1 - University of Illinois at Chicago
An Application of a Mixed-Effects Location Scale Model for Analysis of Ecological Momentary Assessment (EMA) Data Don Hedeker, Robin Mermelstein, & Hakan Demirtas University of Illinois at Chicago [email protected]
Biometrics, in press Supported by National Cancer Institute grant 5PO1 CA98262 (Mermelstein, PI)
1
Ecological Momentary Assessment (EMA) data aka experience sampling and diary methods • Subjects provide frequent reports on events and experiences of their daily lives (e.g., 30-40 responses per subject collected over the 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 more traditional designs e.g., allow investigation of phenomena as they happen over time • Reports could be time-based, following a fixed-schedule, randomly triggered, event-triggered 2
Data are rich and offer many modeling possibilities! • person-level and occasion-level determinants of occasion-level responses ⇒ potential influence of context and/or environment e.g., subject response might vary when alone vs with others • allows examination of why subjects differ in variability rather than just mean level – between-subjects variance e.g., subject heterogeneity could vary by gender or age – within-subjects variance e.g., subject degree of stability could vary by gender or age
3
3
Ecological Momentary Assessment (EMA) Study of Adolescent Smokers (Mermelstein) • 461 adolescents (9th and 10th graders); former and current smoking experimenters, and regular smokers • Carry PDA for a week, answer questions when prompted average = 30 answered prompts (range = 7 to 71) • PN i ni = 14, 105 total number of observations Outcomes: positive and negative affect Interest: characterizing determinants of affect level, as well as BS and WS affect heterogeneity 4
Mixed-effects regression model for measurement y of subject i (i = 1, 2, . . . , N ) on occasion j (j = 1, 2, . . . , ni) yij = x0ij β + υi + ij xij = p × 1 vector of regressors (including a column of ones) β = p × 1 vector of regression coefficients υi ∼ N (0, συ2 ) BS variance ij ∼ N (0, σ2) WS variance
5
Log-linear models for variances BS variance
WS variance
συ2i = exp(u0iα) σ2ij = exp(w0ij τ )
or log(συ2i ) = u0iα or log(σ2ij ) = w0ij τ
• ui and wij include covariates (and 1) • subscripts i and j on variances indicate that these change depending 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 so resulting variances are positive 6
WS variance varies across subjects σ2ij = exp(w0ij τ + ωi)
where
ωi ∼ N (0, σω2 )
log(σ2ij ) = w0ij τ + ωi • ωi are log-normal subject-specific perturbations of WS variance • ωi are “scale” random effects - how does a subject differ in terms of the variation in their data • υi are “location” random effects - how does a subject differ in terms of the mean of their data
7
Literature on Random Scale Effects Summary Cleveland, Denby, & Liu (2002), Bell Labs technical report Frequentist Chinchilli, Esinhart, & Miller (1995), Biometrics James, Venables, Dry, & Wiskich (1994), Biometrika Lin, Raz, & Harlow (1997), Biometrics Bayesian Leonard (1975), Technometrics Myles, Price, Hunter, Day, & Duffy (2003), Public Health Nutrition • Most use (square root) inverse gamma for random scale distribution (for computational reasons) • Do not allow random location and scale effects to be correlated 8
Location random effects for two subjects 9
Location and scale random effects for two subjects 10
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 11
Standardize the random effects via the Cholesky factorization
υi = ωi
συi 0v u 2 /σ 2 συω /συi utσω2 − συω υi
θ1i = θ2i
s1i 0 θ1i s2i s3i θ2i
The model is now, with θ1i, θ2i, eij all N (0, 1) yij = x0ij β + συi θ1i + σij eij
BS std dev
1 0 συi = exp uiα 2
WS std dev
σij
1 0 = exp exp(wij τ + s2iθ1i + s3iθ2i) 2
"
12
#
• E(yij ) = x0ij β
• V (yij ) = exp(u0iα) + exp w0ij τ + 12 σω2 BS variance WS variance !
