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Multilevel Models for Estimating the Number of Deaths in Armed - - PowerPoint PPT Presentation

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Multilevel Models for Estimating the Number of Deaths in Armed Conflict (in Colombia) Shira Mitchell JSM 2014 Collaborators: Brent Coull, Alan Zaslavsky, Al Ozonoff, Kristian Lum, Patrick Ball, Megan Price August 5, 2014 Shira Mitchell

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Multilevel Models for Estimating the Number of Deaths in Armed Conflict (in Colombia)

Shira Mitchell

JSM 2014

Collaborators: Brent Coull, Alan Zaslavsky, Al Ozonoff, Kristian Lum, Patrick Ball, Megan Price August 5, 2014

Shira Mitchell (Columbia) August 5, 2014 1 / 27

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Shira Mitchell (Columbia) August 5, 2014 2 / 27

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Shira Mitchell (Columbia) August 5, 2014 3 / 27

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Data from the Human Rights Data Analysis Group (HRDAG)

NGOs and govt groups provide lists of killings in 1998-2007, Casanare, Colombia: department of Colombia, population 300,000, BP oil pipeline, much corruption and violence. Goal: Estimate the number of killings in Casanare in years 1998-2007.

Shira Mitchell (Columbia) August 5, 2014 4 / 27

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nk1k2 ∼ Pois(µk1k2) independent

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log(µk1k2) = λ0 + λ1k1 + λ2k2 ⇒ E[N]MLE = n1+n+1 n11

Shira Mitchell (Columbia) August 5, 2014 8 / 27

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General number of lists k = (k1, k2, ..., kJ) for example, if in lists 3, 4, and 6 = (0, 0, 1, 1, 0, 1) Independence model: log(µk) = λ0 + λ1k1 + ... + λJkJ

Shira Mitchell (Columbia) August 5, 2014 9 / 27

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Data - from HRDAG

Matching:

Commission of Jurists Year Gender location . . . 1998 male TAMARA . . . . . . National Police Year Perpetrator Gender . . . . . . . . . 1998 FARC male

6 lists contain among them 2619 observed killings.

Shira Mitchell (Columbia) August 5, 2014 11 / 27

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Data - from HRDAG

year IMLM (govt) PN0 (govt) VP (govt) CCJ (NGO) CIN (NGO) CCE (NGO) 1998 1 14 13 3 1999 2 6 8 2 2000 213 5 22 23 2001 262 2 21 12 2002 268 1 33 9 2003 348 274 2 12 11 2004 412 324 295 14 11 1 2005 210 155 138 8 13 16 2006 104 71 26 3 2 15 2007 54 33 27 36 35

Shira Mitchell (Columbia) August 5, 2014 12 / 27

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We’re far from the independence model Heterogeneity of a person’s recordability Groups collecting data interact Want yearly estimates, but very little data exist in some years Groups operating in different but overlapping time periods

Shira Mitchell (Columbia) August 5, 2014 13 / 27

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Heterogeneity of a person’s recordability

Pj(θ) = P(person with recordability θ is recorded on list j) log

  • Pj(θgovt)

1 − Pj(θgovt)

  • = θgovt + λj for j ∈ govts

log

  • Pj(θNGO)

1 − Pj(θNGO)

  • = θNGO + λj for j ∈ NGOs

Shira Mitchell (Columbia) August 5, 2014 14 / 27

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Heterogeneity of a person’s recordability

Let (θNGO, θgovt) ∼ p(θNGO, θgovt). Then log(µk) = λ0 + λ1k1 + ... + λ6k6 + γ(kNGO

+

, kgovt

+

)

Shira Mitchell (Columbia) August 5, 2014 15 / 27

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log(µk) = λ0 + λ1k1 + ... + λ6k6+

  • j,j′∈NGOs

ωNGOkjkj′ +

  • j,j′∈govts

ωgovtkjkj′ +

  • j∈NGOs,j′∈govts

ωmixkjkj′

Heterogeneity of a person’s recordability Groups collecting data interact Want yearly estimates, but very little data exist in some years Groups operating in different but overlapping time periods

Shira Mitchell (Columbia) August 5, 2014 16 / 27

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log

  • µk

(t)

