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Imputation of Incomplete Covariates in Longitudinal Data Can Bayesian non-parametric methods prevent model-misspecification? Nicole Erler and Dimitris Rizopoulos Erasmus Medical Center, Rotterdam 15 July 2019 Nicole Erler and Dimitris

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Imputation of Incomplete Covariates in Longitudinal Data

Can Bayesian non-parametric methods prevent model-misspecification? Nicole Erler and Dimitris Rizopoulos

Erasmus Medical Center, Rotterdam

15 July 2019

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 1

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Motivation

What are risk factors for diabetic retinopathy?

important predictors: blood pressure haemoglobin A1c (HbA1c)

  • ther covariates:

age at baseline gender diabetes duration smoking history & status

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 2

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Motivation

Challenge:

Missing values retinopathy grade: 43% blood pressure: 20% HbA1c: 20% diabetes duration: 11% smoking history: 33% smoking status: 28%

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 3

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Motivation

Challenge:

Missing values retinopathy grade: 43% blood pressure: 20% HbA1c: 20% diabetes duration: 11% smoking history: 33% smoking status: 28%

Solution?

Multiple Imputation

MICE / FCS Joint Model (e.g. multivariate normal)

Fully Bayesian ...

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 3

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Fully Bayesian Analysis & Imputation

Joint distribution p(y | X, b, θ)

  • analysis

model p(X | θ)

  • imputation

part p(b | θ)

  • random

effects p(θ)

  • priors

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 4

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Fully Bayesian Analysis & Imputation

Joint distribution p(y | X, b, θ)

  • analysis

model p(X | θ)

  • imputation

part p(b | θ)

  • random

effects p(θ)

  • priors

Imputation part p(

X

  • x1, . . . , xp, Xcompl. | θ)

= p(x1 | Xcompl., θ) p(x2 | x1, Xcompl., θ) p(x3 | x1, x2, Xcompl., θ) . . .

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 4

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Fully Bayesian Analysis & Imputation

Joint distribution p(y | X, b, θ)

  • analysis

model p(X | θ)

  • imputation

part p(b | θ)

  • random

effects p(θ)

  • priors

Imputation part p(

X

  • x1, . . . , xp, Xcompl. | θ)

= p(x1 | Xcompl., θ) p(x2 | x1, Xcompl., θ) p(x3 | x1, x2, Xcompl., θ) . . . Software Implemented in the R package JointAI

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 4

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Handling Missing Values

Assumptions about association structure ➡ linear, additive conditional distribution ➡ normal (for continuous) missingness process ➡ ignorable

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 5

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Handling Missing Values

Assumptions about association structure ➡ linear, additive conditional distribution ➡ normal (for continuous) missingness process ➡ ignorable Violation of the implied assumptions may result in bias!

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 5

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Real Data

Non-linear evolutions over time

HbA1c retinopathy SBP

5 10 15 20 5 10 15 20 5 10 15 20 142.5 145.0 147.5 150.0 152.5 −1 1 2 60 62 64 66

follow−up time (years) fitted value (lin. predictor)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 6

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Real Data

Non-linear associations among variables

retinopathy SBP

50 100 150 50 100 150 130 140 150 −6 −4 −2

HbA1 c fitted value (lin. predictor)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 7

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Bayesian P-Splines

Instead of y ∼ β0 + β1x1 + . . . we assume y ∼ β0 +

d

  • ℓ=1

βℓBℓ(x1) + . . .

x1 y

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 8

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Bayesian P-Splines

Instead of y ∼ β0 + β1x1 + . . . we assume y ∼ β0 +

d

  • ℓ=1

βℓBℓ(x1) + . . .

x1 y

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 8

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Bayesian P-Splines

Instead of y ∼ β0 + β1x1 + . . . we assume y ∼ β0 +

d

  • ℓ=1

βℓBℓ(x1) + . . .

x1 y

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 8

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Bayesian P-Splines

How many Bℓ’s do we need? y ∼ β0 +

d=4

  • ℓ=1

βℓBℓ(x1) + . . .

x1 y

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 9

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Bayesian P-Splines

How many Bℓ’s do we need? y ∼ β0 +

d=30

  • ℓ=1

βℓBℓ(x1) + . . .

x1 y

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 9

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Bayesian P-Splines

Idea: Use many functions but restrict neighboring β’s to be similar: (β1, . . . , βd) ∼ MVN(0, 1/σ2DTD), with penalty matrix D, for example: D =

       

1 −2 1 · · · 1 −2 1 · · · ... ... ... ... ... · · · 1 −2 1 · · · 1 −2 1

       

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 10

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Simulation

Analysis model: y ∼ β0 + β1x1 + β2x2 + . . . Quadratic association between covariates: x1 ∼ α0 + α1x2 + α2x2

2 + . . .