• C(yij , yij 0 ) = συ2i = exp u0iα
for j 6= j 0
exp(u0iα) ! • rij = 1 0 0 exp(uiα) + exp w ij τ + 2 σω2
⇒ ICC varies as a function of BS covariates (α), WS covariates (τ ), and variance of random scale effects (σω2 )
13
Estimation Model for the ni × 1 vector of responses, yi, of subject i
1 y i = X iβ + 1is1iθ1i + exp [W iτ + 1is2iθ1i + 1is3iθ2i] ei 2
The marginal density of y i in the population Z
h(y i) = θ f (yi | θi) g(θ) dθ The marginal log-likelihood from the sample of N subjects log L =
N X i
log h(y i)
⇒ can be solved using SAS PROC NLMIXED 14
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;
15
Simulation Study 1000 datasets of size N =200 and ni between 10 and 30 covariates x1ij and x2i as standard normals Intercept β0 x1ij effect β1 x2i effect β2 WS variance (ln) τ0 BS variance (ln) α0
true value 0.0 0.1 0.2 0.2 0.2
estimate 0.00071 0.09928 0.20078 0.19949 0.18820
bias 0.00071 -0.00073 0.00078 -0.00051 -0.01180
coverage 0.950 0.945 0.952 0.953 0.938
Intercept β0 x1ij effect β1 x2i effect β2 WS variance (ln) τ0 BS variance (ln) α0 BS var of WS var σω2 covariance συω
0.0 0.1 0.2 0.2 0.2 0.0 0.0
-0.00066 0.09802 0.20147 0.19698 0.19008 0.00462 0.00117
-0.00066 -0.00198 0.00147 -0.00303 -0.00992 0.00462 0.00117
0.950 0.954 0.964 0.972 0.911 0.986 0.966
latter model converges 562 of 1000 LR rejection rate = .0445 of true model in favor of latter (2 df, 1-tailed .05) 16
Intercept β0 x1ij effect β1 x2i effect β2 WS variance (ln) τ0 WS x1ij effect τ1 WS x2i effect τ2 BS variance (ln) α0 BS x2i effect α1 BS var of WS var σω2 covariance συω
true value 0.00 0.10 0.20 0.20 0.10 0.20 0.20 0.20 0.25 0.20
estimate -0.00005 0.09914 0.20032 0.19826 0.09915 0.19892 0.17807 0.19385 0.22853 0.19853
bias -0.00005 -0.00086 0.00032 -0.00175 -0.00085 -0.00108 -0.02193 -0.00615 -0.02148 -0.00147
coverage 0.948 0.953 0.944 0.951 0.944 0.941 0.939 0.954 0.859 0.946
Intercept β0 x1ij effect β1 x2i effect β2 WS variance (ln) τ0 BS variance (ln) α0
0.00 0.10 0.20 0.20 0.20
0.00066 0.09944 0.20082 0.34314 0.20649
0.00066 -0.00056 0.00082 0.14314 0.00649
0.948 0.946 0.947 0.018 0.937
first model converged 987 of 1000, average model deviance = 12960 latter model converged all 1000, average model deviance = 13298
17
Ecological Momentary Assessment (EMA) Study of Adolescent Smokers (Mermelstein) • 461 adolescents (9th and 10th graders); former and current smoking experimenters, and regular smokers • Carry PDA for a week, answer questions when prompted average = 30 answered prompts (range = 7 to 71) • PN i ni = 14, 105 total number of observations Outcomes: positive and negative affect Interest: characterizing determinants of affect level, as well as BS and WS affect heterogeneity 18
Dependent Variables • Positive Affect mood scale (mean=6.797 and sd=1.935) – Before – Before – Before – Before – Before
signal: signal: signal: signal: signal:
I I I I I
felt felt felt felt felt
Happy Relaxed Cheerful Confident Accepted by Others
• Negative Affect mood scale (mean=3.455 and sd=2.253) – Before – Before – Before – Before – Before
signal: signal: signal: signal: signal:
I I I I I
felt felt felt felt felt
Sad Stressed Angry Frustrated Irritable
⇒ items rated on 1 (not al all) to 10 (very much) scale 19
Positive and Negative Affect - ML ests and std errs Positive Affect Negative Affect estimate se estimate se
parameter β0 WS variance τ0
6.779 .058 .622 .036
3.482 .741
.071 .047
.367 .069
.793
.069
.518 .039
.963
.069
covariance συ ω
-.386 .048
.765
.080
