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A COM-Poisson Mixed Model for
Clustered Count Data
Darcy Steeg Morris, U.S. Census Bureau
Kimberly Sellers, Georgetown University and U.S. Census Bureau
Conference on Multivariate Count AnalysisJuly 5, 2018
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Disclaimer
This presentation is intended to inform interested parties of ongoingresearch and to encourage discussion of work in progress. Any viewsexpressed on statistical, methodological, technical, or operational issuesare those of the authors and not necessarily those of the U.S. CensusBureau.
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Outline
Introduction
COM-Poisson Distribution and Regression
COM-Poisson Mixed ModelCOM-Poisson-Normal ModelCOM-Poisson-Conjugate Model
Analysis: Simulated Data
Analysis: Epilepsy Data
Discussion
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Clustered Count Data
I Count data may exhibit over- or under-dispersion.
I Positive correlation between responses is one cause of over-dispersion(Hilbe 2008).
I Clustered data has inherent correlation within a cluster.
I Can account for the correlation by incorporating random effects in acount model.
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Random Intercept Poisson Model
I Poisson distribution assumes equi-dispersion.
I Random effects loosen this assumption to capture additional variabilityinduced by the correlation of measurements within a cluster.
I Model Assumptions:
yij |αi ∼ Poi(λ∗ij)
log(λ∗ij
)= β1xij1 + · · ·+ βpxijp + αi
αi ∼ g(αi |θ)
where yij is the count outcome for cluster i = 1, . . . ,N at occurrence
j = 1, . . . , Ji ; xij1, . . . , xijp are the p covariates for cluster i at occurrence j ; and
αi is the cluster-specific random intercept.
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Random Intercept Poisson Model
L(β,θ) =N∏i=1
∫ Ji∏j=1
e−λ∗ijλ∗yijij
yij !
g(αi |θ) dαi
I Random intercept distributional assumption:
1. g(αi |θ) = N(µ, σ2)⇒ Poisson-normal model, or
2. g(ui |θ) =gamma(a, c), where ui = eαi ⇒ Poisson-gamma model.
I Assumption (1) requires numerical integration.
I Assumption (2) reduces to a tractable form of the density.
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Why use COM-Poisson?
I Dispersion may also exist from underlying count process mechanism(mean-variance relationship) - not adequately modeled by clusterrandom effects (Booth et. al. 2003, Molenberghs et. al. 2007, 2010).
I The Conway-Maxwell-Poisson (COM-Poisson) distribution is a flexiblecount distribution that allows for under- and over-dispersion.
I COM-Poisson model for clustered data allows modeling of additionalvariability due to (1) the within-cluster correlation and (2) thedispersion from the underlying count process.
I Marginal COM-Poisson models have been proposed (Khan andJowaheer 2013, Choo-Wosoba et. al. 2016).
I We study a COM-Poisson mixed (i.e. conditional) model(Choo-Wosoba and Datta 2018, Choo-Wosoba et. al. 2018).
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Outline
Introduction
COM-Poisson Distribution and Regression
COM-Poisson Mixed ModelCOM-Poisson-Normal ModelCOM-Poisson-Conjugate Model
Analysis: Simulated Data
Analysis: Epilepsy Data
Discussion
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COM-Poisson Distribution
I The COM-Poisson pmf takes the form (Shmueli et. al., 2005)
P(Y = y | λ, ν) =λy
(y !)νZ (λ, ν), y = 0, 1, 2, . . .
for a random variable Y , where Z (λ, ν) =∑∞
s=0λs
(s!)ν is a normalizingconstant.
I Dispersion parameter ν ≥ 0:
ν = 1 ⇒ equi-dispersion
ν > 1 ⇒ under-dispersion
ν < 1 ⇒ over-dispersion
I Special Cases: Poisson (ν = 1), geometric (ν = 0, λ < 1) andBernoulli (ν →∞ with probability λ
1+λ).
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COM-Poisson Distribution: Mean
Moments are not of closed form, but mean can be approximated:
E (Y ) = λ∂ logZ (λ, ν)
∂λ≈ λ1/ν − ν − 1
2ν
for ν ≤ 1 or λ > 10ν (Shmueli et. al., 2005). Or more generally?
