mixed models: misconceptions, pitfalls, and many opportunities · • misconceptions • problems...

Mixed models:

Misconceptions, pitfalls, and many opportunities

4th Channel Network Conference: St-Andrews, July 3-5, 2013

Geert Verbeke

I-Biostat: International Institute for Biostatistics and statistical Bioinformatics

Katholieke Universiteit Leuven & Universiteit Hasselt, Belgium

http://perswww.kuleuven.be/geert verbeke

Outline

• Mixed models in action

• Random-effects assumptions

• Mixed models in large data sets

4th Channel Network Conference: St-Andrews, July 3-5, 2013 1

I will focus on . . .

• Model formulation

• Parameter interpretation

• Misconceptions

• Problems often encountered in practice

• Random-effects distributional assumptions

• Issues with large data sets


I will NOT talk about . . .

• Estimation methods

• Inferential procedures

• Asymptotics

• Algorithms

• Model selection

• . . .


Mixed models in action


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


• 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.


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 )


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”


Logistic random-intercepts model

E[Yijk|ai, bj(i)

]= πijk =

exp[β0 + β1tk + ai + bj(i)

]

1 + exp[β0 + β1tk + ai + bj(i)

]


Average subject treated by average GP

E[Yijk|ai = 0, bj(i) = 0

]=

exp [β0 + β1tk + 0 + 0]

1 + exp [β0 + β1tk + 0 + 0]


Average evolution

E[Yijk

]= E


]


]


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∫∫


]


]

f (ai)f (bj(i)) daidbj(i)


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 !!!


Random effects predictions


Scatterplot of random effects predictions


• 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


The reverse ?

• Simulation of 1000 subjects with 5 measurements each

• Histogram of true random intercepts:


• Histogram of predictions assuming normality:

• The normal “prior” forces the predictions to satisfy normality


Conclusion

(Verbeke & Lesaffre 1996)

The normality assumption for random effectscannot be tested using their predictions

⇓

Other techniques needed


Random-effects assumptions


The general mixed model

• Conditional model for vector yi of repeated measurements:

yi|bi ∼ Fi(bi), with density fi(yi|bi),

possibly depending on unknown parameters.

• Marginal model, assuming mixing distribution G for bi:

yi with density fi(yi|G) =

∫fi(yi|b)dG(b)

• Inference based on marginal log-likelihood:

`(G) =

N∑

i=1

ln[fi(yi|G)]


The importance of G: Literature

• Neuhaus, Hauck, and Kalbfleisch (1992)• Butler and Louis (1992)• Magder and Zeger (1996)• Verbeke and Lesaffre (1996, 1997)• Heagerty and Kurland (2001)• Zhang and Davidian (2001)• Chen, Zhang, and Davidian (2002)• Agresti, Caffo, and Ohman-Strickland (2004)• Ghidey, Lesaffre, and Eilers (2004)• Ritz (2004)• Pan and Lin (2005)• Tchetgen and Coull (2006)• Litiere, Alonso, and Molenberghs (2007, 2008)• Huang (2008)• Muthen and Asparouhov (2009)• Tsonaka, Verbeke, and Lesaffre (2009)• . . .


The importance of G: Conclusions

• Mixed feelings: Sometimes important, sometimes not

• Some aspects of inference sensitive, some not

• Some models sensitive, some not

• Proposed solutions:

. ‘Omnibus’ goodness-of-fit testsProblem: No specific alternative

. Compare model to extended versionsProblem: Lack of software


Our aim

To develop a simple, widely applicable, diagnostic tool tocheck the random-effects assumption in any mixed model

To develop a simple, widely applicable, diagnostictool which indicates how the random-effects

model can be improved

Focus on models with (multivariate) normal random effects,but equally well applicable to other latent variable models


General idea

• Suppose a mixed model with specific parametric assumption for G hasbeen fitted, maximizing the marginal log-likelihood `(G)

• Question:

Can `(G) be increased considerably by replacing G byanother mixing distribution H ?

• Ideal situation:

G maximizes `(G) over all possible mixing distributions G


The gradient function and its properties

∆(G, b) =1

N

∑

i

fi(yi|b)

fi(yi|G)

G maximizes `(G) over all G

if and only if

∆(G, b) does not exceed 1.

Moreover, ∆(G, b) equals 1 in support points of G


Implications for checking normality

(Verbeke & Molenberghs, 2013)

• Graphical check for goodness-of-fit:

(1) Fit mixed model under normality for G

(2) Check whether ∆(G, b) = 1N

∑i

fi(yi|b)

fi(yi|G)≡ 1

• Attention can be restricted to region I of unique modes of all fi(yi|b)

• Pointwise confidence band around ∆(G, b) possible using CLT


Application: The onychomycosis trial

• De Backer et al (Brit. J. Dermatology, 1996), Verbeke and Molenberghs(Springer, 2000), Molenberghs and Verbeke (Springer, 2005).