BS variance = exp(α0)
1.443
2.210
WS variance = exp(τ0 + .5σω2 ) ICC
2.413 .374
3.396 .394
BS var of location α0 (ln var of υi) BS variance of scale σω2
20
Correlation of Empirical Bayes PA and NA estimates • mean estimates (ˆ υi) are correlated at -.573 ⇒ subjects higher on PA are lower on NA • scale estimates (ˆ ωi) are correlated at .641 ⇒ subjects consistent on PA are also consistent on NA, or ⇒ subjects inconsistent on PA are also inconsistent on NA
21
Observed vs model-based subject means of PA y¯i = observed mean of responses for subject i yˆi = model-based mean for subject i ( = βˆ0 + υˆi) mean y¯i yˆi
std dev y¯i yˆi
sample
N
overall
461
6.78
6.78
1.24
ni ≤ 15 ni ≥ 43
21 17
6.63 6.50
6.63 6.49
erratic consistent
13 16
5.63 8.36
5.66 8.36
22
corr
slope
1.18
.999
.95
1.04 1.39
.90 1.35
.994 .999
.86 .97
.92 1.11
.74 1.10
.998 .999
.80 .99
Observed vs model-based subject sds of PA Syi = observed std dev of responses for subject i σˆyi = model-based std dev for subject i
s
!
= exp[ˆ τ0 + ω ˆ i]
sample
N
mean Syi σˆyi
overall
461
1.44 1.40
std dev Syi σˆyi .51 .44
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
.08
.08
.949
.95
.49
.55
23
corr
slope
.995
.86
mean std dev Subject-level independent variables Smoker .508 .500 NovSeek 2.52 .654 NegMoodReg 2.46 .680 Male .449 .498 Grade10 .527 .500 AloneBS .517 .196
min
max
0 0 0 0 0 .024
1 4 4 1 1 .950
Prompt-level independent variables AloneWS 0 .461 Tuesday .143 .350 Wednesday .144 .351 Thursday .145 .352 Friday .145 .352 Saturday .136 .343 Sunday .145 .352
-.950 0 0 0 0 0 0
.976 1 1 1 1 1 1
24
Model of Positive Affect, ML estimates (standard errors) mean BS variance WS variance Intercept 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
25
mean AloneWS -.364 ∗∗∗ (.023) Tuesday -.036 (.036) Wednesday -.065 (.038) Thursday -.096 ∗ (.039) Friday .002 (.039) Saturday .174 ∗∗∗ (.038) Sunday .149 ∗∗∗ (.036) Random WS variation σω2
BS variance WS variance .070 (.028) .039 (.050) .137 (.050) .227 (.050) .259 (.050) .152 (.050) -.017 (.050) .461 (.036) Random WS covariance συω -.306 (.040) ∗∗∗ = p < .001, ∗∗ = p < .01, ∗ = p < .05 26
∗
∗∗ ∗∗∗ ∗∗∗ ∗∗
∗∗∗ ∗∗∗
Model of Negative Affect, ML estimates (standard errors) mean BS variance WS variance Intercept 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
27
∗ ∗ ∗∗ ∗∗∗ ∗∗∗
mean AloneWS .173 ∗∗∗ (.021) Tuesday .058 (.031) Wednesday .127 ∗∗∗ (.032) Thursday .123 ∗∗∗ (.032) Friday .083 ∗ (.032) Saturday -.022 (.033) Sunday -.037 (.030) Random WS variation σω2
BS variance WS variance .070 (.029) .037 (.051) .098 (.051) .180 (.051) .249 (.051) .264 (.053) .027 (.051) .812 (.059) Random WS covariance συω .527 (.061) ∗∗∗ = p < .001, ∗∗ = p < .01, ∗ = p < .05 28
∗
∗∗∗ ∗∗∗ ∗∗∗
∗∗∗ ∗∗∗
Consistent Results for both PA and NA • mean PA : positive effects (NegMoodReg) PA : negative effects (AloneBS and WS, Thursday) reversed for NA • WS variance positive 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 for PA and NA) 29
Summary • More applications for this class of models • Only single random location effect considered here, but this could be 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 data collection procedures are good for this • Simulations with small datasets (e.g., 20 subjects with 5 observations) often leads to non-convergence; this improves as numbers increase
30
Biometrics, in press Supported by National Cancer Institute grant 5PO1 CA98262 (Mermelstein, PI)
1