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COM-Poisson Regression
I Sellers and Shmueli (2010) extend the COM-Poisson distribution toregression.
I Allows varying λ for each observation i .
I Model Assumptions:
yi ∼ CMP(λi , ν)
log (λi ) = β0 + β1xi1 + · · ·+ βpxip
I Indirectly models the relationship between the mean and the linearpredictor.
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COM-Poisson Regression
I Likelihood:
L(β, ν) =N∏i=1
λyii(yi !)
ν Z (λi , ν)
I Loglikelihood:
log L(β, ν) =N∑i=1
yi log λi − νN∑i=1
log yi !−N∑i=1
logZ (λi , ν)
I Sellers and Shmueli (2010): maximum likelihood estimation.
I Guikema and Coffelt (2008): Bayesian estimation of are-parameterized version.
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Outline
Introduction
COM-Poisson Distribution and Regression
COM-Poisson Mixed ModelCOM-Poisson-Normal ModelCOM-Poisson-Conjugate Model
Analysis: Simulated Data
Analysis: Epilepsy Data
Discussion
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Random Intercept COM-Poisson Model
I Extend Sellers and Shmueli (2010) COM-Poisson regression model toinclude a random effect.
I Model Assumptions:
yij |αi ∼ CMP(λ∗ij , ν)
log(λ∗ij
)= log (uiλij) = β1xij1 + · · ·+ βpxijp + αi
αi ∼ g(αi |θ)
I αi assumed to capture all within-cluster correlation so that
yij ⊥ yik |αi for j 6= k .
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Random Intercept COM-Poisson Model
L(β, ν,θ) =N∏i=1
∫ Ji∏j=1
λ∗yijij
(yij !)ν1
Z (λ∗ij , ν))
g(αi |θ) dαi
I Random intercept distributional assumption:
1. g(αi |θ) = N(µ, σ2)⇒ COM-Poisson-normal model, or
2. g(ui |θ) ∝ ua−1i Z−c(ui , ν)⇒ COM-Poisson-conjugate model.
I Assumption (1) and (2) BOTH require numerical integration.
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Outline
Introduction
COM-Poisson Distribution and Regression
COM-Poisson Mixed ModelCOM-Poisson-Normal ModelCOM-Poisson-Conjugate Model
Analysis: Simulated Data
Analysis: Epilepsy Data
Discussion
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CMP-normal Model: Loglikelihood
I CMP-normal loglikelihood involves an intractable integral:
log L(β, ν, µ, σ2) = log
N∏i=1
∫ Ji∏j=1
f (yij |αi )︷ ︸︸ ︷(λ∗ij )
yij
(yij !)ν1
Z(λ∗ij , ν)
g(αi )︷ ︸︸ ︷[
1
σ√2π
e− (αi−µ)2
2σ2
]dαi
=
N∑i=1
Ji∑j=1
yij log(λij)− νN∑i=1
Ji∑j=1
log(yij !)−N∑i=1
log(σ√2π)
+N∑i=1
log
∫ eαi
∑Jij=1 yij−
(αi−µ)2
2σ2
(Ji∏j=1
Z(eαiλij , ν)
)−1
dαi
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CMP-normal Model: MLE
Obtain maximum likelihood estimates in R (with help Rcpp!) using:
1 numerical integration (the integrate function) to obtain anapproximation of the marginal loglikelihood,
2 optimization (the nlminb function) to maximize the approximatemarginal loglikelihood.
Maximum likelihood estimates of the CMP-normal model can similarly beobtained in SAS R© using the NLMIXED procedure (Morris et. al. 2017).
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Outline
Introduction
COM-Poisson Distribution and Regression
COM-Poisson Mixed ModelCOM-Poisson-Normal ModelCOM-Poisson-Conjugate Model
Analysis: Simulated Data
Analysis: Epilepsy Data
Discussion
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COM-Poisson Conjugate (Kadane et. al. 2005)
I Conjugate prior for COM-Poisson (“extended bivariate gamma”):
h(λ, ν) = λa−1e−νbZ−c(λ, ν) κ(a, b, c)
where κ(a, b, c) is the integration constant.
I Associated conditional distribution of λ:
h(λ|ν) = λa−1Z−c(λ, ν) κ(a, c)
where κ(a, c) is the integration constant.