= λ0t + λ1,tk1 + ... + λ6,tk6+

  • j,j′∈NGOs

ωNGOkjkj′ +

  • j,j′∈govts

ωgovtkjkj′ +

  • j∈NGOs,j′∈govts

ωmixkjkj′

Heterogeneity of a person’s recordability Groups collecting data interact Want yearly estimates, but very little data exist in some years Groups operating in different but overlapping time periods

Shira Mitchell (Columbia) August 5, 2014 17 / 27

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log

  • µk

(t)

= λ0t + λ1,tk1 + ... + λ6,tk6+

  • j,j′∈NGOs

ωNGOkjkj′ +

  • j,j′∈govts

ωgovtkjkj′ +

  • j∈NGOs,j′∈govts

ωmixkjkj′

λj,t ∼ N(µj, τ2) for j = 1, ..., 6 Heterogeneity of a person’s recordability Groups collecting data interact Want yearly estimates, but very little data exist in some years Groups operating in different but overlapping time periods

Shira Mitchell (Columbia) August 5, 2014 18 / 27

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AR1 Model

    λj,1 . . . λj,T     | µj, ρ, τ2 ∼ N               µj . . . . . . µj        ,        1 ρ . . . ρT−1 ρ ... . . . . . . . . . ρT−1 1        τ2        . Heterogeneity of a person’s recordability Groups collecting data interact Want yearly estimates, but very little data exist in some years Groups operating in different but overlapping time periods

Shira Mitchell (Columbia) August 5, 2014 19 / 27

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Mixture Model

λj,t | γj,t ∼ (1 − γj,t)N(µinactive, σ2

inactive) + γj,tN(µj, τ2) for j = 1, ..., 6

γj,t ∼ Bern(p) independent p ∼ Unif(0, 1)

Heterogeneity of a person’s recordability Groups collecting data interact Want yearly estimates, but very little data exist in some years Groups operating in different but overlapping time periods

Shira Mitchell (Columbia) August 5, 2014 20 / 27

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Missing Data

Consider inactive lists as missing data [Zwane et al., 2004]. In year t, lists 3 and 4 are inactive. Treat n(t)

01000 as margin n(t) 01++0, and cells n(t) 01000, n(t) 01010, n(t) 01100, n(t) 01110 as

missing data. Zeros from missing data (ZM) vs Zeros from sampling (ZS)

Shira Mitchell (Columbia) August 5, 2014 21 / 27

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Zeros from missing data (ZM) Zeros from sampling (ZS) unpooled main effects (U) U-ZM U-ZS Multilevel model (M) M-ZM M-ZS AR1-ZS

Shira Mitchell (Columbia) August 5, 2014 22 / 27

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1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 500 1000 1500 2000 2500 Casanare Data Results Year Estimated Killings

  • Separate each year

U−ZS M−ZS AR1−ZS U−ZM M−ZM Shira Mitchell (Columbia) August 5, 2014 23 / 27

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Posterior Predictive Checks: M-ZS

Number recorded once Frequency 1600 1800 2000 500 1500 2500 p = 0.46 Number recorded twice Frequency 450 550 650 500 1000 1500 p = 0.35 Maximum number of recordings Frequency 4.0 4.5 5.0 5.5 6.0 1000 3000 p = 0.4

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Simulations Simulate from posterior predictive distribution of M-ZM, M-ZS, and AR1-ZS fit to Casanare data. Fit all the models. Coverage is similar for all models. Multilevel models have narrower intervals, and lower bias.

Shira Mitchell (Columbia) August 5, 2014 25 / 27

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Recommendations

In many applications, lists concentrate effort in different years, locations, or demographics. If these groups are overlapping ⇒ fit joint models, to be able to model more list interactions, and to borrow information across strata.

Shira Mitchell (Columbia) August 5, 2014 26 / 27

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We recommend Multilevel Models

In years with little data, we might not trust unpooled estimates - high variance, likely to get extreme estimates. Exchangeability and normality can be assessed via posterior predictive checks, relaxed by expanding the model. If we want monthly estimates at municipality-level, less and less data per stratum.

Colombia (2003-2011) Syria (2011-2013)

Shira Mitchell (Columbia) August 5, 2014 27 / 27

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References I

F Dominici. Combining contingency tables with missing dimensions. Biometrics, 56(2):546–553, 2000. A Gelman, A Jakulin, M G Pittau, and Y Su. A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics, 2(4):1360–1383, 2008. E N Zwane, K van der Pal-de Bruin, and P G M van der Heijden. The multiple-record systems estimator when registrations refer to different but overlapping populations. Statistics in Medicine, 23:2267–2281, 2004.

