Models for Longitudinal and Incomplete Data Geert Molenberghs [email protected]Geert Verbeke [email protected]Interuniversity Institute for Biostatistics and statistical Bioinformatics (I-BioStat) Katholieke Universiteit Leuven & Universiteit Hasselt, Belgium www.kuleuven.ac.be/biostat/ & www.censtat.uhasselt.be Interuniversity Institute for Biostatistics and statistical Bioinformatics ESALQ, Piracicaba, November 24–28, 2014
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Models for Longitudinal and Incomplete Data...Models for Longitudinal and Incomplete Data Geert Molenberghs [email protected] Geert Verbeke [email protected]
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• Tan, M.T., Tian, G.-L., and Ng, K.W. (2010). Bayesian Missing Data Problems.Boca Raton: Chapman & Hall/CRC.
• van Buuren, S. (2012). Flexible Imputation of Missing Data. Boca Raton:Chapman & Hall/CRC.
• Verbeke, G. and Molenberghs, G. (1997). Linear Mixed Models In Practice: ASAS Oriented Approach, Lecture Notes in Statistics 126. New-York: Springer.
• Verbeke, G. and Molenberghs, G. (2000). Linear Mixed Models for LongitudinalData. Springer Series in Statistics. New-York: Springer.
• Vonesh, E.F. and Chinchilli, V.M. (1997). Linear and Non-linear Models for theAnalysis of Repeated Measurements. Basel: Marcel Dekker.
ESALQ Course on Models for Longitudinal and Incomplete Data 6
• Weiss, R.E. (2005). Modeling Longitudinal Data. New York: Springer.
• West, B.T., Welch, K.B., and Ga lecki, A.T. (2007). Linear Mixed Models: APractical Guide Using Statistical Software. Boca Raton: Chapman & Hall/CRC.
• Wu, H. and Zhang, J.-T. (2006). Nonparametric Regression Methods forLongitudinal Data Analysis. New York: John Wiley & Sons.
• Wu, L. (2010). Mixed Effects Models for Complex Data. Boca Raton: Chapman& Hall/CRC.
ESALQ Course on Models for Longitudinal and Incomplete Data 7
Part I
Continuous Longitudinal Data
ESALQ Course on Models for Longitudinal and Incomplete Data 8
Chapter 1
Introduction
. Repeated Measures / Longitudinal data
. Examples
ESALQ Course on Models for Longitudinal and Incomplete Data 9
1.1 Repeated Measures / Longitudinal Data
Repeated measures are obtained when a responseis measured repeatedly on a set of units
• Units:
. Subjects, patients, participants, . . .
. Animals, plants, . . .
. Clusters: families, towns, branches of a company,. . .
. . . .
• Special case: Longitudinal data
ESALQ Course on Models for Longitudinal and Incomplete Data 10
1.2 Rat Data
• Research question (Dentistry, K.U.Leuven):
How does craniofacial growth depend ontestosteron production ?
• Randomized experiment in which 50 male Wistar rats are randomized to:
. Control (15 rats)
. Low dose of Decapeptyl (18 rats)
. High dose of Decapeptyl (17 rats)
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• Treatment starts at the age of 45 days; measurements taken every 10 days, fromday 50 on.
• The responses are distances (pixels) between well defined points on x-ray picturesof the skull of each rat:
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• Measurements with respect to the roof, base and height of the skull. Here, weconsider only one response, reflecting the height of the skull.
• Individual profiles:
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• Complication: Dropout due to anaesthesia (56%):
# Observations
Age (days) Control Low High Total
50 15 18 17 50
60 13 17 16 46
70 13 15 15 43
80 10 15 13 38
90 7 12 10 29
100 4 10 10 24
110 4 8 10 22
• Remarks:
. Much variability between rats, much less variability within rats
. Fixed number of measurements scheduled per subject, but not allmeasurements available due to dropout, for known reason.
. Measurements taken at fixed time points
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1.3 Prostate Data
• References:
. Carter et al (1992, Cancer Research).
. Carter et al (1992, Journal of the American Medical Association).
. Morrell et al (1995, Journal of the American Statistical Association).
. Pearson et al (1994, Statistics in Medicine).
• Prostate disease is one of the most common and most costly medical problems inthe United States
• Important to look for markers which can detect the disease at an early stage
• Prostate-Specific Antigen is an enzyme produced by both normal and cancerousprostate cells
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• PSA level is related to the volume of prostate tissue.
• Problem: Patients with Benign Prostatic Hyperplasia also have an increased PSAlevel
• Overlap in PSA distribution for cancer and BPH cases seriously complicates thedetection of prostate cancer.
• Research question (hypothesis based on clinical practice):
Can longitudinal PSA profiles be used todetect prostate cancer in an early stage ?
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• A retrospective case-control study based on frozen serum samples:
. 16 control patients
. 20 BPH cases
. 14 local cancer cases
. 4 metastatic cancer cases
• Complication: No perfect match for age at diagnosis and years of follow-uppossible
• Hence, analyses will have to correct for these age differences between thediagnostic groups.
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• Individual profiles:
ESALQ Course on Models for Longitudinal and Incomplete Data 18
• Remarks:
. Much variability between subjects
. Little variability within subjects
. Highly unbalanced data
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Chapter 2
A Model for Longitudinal Data
. Introduction
. The 2-stage model formulation
. Example: Rat data
. The general linear mixed-effects model
. Hierarchical versus marginal model
. Example: Rat data
. Example: Bivariate observations
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2.1 Introduction
• In practice: often unbalanced data:
. unequal number of measurements per subject
. measurements not taken at fixed time points
• Therefore, multivariate regression techniques are often not applicable
• Often, subject-specific longitudinal profiles can be well approximated by linearregression functions
• This leads to a 2-stage model formulation:
. Stage 1: Linear regression model for each subject separately
. Stage 2: Explain variability in the subject-specific regression coefficients usingknown covariates
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2.2 A 2-stage Model Formulation
2.2.1 Stage 1
• Response Yij for ith subject, measured at time tij, i = 1, . . . ,N , j = 1, . . . , ni
• Response vector Yi for ith subject: Yi = (Yi1, Yi2, . . . , Yini)′
• Stage 1 model:
Yi = Ziβi + εi
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• Zi is a (ni × q) matrix of known covariates
• βi is a q-dimensional vector of subject-specific regression coefficients
• εi ∼ N (0,Σi), often Σi = σ2Ini
• Note that the above model describes the observed variability within subjects
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2.2.2 Stage 2
• Between-subject variability can now be studied from relating the βi to knowncovariates
• Stage 2 model:
βi = Kiβ + bi
• Ki is a (q × p) matrix of known covariates
• β is a p-dimensional vector of unknown regression parameters
• bi ∼ N (0, D)
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2.3 Example: The Rat Data
• Individual profiles:
ESALQ Course on Models for Longitudinal and Incomplete Data 25
• Transformation of the time scale to linearize the profiles:
Ageij −→ tij = ln[1 + (Ageij − 45)/10)]
• Note that t = 0 corresponds to the start of the treatment (moment ofrandomization)
• Note that the model implicitly assumes that the variance function is quadraticover time, with positive curvature d22.