−2 2 −2 2 −2 2 −10 10 20 30

x2 (complete covariate) x1 (incompl. covariate)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 11

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Simulation

Analysis model: y ∼ β0 + β1x1 + β2x2 + . . . Quadratic association between covariates: x1 ∼ α0 + α1x2 + α2x2

2 + . . .

−2 2 −2 2 −2 2 −10 10 20 30

x2 (complete covariate) x1 (incompl. covariate) default p−spline default p−spline default p−spline

1.00 1.25 1.50 1.75

relative bias

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 11

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Real Data

Non-normal continuous distributions

diabetes duration HbA1c

10 20 30 40 40 80 120 200 400 600 500 1000 1500 2000 2500

value count

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 12

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Mixture of normal distributions

diabetes duration HbA1c value density

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 13

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Mixture of normal distributions

diabetes duration HbA1c value density

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 13

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Dirichlet Process Mixture Model

x1i | θi ∼ F(θi) θi | G ∼ G =

∞

  • k=1

πkδθ∗

k

G | α, G0 ∼ DP

↓ (α, G0)

stick-breaking construction

e.g. x1i ∼ N(µk, σ2

k)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 14

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Dirichlet Process Mixture Model

x1i | θi ∼ F(θi) θi | G ∼ G =

∞

  • k=1

πkδθ∗

k

G | α, G0 ∼ DP

↓ (α, G0)

stick-breaking construction

e.g. x1i ∼ N(µk, σ2

k)

xi ∼ N(ηi+µk

↓

, σ2

k ↓

),

p(µk) p(σ2

k)

with ηi = α1x2i + α2x3i + . . .

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 14

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Dirichlet Process Mixture Model

x1i | θi ∼ F(θi) θi | G ∼ G =

∞

  • k=1

πkδθ∗

k

G | α, G0 ∼ DP

↓ (α, G0)

stick-breaking construction

e.g. x1i ∼ N(µk, σ2

k)

xi ∼ N(ηi+µk

↓

, σ2

k ↓

),

p(µk) p(σ2

k)

with ηi = α1x2i + α2x3i + . . . very flexible

  • ➡

➡

little contribution

  • Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven

14

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Mixture of Polya Trees

0.0 0.1 0.2 0.3 0.4 10 20 30 40 50

diabetes duration density

Beta(a0, a0)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 15

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Mixture of Polya Trees

0.0 0.1 0.2 0.3 0.4 10 20 30 40 50

diabetes duration density

Beta(a0, a0) Beta(a1, a1) Beta(a1, a1)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 15

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Mixture of Polya Trees

0.0 0.1 0.2 0.3 0.4 10 20 30 40 50

diabetes duration density

Beta(a0, a0) Beta(a1, a1) Beta(a1, a1) Beta(a2, a2) Beta(a2, a2)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 15

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Mixture of Polya Trees

0.0 0.1 0.2 0.3 0.4 10 20 30 40 50

diabetes duration density

Beta(a0, a0) Beta(a1, a1) Beta(a1, a1) Beta(a2, a2) Beta(a2, a2) Beta(a3, a3) Beta(a3, a3)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 15

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Mixture of Polya Trees

0.0 0.1 0.2 0.3 0.4 10 20 30 40 50

diabetes duration density

Beta(a0, a0) Beta(a1, a1) Beta(a1, a1) Beta(a2, a2) Beta(a2, a2) Beta(a3, a3) Beta(a3, a3)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 15

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Mixture of Polya Trees

0.0 0.1 0.2 0.3 0.4 10 20 30 40 50

diabetes duration density

Beta(a0, a0) Beta(a1, a1) Beta(a1, a1) Beta(a2, a2) Beta(a2, a2) Beta(a3, a3) Beta(a3, a3)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 15

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Mixture of Polya Trees

0.0 0.1 0.2 0.3 0.4 10 20 30 40 50

diabetes duration density

Beta(a0, a0) Beta(a1, a1) Beta(a1, a1) Beta(a2, a2) Beta(a2, a2) Beta(a3, a3) Beta(a3, a3)

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 15

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Practical Issues & Ideas

flexible fit needs observed data everywhere

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 16

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Practical Issues & Ideas

flexible fit needs observed data everywhere

covariate incomplete covariate

missing

  • bserved

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 16

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Practical Issues & Ideas

flexible fit needs observed data everywhere

covariate incomplete covariate

missing

  • bserved

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 16

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Practical Issues & Ideas

flexible fit needs observed data everywhere computational time

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 16

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Practical Issues & Ideas

flexible fit needs observed data everywhere computational time Ideas: posterior predictive checks

χ2 type of tests Kolmogorov-Smirnoff test? discordance tests?

feasibility checks before running the model?

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 16

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Preliminary Conclusion Can Bayesian non-parametric methods prevent reduce model-misspecification?

Nicole Erler and Dimitris Rizopoulos, ISCB 2019, Leuven 17

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Thank you for your attention.

  • n.erler@erasmusmc.nl
  • N_Erler
  • NErler
  • www.nerler.com