Dealing with Missing Values in Multivariate Joint Models for - - PowerPoint PPT Presentation

Dealing with Missing Values in Multivariate Joint Models for Longitudinal and Survival Data

ISCB 2020

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

Department of Biostatistics, Erasmus Medical Center

Nicole Erler

Chronic Hepatitis C

Image: https://www.hepatitisc.uw.edu/go/evaluation-staging-monitoring/natural-history/core-concept/all 1

Longitudinal Covariates

log(bilirubin) albumin GGT log(AST) log(ALT) platelets 10 20 30 10 20 30 10 20 30 500 1000 1000 2000 3000 2 4 6 8 20 40 60 2.5 5.0 7.5 10.0 2 4 6

years since diagnosis

2

Baseline Covariates

No Yes NA

alcohol (4.2% NA)

frequency 100 200 300 400 Negative Positive NA

anti−HBc (10.7% NA)

frequency 50 100 200 300

BMI (19.1% NA)

frequency 15 20 25 30 35 40 45 10 20 30 40 50 60 No Yes NA

diabetes (0.1% NA)

frequency 100 200 300 400 500 600 Positive Negative NA

smoking (15.1% NA)

frequency 50 100 150 200 250 300 Male Female

sex

frequency 100 200 300 400

age

frequency 20 30 40 50 60 70 80 90 10 20 30 40 50

year

frequency 1985 1995 2005 10 20 30 40

3

Missing Values in Longitudinal Covariates

patient 5 patient 6 patient 7 patient 1 patient 2 patient 3 patient 4 10 20 30 5 10 10 15 5 10 0.0 0.5 1.0 1.5 5 10 15 0.0 2.5 5.0 7.5 10.0

follow−up (years) scaled biomarker value biomarker

log(bilirubin) log(AST) log(ALT) platelets albumin GGT

4

Multivariate Joint Model

Proportional hazards model for time until event: hi(t) = h0(t) exp  x⊤

i β(tc)

+

K

• k=1

ηki(t)⊤β(tv)

k

• 



time constant time varying 5

Multivariate Joint Model

Proportional hazards model for time until event: hi(t) = h0(t) exp  x⊤

i β(tc)

+

K

• k=1

ηki(t)⊤β(tv)

k

• 



time constant time varying

Longitudinal (mixed) model for each biomarker k = 1, ...K: E(yki(t) | bki) = ηki(t) = xki(t)⊤β(k)

• fixed

effects

+ zki(t)⊤bki

• random

effects 5

Multivariate Joint Model

Proportional hazards model for time until event: hi(t) = h0(t) exp

• x⊤

i β(tc)

+

K

• k=1

ηki(t)⊤β(tv)

k

• time

constant time varying

Longitudinal (mixed) model for each biomarker k = 1, ...K: E(yki(t) | bki) = ηki(t) = xki(t)⊤β(k)

• fixed

effects

+ zki(t)⊤bki

• random

effects

Missing values in (baseline) covariates.

5

Imputation of Missing Covariates

Imputation of a (baseline) variable xi: ➡ sample from the predictive distribution of the missing values given the observed values

6

Imputation of Missing Covariates

Imputation of a (baseline) variable xi: ➡ sample from the predictive distribution of the missing values given the observed values p(xi | everything else

• )
• ther baseline variables

repeatedly measured variables (incl. outcomes) survival outcome

6

Imputation of Missing Covariates

Imputation of a (baseline) variable xi: ➡ sample from the predictive distribution of the missing values given the observed values p(xi | everything else

• )
• ther baseline variables

repeatedly measured variables (incl. outcomes) survival outcome ➡ We cannot directly specify the (correct) imputation model!

6

Imputation of Missing Covariates

Idea: ◮ specify the joint distribution p(everything) ◮ derive p(xi | everything else) from p(everything)

7

Imputation of Missing Covariates

Idea: ◮ specify the joint distribution p(everything) ◮ derive p(xi | everything else) from p(everything) But: p(everything) = p(survival outcome, longitudinal outcomes, longitudinal covariates, baseline covariates, random effects, parameters) = p(T, D, y, X, b, θ) Does this really solve anything?

7

Imputation of Missing Covariates

Idea: ◮ specify the joint distribution p(everything) ◮ derive p(xi | everything else) from p(everything) But: p(everything) = p(survival outcome, longitudinal outcomes, longitudinal covariates, baseline covariates, random effects, parameters) = p(T, D, y, X, b, θ) Does this really solve anything? Yes, it does!

7

Fully Bayesian Analysis & Imputation

From probability theory: p(A, B) = p(A | B) p(B)

8

Fully Bayesian Analysis & Imputation

From probability theory: p(A, B) = p(A | B) p(B) Joint distribution p(T, D, y, X, b, θ) = p(T, D | X, b, θ) p(y | X, b, θ) p(X | θ) p(b | θ) p(θ)

8

Fully Bayesian Analysis & Imputation

From probability theory: p(A, B) = p(A | B) p(B) Joint distribution p(T, D, y, X, b, θ) = p(T, D | X, b, θ)

• survival

model

p(y | X, b, θ) p(X | θ) p(b | θ) p(θ)

8

Fully Bayesian Analysis & Imputation

From probability theory: p(A, B) = p(A | B) p(B) Joint distribution p(T, D, y, X, b, θ) = p(T, D | X, b, θ)

• survival

model

p(y | X, b, θ)

• multivariate

longitudinal model

p(X | θ) p(b | θ) p(θ)

