Regression Analysis for Probabilistic Cause-of-Disease Assignment - - PowerPoint PPT Presentation
Background Models Regression Simulations Results Discussion Regression Analysis for Probabilistic Cause-of-Disease Assignment Using Case-Control Diagnostic Tests Zhenke Wu Assistant Professor of Biostatistics Research Assistant Professor
Background Models Regression Simulations Results Discussion
Zhenke Wu
Assistant Professor of Biostatistics Research Assistant Professor of Michigan Institute for Data Science (MIDAS) University of Michigan, Ann Arbor Twitter handle: @ZhenkeWu
R package “baker”: https://github.com/zhenkewu/baker
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 1 / 55
Background Models Regression Simulations Results Discussion
Pneumonia Etiology Research for Child Health (PERCH) (PERCH Study Group, Lancet 2019, In Press)
Background:
Goal:
Study details:
Sites at Sub-Saharan Africa and South Asia
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 2 / 55
Background Models Regression Simulations Results Discussion
cases
(~5,000)
controls
(~5,000)
Lung Infection
NA
NA NA
𝐽𝑀
𝑗
Nasopharyngeal PCR
Blood Culture Lung aspirate 𝑁𝑗
𝑇
Bronze- Standard (BrS) Silver- Standard (SS) Gold- Standard (GS)
(𝜄, 𝜔)
Latent Health State Measurements Measurement Precisions Specimen (S)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 3 / 55
Background Models Regression Simulations Results Discussion
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 4 / 55
Background Models Regression Simulations Results Discussion
Summary
Problem:
(CSCFs) Features:
No method has effectively estimated the etiologic distribution (“pie”) using such data.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 5 / 55
Background Models Regression Simulations Results Discussion
A Selected Review
for areas without medical death certification; Kernel smoothing for estimating sparse probability contingency table Pr[MBrS | I] (King and Lu, 2008, Stat. Sci.)
(Hoff, 2004, Biometrics); Subset clustering (Friedman and Meulman, 2004, JRSS-B). Both no pre-defined cluster labels.
sensitivities: Attributable fraction method (Bruzzi et al., 1985, AJE) based on logistic regression logit Pr[Yi = 1 | MBrS
i
, Xi] =
J
βjMBrS
ij
+ X ′
i γ
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 6 / 55
Background Models Regression Simulations Results Discussion
Joint Distribution of (Health State, Measurements)
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Background Models Regression Simulations Results Discussion
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Background Models Regression Simulations Results Discussion
Notation
1, case
i =
0, control 1, pathogen 1 ... L, pathogen L
i = (MS i1, ..., MS iJS)′ - Measurement vector
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 9 / 55
Background Models Regression Simulations Results Discussion
Model Structure (Bronze-Standard Data Only)
partial identifiability statistical information cases controls A B C D E 𝜌A 𝜌B 𝜌C 𝜌𝐸 𝜌𝐹
𝜔1
(𝐵)
𝜔1
(𝐶)
𝜔1
(𝐷)
𝜔1
(𝐸)
𝜔1
(𝐹)
𝜄1
(𝐵)
𝜔1
(𝐶)
𝜔1
(𝐷)
𝜔1
(𝐸)
𝜔1
(𝐹)
𝜔1
(𝐵)
𝜄1
(𝐶)
𝜔1
(𝐷)
𝜔1
(𝐸)
𝜔1
(𝐹)
𝜔1
(𝐵)
𝜔1
(𝐶)
𝜄1
(𝐷)
𝜔1
(𝐸)
𝜔1
(𝐹)
𝜔1
(𝐵)
𝜔1
(𝐶)
𝜔1
(𝐷)
𝜄1
(𝐸)
𝜔1
(𝐹)
𝜔1
(𝐵)
𝜔1
(𝐶)
𝜔1
(𝐷)
𝜔1
(𝐸)
𝜄1
(𝐹)
False positive rate (FPR) True positive rate (TPR) Population etiology (𝝆) disease “class” Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 10 / 55
Background Models Regression Simulations Results Discussion
pLCM
P(MBrS
[−(j,j′)] | I L = j, Y = 1)
= P(MBrS
[−(j,j′)] | I L = j′, Y = 1),
j, j′ = 1, ..., J. P(MBrS
[−j] | Y = 0)
= P(MBrS
[−j] | I L = j, Y = 1),
j = 1, ..., J
i )
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 11 / 55
Background Models Regression Simulations Results Discussion
pLCM
P0,BrS
i
=
J
j=1
j
mj 1−ψBrS
j
1−mj , P1,BrS
i′
=
J
j=1 πj·
j
mj 1−θBrS
j
1−mj
l=j
l
ml 1−ψBrS
l
1−ml , m=mBrS
i′
P1,SS
i′
=Pr(MSS
i′
=m|π,θSS )=J′
j=1 πj·
j
mj 1−θSS
j
1−mj 1{
J′ l=1 ml ≤1}, m=mSS i′
P1,GS
i′
=Pr
i′
=m|π
J
j=1 π 1{mj =1} j
1{
j mj =1}, m=mGS i′
.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 12 / 55
Background Models Regression Simulations Results Discussion
Necessity of Informative Priors on True Positive Rate
Model structure
Pr
ij
= 1
j
+ (1 − πj)ψBrS
j
matrix of model parametrization
2015, JRSS-C)
not shrinking to point mass as sample size grows
econometrics, sociology (Cf. Greenland, 2005, JRSS-A; Gustafson, 2009, JASA; Gustafson, 2005, Stat. Sci.; Manski, 2010, PNAS)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 13 / 55
Background Models Regression Simulations Results Discussion
pLCM
j
∼ Beta(c1j, c2j) - true positive rates for BrS data
j
∼ Beta(d1j, d2j) - true positive rates for SS data
j
∼ Beta(1, 1) - false positive rates for BrS data
Joint prior for γ = (π, ψBrS , θBrS , θSS)′, a priori independent: [γ] = [π][ψBrS ][θBrS ][θSS]
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 14 / 55
Background Models Regression Simulations Results Discussion
shape of joint posterior distribution of the unknowns
i -latent health state; for case i
approximate by Pr(I L
i = j | m∗, D)
=
i = j | m∗, γ) Pr(γ | m∗, D)dγ,
j = 1, ...J
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Background Models Regression Simulations Results Discussion
i = j versus I L i = ℓ given others is Rjℓ = log πj πℓ
θBrS
j
ψBrS
j
m∗j
1 − θBrS
j
1 − ψBrS
j
1−m∗j
+ log
ℓ
θBrS
ℓ
m∗ℓ 1 − ψBrS
ℓ
1 − θBrS
ℓ
1−m∗ℓ
i = j. Averaging over m∗: E[Rjℓ] = log
I(θBrS
j
; ψBrS
j
) + I(ψBrS
ℓ
; θBrS
ℓ
)
& positive if the arguments are discrepant Model structure
for discriminating against m∗j ∼ Bern(v2).