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CMP Conditional Conjugate: Special Cases
I Conditional conjugate distribution h(λ|ν) special cases:
1. ν = 1⇒ gamma(a, c),
2. ν = 0⇒ beta(a, c + 1), and
3. ν →∞⇒ ac−aF (2a, 2(c − a)), c > a ≡ λ
1+λ ∼ beta(a, c − a).
I COM-Poisson conjugate relationship special cases:
1. ν = 1⇒ Poisson-gamma,
2. ν = 0⇒ geometric-beta, and
3. ν →∞⇒ Bernoulli-beta.
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CMP Conjugate: Parameter Constraints
I Joint conjugate h(λ, ν): κ−1(a, b, c) is finite when (Kadane et. al.2005)
b
c> log(ba/cc!) + (a/c − ba/cc) log(ba/cc+ 1)
I Conditional conjugate h(λ|ν):
κ−1(a, c) =
∫λa−1Z−c(λ, ν)dλ =
∫λa−1
[ ∞∑k=0
λk
(k!)ν
]−cdλ
divergent when a > c and ν large.
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Empirical Study of Parameter Constraints
κ−1(a, c) evaluated over a ∈ (.1, 5), c ∈ (.1, 5) by .1 and ν ∈ (0, 30) by .05 for a > c.
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CMP-conjugate Model: Loglikelihood
I Poisson-gamma model: conjugate distribution ⇒ closed form.
I Unfortunately CMP-conjugate loglikelihood involves intractableintegrals (violates strong conjugacy, Molenberghs et. al. 2010):
log L(β, ν, a, c) = log
N∏i=1
∫ Ji∏j=1
f (yij |ui )︷ ︸︸ ︷(λ∗ij )
yij
(yij !)ν1
Z(λ∗ij , ν)
g(ui )︷ ︸︸ ︷[
ua−1i Z−c(ui , ν) κ(a, c)
]dui
=
N∑i=1
Ji∑j=1
yij log(λij)− νN∑i=1
Ji∑j=1
log(yij !) +N∑i=1
log (κ(a, c))
+N∑i=1
log
∫ui
ua−1+
∑Jij=1 yij
i
(Z c(ui , ν)
Ji∏j=1
Z(uiλij , ν)
)−1
dui
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CMP-conjugate Model: MLE
Obtain maximum likelihood estimates in R (with help Rcpp!) from using:
1 numerical integration (the integrate function) to obtain anapproximation of the marginal loglikelihood,
2 optimization (the nlminb function) to maximize the approximatemarginal loglikelihood.
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Outline
Introduction
COM-Poisson Distribution and Regression
COM-Poisson Mixed ModelCOM-Poisson-Normal ModelCOM-Poisson-Conjugate Model
Analysis: Simulated Data
Analysis: Epilepsy Data
Discussion
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Simulated Data Generating Process
I N = 100 clusters & Ji = 5 ∀ i (500 observations) & 50 replications.
I Distributional Assumptions:
yij |ui ∼ f (yij |λ∗ij , {ν})
log(λ∗ij
)= β1xi + αi
xi ∼ N(0, .1) and αi ∼ N(.5, .5) and β1 = .5
where f (yij |λ∗ij , {ν}) is:
I Poi(λ∗ij) ≡ CMP(λ∗ij , 1
)I Bern
(λ∗ij
1+λ∗ij
)≡ CMP
(λ∗ij ,∞
)I geom
(1
1+λ∗ij
)∼= CMP
(λ∗ij , 0
)I CMP
(λ∗ij , 5
)I CMP
(λ∗ij , .75
)
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Simulated Data: Mean Link
Recall
E (Y ) = λ∗∂ logZ (λ∗, ν)
∂λ∗
I For Poisson data:
E (Y ) = λ∗ ≡ E (Y ) = λ∗∂ log
(eλ∗)
∂λ∗= λ∗
I For Bernoulli data:
E (Y ) = p =λ∗
1 + λ∗≡ E (Y ) = λ∗
∂ log (1 + λ∗)
∂λ∗=
λ∗
1 + λ∗
I For geometric data:
E (Y ) =1− p
p= λ∗ 6≡ E (Y ) = λ∗
∂ log (1− λ∗)−1
∂λ∗=
λ∗
1− λ∗
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Simulated Data: Misspecification
I Fit 4 models: Poisson-normal, NB-normal, CMP-normal,CMP-conjugate.