8

Fully Bayesian Analysis & Imputation

From probability theory: p(A, B) = p(A | B) p(B) Joint distribution p(T, D, y, X, b, θ) = p(T, D | X, b, θ)

• survival

model

p(y | X, b, θ)

• multivariate

longitudinal model

• analysis model

p(X | θ) p(b | θ)

• random

effects

p(θ)

• priors

8

Fully Bayesian Analysis & Imputation

From probability theory: p(A, B) = p(A | B) p(B) Joint distribution p(T, D, y, X, b, θ) = p(T, D | X, b, θ)

• survival

model

p(y | X, b, θ)

• multivariate

longitudinal model

• analysis model

p(X | θ)

• imputation

part

p(b | θ)

• random

effects

p(θ)

• priors

8

Fully Bayesian Analysis & Imputation

Imputation part p(X | θ) = p(x1, . . . , xp, Xcompl. | θ) = p(x1 | Xcompl., x2, x3, . . . , xp, θ) p(x2 | Xcompl., x3, . . . , xp, θ) . . . p(xp | Xcompl., θ)

9

Fully Bayesian Analysis & Imputation

Imputation part p(X | θ) = p(x1, . . . , xp, Xcompl. | θ) = p(x1 | Xcompl., x2, x3, . . . , xp, θ) p(x2 | Xcompl., x3, . . . , xp, θ) . . . p(xp | Xcompl., θ) Estimation: via MCMC ➡ Gibbs sampling (using Metropolis-Hastings, ...)

9

Fully Bayesian Analysis & Imputation

Imputation part p(X | θ) = p(x1, . . . , xp, Xcompl. | θ) = p(x1 | Xcompl., x2, x3, . . . , xp, θ) p(x2 | Xcompl., x3, . . . , xp, θ) . . . p(xp | Xcompl., θ) Estimation: via MCMC ➡ Gibbs sampling (using Metropolis-Hastings, ...) Software: Implemented in the R package JointAI (using JAGS)

9

In Practice: Analysis of the HCV Data

library("JointAI") library("splines") fmla <- list( # formula for survival model Surv(etime, event) ~ age + sex + alc + smoke + BMI + DM + year + logBili + logALT + logAST + Plt, # formulas for the longitudinal outcomes logBili ~ age + sex + time + (time | id), logAST ~ age + sex + ns(time, df = 5) + (ns(time, df = 5) | id), logALT ~ age + sex + ns(time, df = 3) + (ns(time, df = 3) | id), Plt ~ age + sex + ns(time, df = 3) + (ns(time, df = 3) | id) )

10

In Practice: Analysis of the HCV Data

library("JointAI") library("splines") fmla <- list( # formula for survival model Surv(etime, event) ~ age + sex + alc + smoke + BMI + DM + year + logBili + logALT + logAST + Plt, # formulas for the longitudinal outcomes logBili ~ age + sex + time + (time | id), logAST ~ age + sex + ns(time, df = 5) + (ns(time, df = 5) | id), logALT ~ age + sex + ns(time, df = 3) + (ns(time, df = 3) | id), Plt ~ age + sex + ns(time, df = 3) + (ns(time, df = 3) | id) ) mod <- JM_imp(fmla, data = HCVdata, timevar = "time", n.iter = 2000)

10

In Practice: Analysis of the HCV Data

Additional options: ◮ covariate model types ◮ hyper-parameters ◮ number of chains & thinning interval ◮ ... Additional features: ◮ use of auxiliary variables ◮ use of ridge shrinkage priors ◮ multi-level settings (e.g., multi-center) ◮ ... For more info, see https://nerler.github.io/JointAI

11

Connecting Models

Longitudinal ➡ Survival Longitudinal ➡ Longitudinal

12

Connecting Models

Longitudinal ➡ Survival type of association ◮ underlying value ηki(t) ◮ slope ◮ cumulative effect ◮ time-lag ◮ ... ◮ combination of the above Longitudinal ➡ Longitudinal

12

Connecting Models

Longitudinal ➡ Survival type of association ◮ underlying value ηki(t) ◮ slope ◮ cumulative effect ◮ time-lag ◮ ... ◮ combination of the above Longitudinal ➡ Longitudinal ◮ independent

12

Connecting Models

Longitudinal ➡ Survival type of association ◮ underlying value ηki(t) ◮ slope ◮ cumulative effect ◮ time-lag ◮ ... ◮ combination of the above Longitudinal ➡ Longitudinal ◮ independent ◮ correlated random effects

endogenous dimensionality (136 elements!) linear association

12

Connecting Models

Longitudinal ➡ Survival type of association ◮ underlying value ηki(t) ◮ slope ◮ cumulative effect ◮ time-lag ◮ ... ◮ combination of the above Longitudinal ➡ Longitudinal ◮ independent ◮ correlated random effects

endogenous dimensionality (136 elements!) linear association

◮ fixed effects

potentially non-linear exogenous

12

(Interim) Conclusion: Does it work?

◮ Theoretically:

(if no assumptions are violated)

13

(Interim) Conclusion: Does it work?

◮ Theoretically:

(if no assumptions are violated)

◮ Practically:

available in software computationally feasible:

713 ∼20,000 rows complex model (4 dependent outcomes) 10,000 iterations 6.5h

13

(Interim) Conclusion: Does it work?

◮ Theoretically:

(if no assumptions are violated)

◮ Practically:

available in software computationally feasible:

713 ∼20,000 rows complex model (4 dependent outcomes) 10,000 iterations 6.5h

◮ Empirically: to be continued...

13

Thank you for your attention.

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