Subject Index
A
from conventional, baseline, and survivor approaches, 240
fixed-effect estimates, 214
heterogeneous patterns, 237
longitudinal trajectory, 213
marginal mean, 239
model-based predictions, 237
pattern of change over time, 241
population-averaged, 434
predicted values for, 228
repeated measurements, 428
six patterns/six time points, 238
standard errors, 238
time plots
of adjusted, 88
two-step parametric/nonparametric mixed models, analytic results of, 468
applied to compute fixed and random effects for, 319
default integration method, 330
empirical Bayes estimate, 262
integral approximation, 263
integral of likelihood over random effects, 328
log-likelihood function, evaluation, 261
recommended approach, 263
ADL trajectory curve, 470
Age-specific transition probabilities, 409
AHEAD baseline survey, 90
AHEAD cohort, 468
analyses of, 222
survivors, percent distribution of, 239
output, 485
pattern-mixture modeling to analyze, 234
sample size of, 198
TOEP covariance pattern model, 163
American Psychological Association (APA) Task Force on Statistical Inference, 27
classical model, 7
mean square statistics, 43
null hypothesis, 37
empirical illustration, PCL revisited, acupuncture treatment, 45
two-factor, 42
Approximation method, empirical application of, 394
Associated missing-data indicator random vector, 460
B
Baseline mixture, 241
Bayes’ rule in, 97
probability model, 98
REML estimator, 104
to specify marginal mean of response probability, 319
Bayesian methods, 99
Bayes minimization, 263
approximation methods, 4
complex simulating procedures, 298
of between-subjects random effects, 363
expression, 105
shrinkage estimates, 115
statistic, 423
theory, 447
BDI-II score, See Beck Depression Inventory-II (BDI-II) score
Beck Depression Inventory-II (BDI-II) score, 53
approximation
empirical Bayes, 269
in linear mixed models, 267
covariance estimator, 270
estimation, 114
generalized least squares estimator, 112
linear random coefficient model, 115
model-based least squares, 217
nonlinear predictions, 116
reduced-form empirical approach, 331
retransformation method, 376
variance–covariance matrix, 112
Between-sample variations, 38
Between-subjects
random error, 317
variability, 298
Binary data, pair-wise correlations, 293
Binary longitudinal data
conditional effect, computation, 323
conventional logistic, overview of, 310
empirical illustration
analytic plan, 328
graphical results, 339
marital status on probability of disability among older Americans, 327
nonlinear predictions, 335
SAS programs, analytic steps, 328
three logit models, analytic results of, 333
mixed-effects logit model, inference of, 318
random intercept logistic regression model, specification of, 313
Block design, 6
data structure for analysis, 86
Bootstrapping, 393
Borrowing of strength approach, 79
C
Case-by-case base, 13
Categorical response data, longitudinal transition models, 379
Chi-square value, 430
Cholesky decomposition application, on variance–covariance matrix, 176
Classical ANOVA model, 7
CLASS statement, 211
Clinical experimental studies, 26
Coding schemes, for classification factor, 147
Cohen's d, 24
Column vector, 13
Common variance-covariance structure, 237
Compound symmetry (CS), 135
residual covariance structure, 135
variance–covariance structure, 135
Computer programming, 426
Conditional effect, 324
computation, 323
conditional log OR on logit, 326
statistical significance, 408
on probability scale, 325
covariates on, 355
Wald chi-square statistic, 337
Confidence intervals (CI), 35
Conventional logistic, overview of, 310
for covariate, 323
in longitudinal data analysis, 317
Correlation matrix, 287
Corresponding variance–covariance matrix, 425
Count data, 275
Covariance matrices, 417
of parameter estimates, 371
standard GEE procedure, 291
Covariance pattern model, 287
Covariates, 327
Cox model, proportional hazard rate model, 277
Cramér–Rao inequality, 480
Cross-sectional data analysis, 2
Cubit polynomial function, J -shaped time trend, 82
