni ≥ 43 17 1.56 1.52 .62 .55 .999 .89

erratic 13 2.71 2.48 .21 .19 .890 .81

consistent 16 .49 .55 .08 .08 .949 .95

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mean std dev min maxSubject-level independent variablesSmoker .508 .500 0 1NovSeek 2.52 .654 0 4NegMoodReg 2.46 .680 0 4Male .449 .498 0 1Grade10 .527 .500 0 1AloneBS .517 .196 .024 .950

Prompt-level independent variablesAloneWS 0 .461 -.950 .976Tuesday .143 .350 0 1Wednesday .144 .351 0 1Thursday .145 .352 0 1Friday .145 .352 0 1Saturday .136 .343 0 1Sunday .145 .352 0 1

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Model of Positive Affect, ML estimates (standard errors)

mean BS variance WS varianceIntercept 5.861 ∗∗∗ .627 .535 ∗

(.318) (.371) (.210)Smoker -.121 .017 .072

(.101) (.124) (.069)NovSeek .054 -.261∗∗ .130 ∗

(.081) (.092) (.053)NegMoodReg .585 ∗∗∗ -.086 -.167 ∗∗

(.077) (.098) (.052)Male .187 -.133 -.233 ∗∗

(.106) (.130) (.073)Grade10 -.014 -.290∗ -.151∗

(.103) (.122) (.069)AloneBS -1.316 ∗∗∗ 1.103∗∗∗ .343

(.268) (.312) (.180)∗∗∗ = p < .001, ∗∗ = p < .01, ∗ = p < .05

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mean BS variance WS varianceAloneWS -.364 ∗∗∗ .070 ∗

(.023) (.028)Tuesday -.036 .039

(.036) (.050)Wednesday -.065 .137 ∗∗

(.038) (.050)Thursday -.096 ∗ .227 ∗∗∗

(.039) (.050)Friday .002 .259 ∗∗∗

(.039) (.050)Saturday .174 ∗∗∗ .152 ∗∗

(.038) (.050)Sunday .149 ∗∗∗ -.017

(.036) (.050)

Random WS variation σ2ω .461 ∗∗∗

(.036)Random WS covariance συω -.306 ∗∗∗

(.040)∗∗∗ = p < .001, ∗∗ = p < .01, ∗ = p < .05

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Model of Negative Affect, ML estimates (standard errors)

mean BS variance WS varianceIntercept 4.382 ∗∗∗ 1.244∗∗∗ .617 ∗

(.383) (.359) (.269)Smoker .242 .120 .211 ∗

(.124) (.119) (.088)NovSeek .190 -.145 .222 ∗∗

(.097) (.087) (.068)NegMoodReg -.765 ∗∗∗ -.245 ∗ -.277 ∗∗∗

(.095) (.095) (.067)Male -.366 ∗∗ -.217 -.375 ∗∗∗

(.130) (.126) (.094)Grade10 .094 .020 -.074

(.124) (.120) (.089)AloneBS .848 ∗∗ .542 .415

(.321) (.306) (.231)∗∗∗ = p < .001, ∗∗ = p < .01, ∗ = p < .05

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mean BS variance WS varianceAloneWS .173 ∗∗∗ .070 ∗

(.021) (.029)Tuesday .058 .037

(.031) (.051)Wednesday .127 ∗∗∗ .098

(.032) (.051)Thursday .123 ∗∗∗ .180 ∗∗∗

(.032) (.051)Friday .083 ∗ .249 ∗∗∗

(.032) (.051)Saturday -.022 .264 ∗∗∗

(.033) (.053)Sunday -.037 .027

(.030) (.051)

Random WS variation σ2ω .812 ∗∗∗

(.059)Random WS covariance συω .527 ∗∗∗

(.061)∗∗∗ = p < .001, ∗∗ = p < .01, ∗ = p < .05

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Consistent Results for both PA and NA

• meanPA : positive effects (NegMoodReg)PA : negative effects (AloneBS and WS, Thursday)reversed for NA

• WS variancepositive effects (NovSeek, AloneWS, Thursday-Saturday)negative effects (NegMoodReg, Male)

• BS variance (none signif. on both; several with same direction)positive effects (AloneBS)negative effects (NegMoodReg, NovSeek)

• Highly significant BS variance of WS variation (scale)

• Highly significant covariance of random effects (opposite sign forPA and NA)

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Summary

• More applications for this class of models

• Only single random location effect considered here, but this couldbe generalized (e.g., random intercept and trend model)

• Other kinds of outcomes, especially ordinal

• Need a fair amount of BS and WS data, but modern datacollection procedures are good for this

• Simulations with small datasets (e.g., 20 subjects with 5observations) often leads to non-convergence; this improves asnumbers increase

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