I Sources of misspecification.
I Random effect distribution.
I Special Cases: CMP-conjugate.
I COM-Poisson Data: CMP-conjugate.
I Link to linear predictor.
I Special Cases: CMP-normal/conjugate for geometric data.
I COM-Poisson Data: Poisson-normal, NB-normal.
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Simulation Study Results: Special Cases
Special Case Simulated Data Mean Estimates
Simulated Model
Dataset Est. Poisson NB CMP-normal CMP-conjugate
Poisson Disp. k = 0.00 ν = 1.02 ν = 0.99Var. σ2 = 0.49 σ2 = 0.49 σ2 = 0.51 a = 2.12, c = 1.01
min AIC 0.96 0.72 0.96 0.12max ` 0.00 0.04 0.76 0.20
Bernoulli** Disp. k = 0.00 ν = 37.9 ν = 34.8Var. σ2 = 0.00 σ2 = 0.00 σ2 = 0.53 a = 7.26, c = 11.75
min AIC 0.00 0.00 1.00 1.00max ` 0.00 0.00 0.55 0.45
Geometric Disp k = 1.01 ν = 0.02 ν = 0.02Var. σ2 = 0.67 σ2 = 0.45 σ2 = 0.04 a = 6.70, c = 3.33
min AIC 0.00 0.98 0.22 0.34max ` 0.00 0.64 0.02 0.34
Note: min AIC is the proportion of replications where AIC ≤ min(AIC) + 2 and max `is proportion of replications where ` is largest.
** The random intercept logistic model results/estimates for the simulated Bernoullidata are: σ2 = 0.45, min AIC = 1.00, and max ` = 0.00.
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Simulation Study Results: Special Cases
I COM-Poisson models have better/comparable model fit to specialcases.
I COM-Poisson model recognizes special cases:
I ν = 1.02, 0.99 ≈ 1 for Poisson,
I ν = 37.9, 34.8 is large for Bernoulli, and
I ν = 0.02, 0.02 ≈ 0 for geometric.
I Cluster variability.
I Captured by Poisson and NB for over-dispersed data: σ2 > 0.
(NB recognizes geometric special case: k ≈ 1.)
I Not captured by Poisson or NB for under-dispersed data: σ2 = 0.
I Captured by COM-Poisson models: σ2 > 0 and ...
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Simulation Study Results: Special Cases
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
Estimated Random Effect Distribution: Poisson Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71Lognormal (CMP): E(u) = 2.17 , SD(u) = 1.76Conjugate (CMP): E(u) = 2.09 , SD(u)= 1.43
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
Estimated Random Effect Distribution: Bernoulli Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71Lognormal (CMP): E(u) = 2.12 , SD(u) = 1.61Conjugate (CMP): E(u) = 2.08 , SD(u)= 1.61
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Simulation Study Results: Special Cases
0 2 4 6 8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Estimated Random Effect Distribution: Geometric Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71True (adjusted): E(u) = 0.83 , SD(u) = 0.12Lognormal (CMP): E(u) = 0.63 , SD(u) = 0.13Conjugate (CMP): E(u) = 0.62 , SD(u)= 0.15
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Simulation Study Results: COM-Poisson Data
COM-Poisson Simulated Data Mean Estimates
Simulated Model
Dataset Est. Poisson NB CMP-normal CMP-conjugate
CMP Disp. k = 0.00 ν = 5.05 ν = 5.08(under) Var. σ2 = 0.00 σ2 = 0.00 σ2 = 0.50 a = 6.00, c = 8.83
min AIC 0.00 0.00 1.00 0.97max ` 0.00 0.00 0.58 0.42
CMP Disp k = 0.00 ν = 0.77 ν = 0.74(over) Var. σ2 = 0.76 σ2 = 0.75 σ2 = 0.51 a = 1.76, c = 0.56
min AIC 0.17 0.19 0.93 0.19max ` 0.00 0.10 0.79 0.12
Note: min AIC is the proportion of replications where AIC ≤ min(AIC) + 2 and max `
is proportion of replications where ` is largest.