Cumulative standard normal distribution function, 312
D
Data available, at single point of time, 2
Degrees of freedom (df), 39
Department of Defense (DoD), 17
Deployment Health Clinical Center (DHCC), 17
DESCENDING option, 465
DFFITS score, 181
effectiveness of acupuncture treatment on PTSD, 85
longitudinal data of, 451
Dichotomous variables, 146
conditional density function on, 457
model-based prediction of, 462
terms of joint distribution, 461
truncated linear mixed model, 459
Dispersion/scale parameter, 245
Distinctive mixture groups, 231
E
Economic capability, 216
Effect size, 23
EM algorithms, See Expectation-Maximization (EM) algorithms
Empirical-Bayes methods
development of the retransformation method, 272
estimation, 266
Empirical BLUP, 362
Erroneous model-based predictions, 429
Estimator of variances, 247
Euclidean distance, 140
Exogenous latent variables, 413
longitudinal data analysis, 108
random effect, 108
F
Family history of diseases, 216
First order, autoregressive processes, 138
Fixed-effect estimates, 237
Fixed-effects multinomial logit model, 367
Flexible mixed-effects multinomial logit models, 364
Follow-up time point, 9
G
Gamma distribution, 225
Gauss–Hermite abscissas, 263
Gauss–Hermite quadrature, 262
Gaussian distributions methods, 221
random effects, 222
Gauss–Newton method, 286
approaches
advantages of, 285
comparison, 295
working correlation matrix, 287
empirical illustration
marital status on disability severity in older Americans, 299
inferences, 282
logit link, 299
marginal regression models, 295
naïve model with independence hypothesis, 282
procedure, 291
quasi-likelihood information criteria, 288
random-effects models, 295
Zhao method, 291
Generalized least squares (GLS) equation, 111
Generalized linear mixed models (GLMMs), 244
analytic convenience of, 250
between-subjects random effects, 251
BLUP procedure, 267
complexity of data structures, 253
deriving parameter estimates, 261
empirical Bayes BLUP, 268
estimating procedures, 254
GLMs, probability distribution of, 251
hypothesis testing on fixed effects, 255
likelihood functions, 251
linearization, best linear unbiased prediction, 267
linear predictor of, 250
link function, 267
marginal quasi-likelihood method, 258
methods of estimating parameters, 255
adaptive Gaussian quadrature methods, 261
Gaussian quadrature, 261
Laplace method, 259
marginal quasi-likelihood method, 258
Markov chain Monte Carlo methods, 264
penalized quasi-likelihood (PQL) method, 256
mixed-effects logistic regression model, 273
mixed-effects multinomial logit regression models, 275
mixed-effects ordered logistic model, 274
mixed-effects Poisson regression model, 275
mixture of distributions, 255
nonlinear distribution of, 272
nonlinear predictions, 266
nonlinear response, 268
nonnormal distributions, 272
overview of, 244
pseudo-error term, 257
pseudo-response variable, 257
quasi-likelihood functions, 258
random components, retransformation, 266
random effects, 251
parameters, 254
variance-covariance components of, 255
vector, 249
retransformation method, 269
statistical inferences, 248
basic specifications, 249
hypothesis testing
procedures on fixed effects, 253
on variance components, 255
maximization procedures on fixed effects, 253
survival models, 276
variance function, 249
within-subject random errors in nonlinear predictions, 271
within-subject variance, 258
Generalized linear models (GLMs), 243
approach, 284
coding scheme, 149
estimates, 291
independence hypothesis, 248
log-likelihood function, 247
probability distribution, 245
regression coefficient of covariate, 246
statistical expression, 246
statistical inferences, 245
systematic component, 246
user-friendly procedures, 312
variance–covariance matrix for estimates, 248
Genetic predisposition, 216
Glass’s effect, 24
GLS equation, See Generalized least squares (GLS) equation
Group-based model, 422
conditional independence hypothesis, 423
development, 422