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Background Models Regression Simulations Results Discussion
Simulation: 3 Pathogens; 500 Cases/Controls; 5 Cases with GS Measure
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Background Models Regression Simulations Results Discussion
Relax the LI and Non-interference Assumption
(Mi1, ..., MiJ)′
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Background Models Regression Simulations Results Discussion
Relax the LI and Non-interference Assumption
(Mi1, ..., MiJ)′
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 18 / 55
Background Models Regression Simulations Results Discussion
Relax the LI and Non-interference Assumption
(Mi1, ..., MiJ)′
Janes, 2007, Biostatistics; Albert et al., 2001, Biometrics)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 18 / 55
Background Models Regression Simulations Results Discussion
Relax the LI and Non-interference Assumption
(Mi1, ..., MiJ)′
Janes, 2007, Biostatistics; Albert et al., 2001, Biometrics)
probability contingency table P[Mi1 = m1, ..., MiJ = mJ | Ii], ∀m = (m1, ..., mJ)′
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 18 / 55
Background Models Regression Simulations Results Discussion
Relax the LI and Non-interference Assumption
(Mi1, ..., MiJ)′
Janes, 2007, Biostatistics; Albert et al., 2001, Biometrics)
probability contingency table P[Mi1 = m1, ..., MiJ = mJ | Ii], ∀m = (m1, ..., mJ)′
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 18 / 55
Background Models Regression Simulations Results Discussion
Relax the LI and Non-interference Assumption
(Mi1, ..., MiJ)′
Janes, 2007, Biostatistics; Albert et al., 2001, Biometrics)
probability contingency table P[Mi1 = m1, ..., MiJ = mJ | Ii], ∀m = (m1, ..., mJ)′
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 18 / 55
Background Models Regression Simulations Results Discussion
Relax the LI and Non-interference Assumption
(Mi1, ..., MiJ)′
Janes, 2007, Biostatistics; Albert et al., 2001, Biometrics)
probability contingency table P[Mi1 = m1, ..., MiJ = mJ | Ii], ∀m = (m1, ..., mJ)′
Bhattacharya and Dunson, 2012, JASA)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 18 / 55
Background Models Regression Simulations Results Discussion
Relax the LI and Non-interference Assumption
(Mi1, ..., MiJ)′
Janes, 2007, Biostatistics; Albert et al., 2001, Biometrics)
probability contingency table P[Mi1 = m1, ..., MiJ = mJ | Ii], ∀m = (m1, ..., mJ)′
Bhattacharya and Dunson, 2012, JASA)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 18 / 55
Background Models Regression Simulations Results Discussion
Example: 5 Pathogens, 2 Subclasses; BrS Data Only
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Background Models Regression Simulations Results Discussion
Example: 5 Pathogens, 3 Subclasses; BrS Data Only
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Background Models Regression Simulations Results Discussion
Vj ∼ Beta(1, α); Example: K = 10, α = 1
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Background Models Regression Simulations Results Discussion
BrS Data Only
P0(Mi = m) =
K
νk
J
k
mj 1 − ψ(j)
k
1−mj , P1(Mi = m) =
J
πj
K
ηk
k
mj 1 − θ(j)
k
1−mj
ℓ=j
k
mℓ 1 − ψ(j)
k
1−mℓ ,
π ∼ Dirichlet(.5, . . . , .5), ψ(j)
k
∼ Beta(1, 1), θk ∼ Beta(c1kj, c2kj), j = 1, ..., J; k = 1, ..., ∞, Zi′ | I L
i′ = j
∼
∞
Uk
[1 − Uℓ] δk, Uk ∼ Beta(1, α0), for all cases, Zi ∼
∞
Vk
[1 − Vℓ]δk, Vk ∼ Beta(1, α0), for all controls, α0 ∼ Gamma(0.25, 0.25),
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 22 / 55
Background Models Regression Simulations Results Discussion
Simulation: LD Truth (npLCM) Estimated by Working LI Models (pLCM)
A B C D E
(A,B)
0.5 1
A B C D E
(A,C)
0.5 1
A B C D E
(A,D)
0.5 1
A B C D E
(A,E)
0.5 1
(A,B)
0.2 0.5 1 2 4
A B C D E
(B,C)
A B C D E
(B,D)
A B C D E
(B,E) (A,C)
0.2 0.5 1 2 4
Odds Ratio (log−scale)
(B,C)
CASES
C O N T R O L S
A B C D E
(C,D)
A B C D E
(C,E) (A,D)
0.2 0.5 1 2 4
(B,D) (C,D)
A B C D E
(D,E) (A,E)
0.2 0.5 1 2 4 0.5 1
(B,E)
0.5 1
(C,E)
0.5 1
(D,E)
0.5 1
A B C D E
(A,B)
0.5 1
A B C D E
(A,C)
0.5 1
A B C D E
(A,D)
0.5 1
A B C D E
(A,E)
0.2 0.5 1 2 4 0.5 1
(A,B)
A B C D E
(B,C)
A B C D E
(B,D)
A B C D E
(B,E)
0.2 0.5 1 2 4
(A,C) (B,C)
CASES
C O N T R O L S
A B C D E
(C,D)
A B C D E
(C,E)
0.2 0.5 1 2 4
(A,D) (B,D) (C,D)
A B C D E
(D,E)
0.2 0.5 1 2 4
(A,E)
0.5 1
(B,E)
0.5 1
(C,E)
0.5 1
(D,E)
0.5 1
(I) (II)
smoothed_mat
A B C D E −120 −100 −80 −60 −40 −20 20 40 60 80 100 120
Percent Relative Asymptotic Bias (PRAB)
(
0.25 0.5 0.75 1
smoothed_mat
A B C D E 0.25 0.5 0.75 1
Cases' First Subclass Weight (η1)
(a)
Marginal Class A Class B Class C Class D Class E
Controls:
Marginal
(I: weak LD) (II: strong LD)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 23 / 55
For simplicity, we assume “single-pathogen causes”, or a single relevant feature per cluster, or more visually, ”one row of green boxes per disease class”
Background Models Regression Simulations Results Discussion
Three components of a likelihood function:
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Background Models Regression Simulations Results Discussion
Three components of a likelihood function:
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 25 / 55
Background Models Regression Simulations Results Discussion
Three components of a likelihood function:
{πℓ = P(I = ℓ | Y = 1), ℓ = 1, . . . , L} ∈ SL−1;
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 25 / 55
Background Models Regression Simulations Results Discussion
Three components of a likelihood function:
{πℓ = P(I = ℓ | Y = 1), ℓ = 1, . . . , L} ∈ SL−1;
probabilities of making J binary observations M = m in a case class ℓ = 0;
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 25 / 55
Background Models Regression Simulations Results Discussion
Three components of a likelihood function:
{πℓ = P(I = ℓ | Y = 1), ℓ = 1, . . . , L} ∈ SL−1;
probabilities of making J binary observations M = m in a case class ℓ = 0;
