mixed models with applications to large data sets€¦ · • non-linear models (pk/pd data) •...
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Mixed models
with applications to large data sets
Geert Verbeke
L-Biostat: Leuven Biostatistics and statistical Bioinformatics Centre
Katholieke Universiteit Leuven, Belgium
http://perswww.kuleuven.be/geert verbeke
Outline
• Mixed models, multi-level models, or something else ?
• Mixed models in action: The Diabetes Project Leuven
• Mixed models in large data sets
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I will focus on . . .
• Model formulation
• Parameter interpretation
• Misconceptions
• Problems often encountered in practice
• Issues with large data sets
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I will NOT talk about . . .
• Estimation methods
• Inferential procedures
• Model selection
• Diagnostics
• . . .
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Growth curves
(Goldstein 1979)
Mothers height Children
Small mothers < 155 cm 1 → 6
Medium mothers [155cm; 164cm] 7 → 13
Tall mothers > 164 cm 14 → 20
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A multi-level model
• Yij is jth measurement for the ith child, taken at time tj (age)
• Level 1:
Yij = β1i + β2itj + εij
• Level 2:
β1i = β1Si + β3Mi + β5Ti + b1i
β2i = β2Si + β4Mi + β6Ti + b2i
• Assumptions: εij ∼ N (0, σ2res) and bi = (b1i, b2i)
′ ∼ N (0, D)
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Level 1 + Level 2 = Mixed model
Yij = (β1Si + β3Mi + β5Ti + b1i)
+(β2Si + β4Mi + β6Ti + b2i)tj+εij
⇓Linear regression model with fixed and random parameters
m“Mixed” model
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The general model
• Mixed / multi-level model:
Yi = Xiβ + Zibi + εi
• Implied marginal model:
Yi ∼ N (Xiβ, ZiDZ′i + σ2
resI)
• Fixed effects model systematic trendsRandom effects generate association
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Flexibilities & extensions
• Unbalanced data
• More than 2 levels:
E.g., 10 cities→ In each: 5 schools
→ In each: 2 classes→ In each: 5 students
→ Each student given a test twice
• Linear −→ • generalized linear models (logistic, count, . . . )• non-linear models (PK/PD data)• generalized non-linear models
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The Diabetes Project Leuven
(Borgermans et al. 2009)
• The impact of offering GP’s assistance of a diabetes care team,consisting of a nurse educator, a dietician, an ophthalmologist and aninternal medicine doctor, for the treatment of their diabetes patients
• GP’s randomized to one of two programs:
. LIP: Low Intervention Program (group A)
. HIP: High Intervention Program (group R)
• We consider the HIP group only
. 61 GP’s with 1577 patients
. # patients per GP between 5 and 138, with median of 47
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• Patients were measured twice:
. When the program was initiated (time T0)
. After one year (time T1)
• HbA1c: glycosylated hemoglobin:
. Molecule in red blood cells that attaches to glucose (blood sugar)
. High values reflect more glucose in blood
. Gives a good estimate of how well diabetes has been managed overlast 2 or 3 months
. Non-diabetics have values between 4% and 6%
. HbA1c above 7% means diabetes is poorly controlled, implyinghigher risk for long-term complications.
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A logistic mixed model
• Dichotomized version of HbA1c:
Y =
1 if HbA1c < 7%
0 if HbA1c ≥ 7%
• A three-level logistic mixed model:
Yijk ∼ Bernoulli(πijk)
logit(πijk) = log(
πijk
1−πijk
)= β0 + β1tk + ai + bj(i),
ai ∼ N (0, σ2GP ), bj(i) ∼ N (0, σ2
PAT )
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Fixed effects
Effect Estimate (se) p-value
Intercept β0: 0.1662 (0.0796) 0.0410
Time β1: 0.6240 (0.0812) < .0001
“Fixed effects model systematic trends”
6=“Fixed effects model average trends”
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Logistic random-intercepts model
E[Yijk|ai, bj(i)
]= πijk =
exp[β0 + β1tk + ai + bj(i)
]
1 + exp[β0 + β1tk + ai + bj(i)
]
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Average subject treated by average GP
E[Yijk|ai = 0, bj(i) = 0
]=
exp [β0 + β1tk + 0 + 0]
1 + exp [β0 + β1tk + 0 + 0]
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Average evolution
E[Yijk
]= E
exp[β0 + β1tk + ai + bj(i)
]
1 + exp[β0 + β1tk + ai + bj(i)
]
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Conclusion
Average evolution 6= Evolution average subject
• Parameters in the mixed model have a subject-specific interpretation,not a population-averaged one.
• Calculation of the marginal average population requires computation of∫∫
exp[β0 + β1tk + ai + bj(i)
]
1 + exp[β0 + β1tk + ai + bj(i)
]
f (ai)f (bj(i)) daidbj(i)
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Variance components
Effect Estimate (se) p-value
Between GP variance σ2GP : 0.1399 (0.0528) ?
Between patient variance σ2PAT : 1.1154 (0.1308) ?
!!! Tests for variance components not standard !!!