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• A model which assumes that all variability in subject-specific slopes can beascribed to treatment differences can be obtained by omitting the random slopesb2i from the above model:
Yij = (β0 + b1i) + (β1Li + β2Hi + β3Ci)tij + εij
=
β0 + b1i + β1tij + εij, if low dose
β0 + b1i + β2tij + εij, if high dose
β0 + b1i + β3tij + εij, if control.
• This is the so-called random-intercepts model
• The same marginal mean structure is obtained as under the model with randomslopes
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• Hence, the implied covariance matrix is compound symmetry:
. constant variance d11 + σ2
. constant correlation ρI = d11/(d11 + σ2) between any two repeatedmeasurements within the same rat
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2.7 Example: Bivariate Observations
• Balanced data, two measurements per subject (ni = 2), two models:
Model 1:Random intercepts
+heterogeneous errors
V =
1
1
(d) (1 1) +
σ21 0
0 σ22
=
d + σ21 d
d d+ σ22
Model 2:Uncorrelated intercepts and slopes
+measurement error
V =
1 0
1 1
d1 0
0 d2
1 1
0 1
+
σ2 0
0 σ2
=
d1 + σ2 d1
d1 d1 + d2 + σ2
ESALQ Course on Models for Longitudinal and Incomplete Data 37
• Different hierarchical models can produce the same marginal model
• Hence, a good fit of the marginal model cannot be interpreted as evidence for anyof the hierarchical models.
• A satisfactory treatment of the hierarchical model is only possible within aBayesian context.
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Chapter 3
Estimation and Inference in the Marginal Model
. ML and REML estimation
. Fitting linear mixed models in SAS
. Negative variance components
. Inference
ESALQ Course on Models for Longitudinal and Incomplete Data 39
3.1 ML and REML Estimation
• Recall that the general linear mixed model equals
Yi = Xiβ + Zibi + εi
bi ∼ N (0, D)
εi ∼ N (0,Σi)
independent
• The implied marginal model equals Yi ∼ N (Xiβ, ZiDZ′i + Σi)
• Note that inferences based on the marginal model do not explicitly assume thepresence of random effects representing the natural heterogeneity between subjects
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• Notation:
. β: vector of fixed effects (as before)
. α: vector of all variance components in D and Σi
. θ = (β′,α′)′: vector of all parameters in marginal model
• Marginal likelihood function:
LML(θ) =N∏
i=1
(2π)−ni/2 |Vi(α)|−1
2 exp−1
2(Yi −Xiβ)′ V −1
i (α) (Yi −Xiβ)
• If α were known, MLE of β equals
β(α) =
N∑
i=1X ′iWiXi
−1
N∑
i=1X ′iWiyi,
where Wi equals V −1i .
ESALQ Course on Models for Longitudinal and Incomplete Data 41
• In most cases, α is not known, and needs to be replaced by an estimate α
• Two frequently used estimation methods for α:
. Maximum likelihood
. Restricted maximum likelihood
ESALQ Course on Models for Longitudinal and Incomplete Data 42
• In SAS the estimates can be obtained from adding the option ‘solution’ to therandom statement:
random intercept time time2
/ type=un subject=id solution;
ods listing exclude solutionr;
ods output solutionr=out;
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• The ODS statements are used to write the EB estimates into a SAS output dataset, and to prevent SAS from printing them in the output window.
• In practice, histograms and scatterplots of certain components of bi are used to
detect model deviations or subjects with ‘exceptional’ evolutions over time
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• Strong negative correlations in agreement with correlation matrix corresponding tofitted D:
Dcorr =
1.000 −0.803 0.658
−0.803 1.000 −0.968
0.658 −0.968 1.000
• Histograms and scatterplots show outliers
• Subjects #22, #28, #39, and #45, have highest four slopes for time2 andsmallest four slopes for time, i.e., with the strongest (quadratic) growth.
• Subjects #22, #28 and #39 have been further examined and have been shown tobe metastatic cancer cases which were misclassified as local cancer cases.
• Subject #45 is the metastatic cancer case with the strongest growth
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Part II
Marginal Models for Non-Gaussian Longitudinal Data
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Chapter 5
The Toenail Data
• Toenail Dermatophyte Onychomycosis: Common toenail infection, difficult totreat, affecting more than 2% of population.
• Classical treatments with antifungal compounds need to be administered until thewhole nail has grown out healthy.
• New compounds have been developed which reduce treatment to 3 months
• Randomized, double-blind, parallel group, multicenter study for the comparison oftwo such new compounds (A and B) for oral treatment.
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• Research question:
Severity relative to treatment of TDO ?
• 2× 189 patients randomized, 36 centers
• 48 weeks of total follow up (12 months)
• 12 weeks of treatment (3 months)
• measurements at months 0, 1, 2, 3, 6, 9, 12.
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• Frequencies at each visit (both treatments):
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Chapter 6
The Analgesic Trial
• single-arm trial with 530 patients recruited (491 selected for analysis)
• analgesic treatment for pain caused by chronic nonmalignant disease
• treatment was to be administered for 12 months
• we will focus on Global Satisfaction Assessment (GSA)
• GSA scale goes from 1=very good to 5=very bad
• GSA was rated by each subject 4 times during the trial, at months 3, 6, 9, and 12.