individual-level likelihood function, 424
longitudinal sequence of response measurement, 422
multinomial function as, 423
GROUP BY TIME option, 223
H
Handling missing data, 443
Hat-value, 183
Heckman’s classical two-step estimator, 472
Hedges’s d, 25
Hermite polynomials, 262
Hessian of log-likelihood, 247
Heterogeneous linear mixed model, 218
Heterogeneous mixture patterns, classification of, 230
Heterogeneous transition pattern, 386
Household and Retirement Survey (HRS), 18
Hybrid variance–covariance structure, 142
Hypothesis testing, on nonnegative variance, 145
I
IDENTITY option, 55
IIA hypothesis, See Independence from irrelevant alternatives (IIA) hypothesis
Immortal cohort, 234
Independence from irrelevant alternatives (IIA) hypothesis, 346
Individual-level likelihood function, 424
Influence diagnostics, 181
Cook’s distance statistic, 181
empirical illustrations, 190
linear mixed model concerning marital status/disability severity among older Americans, 198
PCL score, acupuncture treatment
linear mixed model, checks, 190
leverage statistic, 183
likelihood displacement statistic approximation, 187
linear mixed model, 173
LMAX statistic, for influential observations identification, 189
INFLUENCE option, 195
Institute for Social Research (ISR), 18
Integral approximation methods, 255
Intraindividual growth patterns, 20
Irregular time trends, 85
L
Laplace approximation, 255
parameters, 260
specifications, 260
Last observation carried forward (LOCF) approach, 444
advantage of, 447
classical method handling missing data, 446
Latent endogenous random variables, 413
group-based model, 422
latent growth modeling, 411
maximum likelihood approach, 420
model covariates, 421
application, 436
assumption of multivariate normality, 418
empirical illustration, marital status effect on ADL count, 425
factor-scoring coefficients, 417
group-based model, 422
linear slope component, 417
model, 419
structural equation modeling, overview of, 412
Latent variable model, 413
covariance matrices, 413
measurement, 413
structural equation model construction, 413
LD statistic, See Likelihood displacement (LD) statistic
Least squares means, 222
Leverage measurement, 183
Likelihood-based method, 79
canonical parameters, 283
for checking the polynomial form of time, 79
ADL count, for three linear mixed models, 201
linear mixed models, 194
Likelihood distance, 192
Likelihood function, 219
Likelihood ratio statistic, 116
asymptotic null distribution of, 116
Gaussian quadrature use of, 263
goodness-of-fit information, 333
log-likelihood function, 77
p -value of, 116
test statistic, 220
Linear discriminant analysis (LDA)
classical approach, 230
random-effects regression models, 230
use of, 230
Linearization-based approaches, 278
Linearization methods, 259
empirical illustrations
applications, 85
baseline score, adjustment, 210
BLUPs vs. least squares means, 222
marital status and disability severity in older Americans, 89
PCL score, acupuncture treatment, 85
fixed effects, inference/estimation of, 73
maximum likelihood methods, 73
missing data, 78
statistical/hypothesis testing, 75
formalization of, 66
variance–covariance components, 71
general specification of, 67
intraindividual correlation, 69
longitudinal data analysis
baseline response, adjustment, 206
Lord’s paradox, 206
one-factor with random intercept, 62
basic theory, 231
empirical illustration of, 234
heterogeneous groups, classification, 229
random effects
assumed distribution, misspecification, 216
in different distributions, 220
heterogeneity linear mixed model, 217
nonnormal random effect distribution, 218
and three covariates, 65
random intercept and random slope, 64
trend analysis, 79
polynomial time functions, 80
numeric checks, 84
reduce collinearity methods, 82
variance–covariance matrix, 69
Linear predictor, 245
Linear random coefficient model, 218
Linear random intercept model, 220
Linear regression estimator, 461
Linear regression model, 14
LMAX approximation, 190
LMAX score, 189