probability table as above but for controls.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 25 / 55
Background Models Regression Simulations Results Discussion
Three components of a likelihood function:
{πℓ = P(I = ℓ | Y = 1), ℓ = 1, . . . , L} ∈ SL−1;
probabilities of making J binary observations M = m in a case class ℓ = 0;
probability table as above but for controls. Cases’ disease classes are unobserved, so the distribution of their measurements is a weighted finite-mixture model: P1 = L
ℓ=1 πℓP1ℓ
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 25 / 55
Background Models Regression Simulations Results Discussion
Three components of a likelihood function:
{πℓ = P(I = ℓ | Y = 1), ℓ = 1, . . . , L} ∈ SL−1;
probabilities of making J binary observations M = m in a case class ℓ = 0;
probability table as above but for controls. Cases’ disease classes are unobserved, so the distribution of their measurements is a weighted finite-mixture model: P1 = L
ℓ=1 πℓP1ℓ
The likelihood: L = L1 · L0 =
L
πℓ · P1ℓ(Mi; Θ, Ψ, η) ×
P0(Mi′; Ψ, ν)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 25 / 55
Background Models Regression Simulations Results Discussion
Setting η1 = 1 and ν1 = 1
Control model for multivariate binary data {Mi : where Yi = 0}:
j=1{ψj}mj{1 − ψj}1−mj = Π(m; ψ)
j=1{sj}mij{1 − sj}1−mij is the probability mass
function for a product Bernoulli distribution given the success probabilities s = (s1, . . . , sJ)⊤, 0 ≤ sj ≤ 1
absent disease, referred to as “false positive rates” (FPRs).
Local Independence: Mij ⊥ Mij′ | I = 0
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Background Models Regression Simulations Results Discussion
Model for the multivariate binary data in case class ℓ = 0
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Background Models Regression Simulations Results Discussion
Model for the multivariate binary data in case class ℓ = 0
P(Mℓ | I = ℓ, Y = 1, θ) = {θℓ}Mℓ{1 − θℓ}1−Mℓ, where θ = (θ1, . . . , θJ)⊤ are “true positive rates” (TPRs), larger than FPRs.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 27 / 55
Background Models Regression Simulations Results Discussion
Model for the multivariate binary data in case class ℓ = 0
P(Mℓ | I = ℓ, Y = 1, θ) = {θℓ}Mℓ{1 − θℓ}1−Mℓ, where θ = (θ1, . . . , θJ)⊤ are “true positive rates” (TPRs), larger than FPRs.
P(Mi[−ℓ] | Ii = ℓ, Yi = 1, ψ[−ℓ]) = Π(M[−ℓ]; ψ[−ℓ]), where a[−ℓ] represents all but the ℓ-th element in a vector a.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 27 / 55
Background Models Regression Simulations Results Discussion
Model for the multivariate binary data in case class ℓ = 0
P(Mℓ | I = ℓ, Y = 1, θ) = {θℓ}Mℓ{1 − θℓ}1−Mℓ, where θ = (θ1, . . . , θJ)⊤ are “true positive rates” (TPRs), larger than FPRs.
P(Mi[−ℓ] | Ii = ℓ, Yi = 1, ψ[−ℓ]) = Π(M[−ℓ]; ψ[−ℓ]), where a[−ℓ] represents all but the ℓ-th element in a vector a.
TPRs θ for L = J causes and J FPRs ψ.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 27 / 55
Background Models Regression Simulations Results Discussion
Model for the multivariate binary data in case class ℓ = 0
P(Mℓ | I = ℓ, Y = 1, θ) = {θℓ}Mℓ{1 − θℓ}1−Mℓ, where θ = (θ1, . . . , θJ)⊤ are “true positive rates” (TPRs), larger than FPRs.
P(Mi[−ℓ] | Ii = ℓ, Yi = 1, ψ[−ℓ]) = Π(M[−ℓ]; ψ[−ℓ]), where a[−ℓ] represents all but the ℓ-th element in a vector a.
TPRs θ for L = J causes and J FPRs ψ.
2a-2b: Local Independence (LI): Mij ⊥ Mij′ | I = ℓ = 0
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 27 / 55
Background Models Regression Simulations Results Discussion
Model for the multivariate binary data in case class ℓ = 0
P(Mℓ | I = ℓ, Y = 1, θ) = {θℓ}Mℓ{1 − θℓ}1−Mℓ, where θ = (θ1, . . . , θJ)⊤ are “true positive rates” (TPRs), larger than FPRs.
P(Mi[−ℓ] | Ii = ℓ, Yi = 1, ψ[−ℓ]) = Π(M[−ℓ]; ψ[−ℓ]), where a[−ℓ] represents all but the ℓ-th element in a vector a.
TPRs θ for L = J causes and J FPRs ψ.
2a-2b: Local Independence (LI): Mij ⊥ Mij′ | I = ℓ = 0 2a-2b. Non-interference: disease-causing pathogen(s) are more frequently detected among cases than controls (θℓ > ψℓ) and the non-causative pathogens are observed with the same rates among cases as in controls
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 27 / 55
Background Models Regression Simulations Results Discussion
In large-scale disease etiology studies:
(CSCF); Think “Pie chart” for cases
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Background Models Regression Simulations Results Discussion
In large-scale disease etiology studies:
(CSCF); Think “Pie chart” for cases
mixing distribution among the cases
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 28 / 55
Background Models Regression Simulations Results Discussion
In large-scale disease etiology studies:
(CSCF); Think “Pie chart” for cases
mixing distribution among the cases
season, a child’s age, HIV status, disease severity
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 28 / 55
Background Models Regression Simulations Results Discussion
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 29 / 55
Background Models Regression Simulations Results Discussion
presence or absence of J pathogens for subject i = 1, . . . , N.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 29 / 55
Background Models Regression Simulations Results Discussion
presence or absence of J pathogens for subject i = 1, . . . , N.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 29 / 55
Background Models Regression Simulations Results Discussion
presence or absence of J pathogens for subject i = 1, . . . , N.
etiologic fractions
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 29 / 55
Background Models Regression Simulations Results Discussion
presence or absence of J pathogens for subject i = 1, . . . , N.
etiologic fractions
different from Xi; may influence control distribution [Mi | Wi, Yi = 0]. For example, healthy controls do not have disease severity information (which can be included in Xi).
