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Tutorial III: Joint Models for Multivariate Longitudinal Data · 2.3 Multivariate Models •...
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Tutorial III:
Joint Models for Multivariate Longitudinal Data
Joint Modeling and BeyondHasselt 2016
Geert Verbeke
Interuniversity Institute for Biostatistics and statistical Bioinformatics
http://perswww.kuleuven.be/geert verbeke
Contents
1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Approaches to Simultaneously Analyze Multiple Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Random-effects Models for High-dimensional Multivariate Longitudinal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Joint Modeling and Beyond: Hasselt 2016 i
Chapter 1
Examples
. Example 1: Hearing Data
. Example 2: Fitness Data
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1.1 Example 1: Hearing Data∗
∗Fieuws & Verbeke, Biometrics 2006
• Threshold sound pressure levels (dB), on both ears,11 frequencies: 125 → 8000 Hz
• Observations from 603 males, with up to 15 obs./subject.
× 603
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• Research questions:
. Is the relation between hearing loss and age the same for all frequencies?
. How are subject-specific evolutions for the different frequencies related?
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1.2 Example 2: Fitness Data∗
∗Fieuws, Verbeke, Boen, & Delecluse, Applied Statistics 2006
• Intervention study on 105 elderly participants
• Randomization:
. classical fitness: 3 weekly visits to gym
. distance coaching program with emphasis on incorporating physical activities indaily life
• Aim is to study the effect on psycho-cognitive functioning
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• Psycho-cognitive functioning: 106 dichotomised items, 7 different questionnaires,each measuring a latent component of psycho-cognitive functioning:
1. Physical well-being (10)
2. Psychological well-being (14)
3. Self-esteem (10)
4. Physical self-perception (30)
5. Degree of opposition to physical activities (21)
6. Perceived self-efficacy towards physical activity (5)
7. Motivation for intervention program (16)
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• Research questions:
. Is there an overall treatment effect?
. How are the various components of psycho-cognitive functioning associated?
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Chapter 2
Approaches to Simultaneously Analyze Multiple Outcomes
. Introduction and notation
. Why joint modeling ?
. Multivariate models
. Conditional models
. Shared-parameter models
. Random-effects models
. Methods based on dimension reduction
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2.1 Introduction and Notation
• Let Y1 and Y2 be two outcomes measured on a number of subjects for which jointmodeling is of scientific interest.
• We focus on settings where multiple measurements are available for both,potentially but not necessarily longitudinal:
. E.g., Hearing threshold at 125 Hz and hearing threshold at 500 Hz
. E.g., Physical well-being and Psychological well-being
• Same ideas can be applied if only one observation is available:
. E.g., Longitudinal outcome modeled jointly with time-to-event
. E.g., Longitudinal outcome modeled jointly with dropout indicator
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• We will discuss various approaches possible to construct a joint density f (y1, y2)of (Y1, Y2)
• Extensions to more than 2 outcomes are (relatively) straightforward
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2.2 Why joint modeling ?
• Joint tests for fixed effects (e.g., common average trend)
• Interest in association structure (e.g., association of evolutions)
• Modelling changes in shape
• Improving classification results
• . . .
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2.3 Multivariate Models
• General idea: Specify f (y1, y2) directly
• Advantages:
. Allows for direct inferences for marginal characteristics of Y1, Y2, and theirassociations
. Symmetric treatment of Y1 and Y2
• Disadvantages:
. Difficult with Y1 and Y2 of a different type
. Difficult for unbalanced data since association between Y1 and Y2 needs to bemodeled directly
. Difficult to extend to higher dimensions
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2.4 Conditional Models
• General idea: Factorize f (y1, y2) as
f (y1, y2) = f (y1|y2)f (y2) = f (y2|y1)f (y1)
• Advantage:
. Modeling tasks reduced to specifying models for each of the outcomesseparately
• Disadvantages:
. With Y1 and Y2, specifying f (y1|y2) requires careful reflection about plausibleassociations between response Y1 and time-varying covariate Y2
. No direct marginal inferences.
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. For example, based on f (y1, y2) = f (y1|y2)f (y2), E(Y1) requires
E(Y1) = E[E(Y1|Y2)] =
∫ [∫y1f (y1|y2) dy1
]f (y2) dy2,
. E(Y1) not necessarily of the same parametric form as E(Y1|Y2)(e.g., logistic)
. Effects on Y1 may be attenuated by conditioning on Y2.