LMAX statistic, 189
LOCF approach, See Last observation carried forward (LOCF) approach
Log-gamma distributed random coefficient model, 220
Log-gamma distributed slopes, 219
Log-gamma linear mixed model, 219
Logistic cumulative distribution function, 274
Logit models, 310, See also Multinomial logit models
analytic results, 334
fixed-effects, 339
Logit regression, 328
expression, complete-data, 421
maximization, 103
ratio statistic, 84
Log-linear probability distribution, 300
Log OR GEE model, 306
Log transformation, 461
Longitudinal clinical controlled trial, 442
Longitudinal courses, 2
ANOVA, repeated measures of, 37
book/data organization, for illustrations, 16
asset and health dynamics among the oldest old (AHEAD), 18
computation of, 27
meta-analysis, 26
defined, 1
empirical illustration, 29
history of, 3
intraindividual correlation, 14
MANOVA, repeated measures of, 47
missing data, patterns and mechanisms, 9
monotone missing pattern, 9
nonmonotone missing data, 9
paired t -test, 21
PTSD symptom, acupuncture treatment effectiveness, 29
randomized controlled clinical trial, 17
structures, See Longitudinal data structures
time plots, of trends, 20
time scale/number of time points, 12
traditional methods, 19
Longitudinal data designs, 2
applications, statistical techniques, 4
risk factors over time points, 2
unbalanced, 8
Longitudinal data structures, 4
balanced/unbalanced, 7
multivariate data, 5
univariate data, 6
Longitudinal modeling, basic expressions of, 13
Longitudinal processes, sources of correlation, 10
Longitudinal regression models, on nonignorable missing data, 456
Longitudinal trajectories
of mortality derived, 377
for probability of disability prediction, 339
Longitudinal transition models
for categorical response data, 379
empirical illustration
measures/models/SAS programs, 395
predicted transition probabilities in functional status/marital status, 395
transition probabilities
effects of marital status, 407
prediction of, 399
mixed-effects multinomial logit transition model, 386
random coefficient, 389
random intercept, 386
separate creation, 394
statistical inference, 390
variance–covariance matrix approximation for transition probabilities, 392
with only fixed effects, 384
two-time multinomial transition modeling, overview of, 380
M
Marginal effect, 356
variance-covariance matrix for, 241
Marginal predictions, 240
Marginal regression model, 286
Marital status
on probability of disability among older Americans, 327
on transition probabilities, 408
approximation method, 392
imputation on variable PCL_SUM, 452
Markov chain process, 264
fixed-effects techniques, 379
hypothesis, 384
response at time point, 389
Markov random variable, 384
Maximized log-likelihoods, 77
Maximizing equation, 345
equations, 254
estimates
factor-scoring coefficients, 430
log-likelihood functions, 107
Fisher information matrix, 248
linear mixed models, 194
for three linear mixed models on ADL count, 201
log-likelihood function, 260
parameter, 247
solution for regression coefficients, 306
Mean PCL scores, time plot of, 33
Mean square (MS)
error, 38
factor, 38
Meta-analysis, 26
estimated effect size, 26–27
medical treatment on PTSD, 26
Methods handling missing data
mixed-effects regression models, 441
not at random, 454
empirical illustration
analytic results/ADL count predictions, 468
pattern of change over time in ADL count, 470
measures/models/SAS programming, 463
nonignorable missing data, 463
impact of nonignorable missing data, 456
nonparametric regression model, on nonignorable missing data, 461
pattern mixture model on MNAR, 460
selection model on MNAR, 458
at random, 444
empirical illustration, analytic results with and without multiple imputations, 451
last observation carried forward (LOCF), 446
multiple imputations (MI), 447
and shrinkage, comparison, 450
simple approaches, 444
Metropolis–Hastings algorithm, 265
Metropolis sampling, 264
Minimum mean square error, of prediction, 220
defined, 443