Shira Mitchell (Columbia) August 5, 2014 28 / 27

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EM-like algorithm

E step: ˆ n(t)

01010 =

T

s=1 µ(s) 01010

T

s=1

  • µ(s)

01000 + µ(s) 01010 + µ(s) 01100 + µ(s) 01110

n(t)

01++0.

M step: Fit log-linear model to completed data {n(t)

k }k=00000,00010,00100,00110.

Bayesian version [Dominici, 2000].

Shira Mitchell (Columbia) August 5, 2014 30 / 27

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Sensitivity Analysis: Choice of µinactive, τ2

inactive

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 500 1000 1500 2000 Year Estimated Killings

  • −13, 3

−9, 1.5 −9, 3 −9, 6 −5, 3 no mix estimate offs

Shira Mitchell (Columbia) August 5, 2014 31 / 27

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Posterior Predictive Checks: AR1-ZS

Number recorded once Frequency 1600 1800 2000 500 1500 2500 p = 0.49 Number recorded twice Frequency 450 550 650 500 1000 p = 0.37 Maximum number of recordings Frequency 4.0 4.5 5.0 5.5 6.0 1000 3000 p = 0.43

Shira Mitchell (Columbia) August 5, 2014 32 / 27

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Posterior Predictive Checks: M-ZM

Number recorded once Frequency 1700 1850 2000 100 200 300 p = 0.53 Number recorded twice Frequency 500 550 600 50 150 250 p = 0.2 Maximum number of recordings Frequency 4.0 4.5 5.0 5.5 6.0 200 400 600 p = 0.36

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Generate data from M-ZM: Coverage

  • Coverage

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM 0.2 0.4 0.6 0.8 1

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Generate data from M-ZS: Coverage

  • Coverage

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM 0.2 0.4 0.6 0.8 1

Shira Mitchell (Columbia) August 5, 2014 35 / 27

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Generate data from AR1-ZS: Coverage

  • Coverage

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM 0.2 0.4 0.6 0.8 1

Shira Mitchell (Columbia) August 5, 2014 36 / 27

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Generate data from M-ZM: Bias

  • −20

20 40 60 80 100 Average Bias

(log scale)

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM

Shira Mitchell (Columbia) August 5, 2014 37 / 27

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Generate data from M-ZS: Bias

  • −20

20 40 60 80 100 Average Bias

(log scale)

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM

Shira Mitchell (Columbia) August 5, 2014 38 / 27

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Generate data from AR1-ZS: Bias

  • −20

20 40 60 80 100 Average Bias

(log scale)

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM

Shira Mitchell (Columbia) August 5, 2014 39 / 27

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Generate data from M-ZM: Interval Width

  • 20

40 60 80 100 Average Interval Width

(log scale)

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM

Shira Mitchell (Columbia) August 5, 2014 40 / 27

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Generate data from M-ZS: Interval Width

  • 20

40 60 80 100 Average Interval Width

(log scale)

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM

Shira Mitchell (Columbia) August 5, 2014 41 / 27

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Generate data from AR1-ZS: Interval Width

  • 20

40 60 80 100 Average Interval Width

(log scale)

Separate each year U−ZS M−ZS AR1−ZS U−ZM M−ZM

Shira Mitchell (Columbia) August 5, 2014 42 / 27

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Continuous Model Expansion

log

  • µ(t)

k

  • = λ0t + λ1,tk1 + ... + λ6,tk6+
  • j,j′∈NGOs

ωNGOkjkj′ +

  • j,j′∈govts

ωgovtkjkj′ +

  • j∈NGOs,j′∈govts

ωmixkjkj′

ωj,j′ ∼ N

  • ωNGO, σ2

NGO

  • Shira Mitchell (Columbia)

August 5, 2014 43 / 27

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Continuous Model Expansion: 3-way log-linear interactions Population heterogeneity ⇒ higher-order interactions. Story for list cooperations?

Shira Mitchell (Columbia) August 5, 2014 44 / 27

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Continuous Model Expansion: 3-way log-linear interactions A Story:

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Continuous Model Expansion: 3-way log-linear interactions Cauchy priors - regularization [Gelman et al., 2008] Exchangeable based on NGO/govt

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