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Simulation Study Results: COM-Poisson Data
I COM-Poisson models outperform for both cases of intermediate levelsof over- and under-dispersion.
I COM-Poisson models recognize over- and under-dispersion:
I ν = 5.05, 5.08 ≈ 5.00 > 1 for COM-Poisson (under), and
I ν = 0.77, 0.74 ≈ 0.75 < 1 for COM-Poisson (over).
I Cluster variability.
I Not captured by Poisson or NB for under-dispersed data: σ2 = 0.
I Captured by COM-Poisson models: σ2 > 0 and ...
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Simulation Study Results: COM-Poisson Data
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
Estimated Random Effect Distribution: CMP Underdispersed Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71Lognormal (CMP): E(u) = 2.26 , SD(u) = 1.81Conjugate (CMP): E(u) = 2.16 , SD(u)= 1.54
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
Estimated Random Effect Distribution: CMP Overdispersed Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71Lognormal (CMP): E(u) = 2.18 , SD(u) = 1.78Conjugate (CMP): E(u) = 2.08 , SD(u)= 1.4
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Simulation Study Results: Model Comparisons
Simulated Data Best Model by AIC
Simulated Model
Dataset Poisson NB CMP-normal CMP-conjugate
Poisson X X X
Bernoulli X X
Geometric X X
CMP (under) X X
CMP (over) X
Note: bolded are misspecified models.
I But what if random effect distribution is misspecified for -normalmodels?
ui ∼ gamma(1.54, 1.37)
assuming same mean and variance as ui ∼ log N(.5, .5).
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Simulated Data: Misspecification
I Fit 4 models: Poisson-normal, NB-normal, CMP-normal,CMP-conjugate.
I Sources of misspecification.
I Random effect distribution.
I Special Cases: All except CMP-conjugate for Poisson data.
I COM-Poisson Data: All.
I Link to linear predictor.
I Special Cases: CMP-normal/conjugate for geometric data.
I COM-Poisson Data: Poisson-normal, NB-normal.
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Simulation Study Results: Special Cases
Special Case Simulated Data Mean Estimates
Simulated Model
Dataset Est. Poisson NB CMP-normal CMP-conjugate
Poisson Disp. k = 0.00 ν = 1.03 ν = 1.00Var. σ2 = 0.68 σ2 = 0.68 σ2 = 0.71 a = 1.67, c = 0.48
min AIC 0.28 0.08 0.22 0.88max ` 0.00 0.00 0.12 0.88
Bernoulli** Disp. k = 0.00 ν = 36.0 ν = 35.6Var. σ2 = 0.00 σ2 = 0.00 σ2 = 0.96 a = 4.47, c = 6.50
min AIC 0.00 0.00 1.00 1.00max ` 0.00 0.00 0.69 0.31
Geometric Disp k = 1.03 ν = 0.00 ν = 0.00Var. σ2 = 0.90 σ2 = 0.66 σ2 = 0.04 a = 5.62, c = 1.59
min AIC 0.00 0.97 0.00 0.45max ` 0.00 0.76 0.00 0.24
Note: min AIC is the proportion of replications where AIC ≤ min(AIC) + 2 and max `is proportion of replications where ` is largest.
** The random intercept logistic model results/estimates for the simulated Bernoullidata are: σ2 = 0.85, min AIC = 1.00, and max ` = 0.00.
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Simulation Study Results: Special Cases
0 2 4 6 8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Estimated Random Effect Distribution: Poisson Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71Lognormal (CMP): E(u) = 3.84 , SD(u) = 3.91Conjugate (CMP): E(u) = 3.53 , SD(u)= 2.73
0 2 4 6 8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Estimated Random Effect Distribution: Bernoulli Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71Lognormal (CMP): E(u) = 4.13 , SD(u) = 5.25Conjugate (CMP): E(u) = 4.38 , SD(u)= 33.31
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Simulation Study Results: Special Cases
0 2 4 6 8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Estimated Random Effect Distribution: Geometric Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71True (adjusted): E(u) = 0.82 , SD(u) = 0.14Lognormal (CMP): E(u) = 0.73 , SD(u) = 0.14Conjugate (CMP): E(u) = 0.73 , SD(u)= 0.17
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Simulation Study Results: COM-Poisson Data
COM-Poisson Simulated Data Mean Estimates
Simulated Model
Dataset Est. Poisson NB CMP-normal CMP-conjugate
CMP Disp. k = 0.00 ν = 5.10 ν = 5.09(under) Var. σ2 = 0.00 σ2 = 0.00 σ2 = 0.79 a = 4.01, c = 5.14
min AIC 0.00 0.00 1.00 1.00max ` 0.00 0.00 0.21 0.79
CMP Disp k = 0.02 ν = 0.78 ν = 0.76(over) Var. σ2 = 1.14 σ2 = 1.14 σ2 = 0.77 a = 1.38, c = 0.23
min AIC 0.06 0.09 0.23 0.86max ` 0.00 0.06 0.14 0.80
Note: min AIC is the proportion of replications where AIC ≤ min(AIC) + 2 and max `
is proportion of replications where ` is largest.