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• Research questions:
. Evolution over time
. Relation with baseline covariates: age, sex, duration of the pain, type of pain,disease progression, Pain Control Assessment (PCA), . . .
. Investigation of dropout
• Frequencies:
GSA Month 3 Month 6 Month 9 Month 12
1 55 14.3% 38 12.6% 40 17.6% 30 13.5%
2 112 29.1% 84 27.8% 67 29.5% 66 29.6%
3 151 39.2% 115 38.1% 76 33.5% 97 43.5%
4 52 13.5% 51 16.9% 33 14.5% 27 12.1%
5 15 3.9% 14 4.6% 11 4.9% 3 1.4%
Tot 385 302 227 223
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• Missingness:
Measurement occasion
Month 3 Month 6 Month 9 Month 12 Number %
Completers
O O O O 163 41.2
Dropouts
O O O M 51 12.91
O O M M 51 12.91
O M M M 63 15.95
Non-monotone missingness
O O M O 30 7.59
O M O O 7 1.77
O M O M 2 0.51
O M M O 18 4.56
M O O O 2 0.51
M O O M 1 0.25
M O M O 1 0.25
M O M M 3 0.76
ESALQ Course on Models for Longitudinal and Incomplete Data 68
Chapter 7
Generalized Linear Models
. The model
. Maximum likelihood estimation
. Examples
. McCullagh and Nelder (1989)
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7.1 The Generalized Linear Model
• Suppose a sample Y1, . . . , YN of independent observations is available
• All Yi have densities f (yi|θi, φ) which belong to the exponential family:
f (y|θi, φ) = exp{φ−1[yθi − ψ(θi)] + c(y, φ)
}
• θi the natural parameter
• Linear predictor: θi = xi′β
• φ is the scale parameter (overdispersion parameter)
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• ψ(·) is a function, generating mean and variance:
E(Y ) = ψ′(θ)
Var(Y ) = φψ′′(θ)
• Note that, in general, the mean µ and the variance are related:
Var(Y ) = φψ′′[
ψ′−1(µ)
]
= φv(µ)
• The function v(µ) is called the variance function.
• The function ψ′−1 which expresses θ as function of µ is called the link function.
• ψ′ is the inverse link function
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7.2 Examples
7.2.1 The Normal Model
• Model:
Y ∼ N (µ, σ2)
• Density function:
f (y|θ, φ) =1√
2πσ2exp
−
1
σ2(y − µ)2
= exp
1
σ2
yµ− µ2
2
+
ln(2πσ2)
2− y2
2σ2
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• Exponential family:
. θ = µ
. φ = σ2
. ψ(θ) = θ2/2
. c(y, φ) = ln(2πφ)2− y2
2φ
• Mean and variance function:
. µ = θ
. v(µ) = 1
• Note that, under this normal model, the mean and variance are not related:
φv(µ) = σ2
• The link function is here the identity function: θ = µ
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7.2.2 The Bernoulli Model
• Model:Y ∼ Bernoulli(π)
• Density function:
f (y|θ, φ) = πy(1− π)1−y
= exp {y ln π + (1− y) ln(1− π)}
= exp
y ln
π
1− π + ln(1− π)
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• Exponential family:
. θ = ln(
π1−π
)
. φ = 1
. ψ(θ) = ln(1− π) = ln(1 + exp(θ))
. c(y, φ) = 0
• Mean and variance function:
. µ = exp θ1+exp θ = π
. v(µ) = exp θ(1+exp θ)2
= π(1− π)
• Note that, under this model, the mean and variance are related:
φv(µ) = µ(1− µ)
• The link function here is the logit link: θ = ln(
µ1−µ
)
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7.3 Generalized Linear Models (GLM)
• Suppose a sample Y1, . . . , YN of independent observations is available
• All Yi have densities f (yi|θi, φ) which belong to the exponential family
• In GLM’s, it is believed that the differences between the θi can be explainedthrough a linear function of known covariates:
θi = xi′β
• xi is a vector of p known covariates
• β is the corresponding vector of unknown regression parameters, to be estimatedfrom the data.
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7.4 Maximum Likelihood Estimation
• Log-likelihood:
`(β, φ) =1
φ
∑
i[yiθi − ψ(θi)] +
∑
ic(yi, φ)
• The score equations:
S(β) =∑
i
∂µi
∂βv−1
i (yi − µi) = 0
• Note that the estimation of β depends on the density only through the means µi
and the variance functions vi = v(µi).
ESALQ Course on Models for Longitudinal and Incomplete Data 77
• The score equations need to be solved numerically:
. iterative (re-)weighted least squares
. Newton-Raphson
. Fisher scoring
• Inference for β is based on classical maximum likelihood theory:
. asymptotic Wald tests
. likelihood ratio tests
. score tests
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7.5 Illustration: The Analgesic Trial
• Early dropout (did the subject drop out after the first or the second visit) ?
• Binary response
• PROC GENMOD can fit GLMs in general
• PROC LOGISTIC can fit models for binary (and ordered) responses
• SAS code for logit link:
proc genmod data=earlydrp;
model earlydrp = pca0 weight psychiat physfct / dist=b;
run;
proc logistic data=earlydrp descending;
model earlydrp = pca0 weight psychiat physfct;
run;
ESALQ Course on Models for Longitudinal and Incomplete Data 79
• SAS code for probit link:
proc genmod data=earlydrp;
model earlydrp = pca0 weight psychiat physfct / dist=b link=probit;
run;
proc logistic data=earlydrp descending;
model earlydrp = pca0 weight psychiat physfct / link=probit;
Ti β1 0.024(0.160;0.251) 0.011(0.196;0.262) 0.036(0.242;0.242)
tij β2 -0.177(0.025;0.030) -0.177(0.022;0.031) -0.204(0.038;0.034)
Ti · tij β3 -0.078(0.040:0.055) -0.089(0.038;0.057) -0.106(0.058;0.058)
estimate (model-based s.e.; empirical s.e.)