• A randomized, double-blind, parallel group, multicenter longitudinalstudy for the comparison of two oral treatments (A and B) for toenaildermatophyte onychomycosis (TDO)

• 146 and 148 subjects, measured at time-points:

. During treatment: Month 0, 1, 2, 3

. After treatment: Month 6, 9, 12

• Outcome: Severity of infection (0: not severe, 1: severe)


Observed proportions


Logistic mixed model

yij ∼ Bernoulli(πij), logit(πij) = β0 + bi + β1Ti + β2tj + β3Titj

Normal model

Linear predictor: β0 -1.6306 (0.4345)

β1 -0.1146 (0.5852)

β2 -0.4041 (0.0460)

β3 -0.1613 (0.0718)

Mixing distribution: σ 4.0133 (0.3763)

−2` 1247.8


Gradient function


Logistic mixture mixed model

• Clear evidence for non-normality

• A more flexible model:

bi ∼ π1N (µ1, σ2) + π2N (µ2, σ

2) + π3N (µ3, σ2)

with π1 + π2 + π3 = 1 and π1µ1 + π2µ2 + π3µ3 = 0

• Multimodal as well as unimodal, symmetric as well as skewed


Gradient function


Results

Normal model Mixture model

Linear predictor: β0 -1.6306 (0.4345) -1.5160 (0.4854)

β1 -0.1146 (0.5852) 0.4479 (0.4306)

β2 -0.4041 (0.0460) -0.3992 (0.0466)

β3 -0.1613 (0.0718) -0.1562 (0.0758)

Mixing distribution: σ 4.0133 (0.3763) 0.8561 (0.1889)

µ1 -2.5617 (0.4831)

µ2 2.7744 (0.3146)

µ3 9.5282 (1.2788)

π1 0.5770 (0.0422)

π2 0.3779 (0.0426)

π3 0.0451 (0.0129)

−2` 1247.8 1219.5


Fitted mixing distributions G


Components in mixing distribution

• Component 1: µ1 = −2.5617, π1 = 57.70%

Patients with no severe infections at all, 163/294 = 55.44%

• Component 2: µ2 = 2.7744, π2 = 37.79%

Patients with non-constant profiles, 115/294 = 39.12%

• Component 3: µ3 = 9.5282, π3 = 4.51%

Patients with only severe infections, 16/294 = 5.44%


Gradient function to check G

• Applicable to any type of mixed model

• Applicable to check any G (mixtures, latent class, . . . )

• No computations needed additional to the fitting of the mixed model

• Straightforward multivariate extension

• Ongoing:

. Construction of a formal test based on the gradient

. Studying operating characteristics


Mixed models in large data sets


Large data sets

Measurements → 1 2 3 4 n

↓ Subjects#1 • • • • • • • • • • • • • • • • • • •#2 • • • • • • • • • • • • • • • • • • •#3 • • • • • • • • • • • • • • • • • • •#4 • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •

#N • • • • • • • • • • • • • • • • • • •

N large, or n large, or both ?


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


Large N


↓ Subjects#1 • • • • • • • • • • • • • • • • • • •#2 • • • • • • • • • • • • • • • • • • •#3 • • • • • • • • • • • • • • • • • • •#4 • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •

#N • • • • • • • • • • • • • • • • • • •

{

{{

=⇒ Independent sub-samples


Large n


↓ Subjects#1 • • • • • • • • • • • • • • • • • • •#2 • • • • • • • • • • • • • • • • • • •#3 • • • • • • • • • • • • • • • • • • •#4 • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •

#N • • • • • • • • • • • • • • • • • • •︸︷︷︸︸︷︷︸︸︷︷︸

=⇒ Dependent sub-samples


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


Pseudo likelihood

(Arnold & Strauss 1991)

• (Log-)Likelihood:

`(Θ) =∑

i

`(yi|Θ), Θd→ N (Θ, I−1

0 )

• Pseudo (log-)likelihood:

p`(Θ) =∑

i

∑

s

δs `(yi(s)|Θ), Θ

d→ N (Θ, I−1

0 I1I−10 )


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


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)


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!


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


Overlapping sub-samples

Measurements → 1 2 3 n

↓ Subjects#1 • • • • • • • • • • • • • • • • • • •#2 • • • • • • • • • • • • • • • • • • •#3 • • • • • • • • • • • • • • • • • • •#4 • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •

#N • • • • • • • • • • • • • • • • • • •︸︷︷︸︸︷︷︸︸︷︷︸

︸︷︷︸︸︷︷︸


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


Hearing data: Association of evolutions

• Association between underlying random effects: D44×44 of interest

• PCA on correlation matrix of random slopes, left side:


Overall 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

. Computational issues, especially in large data sets

. Model assessment more involved


Thanks !4th Channel Network Conference: St-Andrews, July 3-5, 2013 54

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