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 29 / 55
Background Models Regression Simulations Results Discussion
presence or absence of J pathogens for subject i = 1, . . . , N.
etiologic fractions
different from Xi; may influence control distribution [Mi | Wi, Yi = 0]. For example, healthy controls do not have disease severity information (which can be included in Xi).
Wi, respectively.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 29 / 55
Background Models Regression Simulations Results Discussion
Data : 494 cases and 944 controls from one site
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 30 / 55
Background Models Regression Simulations Results Discussion
Data : 494 cases and 944 controls from one site Goal a. : Estimate CSCFs at all covariate values, and assign cause-specific probabilities for each case
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 30 / 55
Background Models Regression Simulations Results Discussion
Data : 494 cases and 944 controls from one site Goal a. : Estimate CSCFs at all covariate values, and assign cause-specific probabilities for each case Goal b. : Quantify overall cause-specific disease burdens in a population, i.e., overall CSCFs π∗ = (π∗
1, . . . , π∗ L)⊤ as an
empirical average of the stratum-specific CSCFs (by X); Of policy interest (vaccine/antibiotics development and manufacture)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 30 / 55
Background Models Regression Simulations Results Discussion
Data : 494 cases and 944 controls from one site Goal a. : Estimate CSCFs at all covariate values, and assign cause-specific probabilities for each case Goal b. : Quantify overall cause-specific disease burdens in a population, i.e., overall CSCFs π∗ = (π∗
1, . . . , π∗ L)⊤ as an
empirical average of the stratum-specific CSCFs (by X); Of policy interest (vaccine/antibiotics development and manufacture) Model :
nose
(NoS) cause; So L = J + 1
cases (severe or very severe), HIV status (+/-)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 30 / 55
Background Models Regression Simulations Results Discussion
Table: The observed count (frequency) of cases and controls by age, disease severity and HIV status (1: yes; 0: no). The marginal fractions among cases and controls for each covariate are shown at the bottom. Regression results will be shown for the first two strata.
age ≥ 1 very severe (VS) HIV positive # cases (%) # controls (%) (case-only) total: 524 (100) total: 964 (100) 208 (39.7) 545 (56.5) 1 72 (13.7) 278 (28.8) 1 116 (22.1)
1 33 (6.3)
37 (7.1) 85 (8.8) 1 1 24 (4.5) 51 (5.3) 1 1 25 (4.8)
1 1 3 (0.6)
34.5% 17.0% control: 34.3%
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Background Models Regression Simulations Results Discussion
in each covariate stratum.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 32 / 55
Background Models Regression Simulations Results Discussion
in each covariate stratum. Like pLCM, the npLCM is partially-identified in each stratum, necessitating multiple sets of independent informative priors across multiple strata. Two primary issues:
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 32 / 55
Background Models Regression Simulations Results Discussion
in each covariate stratum. Like pLCM, the npLCM is partially-identified in each stratum, necessitating multiple sets of independent informative priors across multiple strata. Two primary issues: Gap 1a Unstable CSCF estimates due to sparsely-populated strata.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 32 / 55
Background Models Regression Simulations Results Discussion
in each covariate stratum. Like pLCM, the npLCM is partially-identified in each stratum, necessitating multiple sets of independent informative priors across multiple strata. Two primary issues: Gap 1a Unstable CSCF estimates due to sparsely-populated strata. Gap 1b Informative TPR priors are often elicited for a case population and rarely for each stratum; Reusing independent prior distributions of the TPRs across all the strata will lead to
decisions.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 32 / 55
Background Models Regression Simulations Results Discussion
More focus on model formulation; Inference done by ‘baker‘
Extend the npLCM to perform regression analysis in case-control disease etiology studies that
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 33 / 55
Background Models Regression Simulations Results Discussion
More focus on model formulation; Inference done by ‘baker‘
Extend the npLCM to perform regression analysis in case-control disease etiology studies that (a) incorporates controls to estimate the CSCFs (π),
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 33 / 55
Background Models Regression Simulations Results Discussion
More focus on model formulation; Inference done by ‘baker‘
Extend the npLCM to perform regression analysis in case-control disease etiology studies that (a) incorporates controls to estimate the CSCFs (π), (b) specifies parsimonious functional dependence of π upon covariates such as additivity, and
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 33 / 55
Background Models Regression Simulations Results Discussion
More focus on model formulation; Inference done by ‘baker‘
Extend the npLCM to perform regression analysis in case-control disease etiology studies that (a) incorporates controls to estimate the CSCFs (π), (b) specifies parsimonious functional dependence of π upon covariates such as additivity, and (c) correctly assesses the posterior uncertainty of the CSCF functions and the overall CSCFs π∗ by applying the TPR priors just once.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 33 / 55
Background Models Regression Simulations Results Discussion
Let three sets of parameters in an npLCM (pg.17) depend on the
which is of primary scientific interest
in controls: P0(m; w) = [M = m | W = w, I = 0],
P1ℓ(m; w) = [M = m | W = w, I = ℓ], ℓ = 1, . . . , L note Keep the specifications for the TPRs and FPRs (Θ, Ψ) as in the original npLCM.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 35 / 55
Background Models Regression Simulations Results Discussion
πℓ(X) is the primary target of inference.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 36 / 55
Background Models Regression Simulations Results Discussion
πℓ(X) is the primary target of inference.
pathogen ℓ.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 36 / 55
Background Models Regression Simulations Results Discussion
πℓ(X) is the primary target of inference.
pathogen ℓ.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 36 / 55
Background Models Regression Simulations Results Discussion
πℓ(X) is the primary target of inference.
pathogen ℓ.