ESALQ Course on Models for Longitudinal and Incomplete Data 114
10.5.6 Discussion
• GEE1: All empirical standard errors are correct, but the efficiency is higher for themore complex working correlation structure, as seen in p-values for Ti · tij effect:
Structure p-value
IND 0.1515
EXCH 0.1208
UN 0.0275
Thus, opting for reasonably adequate correlation assumptions still pays off, inspite of the fact that all are consistent and asymptotically normal
• Similar conclusions for linearization-based method
ESALQ Course on Models for Longitudinal and Incomplete Data 115
• Model-based s.e. and empirically corrected s.e. in reasonable agreement for UN
• Typically, the model-based standard errors are much too small as they are basedon the assumption that all observations in the data set are independent, herebyoverestimating the amount of available information, hence also overestimating theprecision of the estimates.
• ALR: similar inferences but now also α part of the inferences
ESALQ Course on Models for Longitudinal and Incomplete Data 116
Part III
Generalized Linear Mixed Models for Non-GaussianLongitudinal Data
ESALQ Course on Models for Longitudinal and Incomplete Data 117
Chapter 11
Generalized Linear Mixed Models (GLMM)
. Introduction: LMM Revisited
. Generalized Linear Mixed Models (GLMM)
. Fitting Algorithms
. Example
ESALQ Course on Models for Longitudinal and Incomplete Data 118
11.1 Introduction: LMM Revisited
• We re-consider the linear mixed model:
Yi|bi ∼ N (Xiβ + Zibi,Σi), bi ∼ N (0,D)
• The implied marginal model equals Yi ∼ N (Xiβ, ZiDZ′i + Σi)
• Hence, even under conditional independence, i.e., all Σi equal to σ2Ini, a marginalassociation structure is implied through the random effects.
• The same ideas can now be applied in the context of GLM’s to model associationbetween discrete repeated measures.
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11.2 Generalized Linear Mixed Models (GLMM)
• Given a vector bi of random effects for cluster i, it is assumed that all responsesYij are independent, with density
f (yij|θij, φ) = exp{φ−1[yijθij − ψ(θij)] + c(yij, φ)
}
• θij is now modelled as θij = xij′β + zij
′bi
• As before, it is assumed that bi ∼ N (0, D)
• Let fij(yij|bi,β, φ) denote the conditional density of Yij given bi, the conditionaldensity of Yi equals
fi(yi|bi,β, φ) =ni∏
j=1fij(yij|bi,β, φ)
ESALQ Course on Models for Longitudinal and Incomplete Data 120
• The marginal distribution of Yi is given by
fi(yi|β, D, φ) =∫fi(yi|bi,β, φ) f (bi|D) dbi
=∫ ni∏
j=1fij(yij|bi,β, φ) f (bi|D) dbi
where f (bi|D) is the density of the N (0,D) distribution.
• The likelihood function for β, D, and φ now equals
L(β, D, φ) =N∏
i=1fi(yi|β, D, φ)
=N∏
i=1
∫ ni∏
j=1fij(yij|bi,β, φ) f (bi|D) dbi
ESALQ Course on Models for Longitudinal and Incomplete Data 121
• Under the normal linear model, the integral can be worked out analytically.
• In general, approximations are required:
. Approximation of integrand
. Approximation of data
. Approximation of integral
• Predictions of random effects can be based on the posterior distribution
f (bi|Yi = yi)
• ‘Empirical Bayes (EB) estimate’:Posterior mode, with unknown parameters replaced by their MLE
ESALQ Course on Models for Longitudinal and Incomplete Data 122
11.3 Laplace Approximation of Integrand
• Integrals in L(β,D, φ) can be written in the form I =∫eQ(b)db
• Second-order Taylor expansion of Q(b) around the mode yields
Q(b) ≈ Q(b) +1
2(b−
b)′Q′′(b)(b − b),
• Quadratic term leads to re-scaled normal density. Hence,
I ≈ (2π)q/2∣∣∣∣−Q′′(b)
∣∣∣∣−1/2
eQ(b).
• Exact approximation in case of normal kernels
• Good approximation in case of many repeated measures per subject
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11.4 Approximation of Data
11.4.1 General Idea
• Re-write GLMM as:
Yij = µij + εij = h(x′ijβ + z′ijbi) + εij
with variance for errors equal to Var(Yij|bi) = φv(µij)
• Linear Taylor expansion for µij:
. Penalized quasi-likelihood (PQL): Around current β and
bi
. Marginal quasi-likelihood (MQL): Around current β and bi = 0
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ESALQ Course on Models for Longitudinal and Incomplete Data 133
• Conclusions:
. (Log-)likelihoods are not comparable
. Different Q can lead to considerable differences in estimates and standarderrors
. For example, using non-adaptive quadrature, with Q = 3, we found nodifference in time effect between both treatment groups(t = −0.09/0.05, p = 0.0833).
. Using adaptive quadrature, with Q = 50, we find a significant interactionbetween the time effect and the treatment (t = −0.16/0.07, p = 0.0255).
. Assuming that Q = 50 is sufficient, the ‘final’ results are well approximatedwith smaller Q under adaptive quadrature, but not under non-adaptivequadrature.
ESALQ Course on Models for Longitudinal and Incomplete Data 134
• Comparison of fitting algorithms:
. Adaptive Gaussian Quadrature, Q = 50
. MQL and PQL
• Summary of results:
Parameter QUAD PQL MQL
Intercept group A −1.63 (0.44) −0.72 (0.24) −0.56 (0.17)
Intercept group B −1.75 (0.45) −0.72 (0.24) −0.53 (0.17)
Slope group A −0.40 (0.05) −0.29 (0.03) −0.17 (0.02)
Slope group B −0.57 (0.06) −0.40 (0.04) −0.26 (0.03)
Var. random intercepts (τ 2) 15.99 (3.02) 4.71 (0.60) 2.49 (0.29)
• Severe differences between QUAD (gold standard ?) and MQL/PQL.
• MQL/PQL may yield (very) biased results, especially for binary data.