mathematical definitions of, 441
missing-data mechanism, 444
assumption, 444
defined, 442
hypothesis, 429
longitudinal data analysis, 443
mathematical definitions of, 441
Missing data, See also Methods handling missing data
classification, natural extension of, 230
patterns, 9
classification standards for, 229
monotone missing pattern, 9
nonmonotone missing data, 9
patterns and mechanisms, 9
well-assumed prior distribution of, 448
hypothesis, 115
mathematical definitions of, 441
mechanisms, 229
nonignorable missing data, 444
statistical models, 73
Mixed-effects models, 61
analytic results/nonlinear predictions, 364
analytic steps with SAS programs, 359
data, measures, and models, 358
graphical analysis, on nonlinear predictions, 374
inference of, 318
marital status
conditional effects of, 369
longitudinal trajectories of disability/mortality, 357
approximation method, empirical application of, 394
covariates’ conditional effects on probability scale, 355
fixed/random effects, estimation of, 351
longitudinal data analysis, 343
and nonlinear predictions, 347
random components, 343
regression model, 275
variance–covariance matrix approximation on probabilities, 353
regression model, 273
ordered logistic model, 274
Poisson regression model, 275
probit model, 273
Model-based longitudinal trajectory, 472
Modeling nonignorable missing data, 471
Modeling normal longitudinal data, 150
MQL estimates, 259
MQL technique, See Marginal quasi-likelihood (MQL) technique
on health states, 365
inverse of, 345
marginal means of
approximate variance–covariance matrix, 401
covariates’ conditional effects on probability scale, 355
empirical illustration
analytic results/nonlinear predictions, 364
analytic steps with SAS programs, 359
data, measures, and models, 358
graphical analysis, on nonlinear predictions, 374
marital status, conditional effects of, 369
marital status/longitudinal trajectories of disability/mortality, 357
fixed/random effects, estimation of, 351
longitudinal data analysis, 343
and nonlinear predictions, 347
random components, 343
variance–covariance matrix approximation on probabilities, 353
regression model
likelihood function for, 345
overview of, 344
parameters, 382
handling missing data, 450
and shrinkage, comparison, 450
application of, 49
constant error variance–covariance matrix, 51
distinctive disadvantages, 49
empirical illustration
psychiatric disorders, acupuncture treatment, 53
general uses, 47
hypothesis testing, 49
Response*Time Effect, 56
total sums of squares, 47
Wilks’ lambda distribution, 49
within-group matrix, 48
Multivariate data
PCL_SUM imputed on, 451
of repeated measurements, 5
structure, 5
distinctive disadvantages, 6
vs. univariate longitudinal data matrix, 7
N
Naïve model, 286
Naïve variance estimator, in longitudinal data analysis, 284
National Death Index (NDI), 18
Newton-Raphson (NR)
scoring method, 248
NLMIXED procedure, 370
Nonignorable missing data, 472
clinical experimental studies, 455
regression models, 456
multivariate regression models, 472
nonparametric regression model, 461
selection model to handle, 472
Nonlinear longitudinal data, generalized linear models, 243
Nonlinear predictions, 347
graphical analysis, 374
mixed-effects multinomial logit model, 347
probability of disability, 335
of random components, 266
retransformation method, 339
transition probabilities, 391
within-subject random errors, 270
Nonmonotone missing-data pattern, 450
Nonparametric mixed-effects model, 462
Nonrandom factor-scoring coefficients, 431
Null hypothesis (Type-I error), 37
O
Odds ratios (OR)
interpretability of, 322
parameterization, 292
quadratic formula, 293
probability of disability and conditional, 338
standard error, 326
ODS Graphics, 191
ODS OUTPUT statement, 157
Offset, 276
OLS-type residuals, 178
ONLY suboption, 195
Orthogonal polynomials, 475–476
convenient curvilinear expression, 475
least-square estimators, 476
regression law, 476
P
Paired t-test, 21
Panel data, 4