πiℓ = πℓ(Xi) = exp{φℓ(Xi)}/L
ℓ′=1 exp{φℓ′(Xi)}, ℓ = 1, ..., L,
where φℓ(Xi) − φL(Xi) is the log odds of case i in disease class ℓ relative to L: log πiℓ/πiL.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 36 / 55
Background Models Regression Simulations Results Discussion
πℓ(X) is the primary target of inference.
pathogen ℓ.
πiℓ = πℓ(Xi) = exp{φℓ(Xi)}/L
ℓ′=1 exp{φℓ′(Xi)}, ℓ = 1, ..., L,
where φℓ(Xi) − φL(Xi) is the log odds of case i in disease class ℓ relative to L: log πiℓ/πiL.
classes symmetrically which simplifies prior specification.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 36 / 55
Background Models Regression Simulations Results Discussion
πℓ(X) is the primary target of inference.
pathogen ℓ.
πiℓ = πℓ(Xi) = exp{φℓ(Xi)}/L
ℓ′=1 exp{φℓ′(Xi)}, ℓ = 1, ..., L,
where φℓ(Xi) − φL(Xi) is the log odds of case i in disease class ℓ relative to L: log πiℓ/πiL.
classes symmetrically which simplifies prior specification.
ℓ ) = p1 j=1 f π ℓj (xj; βπ ℓj) +
x⊤γπ
ℓ
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 36 / 55
Background Models Regression Simulations Results Discussion
πℓ(X) is the primary target of inference.
pathogen ℓ.
πiℓ = πℓ(Xi) = exp{φℓ(Xi)}/L
ℓ′=1 exp{φℓ′(Xi)}, ℓ = 1, ..., L,
where φℓ(Xi) − φL(Xi) is the log odds of case i in disease class ℓ relative to L: log πiℓ/πiL.
classes symmetrically which simplifies prior specification.
ℓ ) = p1 j=1 f π ℓj (xj; βπ ℓj) +
x⊤γπ
ℓ
ℓj (·) and use
P-spline for estimating smooth functions.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 36 / 55
Background Models Regression Simulations Results Discussion
πℓ(X) is the primary target of inference.
pathogen ℓ.
πiℓ = πℓ(Xi) = exp{φℓ(Xi)}/L
ℓ′=1 exp{φℓ′(Xi)}, ℓ = 1, ..., L,
where φℓ(Xi) − φL(Xi) is the log odds of case i in disease class ℓ relative to L: log πiℓ/πiL.
classes symmetrically which simplifies prior specification.
ℓ ) = p1 j=1 f π ℓj (xj; βπ ℓj) +
x⊤γπ
ℓ
ℓj (·) and use
P-spline for estimating smooth functions. 5b. x is the subvector of the predictors x; Γπ
ℓ = (βπ ℓj, γπ ℓ ).
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 36 / 55
Background Models Regression Simulations Results Discussion
Desirable properties Model Specification:
interpretation with increasing dimensions or complex patterns
Estimation:
problem, e.g., model residual dependence [M | W , I = 0] only if necessary; model the effect of covariates only if necessary
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 37 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls
Pr(Mi = m | Wi, Ii = 0) = K
k=1 νk(Wi)Π(m; Ψk),
Ψk = (ψ(1)
k , . . . , ψ(J) k )′
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 38 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls
Pr(Mi = m | Wi, Ii = 0) = K
k=1 νk(Wi)Π(m; Ψk),
Ψk = (ψ(1)
k , . . . , ψ(J) k )′
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 38 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls
Pr(Mi = m | Wi, Ii = 0) = K
k=1 νk(Wi)Π(m; Ψk),
Ψk = (ψ(1)
k , . . . , ψ(J) k )′
j=1{sj}mij(1 − sj)1−mij
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 38 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls
Pr(Mi = m | Wi, Ii = 0) = K
k=1 νk(Wi)Π(m; Ψk),
Ψk = (ψ(1)
k , . . . , ψ(J) k )′
j=1{sj}mij(1 − sj)1−mij
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 38 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls
Pr(Mi = m | Wi, Ii = 0) = K
k=1 νk(Wi)Π(m; Ψk),
Ψk = (ψ(1)
k , . . . , ψ(J) k )′
j=1{sj}mij(1 − sj)1−mij
sample subclass indicator : Zi | Wi ∼ CategoricalK(ν(Wi)) generate measurements : Mij | Zi = k ∼ Bernoulli(ψ(j)
k ),
independently for j = 1, ..., J.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 38 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls Stick-breaking parametrization of weight functions νk(Wi) = P(Zi = k | Wi) by hk(Wi; Γν
k)
=
ik) s<k {1 − g(αν is)} ,
if k < K,
is)} ,
if k = K,
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 39 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls Stick-breaking parametrization of weight functions νk(Wi) = P(Zi = k | Wi) by hk(Wi; Γν
k)
=
ik) s<k {1 − g(αν is)} ,
if k < K,
is)} ,
if k = K, g(·) = 1/(1 + exp{−(·)})
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 39 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls Stick-breaking parametrization of weight functions νk(Wi) = P(Zi = k | Wi) by hk(Wi; Γν
k)
=
ik) s<k {1 − g(αν is)} ,
if k < K,
is)} ,
if k = K, g(·) = 1/(1 + exp{−(·)}) . We specify αν
ik via additive models:
αν
ik = µk0 + q1
fkj(Wij; βν
kj) +
W ⊤
i γν k , k = 1, . . . , K − 1.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 39 / 55
Background Models Regression Simulations Results Discussion
Regression model for controls Stick-breaking parametrization of weight functions νk(Wi) = P(Zi = k | Wi) by hk(Wi; Γν
k)
=
ik) s<k {1 − g(αν is)} ,
if k < K,
is)} ,
if k = K, g(·) = 1/(1 + exp{−(·)}) . We specify αν
ik via additive models:
αν
ik = µk0 + q1
fkj(Wij; βν
kj) +
W ⊤
i γν k , k = 1, . . . , K − 1.