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Chapter 12
Fitting GLMM’s in SAS
. Overview
. Example: Toenail data
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12.1 Overview
• GLIMMIX: Laplace, MQL, PQL, adaptive quadrature
• NLMIXED: Adaptive and non-adaptive quadrature−→ not discussed here
ESALQ Course on Models for Longitudinal and Incomplete Data 137
12.2 Example: Toenail data
• Re-consider logistic model with random intercepts for toenail data
• SAS code (PQL):
proc glimmix data=test method=RSPL ;
class idnum;
model onyresp (event=’1’) = treatn time treatn*time
/ dist=binary solution;
random intercept / subject=idnum;
run;
• MQL obtained with option ‘method=RMPL’
• Laplace obtained with option ‘method=LAPLACE’
ESALQ Course on Models for Longitudinal and Incomplete Data 138
• Adaptive quadrature with option ‘method=QUAD(qpoints=5)’
• Inclusion of random slopes:
random intercept time / subject=idnum type=un;
ESALQ Course on Models for Longitudinal and Incomplete Data 139
Part IV
Marginal Versus Random-effects Models
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Chapter 13
Marginal Versus Random-effects Models
. Interpretation of GLMM parameters
. Marginalization of GLMM
. Example: Toenail data
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13.1 Interpretation of GLMM Parameters: Toenail Data
• We compare our GLMM results for the toenail data with those from fitting GEE’s(unstructured working correlation):
GLMM GEE
Parameter Estimate (s.e.) Estimate (s.e.)
Intercept group A −1.6308 (0.4356) −0.7219 (0.1656)
Intercept group B −1.7454 (0.4478) −0.6493 (0.1671)
Slope group A −0.4043 (0.0460) −0.1409 (0.0277)
Slope group B −0.5657 (0.0601) −0.2548 (0.0380)
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• The strong differences can be explained as follows:
. Consider the following GLMM:
Yij|bi ∼ Bernoulli(πij), log
πij
1− πij
= β0 + bi + β1tij
. The conditional means E(Yij|bi), as functions of tij, are given by
E(Yij|bi)
=exp(β0 + bi + β1tij)
1 + exp(β0 + bi + β1tij)
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. The marginal average evolution is now obtained from averaging over therandom effects:
E(Yij) = E[E(Yij|bi)] = E
exp(β0 + bi + β1tij)
1 + exp(β0 + bi + β1tij)
6= exp(β0 + β1tij)
1 + exp(β0 + β1tij)
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• Hence, the parameter vector β in the GEE model needs to be interpretedcompletely different from the parameter vector β in the GLMM:
. GEE: marginal interpretation
. GLMM: conditional interpretation, conditionally upon level of random effects
• In general, the model for the marginal average is not of the same parametric formas the conditional average in the GLMM.
• For logistic mixed models, with normally distributed random random intercepts, itcan be shown that the marginal model can be well approximated by again alogistic model, but with parameters approximately satisfying
β
RE
β
M=√c2σ2 + 1 > 1, σ2 = variance random intercepts
c = 16√
3/(15π)
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• For the toenail application, σ was estimated as 4.0164, such that the ratio equals√c2σ2 + 1 = 2.5649.
• The ratio’s between the GLMM and GEE estimates are:
GLMM GEE
Parameter Estimate (s.e.) Estimate (s.e.) Ratio
Intercept group A −1.6308 (0.4356) −0.7219 (0.1656) 2.2590
Intercept group B −1.7454 (0.4478) −0.6493 (0.1671) 2.6881
Slope group A −0.4043 (0.0460) −0.1409 (0.0277) 2.8694
Slope group B −0.5657 (0.0601) −0.2548 (0.0380) 2.2202
• Note that this problem does not occur in linear mixed models:
. Conditional mean: E(Yi|bi) = Xiβ + Zibi
. Specifically: E(Yi|bi = 0) = Xiβ
. Marginal mean: E(Yi) = Xiβ
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• The problem arises from the fact that, in general,
E[g(Y )] 6= g[E(Y )]
• So, whenever the random effects enter the conditional mean in a non-linear way,the regression parameters in the marginal model need to be interpreted differentlyfrom the regression parameters in the mixed model.
• In practice, the marginal mean can be derived from the GLMM output byintegrating out the random effects.
• This can be done numerically via Gaussian quadrature, or based on samplingmethods.
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13.2 Marginalization of GLMM: Toenail Data
• As an example, we plot the average evolutions based on the GLMM outputobtained in the toenail example:
P (Yij = 1)
=
E
exp(−1.6308 + bi − 0.4043tij)
1 + exp(−1.6308 + bi − 0.4043tij)
,
E
exp(−1.7454 + bi − 0.5657tij)
1 + exp(−1.7454 + bi − 0.5657tij)
,
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• Average evolutions obtained from the GEE analyses:
P (Yij = 1)
=
exp(−0.7219− 0.1409tij)
1 + exp(−0.7219− 0.1409tij)
exp(−0.6493− 0.2548tij)
1 + exp(−0.6493− 0.2548tij)
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• In a GLMM context, rather than plotting the marginal averages, one can also plotthe profile for an ‘average’ subject, i.e., a subject with random effect bi = 0:
P (Yij = 1|bi = 0)
=
exp(−1.6308− 0.4043tij)
1 + exp(−1.6308− 0.4043tij)
exp(−1.7454− 0.5657tij)
1 + exp(−1.7454− 0.5657tij)
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13.3 Example: Toenail Data Revisited
• Overview of all analyses on toenail data:
Parameter QUAD PQL MQL GEE
Intercept group A −1.63 (0.44) −0.72 (0.24) −0.56 (0.17) −0.72 (0.17)
Intercept group B −1.75 (0.45) −0.72 (0.24) −0.53 (0.17) −0.65 (0.17)
Slope group A −0.40 (0.05) −0.29 (0.03) −0.17 (0.02) −0.14 (0.03)
Slope group B −0.57 (0.06) −0.40 (0.04) −0.26 (0.03) −0.25 (0.04)
Var. random intercepts (τ 2) 15.99 (3.02) 4.71 (0.60) 2.49 (0.29)
• Conclusion:
|GEE| < |MQL| < |PQL| < |QUAD|
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Part V
Non-linear Models
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Chapter 14
Non-Linear Mixed Models
. From linear to non-linear models
. Orange Tree Example
. Song Bird Example
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14.1 From Linear to Non-linear Models
• We have studied:
. linear models ←→ generalized linear models
. for Gaussian data ←→ non-Gaussian data
. for cross-sectional (univariate) data ←→ for correlated data (clustered data,multivariate data, longitudinal data)
• In all cases, a certain amount of linearity is preserved:
. E.g., in generalized linear models, linearity operates at the level of the linearpredictor, describing a transformation of the mean (logit, log, probit,. . . )
• This implies that all predictor functions are linear in the parameters:
β0 + β1x1i + . . . βpxpi
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• This may still be considered a limiting factor, and non-linearity in the parametricfunctions may be desirable.