Ecological Momentary Assessment (EMA) data aka experience sampling and diary methods • Subjects provide frequent reports on events and experiences of their daily lives (e.g., 30-40 responses per subject collected over the 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 more traditional designs e.g., allow investigation of phenomena as they happen over time • Reports could be time-based, following a fixed-schedule, randomly triggered, event-triggered 2
Data are rich and offer many modeling possibilities! • person-level and occasion-level determinants of occasion-level responses ⇒ potential influence of context and/or environment e.g., subject response might vary when alone vs with others • allows examination of why subjects differ in variability rather than just mean level – between-subjects variance e.g., subject heterogeneity could vary by gender or age – within-subjects variance e.g., subject degree of stability could vary by gender or age
3
3
Ecological Momentary Assessment (EMA) Study of Adolescent Smokers (Mermelstein) • 461 adolescents (9th and 10th graders); former and current smoking experimenters, and regular smokers • Carry PDA for a week, answer questions when prompted average = 30 answered prompts (range = 7 to 71) • PN i ni = 14, 105 total number of observations Outcomes: positive and negative affect Interest: characterizing determinants of affect level, as well as BS and WS affect heterogeneity 4
Mixed-effects regression model for measurement y of subject i (i = 1, 2, . . . , N ) on occasion j (j = 1, 2, . . . , ni) yij = x0ij β + υi + ij xij = p × 1 vector of regressors (including a column of ones) β = p × 1 vector of regression coefficients υi ∼ N (0, συ2 ) BS variance ij ∼ N (0, σ2) WS variance
5
Log-linear models for variances BS variance
WS variance
συ2i = exp(u0iα) σ2ij = exp(w0ij τ )
or log(συ2i ) = u0iα or log(σ2ij ) = w0ij τ
• ui and wij include covariates (and 1) • subscripts i and j on variances indicate that these change depending 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 so resulting variances are positive 6
WS variance varies across subjects σ2ij = exp(w0ij τ + ωi)
where
ωi ∼ N (0, σω2 )
log(σ2ij ) = w0ij τ + ωi • ωi are log-normal subject-specific perturbations of WS variance • ωi are “scale” random effects - how does a subject differ in terms of the variation in their data • υi are “location” random effects - how does a subject differ in terms of the mean of their data
7
Literature on Random Scale Effects Summary Cleveland, Denby, & Liu (2002), Bell Labs technical report Frequentist Chinchilli, Esinhart, & Miller (1995), Biometrics James, Venables, Dry, & Wiskich (1994), Biometrika Lin, Raz, & Harlow (1997), Biometrics Bayesian Leonard (1975), Technometrics Myles, Price, Hunter, Day, & Duffy (2003), Public Health Nutrition • Most use (square root) inverse gamma for random scale distribution (for computational reasons) • Do not allow random location and scale effects to be correlated 8
Location random effects for two subjects 9
Location and scale random effects for two subjects 10
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 11
Standardize the random effects via the Cholesky factorization
υi = ωi
συi 0v u 2 /σ 2 συω /συi utσω2 − συω υi
θ1i = θ2i
s1i 0 θ1i s2i s3i θ2i
The model is now, with θ1i, θ2i, eij all N (0, 1) yij = x0ij β + συi θ1i + σij eij
BS std dev
1 0 συi = exp uiα 2
WS std dev
σij
1 0 = exp exp(wij τ + s2iθ1i + s3iθ2i) 2
"
12
#
• E(yij ) = x0ij β
• V (yij ) = exp(u0iα) + exp w0ij τ + 12 σω2 BS variance WS variance !