Parameter estimates, asymptotic standard errors of, 419
Parametric hazard regression model, 277
conditional distribution, 232
individual pattern indicator, 233
on missing data, 460
pattern-specific parameter estimates, 233
PATTERN, variable, 235
previous model, 233
Patterson’s expression, 105
Pearson-type residuals, 175
approximation techniques, 255
Gaussian quadrature, 352
pseudo-likelihood estimates of model parameters, 256
Physiological senescence parameters, 216
Poisson process, 276
Polynomial time function, longitudinal data analysis, 84
Posttraumatic stress disorder (PTSD) symptom, 29
acupuncture treatment, 153
randomized controlled clinical trial, 17
longitudinal study, 447
severity, 160
PQL method, See Penalized quasi-likelihood (PQL) method
Practical significance, 24
Predicted probabilities, of disability/death, 368
Predicted response probability, variance approximation, 320
Predicted time trend, 81
in functional status and marital status, 395
variance approximates, 404
PREDICT statements, 399
Prentice’s approach, 291
Prentice’s expansion, 290
Pre–post effect size, 28
Pre–post paired t-test, 21
PRESS statistic, defined, 183
Probability density function, 261
Probability of correct model, 424
PROBIT, 465
Probit regression models, overview of, 310
Probit survival model, 458
PROC GENMOD procedure models, 301
logit components, 361
multinomial distribution, 360
parameter estimates, 398
PROC NLMIXED procedure in SAS system, 359
random effect parameters, 263
PROC MI procedure, 453
estimates regression coefficients, 361
RANDOM statement, 333
in SAS is applied to yield parameter estimation, 226
PROC SQL procedure, 128
PROC TTEST procedure, 34
Proportional odds assumption, 274
Pseudo-likelihood estimates, 260
fixed effects of three linear mixed models, 212
four time points, 149
pattern of change over time, 159
prediction
time plots of, 124
time trends of, 161
Q
Q -point Gaussian quadrature rule, 263
high-order polynomial functions, 82
time function, 80
R
linear model, 235
logistic regression model, specification of, 316
model specification/SAS program, 483–485
AHEAD longitudinal data, 485
multinomial logit model on health, 483
multivariate model, 179
Random-effects multinomial logit transition model, maximum likelihood estimates, 392
covariance structure for within-subject, 133
subject-specific, 134
variance–covariance matrix, 133
Random intercept linear model, 176
empirical BLUP approach and retransformation method based on, 328
reduced-form, 328
regression model, specification of, 313
Wald statistics, 337
within-subject error, 315
random, 335
within-subject variability, 314
Random intercept model, 314
Random intercept regression model, 350
transition model, 405
Randomized block design, 39
Raw residuals, 175
Reduced-form fixed-effects multinomial logit model, 364
estimate, 181
interpretability of, 322
asymptotic process, 74
log-likelihood function, 247
on longitudinal data, 281
multinomial logit regression modeling, 391
nonlinear, 330
standardized diagnostic method, 189
REML approach, See Restricted maximum likelihood (REML) approach
Repeated measures ANOVA, 39
Residual covariance structure, patterns, 133
between-subjects variance component, 133
classification factor
coding schemes of time, 149
scaling approaches, 146
scaling of time, 146
comparison of, 143
empirical illustrations
linear regression model, 153
marital status/disability severity among older Americans, 161
PCL score, acupuncture treatment, 161
two linear regression models, estimation, 153
with equal spacing, 135
autoregressive structures (AR), 137
compound symmetry (CS), 135
toeplitz structures (TOEP), 138
unstructured pattern (UN), 136
nonzero off-diagonal elements, 134
with unequal time intervals, 139
hybrid residual covariance model, 142
spatial exponential model, 141
spatial Gaussian pattern model SP(GAU), 141
spatial power model, 140
variance–covariance pattern models, 134
Residual diagnostics, 174
linear mixed model, 173
types of, 174
semivariogram