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Random effects predictions
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Scatterplot of random effects predictions
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• For each GP, we observe at most 7 different patient predictions.
• These correspond to the 7 possible response profiles:0 −→ 0, 0 −→ 1, 1 −→ 1, 0 −→ ·, 1 −→ ·, · −→ 0, and · −→ 1.
• The negative trends are also a side effect of the discrete nature of theoutcomes.
• Two patients, j1 and j2, treated by different GP’s, i1 and i2, with thesame response profile should get identical predicted probabilities
⇒ ai1 + bj1(i1)= ai2 + bj2(i2)
⇒ ai + bj(i) is constant
⇒ Observed non-normality not necessarily problematic
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The reverse ?
• Simulation of 1000 subjects with 5 measurements each
• Histogram of true random intercepts:
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• Histogram of predictions assuming normality:
• The normal “prior” forces the predictions to satisfy normality
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Conclusion
(Verbeke & Lesaffre 1996)
The normality assumption for random effectscannot be tested using their predictions
⇓Model extensions are needed
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Large data sets
Measurements → 1 2 3 4 n
↓ Subjects#1 • • • • • • • • • • • • • • • • • • •#2 • • • • • • • • • • • • • • • • • • •#3 • • • • • • • • • • • • • • • • • • •#4 • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •
#N • • • • • • • • • • • • • • • • • • •
N large, or n large, or both ?
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Situations leading to large data sets
• N large: Observational longitudinal data
• n large: Statistical genetics / functional data analysis
• N and n large: Large multivariate longitudinal data
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Large N
Measurements → 1 2 3 4 n
↓ Subjects#1 • • • • • • • • • • • • • • • • • • •#2 • • • • • • • • • • • • • • • • • • •#3 • • • • • • • • • • • • • • • • • • •#4 • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •
#N • • • • • • • • • • • • • • • • • • •
{
{{
=⇒ Independent sub-samples
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Large n
Measurements → 1 2 3 4 n
↓ Subjects#1 • • • • • • • • • • • • • • • • • • •#2 • • • • • • • • • • • • • • • • • • •#3 • • • • • • • • • • • • • • • • • • •#4 • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •
#N • • • • • • • • • • • • • • • • • • •︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸
=⇒ Dependent sub-samples
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The general split sample idea
(Molenberghs, Verbeke, & Iddi 2011)
• Split sample in M sub-samples
• Analyse each sub-sample separately
• Combine results in appropriate way
• Inference follows from pseudo likelihood ideas
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Pseudo likelihood
(Arnold & Strauss 1991)
• (Log-)Likelihood:
`(Θ) =∑
i
`(yi|Θ), Θ̂d→ N (Θ, I−1
0 )
• Pseudo (log-)likelihood:
p`(Θ) =∑
i
∑
s
δs `(yi(s)|Θ), Θ̂
d→ N (Θ, I−1
0 I1I−10 )
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Example: Multivariate longitudinal data
• 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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Linear mixed models for hearing data
• Linear mixed model for one outcome:
Yi(t) = (β1 + β2 Fagei + β3 Fage2i + 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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Joint model
• Distributional assumptions:
(a1i, a2i, . . . , a22i, b1i, b2i, . . . , b22i)′ ∼ N (0, D44×44)
(ε1i(t), ε1i(t), . . . , ε1i(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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Pairwise approach
(Fieuws & Verbeke 2006)
• Fit all 231 bivariate models using (RE)ML (SAS PROC MIXED):
(Y1, Y2), (Y1, Y3), . . . , (Y1, Y22), (Y2, Y3), . . . , (Y2, Y22), . . . , (Y21, Y22)
• Equivalent to maximizing pseudo (log-)likelihood:
p`(Θ) = `(Y1, Y2|Θ1,2) + `(Y1, Y3|Θ1,3) + . . . + `(Y21, Y22|Θ21,22)
• Inferences follow from pseudo likelihood theory
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Overlapping sub-samples
Measurements → 1 2 3 n
↓ Subjects#1 • • • • • • • • • • • • • • • • • • •#2 • • • • • • • • • • • • • • • • • • •#3 • • • • • • • • • • • • • • • • • • •#4 • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •
#N • • • • • • • • • • • • • • • • • • •︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸
︸ ︷︷ ︸ ︸ ︷︷ ︸
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Hearing data: Joint tests for fixed effects
• Example: Interaction between the linear time effect and age.
• Estimates and standard errors:
χ210 = 90.4, p < 0.0001 χ2
10 = 110.9, p < 0.0001
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Hearing data: Association of evolutions
• Association between underlying random effects: D44×44 of interest
• PCA on correlation matrix of random slopes, left side:
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Conclusions
• Mixed models provide flexible tools for hierarchical data:
. Unbalanced data
. Multiple levels
. Natural way to incorporate association by modeling variability
. Natural extension of ‘standard models’
. Large data sets can be handled (pseudo-likelihood)
• However:
. Parameter interpretation needs careful reflection
. Inference not always standard
. Model assessment more involved
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Thanks !LSD, June 7-8, 2012 38