• C(yij , yij 0 ) = συ2i = exp u0iα
for j 6= j 0
exp(u0iα) ! • rij = 1 0 0 exp(uiα) + exp w ij τ + 2 σω2
⇒ ICC varies as a function of BS covariates (α), WS covariates (τ ), and variance of random scale effects (σω2 )
13
Estimation Model for the ni × 1 vector of responses, yi, of subject i
1 y i = X iβ + 1is1iθ1i + exp [W iτ + 1is2iθ1i + 1is3iθ2i] ei 2
The marginal density of y i in the population Z
h(y i) = θ f (yi | θi) g(θ) dθ The marginal log-likelihood from the sample of N subjects log L =
N X i
log h(y i)
⇒ can be solved using SAS PROC NLMIXED 14
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;
15
Simulation Study 1000 datasets of size N =200 and ni between 10 and 30 covariates x1ij and x2i as standard normals Intercept β0 x1ij effect β1 x2i effect β2 WS variance (ln) τ0 BS variance (ln) α0
true value 0.0 0.1 0.2 0.2 0.2
estimate 0.00071 0.09928 0.20078 0.19949 0.18820
bias 0.00071 -0.00073 0.00078 -0.00051 -0.01180
coverage 0.950 0.945 0.952 0.953 0.938
Intercept β0 x1ij effect β1 x2i effect β2 WS variance (ln) τ0 BS variance (ln) α0 BS var of WS var σω2 covariance συω
0.0 0.1 0.2 0.2 0.2 0.0 0.0
-0.00066 0.09802 0.20147 0.19698 0.19008 0.00462 0.00117
-0.00066 -0.00198 0.00147 -0.00303 -0.00992 0.00462 0.00117
0.950 0.954 0.964 0.972 0.911 0.986 0.966
latter model converges 562 of 1000 LR rejection rate = .0445 of true model in favor of latter (2 df, 1-tailed .05) 16
Intercept β0 x1ij effect β1 x2i effect β2 WS variance (ln) τ0 WS x1ij effect τ1 WS x2i effect τ2 BS variance (ln) α0 BS x2i effect α1 BS var of WS var σω2 covariance συω
true value 0.00 0.10 0.20 0.20 0.10 0.20 0.20 0.20 0.25 0.20
estimate -0.00005 0.09914 0.20032 0.19826 0.09915 0.19892 0.17807 0.19385 0.22853 0.19853
bias -0.00005 -0.00086 0.00032 -0.00175 -0.00085 -0.00108 -0.02193 -0.00615 -0.02148 -0.00147
coverage 0.948 0.953 0.944 0.951 0.944 0.941 0.939 0.954 0.859 0.946
Intercept β0 x1ij effect β1 x2i effect β2 WS variance (ln) τ0 BS variance (ln) α0
0.00 0.10 0.20 0.20 0.20
0.00066 0.09944 0.20082 0.34314 0.20649
0.00066 -0.00056 0.00082 0.14314 0.00649
0.948 0.946 0.947 0.018 0.937
first model converged 987 of 1000, average model deviance = 12960 latter model converged all 1000, average model deviance = 13298
17
Ecological Momentary Assessment (EMA) Study of Adolescent Smokers (Mermelstein) • 461 adolescents (9th and 10th graders); former and current smoking experimenters, and regular smokers • Carry PDA for a week, answer questions when prompted average = 30 answered prompts (range = 7 to 71) • PN i ni = 14, 105 total number of observations Outcomes: positive and negative affect Interest: characterizing determinants of affect level, as well as BS and WS affect heterogeneity 18
Dependent Variables • Positive Affect mood scale (mean=6.797 and sd=1.935) – Before – Before – Before – Before – Before
signal: signal: signal: signal: signal:
I I I I I
felt felt felt felt felt
Happy Relaxed Cheerful Confident Accepted by Others
• Negative Affect mood scale (mean=3.455 and sd=2.253) – Before – Before – Before – Before – Before
signal: signal: signal: signal: signal:
I I I I I
felt felt felt felt felt
Sad Stressed Angry Frustrated Irritable
⇒ items rated on 1 (not al all) to 10 (very much) scale 19
Positive and Negative Affect - ML ests and std errs Positive Affect Negative Affect estimate se estimate se
parameter β0 WS variance τ0
6.779 .058 .622 .036
3.482 .741
.071 .047
.367 .069
.793