• Examples:
. Certain growth curves, especially when they include both a growth spurt aswell as asymptote behavior towards the end of growth
. Dose-response modeling
. Pharmacokinetic and pharmacodynamic models
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14.2 LMM and GLMM
• In linear mixed models, the mean is modeled as a linear function of regressionparameters and random effects:
E(Yij|bi) = xij′β + zij
′bi
• In generalized linear mixed models, apart from a link function, the mean is againmodeled as a linear function of regression parameters and random effects:
E(Yij|bi) = h(xij′β + zij
′bi)
• In some applications, models are needed, in which the mean is no longer modeledas a function of a linear predictor xij
′β + zij′bi. These are called non-linear
mixed models.
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14.3 Non-linear Mixed Models
• In non-linear mixed models, it is assumed that the conditional distribution of Yij,given bi is belongs to the exponential family (Normal, Binomial, Poisson,. . . ).
• The mean is modeled as:
E(Yij|bi) = h(xij,β, zij, bi)
• As before, the random effects are assumed to be normally distributed, with mean0 and covariance D.
• Let fij(yij|bi,β, φ) be the conditional density of Yij given bi, and let f (bi|D) bethe density of the N (0, D) distribution.
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• Under conditional independence, the marginal likelihood equals
L(β,D, φ) =N∏
i=1fi(yi|β,D, φ)
=N∏
i=1
∫ ni∏
j=1fij(yij|bi,β, φ) f (bi|D) dbi
• The above likelihood is of the same form as the likelihood obtained earlier for ageneralized linear mixed model.
• As before, numerical quadrature is used to approximate the integral
• Non-linear mixed models can also be fitted within the SAS procedure NLMIXED.
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14.4 Example: Orange Trees
• We consider an experiment in which the trunk circumference (in mm) is measuredfor 5 orange trees, on 7 different occasions.
• Data:
Response
Day Tree 1 Tree 2 Tree 3 Tree 4 Tree 5
118 30 33 30 32 30
484 58 69 51 62 49
664 87 111 75 112 81
1004 115 156 108 167 125
1231 120 172 115 179 142
1372 142 203 139 209 174
1582 145 203 140 214 177
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• The following non-linear mixed model has been proposed:
Yij =β1 + bi
1 + exp[−(tij − β2)/β3]+ εij
bi ∼ N (0, σ2b ) εij ∼ N (0, σ2)
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• In SAS PROC NLMIXED, the model can be fitted as follows:
proc nlmixed data=tree;
parms beta1=190 beta2=700 beta3=350
sigmab=10 sigma=10;
num = b;
ex = exp(-(day-beta2)/beta3);
den = 1 + ex;
model y ~ normal(num/den,sigma**2);
random b ~ normal(beta1,sigmab**2) subject=tree;
run;
• An equivalent program is given by
proc nlmixed data=tree;
parms beta1=190 beta2=700 beta3=350
sigmab=10 sigma=10;
num = b + beta1;
ex = exp(-(day-beta2)/beta3);
den = 1 + ex;
model y ~ normal(num/den,sigma**2);
random b ~ normal(0,sigmab**2) subject=tree;
run;
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• Selected output:
Specifications
Data Set WORK.TREE
Dependent Variable y
Distribution for Dependent Variable Normal
Random Effects b
Distribution for Random Effects Normal
Subject Variable tree
Optimization Technique Dual Quasi-Newton
Integration Method Adaptive Gaussian
Quadrature
Parameter Estimates
Standard
Parameter Estimate Error DF t Value Pr > |t|
beta1 192.05 15.6577 4 12.27 0.0003
beta2 727.91 35.2487 4 20.65 <.0001
beta3 348.07 27.0798 4 12.85 0.0002
sigmab 31.6463 10.2614 4 3.08 0.0368
sigma 7.8430 1.0125 4 7.75 0.0015
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• Note that the number of quadrature points was selected adaptively to be equal toonly one. Refitting the model with 50 quadrature points yielded identical results.
• Empirical Bayes estimates, and subject-specific predictions can be obtained asfollows:
proc nlmixed data=tree;
parms beta1=190 beta2=700 beta3=350
sigmab=10 sigma=10;
num = b + beta1;
ex = exp(-(day-beta2)/beta3);
den = 1 + ex;
ratio=num/den;
model y ~ normal(ratio,sigma**2);
random b ~ normal(0,sigmab**2) subject=tree out=eb;
predict ratio out=ratio;
run;
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• We can now compare the observed data to the subject-specific predictions
yij =β1 + bi
1 + exp[−(tij − β2)/β3]
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14.5 Example: Song Birds
• At the University of Antwerp, a novel in-vivo MRI approach to discern thefunctional characteristics of specific neuronal populations in a strongly connectedbrain circuitry has been established.
• Of particular interest: the song control system (SCS) in songbird brain.
• The high vocal center (HVC), one of the major nuclei in this circuit, containsinterneurons and two distinct types of neurons projecting respectively to thenucleus robustus archistriatalis, RA or to area X.
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• Schematically,
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14.5.1 The MRI Data
• T1-weighted multi slice SE images were acquired (Van Meir et al 2003).
• After acquisition of a set of control images, MnCl2 was injected in the cannulatedHVC.
• Two sets of 5 coronal slices (one through HVC and RA, one through area X) wereacquired every 15 min for up to 6–7 hours after injection.
• This results in 30 data sets of 10 slices of each bird (5 controls and 5 testosteronetreated).