Expand the smooth functions by B-spline bases with coefficients βν
kj;
w is a subvector of covariates w
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 39 / 55
Background Models Regression Simulations Results Discussion
Proposed Model
interpretability
following two features:
a) Few subclasses with effective weights (in the sense that νk(·) is bounded away from 0 and 1): a novel additive half-Cauchy prior for µk0. b) Smooth weight regression curves νk(·): by Bayesian Penalized-Splines (P-Splines) combined with mixture priors on spline coefficients to sensitively distinguish constant αν
k(·) from flexible smooth curves
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 40 / 55
Background Models Regression Simulations Results Discussion
Proposed Model
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 41 / 55
Background Models Regression Simulations Results Discussion
Proposed Model
j=1 µ∗ j0, µ∗ j0 > 0. A large µk0 for a large k.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 41 / 55
Background Models Regression Simulations Results Discussion
Proposed Model
j=1 µ∗ j0, µ∗ j0 > 0. A large µk0 for a large k.
stop for a large k
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 41 / 55
Background Models Regression Simulations Results Discussion
Proposed Model
j=1 µ∗ j0, µ∗ j0 > 0. A large µk0 for a large k.
stop for a large k
j0 to be heavy-tailed:
µ∗
j0 ∼ Cauchy+(0, sj), j = 1, . . . , K,
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 41 / 55
Background Models Regression Simulations Results Discussion
Proposed Model
j=1 µ∗ j0, µ∗ j0 > 0. A large µk0 for a large k.
stop for a large k
j0 to be heavy-tailed:
µ∗
j0 ∼ Cauchy+(0, sj), j = 1, . . . , K,
k0 and helps stop the stick-breaking at
class k.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 41 / 55
Background Models Regression Simulations Results Discussion
Proposed Model
j=1 µ∗ j0, µ∗ j0 > 0. A large µk0 for a large k.
stop for a large k
j0 to be heavy-tailed:
µ∗
j0 ∼ Cauchy+(0, sj), j = 1, . . . , K,
k0 and helps stop the stick-breaking at
class k.
approximate the observed 2J probability contingency table in finite samples
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 41 / 55
Background Models Regression Simulations Results Discussion
Simulation: with a single continuous covariate; “—”: truth, “—”: posterior samples
X-axis: covariate values Y-axis: weight; 0 to 1.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 42 / 55
Background Models Regression Simulations Results Discussion
Subclass Weight Regression: For Cases
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 43 / 55
Background Models Regression Simulations Results Discussion
Subclass Weight Regression: For Cases
The pmf for cases’ measurements: Pr(Mi = m) = L
ℓ=1 πiℓ
K
k=1 ηikΠ(Mi; pkℓ)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 43 / 55
Background Models Regression Simulations Results Discussion
Subclass Weight Regression: For Cases
The pmf for cases’ measurements: Pr(Mi = m) = L
ℓ=1 πiℓ
K
k=1 ηikΠ(Mi; pkℓ)
kℓ , j = 1, . . . , J} are positive rates for J measurements
in subclass k of disease class ℓ: p(j)
kℓ =
k
I{j=ℓ} ·
k
1−I{j=ℓ}
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 43 / 55
Background Models Regression Simulations Results Discussion
Subclass Weight Regression: For Cases
The pmf for cases’ measurements: Pr(Mi = m) = L
ℓ=1 πiℓ
K
k=1 ηikΠ(Mi; pkℓ)
kℓ , j = 1, . . . , J} are positive rates for J measurements
in subclass k of disease class ℓ: p(j)
kℓ =
k
I{j=ℓ} ·
k
1−I{j=ℓ}
k
for a causative pathogen and the FPR ψ(j)
k
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 43 / 55
Background Models Regression Simulations Results Discussion
Subclass Weight Regression: For Cases
The pmf for cases’ measurements: Pr(Mi = m) = L
ℓ=1 πiℓ
K
k=1 ηikΠ(Mi; pkℓ)
kℓ , j = 1, . . . , J} are positive rates for J measurements
in subclass k of disease class ℓ: p(j)
kℓ =
k
I{j=ℓ} ·
k
1−I{j=ℓ}
k
for a causative pathogen and the FPR ψ(j)
k
stick-breaking: ηik = hk(Wi; Γη
k), k = 1, . . . , K − 1
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 43 / 55
Background Models Regression Simulations Results Discussion
Subclass Weight Regression: For Cases
The pmf for cases’ measurements: Pr(Mi = m) = L
ℓ=1 πiℓ
K
k=1 ηikΠ(Mi; pkℓ)
kℓ , j = 1, . . . , J} are positive rates for J measurements
in subclass k of disease class ℓ: p(j)
kℓ =
k
I{j=ℓ} ·
k
1−I{j=ℓ}
k
for a causative pathogen and the FPR ψ(j)
k
stick-breaking: ηik = hk(Wi; Γη
k), k = 1, . . . , K − 1
ik: GAMs
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 43 / 55
Background Models Regression Simulations Results Discussion
Subclass Weight Regression: For Cases
The pmf for cases’ measurements: Pr(Mi = m) = L
ℓ=1 πiℓ
K
k=1 ηikΠ(Mi; pkℓ)
kℓ , j = 1, . . . , J} are positive rates for J measurements
in subclass k of disease class ℓ: p(j)
kℓ =
k
I{j=ℓ} ·
k
1−I{j=ℓ}
k
for a causative pathogen and the FPR ψ(j)
k
stick-breaking: ηik = hk(Wi; Γη
k), k = 1, . . . , K − 1
ik: GAMs
ik = αη k(Wi; Γη k ) = µk0 + q1 j=1 fkj(Wij; βη kj) +
W ⊤
i γη k, where
Γη
k = {µk0, {βη kj}, γη k} are the regression parameters.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 43 / 55
Background Models Regression Simulations Results Discussion
Subclass Weight Regression: For Cases
The pmf for cases’ measurements: Pr(Mi = m) = L
ℓ=1 πiℓ
K
k=1 ηikΠ(Mi; pkℓ)
kℓ , j = 1, . . . , J} are positive rates for J measurements
in subclass k of disease class ℓ: p(j)
kℓ =
k
I{j=ℓ} ·
k
1−I{j=ℓ}
k
for a causative pathogen and the FPR ψ(j)
k
stick-breaking: ηik = hk(Wi; Γη
k), k = 1, . . . , K − 1
ik: GAMs
ik = αη k(Wi; Γη k ) = µk0 + q1 j=1 fkj(Wij; βη kj) +
W ⊤
i γη k, where
Γη
k = {µk0, {βη kj}, γη k} are the regression parameters.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 43 / 55
Background Models Regression Simulations Results Discussion
The npLCM regression framework is then obtained as:
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 44 / 55
Background Models Regression Simulations Results Discussion
The npLCM regression framework is then obtained as:
Lreg =
i:Yi=0
K
k=1 νikΠ(Mi; Ψk).
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 44 / 55
Background Models Regression Simulations Results Discussion
The npLCM regression framework is then obtained as:
Lreg =
i:Yi=0
K
k=1 νikΠ(Mi; Ψk).