.069
.518 .039
.963
.069
covariance συ ω
-.386 .048
.765
.080
BS variance = exp(α0)
1.443
2.210
WS variance = exp(τ0 + .5σω2 ) ICC
2.413 .374
3.396 .394
BS var of location α0 (ln var of υi) BS variance of scale σω2
20
Correlation of Empirical Bayes PA and NA estimates • mean estimates (ˆ υi) are correlated at -.573 ⇒ subjects higher on PA are lower on NA • scale estimates (ˆ ωi) are correlated at .641 ⇒ subjects consistent on PA are also consistent on NA, or ⇒ subjects inconsistent on PA are also inconsistent on NA
21
Observed vs model-based subject means of PA y¯i = observed mean of responses for subject i yˆi = model-based mean for subject i ( = βˆ0 + υˆi) mean y¯i yˆi
std dev y¯i yˆi
sample
N
overall
461
6.78
6.78
1.24
ni ≤ 15 ni ≥ 43
21 17
6.63 6.50
6.63 6.49
erratic consistent
13 16
5.63 8.36
5.66 8.36
22
corr
slope
1.18
.999
.95
1.04 1.39
.90 1.35
.994 .999
.86 .97
.92 1.11
.74 1.10
.998 .999
.80 .99
Observed vs model-based subject sds of PA Syi = observed std dev of responses for subject i σˆyi = model-based std dev for subject i
s
!
= exp[ˆ τ0 + ω ˆ i]
sample
N
mean Syi σˆyi
overall
461
1.44 1.40
std dev Syi σˆyi .51 .44
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
.08
.08
.949
.95
.49
.55
23
corr
slope
.995
.86
mean std dev Subject-level independent variables Smoker .508 .500 NovSeek 2.52 .654 NegMoodReg 2.46 .680 Male .449 .498 Grade10 .527 .500 AloneBS .517 .196
min
max
0 0 0 0 0 .024
1 4 4 1 1 .950
Prompt-level independent variables AloneWS 0 .461 Tuesday .143 .350 Wednesday .144 .351 Thursday .145 .352 Friday .145 .352 Saturday .136 .343 Sunday .145 .352
-.950 0 0 0 0 0 0
.976 1 1 1 1 1 1
24
Model of Positive Affect, ML estimates (standard errors) mean BS variance WS variance Intercept 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
25
mean AloneWS -.364 ∗∗∗ (.023) Tuesday -.036 (.036) Wednesday -.065 (.038) Thursday -.096 ∗ (.039) Friday .002 (.039) Saturday .174 ∗∗∗ (.038) Sunday .149 ∗∗∗ (.036) Random WS variation σω2
BS variance WS variance .070 (.028) .039 (.050) .137 (.050) .227 (.050) .259 (.050) .152 (.050) -.017 (.050) .461 (.036) Random WS covariance συω -.306 (.040) ∗∗∗ = p < .001, ∗∗ = p < .01, ∗ = p < .05 26
∗
∗∗ ∗∗∗ ∗∗∗ ∗∗
∗∗∗ ∗∗∗
Model of Negative Affect, ML estimates (standard errors) mean BS variance WS variance Intercept 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 AloneWS .173 ∗∗∗ (.021) Tuesday .058 (.031) Wednesday .127 ∗∗∗ (.032) Thursday .123 ∗∗∗ (.032) Friday .083 ∗ (.032) Saturday -.022 (.033) Sunday -.037 (.030) Random WS variation σω2
BS variance WS variance .070 (.029) .037 (.051) .098 (.051) .180 (.051) .249 (.051) .264 (.053) .027 (.051) .812 (.059) Random WS covariance συω .527 (.061) ∗∗∗ = p < .001, ∗∗ = p < .01, ∗ = p < .05 28
∗
∗∗∗ ∗∗∗ ∗∗∗
∗∗∗ ∗∗∗
Consistent Results for both PA and NA • mean PA : positive effects (NegMoodReg) PA : negative effects (AloneBS and WS, Thursday) reversed for NA • WS variance positive 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 for PA and NA) 29
Summary • More applications for this class of models • Only single random location effect considered here, but this could be 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 data collection procedures are good for this • Simulations with small datasets (e.g., 20 subjects with 5 observations) often leads to non-convergence; this improves as numbers increase
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