• The change of relative SI of each time point is calculated from the mean signalintensity determined within the defined regions of interest (area X and RA) and inan adjacent control area.
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14.5.2 Initial Study
• The effect of testosterone (T) on the dynamics of Mn2+ accumulation in RA andarea X of female starling has been established.
• This has been done with dynamic ME-MRI in individual birds injected withManganese in their HVC.
• This was done in a 2-stage approach: The individual SI data, determined as themean intensity of a region of interest, were submitted to a sigmoid curve fittingproviding curve parameters which revealed upon a two-way repeated ANOVAanalysis that neurons projecting to RA and those to area X were affecteddifferently by testosterone treatment.
• However, this approach could be less reliable if the fit-parameters were mutuallydependent.
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• Thus: an integrated non-linear modeling approach is necessary: the MRI signalintensities (SI) need to be fitted to a non-linear mixed model assuming that the SIfollow a pre-specified distribution depending on a covariate indicating the timeafter MnCl2 injection and parameterized through fixed and random (bird-specific)effects.
• An initial model needs to be proposed and then subsequently simplified.
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14.5.3 A Model for RA in the Second Period
• Let RAij be the measurement at time j for bird i.
• The following initial non-linear model is assumed:
RAij =(φ0 + φ1Gi + fi)T
η0+η1Gi+niij
(τ0 + τ1Gi + ti)η0+η1Gi+ni + T η0+η1Gi+ni
ij
+γ0 + γ1Gi + εij
• The following conventions are used:
. Gi is an indicator for group membership (0 for control, 1 for active)
. Tij is a covariate indicating the measurement time
. The fixed effects parameters:
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∗ the “intercept” parameters φ0, η0, τ0, and γ0
∗ the group effect parameters φ1, η1, τ1, and γ1
∗ If the latter are simultaneously zero, there is no difference between bothgroups. If at least one of them is (significantly) different from zero, then themodel indicates a difference between both groups.
. There are three bird-specific or random effects, fi, ni, and ti, following atrivariate zero-mean normal and general covariance matrix D
. The residual error terms εij are assumed to be mutually independent andindependent from the random effects, and to follow a N (0, σ2) distribution.
• Thus, the general form of the model has 8 fixed-effects parameters, and 7 variancecomponents (3 variances in D, 3 covariances in D, and σ2).
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random vm t n ~ normal([0, 0, 0],[d11, d12, d22, d13, d23, d33])
subject=vogel out=m.eb2;
run;
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• Model simplification:
. First, the random ni effect is removed (implying the removal of d11, d12, andd13), using a likelihood ratio test statistic with value 4.08 and null distributionχ2
2:3. The corresponding p = 0.1914.
. Removal of the random ti effect is not possible since the likelihood ratio equals54.95 on 1:2 degrees of freedom (p < 0.0001).
. Removal of the covariance between the random ti and fi effects is not possible(G2 = 4.35 on 1 d.f., p = 0.0371).
. Next, the following fixed-effect parameters were removed: γ0, γ1, η1 and τ1.
. The fixed-effect φ1 was found to be highly significant and therefore could notbe removed from the model (G2 = 10.5773 on 1 d.f., p = 0.0011). Thisindicates a significant difference between the two groups.
• The resulting final model is:
AREAij =(φ0 + φ1Gi + fi)T
η0ij
(τ0 + ti)η0 + T η0
ij
+ εij.
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Estimate (s.e.)
Effect Parameter Initial Final
φ0 0.1118 (0.0333) 0.1035 (0.0261)
φ1 0.1116 (0.0458) 0.1331 (0.0312)
η0 2.4940 (0.5390) 2.3462 (0.1498)
η1 -0.0623 (0.5631)
τ0 3.6614 (0.5662) 3.7264 (0.3262)
τ1 -0.1303 (0.6226)
γ0 -0.0021 (0.0032)
γ1 0.0029 (0.0048)
Var(fi) d11 0.0038 (0.0020) 0.004271 (0.0022)
Var(ti) d22 0.2953 (0.2365) 0.5054 (0.2881)
Var(ni) d33 0.1315 (0.1858)
Cov(fi, ti) d12 0.0284 (0.0205) 0.03442 (0.0229)
Cov(fi, ni) d13 -0.0116 (0.0159)
Cov(ti, ni) d23 -0.0095 (0.1615)
Var(εij) σ2 0.00016 (0.000014) 0.00016 (0.000014)
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Individual curves, showingdata points as well as empir-ical Bayes predictions for thebird-specific curves, showthat the sigmoidal curvesdescribe the data quite well.
0 1 2 3 4 5 6 7
0.0
00
.10
0.2
0
1
Time
SI.
are
a.X
0 1 2 3 4 5 6 7
0.0
00
.10
0.2
0
2
Time
SI.
are
a.X
0 1 2 3 4 5 6 7
0.0
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.10
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SI.
are
a.X
0 1 2 3 4 5 6 7
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.10
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Time
SI.
are
a.X
0 1 2 3 4 5 6 7
0.0
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.10
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5
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SI.
are
a.X
0 1 2 3 4 5 6 7
0.0
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.10
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SI.
are
a.X
0 1 2 3 4 5 6 7
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.10
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SI.
are
a.X
0 1 2 3 4 5 6 7
0.0
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.10
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8
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SI.
are
a.X
0 1 2 3 4 5 6 7
0.0
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.10
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9
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SI.
are
a.X
0 1 2 3 4 5 6 70
.00
0.1
00
.20
10
Time
SI.
are
a.X
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0 2 4 6
0.0
00
.05
0.1
00
.15
0.2
00
.25
0.3
0
Time
SI.
are
a.X
Group 0Group 1Marg. Group 0Marg. Goup 1
• We can also explore all individualas well as marginal average fittedcurves per group. The marginal effectwas obtained using the sampling-basedmethod.
• It is clear that Area.X is higher for mosttreated birds (group 1) compared tothe untreated birds (group 0).