Lreg
1
=
L
πℓ(Xi; Γπ
ℓ )
K
{ηik · Π(Mi; pkℓ)} (2)
k) : The S? ? ? ?-B? ? ? ? parameterization
k)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 44 / 55
Background Models Regression Simulations Results Discussion
The npLCM regression framework is then obtained as:
Lreg =
i:Yi=0
K
k=1 νikΠ(Mi; Ψk).
Lreg
1
=
L
πℓ(Xi; Γπ
ℓ )
K
{ηik · Π(Mi; pkℓ)} (2)
k) : The S? ? ? ?-B? ? ? ? parameterization
k)
The joint likelihood for the regression model can be written as: Lreg = Lreg
1
× Lreg
0 .
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 44 / 55
Background Models Regression Simulations Results Discussion
Unknown parameters:
ℓ }),
k}) and
controls ({Γν
k}),
k }, Ψ = {ψ(j) k }).
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 45 / 55
Background Models Regression Simulations Results Discussion
Unknown parameters:
ℓ }),
k}) and
controls ({Γν
k}),
k }, Ψ = {ψ(j) k }).
To avoid potential overfitting, we a priori introduce:
prior for the intercepts {µk0}
νk(·) and ηk(·) by Bayesian Penalized-splines (Lang, 2004) combined with shrinkage priors on spline coefficients (Ni et.al, 2015) (to encourage towards constant values)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 45 / 55
Background Models Regression Simulations Results Discussion
Use Markov chain Monte Carlo (MCMC) algorithm to approximate joint posterior distribution
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 46 / 55
Background Models Regression Simulations Results Discussion
Use Markov chain Monte Carlo (MCMC) algorithm to approximate joint posterior distribution
functions of model parameters and individual latent variables
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 46 / 55
Background Models Regression Simulations Results Discussion
Use Markov chain Monte Carlo (MCMC) algorithm to approximate joint posterior distribution
functions of model parameters and individual latent variables Fit npLCMs (w/ or w/out covariates using R package baker (https://github.com/zhenkewu/baker)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 46 / 55
Background Models Regression Simulations Results Discussion
Use Markov chain Monte Carlo (MCMC) algorithm to approximate joint posterior distribution
functions of model parameters and individual latent variables Fit npLCMs (w/ or w/out covariates using R package baker (https://github.com/zhenkewu/baker)
al., 2003) from within R
unknowns
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 46 / 55
Background Models Regression Simulations Results Discussion
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 47 / 55
Background Models Regression Simulations Results Discussion
CSCF functions {πℓ(·)} (not shown here)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 47 / 55
Background Models Regression Simulations Results Discussion
CSCF functions {πℓ(·)} (not shown here)
ℓ
(empirical average) to quantify disease burdens in a population (of policy interest)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 47 / 55
Background Models Regression Simulations Results Discussion
ℓ
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 48 / 55
Background Models Regression Simulations Results Discussion
ℓ
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 49 / 55
Background Models Regression Simulations Results Discussion
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 50 / 55
Background Models Regression Simulations Results Discussion
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 51 / 55
Background Models Regression Simulations Results Discussion
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 52 / 55
Background Models Regression Simulations Results Discussion
covariates to CSCFs
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 52 / 55
Background Models Regression Simulations Results Discussion
covariates to CSCFs
controls, which is critical for assigning cause-specific probabilities to a given case
measurements with similar covariate values
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 52 / 55
Background Models Regression Simulations Results Discussion
covariates to CSCFs
controls, which is critical for assigning cause-specific probabilities to a given case
measurements with similar covariate values
etiology uncertainty estimates.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 52 / 55
Background Models Regression Simulations Results Discussion
Context: Modern large-scale etiology studies generate complex measurements of unobserved causes of disease, and have raised the analytic needs of estimating cause-specific case fractions (CSCFs)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 53 / 55
Background Models Regression Simulations Results Discussion
Context: Modern large-scale etiology studies generate complex measurements of unobserved causes of disease, and have raised the analytic needs of estimating cause-specific case fractions (CSCFs) Gap: Despite recent methodological advances, the need of describing the relationship between covariates and CSCFs, remains unmet
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 53 / 55
Background Models Regression Simulations Results Discussion
Context: Modern large-scale etiology studies generate complex measurements of unobserved causes of disease, and have raised the analytic needs of estimating cause-specific case fractions (CSCFs) Gap: Despite recent methodological advances, the need of describing the relationship between covariates and CSCFs, remains unmet Contribution: A general etiology regression framework building on npLCM that is broadly applicable to case-control studies A general framework for a class of statistical problems that can be formulated as estimating covariate-dependent class-mixing weights.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 53 / 55
Background Models Regression Simulations Results Discussion
AOS; Wu 2019);
depends on the latent state in a monotonic way (e.g., we have TPR greater than FPR in the pneumonia example)
identifibility (based on likelihood only).
ICML) and double feature allocation (Ni and Mueller, 2019, JASA)
2019, Biostatistics) and electronic health records (Ni and Mueller, 2019) and verbal autopsy (King and Lu, 2008 Stat Sci; McCormick et al., 2016, JASA)
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 54 / 55
Background Models Regression Simulations Results Discussion
Student Irena Chen Collaborators Scott Zeger Katherine O’Brien Maria Deloria-Knoll Laura Hammitt Funding Patient-Centered Outcome Research Institute [PCORI ME-1408-20318] Bill & Melinda Gates Foundation [48968] Michigan Precision Health Investigator Award National Cancer Institute (P30CA046592, U01CA229437) Some References (More at: zhenkewu.com)
1. Wu Z and Chen I (2019+). Regression Analysis of Dependent Binary Data: Estimating Disease Etiology from Case-Control Studies.
2. PERCH Study Group (2019+). Causes of severe pneumonia re- quiring hospital admission in children without HIV infection from Africa and Asia: the PERCH multi- country case-control study. The Lancet. https: // doi. org/ 10. 1016/ S0140-6736( 19) 30721-4 3. Wu Z, Deloria-Knoll M and Zeger SL (2019+). A Bayesian Approach to Restricted Latent Class Mod- els for Scientifically-Structured Clustering of Multivariate Binary Outcomes.
4. Wu Z, Deloria-Knoll M and Zeger SL (2017). Nested Partially-Latent Class Models for Estimating Disease Etiology from Case-Control Data.