. Compatible specification of f (y1|y2)f (y2) and f (y2|y1)f (y1) often requiresdirect specification of f (y1, y2)
. Higher dimensions: Many possible factorizations
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• Typical example
. Selection models for longitudinal data subject to informative dropout
. Marginal model f (y1) for the longitudinal outcome Y1
. Conditional model for dropout time Y2:
P [Y2 = t | Y1(t), Y1(t − 1), . . .]
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2.5 Shared-parameter Models
• General idea: Latent variables, shared by Y1 and Y2 imply associations
• Let b denote a vector of random effects, with density f (b) (often normal)
• Assume conditional independence: Y1⊥⊥Y2|b
• Joint density of (Y1, Y2) obtained from
f (y1, y2) =
∫f (y1, y2|b) db =
∫f (y1|b)f (y2|b)f (b) db
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• Advantages:
. Y1 and Y2 can be of different type
. Parameters in joint model f (y1, y2) have the same interpretation as in the‘univariate’ models f (y1) and f (y2)
. Extension to higher dimensions very straightforward
• Disadvantage:
. Very strong assumptions about the association between Y1 and Y2
. Assume random-intercepts models for Y1 and Y2:
Y1(t) = β1 + b + β2t + e1(t)
Y2(t) = β3 + γb + β4t + e2(t)
with e1(t) ∼ N (0, σ2
1), e2(t) ∼ N (0, σ2
2), and b ∼ N (0, σ2
b)
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. Implied association structure (s 6= t):
Corr{Y1(s), Y1(t)} =σ2
b
σ2
b + σ21
Corr{Y2(s), Y2(t)} =γ2σ2
b
γ2σ2
b + σ22
Corr{Y1(s), Y2(t)} =γσ2
b√σ2
b + σ2
1
√γ2σ2
b + σ2
2
=√
Corr{Y1(s), Y1(t)}√
Corr{Y2(s)Y2(t)}
. Association between Y1 and Y2 directly follows from association structures forY1 and Y2
. In some cases this is problematic, especially in higher dimensions
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• Typical example:
. Longitudinal outcome Y1 and time-to-event Y2
. Assume a mixed model for Y1:
Y1(t) = (β1 + b1) + (β2 + b2)t + e1(t)
with b = (b1, b2)′ ∼ N (0, D)
. Assume proportional hazard model for Y2 with hazard depending on b:
limh→0
Pr{t ≤ T < t + h|T ≥ t, b} = = λ0(t) exp{αg(b)}
for some pre-specified function g(b).
. Hazard depends on some feature g(b) of the longitudinal trajectories.
. Relation between Y1 and Y2 addressed via inference for α
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2.6 Random-effects Models
• General idea: Separate but correlated latent variables for Y1 and Y2
• Let b1 and b2 denote vectors of random effects, with joint density f (b1, b2)(often normal)
• Assume conditional independence: Y1⊥⊥Y2|(b1, b2)
• Joint density of (Y1, Y2) obtained from
f (y1, y2) =
∫f (y1, y2|b) db =
∫ ∫f (y1|b1)f (y2|b2)f (b1, b2) db1db2
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• Advantages:
. Same as with shared-parameter models
. Less strict assumtptions about the association between Y1 and Y2
. Assume random-intercepts models for Y1 and Y2:
Y1(t) = β1 + b1 + β2t + e1(t)
Y2(t) = β3 + b2 + β4t + e2(t)
with e1(t) ∼ N (0, σ2
1), e2(t) ∼ N (0, σ2
2), and (b1, b2) ∼ N (0, D)
. Similar expressions for Corr{Y1(s), Y1(t)} and Corr{Y2(s), Y2(t)} as before, but
Corr{Y1(s), Y2(t)} = Corr(b1, b2)√
Corr{Y1(s), Y1(t)}√
Corr{Y2(s), Y2(t)}≤
√Corr{Y1(s), Y1(t)}
√Corr{Y2(s), Y2(t)}
does not directly follow from the association structures for Y1 and Y2
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• Disadvantage:
. Dimensionality of b increases with the number of outcomes modeled, hencealso the integration in
f (y1, . . . , yk) =
∫. . .