in linear random coefficient model, 178
in random intercept linear models, 176
Residual log-likelihood function, maximization, 103
pattern models, 139
use of, 161
Bayesian inference, overview of, 96
computational procedures, 106
AHEAD survey, 124
Expectation–Maximization (EM) algorithm, 108
hypothesis testing, on variance component G, 116
justification of, 104
linear mixed models, estimator, 102
approximation of random effects, 111
best linear unbiased prediction (BLUP), 111
empirical illustrations, 117
marital status and disability among older americans, 124
PCL score, acupuncture treatment on, 117
shrinkage/reliability, 113
MLE bias, in variance estimate, 99
ML estimators, comparison, 105
Newton–Raphson (NR) algorithm, 107
REML, in general linear models, 101
Retransformation method, 357
based on random intercept multinomial logit model, 362
Bayesian inference, development, 272
longitudinal trajectories of health probabilities, 359
mean multinomial logit functions, 396
mixed-effects multinomial logit model, 344
nonlinear response, 269
transition probabilities, 405
Retransformation, mixed-effects logistic regression model, 273
RLD scores, 188
Root of mean square error (RMSE), 191
S
Sandwich estimator, 367
SAS PROC GLIMMIX procedure, 328
SAS PROC SGPLOT steps, 30
SAS PROC TRAJ algorithm, 425
SAS programming, 32, 55, 56, 58, 86, 87, 91, 118, 120, 122, 153, 154, 158, 162, 168, 191, 210, 211, 223–226, 235, 236, 300, 302–304, 306, 307, 330, 360–362, 370, 397, 399, 400, 432, 483
covariance parameter, 202
COVB option, 360
reduced-form random intercept multinomial logit model, 363
SAS–STAT software, 425
SAS system, PROC NLMIXED procedure, 359
Scaling techniques, 134
Selection model, 458
Semi-Markov transition process, 389
Semivariogram, 180
in linear random coefficient model, 178
Serial correlation, 11
Shrinkage technique, 114
SOLUTION option, 121
Spatial Gaussian covariance pattern model, 141
Spatial Gaussian pattern model (SPGAU), 141
SQL procedure, 128
Square root of approximated variance, 322
Standard deviation (SD), 35
Statistical models handling missing data, 442
Stochastic missing-data indicator matrix, 231
Stochastic variations, decomposition of, 177
basic null hypothesis, 415
construction of, 413
estimating procedure, 414
maximum likelihood function, 415
types of random variables, 412
Subject-specific random effects, 12
Subject-specific regression coefficients, 314
Sums of squares (SS), 38
T
higher order terms, 477
Taylor’s theorem, 260
t distribution, 21
Thompson’s expression, 105
Time-independent random parameter, 314
Time trends
irregular, 85
J -shaped, 79
in predicted probability of disability, 375
U -shaped, 80
Time-varying covariates, 7
TOEP covariance structure, 144
TOEP residual variance–covariance pattern model, 164
TOEP variance–covariance structure, 164
Transient states, 380
Transition models, two-time transition models, 380
Transition probabilities, 406
TREAT-by-TIME interaction, 57
T -scale, 83
Type-I error, 37
U
Univariate longitudinal data, 7
Unknown distribution, 462
UN pattern model, 137
US Bureau of Census, 446
V
Variance-component models, 3
approximation, 394
defined, 284
generalized linear mixed models (GLMMs), 243
marginal effect, 241
within-subjects random errors, 387
VARIANCE statement, 428
VAR statement, 453
W
conditional effects, 325
delta method, 325
Wald test, 84
Walter Reed National Military Medical Center (WRNMMC), 17
Wedderburn’s theory, 284
Weibull distributional function, 277
Wide table format, 5
Wilcoxon rank test, 23
Within-sample variations, 38
component, random errors, 38
variations of sample data, 38
Within-study effect sizes, 28
local approximations for, 390
multinomial logit regression modeling, 391
probabilities at series of time points, 349
variance-covariance matrix of, 387
Working correlation, 308
Z
Zero-inflated Poisson (ZIP) distribution, 423
ZIP distribution, See Zero-inflated Poisson (ZIP) distribution
Z -test, 75
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