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Simulation Study Results: COM-Poisson Data
0 2 4 6 8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Estimated Random Effect Distribution: CMP Underdispersed Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71Lognormal (CMP): E(u) = 3.73 , SD(u) = 4.09Conjugate (CMP): E(u) = 3.59 , SD(u)= 3.57
0 2 4 6 8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Estimated Random Effect Distribution: CMP Overdispersed Data
ui
Den
sity
True: E(u) = 2.12 , SD(u) = 1.71Lognormal (CMP): E(u) = 3.92 , SD(u) = 4.22Conjugate (CMP): E(u) = 3.53 , SD(u)= 2.68
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Simulation Study Results: Model Comparisons
Simulated Data Best Model by AIC
Simulated Model
Dataset Poisson NB CMP-normal CMP-conjugate
Poisson X X X X
Bernoulli X X
Geometric X X
CMP (under) X X
CMP (over) X X
Note: bolded are misspecified models.
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Outline
Introduction
COM-Poisson Distribution and Regression
COM-Poisson Mixed ModelCOM-Poisson-Normal ModelCOM-Poisson-Conjugate Model
Analysis: Simulated Data
Analysis: Epilepsy Data
Discussion
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Epilepsy Data & Model (Diggle et. al. 1994)
I Number of seizures measured for 59 epileptic patients in an 8-weekbaseline period followed by 4 consecutive 2-week treatment periods.
I Outcome Variable, yij : number of seizures for subject i in timeperiod j .
I Covariates:I xij1: indicator of a period after baseline (weeks 8− 16).I xij2: indicator of receipt of progabide (vs. placebo).I Tij : length of time period t.
I Model: Diggle et. al. (1994) fit a random intercept Poissonregression model.
log E (yij |ui ) = β0 + xij1β1 + xij2β2 + xij1xij2β3 + log(Tij) + ui
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Epilepsy Data Results
Epilepsy Data Estimates
ModelParameter Poisson NB CMP-normal CMP-conjugate
β0 or µ 1.033 1.080 -0.781β1 0.111 0.023 0.793 0.633β2 -0.024 0.073 0.048 0.165β3 -0.104 -0.310 -0.176 -0.198k 0.148ν 0.420 0.541σ2 0.608 0.661 0.143
a, c 2.93, 2.79
AIC 2031.4 1789.5 1754.0 1776.6−` 1010.6 888.7 871.0 882.3
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Epilepsy Data Results
I CMP models have the best fit.
I All models indicate subject variability: σ2 > 0 and ...
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
Estimated Prior Distribution for CMP Models
ui
Den
sity
Lognormal: E(u) = 0.49 , SD(u) = 0.19Conjugate: E(u) = 0.74 , SD(u)= 0.38
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Epilepsy Data Results
I Additional over-dispersion is evident in CMP and NB:
I ν < 1 (ν = 0.420, 0.541) in COM-Poisson, and
I k > 0 (k = 0.148) in negative binomial.
I Both the negative binomial and COM-Poisson models can account forvariability beyond the subject-specific random effect, however theCOM-Poisson captures additional over-dispersion in a way that thenegative binomial model cannot.
I Findings consistent with Molenberghs et. al. (2010).
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Outline
Introduction
COM-Poisson Distribution and Regression
COM-Poisson Mixed ModelCOM-Poisson-Normal ModelCOM-Poisson-Conjugate Model
Analysis: Simulated Data
Analysis: Epilepsy Data
Discussion
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Discussion
I COM-Poisson regression model can be extended to include randomeffects to model clustered data.