∫f (y1|b1) . . . f (yk|bk)f (b1, . . . , bk) db1 . . . dbk
=⇒ Computational problems (addressed later)
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2.7 Methods Based on Dimension Reduction
• General idea: Use a factor-analytic, or principal-component type, analysis tofirst reduce the dimensionality of the response vector.
• In a second stage, the principal factors are analyzed using any of the classical(longitudinal) models.
• Advantage:
. Standard techniques can be used to analyze the principal factors
• Disadvantages:
. Inferences about principal factor(s), not about original outcome variables.
. Very strong restrictions needed in cases of highly unbalanced longitudinal data:
∗ unequal numbers of measurements for different subjects
∗ observations taken at arbitrary time points.
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Chapter 3
Random-effects Models for High-dimensional MultivariateLongitudinal Data
. A random-effects model
. A pairwise model fitting approach
. Applications
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3.1 A Random-effects Model
• We now consider the setting of modeling multivariate longitudinal data
• Let Y1i(t), . . . , Ymi(t) be the m outcomes measured on subject i, at time point t
• Outcomes can be of different types:
. continuous
. binary
. counts
. . . .
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• Our requirements:
. Inferences for original outcomes
. Direct marginal inferences
. Separate ‘univariate’ models areimplied by ‘multivariate’ model
. Different types of outcomes possible
=⇒ Random-effects approach
. No restriction on dimensionality}
=⇒ Computational problem !
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• As an example, re-consider the hearing data:
. Linear mixed model for each outcome separately:
Yi(t) = (β1 + β2 Fagei + β3 Fage2
i + ai)
+ (β4 + β5 Fagei + bi) t + β6 visit1(t) + εi(t)
. Joint model:
Y1i(t) = µ1(t) + a1i + b1it + ε1i(t)
Y2i(t) = µ2(t) + a2i + b2it + ε2i(t)
...
Y22i(t) = µ22(t) + a22i + b22it + ε22i(t)
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. Distributional assumptions:
(a1i, a2i, . . . , a22i, b1i, b2i, . . . , b22i)′ ∼ N (0,D44×44)
(ε1i(t), ε2i(t), . . . , ε22i(t))′ ∼ N (0,Σ22×22) , for all t
. Full multivariate joint model:
∗ 44 × 44 covariance matrix for random effects
∗ 22 × 22 covariance matrix for error components
∗ 990 + 253 = 1243 covariance parameters
=⇒ Computational problems!
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• As an example, re-consider the fitness data:
. Random-effects logistic regression for each outcome:
logit{P (Yij = 1)} = α + βDCi + bi
. Joint model:
logit{P (Yij1 = 1)} = α1 + β1DCi + bi1
logit{P (Yij2 = 1)} = α2 + β2DCi + bi2
...
logit{P (Yij7 = 1)} = α7 + β7DCi + bi7
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. Distributional assumptions:
(bi1, bi2, . . . , bi7)′ ∼ N (0, D7×7)
. Full multivariate joint model:
∗ Only (?) 28 parameters in covariance matrix
∗ Numerical integration over 7-dim. random-effects distribution!
=⇒ Computational problems!
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3.2 A Pairwise Model Fitting Approach
• General idea∗:
. Estimation of all parameters does not require fitting the full multivariate model
. It is sufficient to fit the implied model for all pairs, i.e., all ‘bivariate’ models
• Fit all bivariate models:
(Y1, Y2), (Y1, Y3), . . . , (Y1, Ym), (Y2, Y3), . . . , (Y2, Ym), . . . , (Ym−1, Ym)
• Straightforward using standard software (e.g., SAS)
• Equivalent to maximizing pseudo (log-)likelihood:
p`(Θ) = `(Y1, Y2|Θ1,2) + `(Y1, Y3|Θ1,3) + . . . + `(Ym−1, Ym|Θm−1,m)
∗Fieuws & Verbeke, Biometrics 2006
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• Θp,q is the vector of parameters in the bivariate model for (Yp, Yq)
• Let Θ be the vector obtained from stacking all Θp,q
• Asymptotic properties (from pseudo likelihood theory):√
N(Θ̂ − Θ) ∼ MV N (0, J−1KJ−1)
J and K consist of first and second-order derivatives of p`.
• Some of the Θp,q contain the same parameters.