5. Wu Z, Deloria-Knoll M, Hammitt LL, and Zeger SL, for the PERCH Core Team (2015). Partially Latent Class Models (pLCM) for Case-Control Studies of Childhood Pneumonia Etiology. Journal of the Royal Statistical Society: Series C (Applied Statistics). 65:97-114. Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 55 / 55
S (discrete covariate); Uniformly sample the subjects’ enrollment dates over a period of 300 days.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 56 / 55
ℓ(t, S = s)
0.0 0.2 0.4 0.6 0.8 1.0 positive rateA
case −−> case −−> control−−> control−−>
1)
etiologic fraction50.5%
44% 57.4%
2)
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1case −−>
0.0 0.2 0.4 0.6 0.8 1.0B
26.2%
20.4% 33.3%
<− Overall Pie −> <− 95% CrI −>
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1 0.0 0.2 0.4 0.6 0.8 1.0C
11.8%
7.8% 16.8%
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1 0.0 0.2 0.4 0.6 0.8 1.0D
6.3%
3.1% 10.8%
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1 0.0 0.2 0.4 0.6 0.8 1.0E
1.7%
0.2% 4.1%
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1 0.0 0.2 0.4 0.8 1.0F
1.1%
0.1% 3.1%
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1 0.0 0.2 0.4 0.8 1.0G
0.6%
0% 2.3%
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1 0.0 0.2 0.4 0.6 0.8 1.0H
0.9%
0% 4.1%
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1 0.0 0.2 0.4 0.6 0.8 1.0I
0.9%
0% 2.8%
2010:Feb−01 May−01 Aug−01 Nov−01 0.2 0.4 0.6 0.8 1Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 57 / 55
True K 0 = 2; Model fitted using a working number K = 7
(a) case
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 58 / 55
data sets simulated under 48 different scenarios
assumption, BrS measurements made on Nd cases and Nu controls for each level of X where Nd = Nu = 250 or 500.
CSCF variability across X: i) βi
0 = (0, 0, 0, 0, 0, 0),
βi
1 = (−1.5, 0, −1.5, −1.5, 0, −1.5); ii) βii 0 = (1, 0, 1, 1, 0, 1)
and βii
1 = (−1.5, 1, −1.5, −1.5, 1, −1.5)
k
= 0.95 or 0.8 and FPRs (ψ(j)
1 , ψ(j) 2 ) ∈ {(0.5, 0.05), (0.5, 0.15)}, for j = 1, . . . , J.
(γ10, γ11) = (−0.5, 1.5) and (γ20, γ21) = (1, −1.5).
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 59 / 55
A
level 1 0.5
Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20level 2 0.5
Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20empirical weights
Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20B
Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20C
Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20D
Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20E
Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20F
Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 Density 0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20Figure: Posterior distributions of the stratum-specific (Row 1 and 2) and the
covariate and L = J = 6 causes. The vertical gray lines indicate the 2.5% and 97.5% posterior quantiles, respectively; The truths are indicated by vertical blue dashed lines. Row 1-2) CSCFs by stratum (level = 1,2) and cause (A-F); Bottom) π∗
ℓ : overall population etiologic fraction for cause A-F (empirical
average of the two CSCFs above).
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 60 / 55
pop_frac_scn: 1 Nd: 250 pop_frac_scn: 2 Nd: 500 pop_frac_scn: 1 Nd: 500 pop_frac_scn: 2 psi_scn: 1 theta: 0.8 psi_scn: 2 theta: 0.8 psi_scn: 1 theta: 0.95 psi_scn: 2 theta: 0.95 A B C D E F A B C D E F A B C D E F A B C D E F −100 −50 50 100 −100 −50 50 100 −100 −50 50 100 −100 −50 50 100
cause (posterior mean − truth)/truth*100%
method
No Reg
Figure: NPLCM analyses with or without regression perform similarly in terms of
percent relative bias (top) and empirical coverage rates (bottom) over R = 100 replications in simulations where the case and control subclass weights do not vary by covariates. Each panel corresponds to one of 16 combinations of true parameter values and sample sizes
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 61 / 55
ℓ
pop_frac_scn: 1 Nd: 250 pop_frac_scn: 2 Nd: 500 pop_frac_scn: 1 Nd: 500 pop_frac_scn: 2 psi_scn: 1 theta: 0.8 psi_scn: 2 theta: 0.8 psi_scn: 1 theta: 0.95 psi_scn: 2 theta: 0.95 A B C D E F A B C D E F A B C D E F A B C D E F −100 −50 50 100 −100 −50 50 100 −100 −50 50 100 −100 −50 50 100
cause (posterior mean − truth)/truth*100%
method
No Reg
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 62 / 55
ℓ
pop_frac_scn: 1 Nd: 250 pop_frac_scn: 2 Nd: 500 pop_frac_scn: 1 Nd: 500 pop_frac_scn: 2 psi_scn: 1 theta: 0.8 psi_scn: 2 theta: 0.8 psi_scn: 1 theta: 0.95 psi_scn: 2 theta: 0.95 A B C D E F A B C D E F A B C D E F A B C D E F 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
cause coverage rate (intervals based on R replications)
method
No Reg
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 63 / 55
pop_frac_scn: 1 Nd: 250 pop_frac_scn: 2 Nd: 500 pop_frac_scn: 1 Nd: 500 pop_frac_scn: 2 psi_scn: 1 theta: 0.8 psi_scn: 2 theta: 0.8 psi_scn: 1 theta: 0.95 psi_scn: 2 theta: 0.95 A B C D E F A B C D E F A B C D E F A B C D E F 0.85 0.90 0.95 1.00 0.85 0.90 0.95 1.00 0.85 0.90 0.95 1.00 0.85 0.90 0.95 1.00
cause coverage rate (intervals based on R replications)
method
No Reg
Figure: NPLCM analyses with or without regression perform similarly in terms of
percent relative bias (top) and empirical coverage rates (bottom) over R = 100 replications in simulations where the case and control subclass weights do not vary by covariates. Each panel corresponds to one of 16 combinations of true parameter values and sample sizes
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 64 / 55
RSV individual etiology pie
enrollment date 2011:Sep−01 Dec−01 2012:Mar−01 Jun−01 Sep−01 Dec−01 2013:Mar−01 Jun−01 Sep−01
NoS individual etiology pie enrollment date 2011:Sep−01 Dec−01 2012:Mar−01 Jun−01 Sep−01 Dec−01 2013:Mar−01 Jun−01 Sep−01
Figure: Individual etiology fraction estimates for RSV (left) and NoS
(right) differ by age and season among HIV negative and severe pneumonia cases for whom the seven pathogens were all tested negative in the nasopharyngeal specimens.
Zhenke Wu(zhenkewu@umich.edu) 2019 TAMU 65 / 55