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Part VI
Incomplete Data
ESALQ Course on Models for Longitudinal and Incomplete Data 185
Chapter 15
Setting The Scene
. Artificial example
. Orthodontic growth data
. Depression trial
. Notation
. Taxonomies
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15.1 Incomplete Longitudinal Data
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15.2 Scientific Question
• In terms of entire longitudinal profile
• In terms of last planned measurement
• In terms of last observed measurement
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15.3 Growth Data
• Taken from Potthoff and Roy, Biometrika (1964)
• Research question:
Is dental growth related to gender ?
• The distance from the center of the pituitary to the maxillary fissure was recordedat ages 8, 10, 12, and 14, for 11 girls and 16 boys
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• Individual profiles:
. Much variability between girls / boys
. Considerable variability within girls / boys
. Fixed number of measurements per subject
. Measurements taken at fixed time points
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15.4 The Depression Trial
• Clinical trial: experimental drug versus standard drug
• 170 patients
• Response: change versus baseline in HAMD17 score
• 5 post-baseline visits: 4–8
=Visit
Ch
an
ge
-20
-10
01
02
0
4 5 6 7 8
Visit
Ch
an
ge
-10
-8-6
-4-2
4 5 6 7 8
Standard DrugExperimental Drug
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15.5 Age-related Macular Degeneration Trial
• Pharmacological Therapy for Macular Degeneration Study Group (1997)
• An occular pressure disease which makes patients progressively lose vision
• 240 patients enrolled in a multi-center trial (190 completers)
• Treatment: Interferon-α (6 million units) versus placebo
• Visits: baseline and follow-up at 4, 12, 24, and 52 weeks
• Continuous outcome: visual acuity: # letters correctly read on a vision chart
• Binary outcome: visual acuity versus baseline ≥ 0 or ≤ 0
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• Missingness:
Measurement occasion
4 wks 12 wks 24 wks 52 wks Number %
Completers
O O O O 188 78.33
Dropouts
O O O M 24 10.00
O O M M 8 3.33
O M M M 6 2.50
M M M M 6 2.50
Non-monotone missingness
O O M O 4 1.67
O M M O 1 0.42
M O O O 2 0.83
M O M M 1 0.42
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15.6 Preview: LOCF, CC, or Direct Likelihood?
Data:20 30
10 40
75
25
LOCF:20 30
10 40
75 0
0 25=⇒
95 30
10 65=⇒ θ = 95
200= 0.475 [0.406; 0.544] (biased & too narrow)
CC:20 30
10 40
0 0
0 0=⇒
20 30
10 40=⇒ θ = 20
100= 0.200 [0.122; 0.278] (biased & too wide)
MAR:20 30
10 40
30 45
5 20=⇒
50 75
15 60=⇒ θ = 50
200= 0.250 [0.163; 0.337]
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15.7 Notation
• Subject i at occasion (time) j = 1, . . . , ni
•Measurement Yij
•Missingness indicator Rij =
1 if Yij is observed,
0 otherwise.
• Group Yij into a vector Y i = (Yi1, . . . , Yini)′ = (Y o
i ,Ymi )
Y oi contains Yij for which Rij = 1,
Y mi contains Yij for which Rij = 0.
• Group Rij into a vector Ri = (Ri1, . . . , Rini)′
• Di: time of dropout: Di = 1 + ∑nij=1Rij
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15.8 Framework
f (Y i, Di|θ,ψ)
Selection Models: f (Y i|θ) f (Di|Y oi ,Y
mi ,ψ)
MCAR −→ MAR −→ MNAR
f (Di|ψ) f (Di|Y oi ,ψ) f (Di|Y o
i ,Ymi ,ψ)
Pattern-Mixture Models: f (Y i|Di,θ)f (Di|ψ)
Shared-Parameter Models: f (Y i|bi, θ)f (Di|bi,ψ)
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Chapter 19
Creating Monotone Missingness
• When missingness is non-monotone, one might think of several mechanismsoperating simultaneously:
. A simple (MCAR or MAR) mechanism for the intermittent missing values
. A more complex (MNAR) mechanism for the missing data past the moment ofdropout
• Analyzing such data are complicated, especially with methods that apply todropout only
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• Solution:
. Generate multiple imputations that render the datasets monotone missing, byincluding into the MI procedure:
mcmc impute = monotone;
. Apply method of choice to the so-completed multiple sets of data
• Note: this is different from the monotone method in PROC MI, intended to fullycomplete already monotone sets of data
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Part VII
Topics in Methods and Sensitivity Analysis for IncompleteData
ESALQ Course on Models for Longitudinal and Incomplete Data 260
Chapter 20
An MNAR Selection Model and Local Influence
. The Diggle and Kenward selection model
. Mastitis in dairy cattle
. An informal sensitivity analysis
. Local influence to conduct sensitivity analysis
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20.1 A Full Selection Model
MNAR :∫f (Y i|θ)f (Di|Y i,ψ)dY m
i
f (Y i|θ)
Linear mixed model
Y i = Xiβ + Zibi + εi
f (Di|Y i,ψ)
Logistic regressions for dropout
logit [P (Di = j | Di ≥ j, Yi,j−1, Yij)]
= ψ0 + ψ1Yi,j−1 + ψ2Yij
Diggle and Kenward (JRSSC 1994)
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20.2 Mastitis in Dairy Cattle
• Infectious disease of the udder
• Leads to a reduction in milk yield
• High yielding cows more susceptible?
• But this cannot be measured directly be-cause of the effect of the disease: ev-idence is missing since infected causehave no reported milk yield
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20.2.1 A Version for the Mastitis Data
•Model for milk yield:
Yi1
Yi2
∼ N
µ
µ + ∆
,
σ21 ρσ1σ2
ρσ1σ2 σ21
•Model for mastitis:
logit [P (Ri = 1|Yi1, Yi2)] = ψ0 + ψ1Yi1 + ψ2Yi2
= 0.37 + 2.25Yi1 − 2.54Yi2
= 0.37 − 0.29Yi1 − 2.54(Yi2 − Yi1)
• LR test for H0 : ψ2 = 0 : G2 = 5.11
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20.3 Criticism −→ Sensitivity Analysis
“. . . , estimating the ‘unestimable’ can be accomplished only by makingmodelling assumptions,. . . . The consequences of model misspeci-fication will (. . . ) be more severe in the non-random case.” (Laird1994)