I The flexibility of the COM-Poisson mixed model allows modeling ofvariability in the count outcome beyond that induced by within-clustercorrelation.
I Assuming the conditional conjugate distribution for random effectsallows further flexibility.
I Framework naturally allows random slopes, mixed modeling of thedispersion parameter, etc.
I To do: Mean parametrization?!
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References
I Shmueli, Minka, Kadane, Borle and Boatwright. (2005). “A UsefulDistribution for Fitting Discrete Data: Revival of theConway-Maxwell-Poisson Distribution.” Journal of the Royal StatisticalSociety, Series C, 54: 127-142.
I Sellers and Shmueli. (2010). “A Flexible Regression Model for Count Data.”The Annals of Applied Statistics, 4(2): 943-961.
I Guikema and Coffelt. (2008). “A Flexible Count Data Regression Model forRisk Analysis.” Risk Analysis. 28(1): 213-223.
I Kadane, Shmueli, Minka, Borle and Boatwright, P. (2006). “ConjugateAnalysis of the Conway-Maxwell-Poisson Distribution.” Bayesian Analysis.1(2): 363-374.
I Diggle, Heagerty, Liang and Zeger. (1994). Analysis of Longitudinal Data.Oxford: Clarendon.
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References
I Choo-Wosoba and Datta. (2018).“Analyzing Clustered Count Data with aCluster-specific Random Effect Zero-inflated Conway-Maxwell-PoissonDistribution.” Journal of Applied Statistics, 45(5): 799-814.
I Choo-Wosoba, Gaskins, Levy and Datta. (2018). “A Bayesian approach foranalyzing zero-inflated clustered count data with dispersion: BayesianConway-Maxwell-Poisson.” Statistics in Medicine, 37(1): 801-812.
I Choo-Wosoba, Levy and Datta. (2016). “Marginal Regression Models forClustered Count Data Based on Zero-Inflated Conway-Maxwell-PoissonDistribution with Applications.” Biometrics, 72(2): 606-618.
I Morris, Sellers and Menger. (2017). “Fitting a Flexible Model forLongitudinal Count Data Using the NLMIXED Procedure,” In SAS GlobalForum Proceedings. Cary, NC: SAS Institute.
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References
I Booth, Casella, Friedl and Hobert. (2003). “Negative Binomial LoglinearMixed Models.” Statistical Modelling, 3: 179-191.
I Molenberghs, Verbeke, Demetrio and Vieira. (2010). “A Family ofGeneralized Linear Models for Repeated Measures with Normal andConjugate Random Effects.” Statistical Science, 25(3): 325-247.
I Molenberghs, Verbeke and Demetrio. (2007). “An Extended Random-EffectsApproach to Modeling Repeated, Overdispersed Count Data.” Lifetime DataAnalysis, 13: 513-531.
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Simulated Data HistogramsPoisson
Count
Fre
quen
cy
0 5 10 15 20
050
150
250
Bernoulli
Count
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
010
020
030
0
Geometric
Count
Fre
quen
cy
0 5 10 15 20
010
020
030
0
CMP−Under
Count
Fre
quen
cy
0.0 0.5 1.0 1.5 2.0
010
020
0
CMP−Over
Count
Fre
quen
cy
0 5 10 15 20
050
150
250
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True Random Effects Distribution
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
True Random Effect Distribution
ui
Den
sity
LognormalGamma
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CMP Conditional Conjugate
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 0.5 and c = 0.5
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10ν = 30
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 0.5 and c = 1
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10ν = 30
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 0.5 and c = 2
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10ν = 30
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CMP Conditional Conjugate
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 1 and c = 0.5
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 1 and c = 1
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10ν = 30
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 1 and c = 2
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10ν = 30
53 / 53
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CMP Conditional Conjugate
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 2 and c = 0.5
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 2 and c = 1
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 2 and c = 2
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10ν = 30
53 / 53
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CMP Conditional Conjugate
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 3 and c = 0.5
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 3 and c = 1
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a = 3 and c = 2
λ
Den
sity
ν = 0ν = 0.5ν = 1ν = 2ν = 10
53 / 53