• Estimates for these parameters are obtained by averaging pair-specific estimates
• Inference is based on:√
N(A′Θ̂ − A′
Θ) ∼ MV N (0, A′J−1KJ−1A)
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• Properties of pairwise estimators:
. Consistent, high agreement with MLE (ICC > 0.95)
. Correct estimation of sampling variability, corrected for misspecified associationstructure
. Relative efficiency versus MLE:
∗ In general, minor loss of efficiency (RE > 0.9), unless with shared parameters
∗ RE independent of number of outcomes
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3.3 Example 1: Hearing Data
• Example: Interaction between the linear time effect and age.
• Estimates and standard errors:
χ2
10= 90.4, p < 0.0001 χ2
10= 110.9, p < 0.0001
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• Association between underlying random effects: D44×44 of interest
• PCA on correlation matrix of random slopes, left side:
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3.4 Example 2: Fitness Data
• Treatment effects (s.e.’s) from univariate and multivariate models:
Univariate Multivariate
Models Model
Physical well-being −0.13 (0.37) −0.12 (0.37)
Psychological well-being 1.22 (0.61) 1.00 (0.68)
Self-esteem 0.43 (0.42) 0.49 (0.39)
Physical self-perception 0.58 (0.24)∗ 0.52 (0.25)∗
Degree of opposition 0.06 (0.24) 0.07 (0.24)
Self-efficacy −0.24 (0.33) −0.22 (0.33)
Motivation −0.35 (0.16)∗ −0.34 (0.16)∗
∗p < 0.05
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• No gain from multivariate analysis if interest is in inferences for each outcomeseparately.
• Wald test for overall treatment effect: χ26
= 16.66, p = 0.011
• Correlation matrix (with variances) of random intercepts:
Physical well-being: 2.55
Psychological well-being: 0.75 4.41
Self-esteem: 0.55 0.76 3.43
Physical self-perception: 0.66 0.46 0.53 1.83
Degree of opposition: 0.19 0.12 0.23 0.38 1.16
Self-efficacy: 0.29 0.24 0.25 0.36 0.23 1.33
Motivation: 0.42 0.31 0.28 0.40 0.47 0.30 0.33
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• Special association models:
. Original model:
logit{P (Yij1 = 1)} = α1 + β1DCi + bi1
· · ·logit{P (Yij7 = 1)} = α7 + β7DCi + bi7
. Special case 1:
logit{P (Yij1 = 1)} = α1 + β1DCi + bi
logit{P (Yij2 = 1)} = α2 + β2DCi + γ2bi
· · ·logit{P (Yij7 = 1)} = α7 + β7DCi + γ7bi
(∆dev. = 533.7)
. Special case 2:
logit{P (Yij1 = 1)} = α1 + β1DCi + bi
· · ·logit{P (Yij7 = 1)} = α7 + β7DCi + bi
(∆dev. = 758.4)
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Chapter 4
Conclusions
• Random-effects models provide flexible tools for joint models:
. Inferences for ‘univariate’ outcomes with classical ‘univariate’ models
. Direct marginal inferences, no conditioning required
. Different types of outcomes, different types of models
. High dimensions can be handled
• Pairwise approach allows fitting of high-dimensional models
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Chapter 5
Related Work
. Fieuws S. and Verbeke G. (2004), ‘Joint modelling of multivariate longitudinalprofiles: Pitfalls of the random-effects approach,’ Statistics in Medicine, 23,3093-3104.
. Fieuws S. and Verbeke G. (2006), ‘Pairwise fitting of mixed models for the jointmodelling of multivariate longitudinal profiles,’ Biometrics, 62, 424-431
. Fieuws S., Verbeke G., Boen F., and Delecluse C. (2006) ‘High DimensionalMultivariate Mixed Models for Binary Questionnaire Data,’ Applied Statistics, 55,449-460.
. Fieuws S., Verbeke G., and Molenberghs G. (2007) ‘Random-effects models formultivariate repeated measures,’ Statistical Methods in Medical Research, 16,387-398.
. Fieuws S. and Verbeke G., Maes B., and Vanrenterghem Y. (2008) ‘Predictingrenal graft failure using multivariate longitudinal profiles,’ Biostatistics, 9, 419-431.
Joint Modeling and Beyond: Hasselt 2016 39
Thanks !
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