Battlestar GalacticaGalactica

Outline Outline ƒ BSG ƒ BSG • Statistics and ƒ Basics ƒ Basics Battlestar Galactica ƒ Estimation Human or Cylon? ƒ Estimation • The story so far… ƒ Identification Group testing on the ƒ Identification ƒ Covariates Group testing on the ƒ Covariates – Video ƒ NIH grant Battlestar Galactica ƒ NIH grant

Christopher R. Bilder Department of Statistics University of Nebraska-Lincoln www.chrisbilder.com [email protected]

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Battlestar GalacticaGalactica Battlestar GalacticaGalactica

Outline Outline ƒ BSG • Statistics and ƒ BSG •Dr. Gaius Baltar ƒ Basics Battlestar Galactica ƒ Basics – Asked to develop a ƒ Estimation • The story so far… ƒ Estimation Cylon detector ƒ Identification ƒ Identification ƒ Covariates –Video ƒ Covariates • Season 1’s Bastille ƒ NIH grant • Cylons ƒ NIH grant Day episode –Centurion – # of Cylons in fleet is – Humanoid form expected to be small (new) – 47,905 individuals to • How can you test! distinguish a human from a Cylon?

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Outline Outline ƒ BSG • Dr. Gaius Baltar (continued) ƒ BSG • Dr. Gaius Baltar (continued) ƒ Basics – Season 1’s Tigh me up and Tigh me down ƒ Basics – Season 1’s Tigh me up and Tigh me down ƒ Estimation ƒ Estimation ƒ Identification ƒ Identification ƒ Covariates ƒ Covariates ƒ NIH grant ƒ NIH grant

– Video –Video

Slide 5 of 37 Slide 6 of 37 – (47,905 blood tests)∗(11 hours each) = 21,956 days www.chrisbilder.com www.chrisbilder.com

Battlestar GalacticaGalactica Battlestar GalacticaGalactica

Outline Outline ƒ BSG • Individual testing ƒ BSG • Group testing ƒ Basics ƒ Basics ƒ Estimation ƒ Estimation ƒ Identification ƒ Identification ƒ Covariates ƒ Covariates ƒ NIH grant ƒ NIH grant + or - + or - + or - + or - + or - + or - + or - + or - + or -

• If a GROUP is negative, then all 4 individuals are not + or - + or - + or - + or - + or - + or - • Problems: Cylons –Time • If the GROUP is positive, then at least ONE of the 4 individuals is a Cylon Slide 7 of 37 – Limited resources Slide 8 of 37 www.chrisbilder.com www.chrisbilder.com – “Retesting” can be done to determine who is a Cylon Battlestar GalacticaGalactica Other examples

Outline Outline ƒ BSG • Group testing (continued) ƒ BSG • Screening blood donations ƒ Basics – Time savings ƒ Basics – American Red Cross uses groups of size 16 ƒ Estimation – Save resources ƒ Estimation – HIV, Hepatitis B, Hepatitis C, … ƒ Identification ƒ Identification ƒ Covariates – Strategy works well when prevalence of a “trait” is ƒ Covariates – Screen about 6 million a year ƒ NIH grant small ƒ NIH grant • Source: Roger Dodd, Executive Director of Blood • If prevalence is large, all groups may test positive Services R & D at ARC • See Dodd et al. (Transfusion, 2002) • Drug discovery experiments – Screen hundreds of thousands of chemical compounds to look for potentially good ones – Remlinger et al. (Technometrics, 2006)

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Other examples NotationNotation

Outline Outline ƒ BSG • Multiple vector transfer design experiments ƒ BSG • Individual responses ƒ Basics ƒ Basics th th – Estimate probability an insect vector – Yik = 1 if the i item in the k group has the “trait” ƒ Estimation transfers a pathogen to a plant ƒ Estimation (positive) and ƒ Identification ƒ Identification Yik = 0 otherwise (negative) for i=1, …, I and k=1, …, K ƒ Covariates – Swallow (Phytopathology, 1985, 1987) ƒ Covariates ƒ NIH grant • Veterinary ƒ NIH grant – Yik are independent Bernoulli(p) random variables – Bovine viral diarrhea in cattle (Peck, Beef, 2006) • p = P(Yik = 1) – Avian pneumovirus (APV) in turkeys (Maherchandani • Homogenous population et al., J. Veterinary Diagnostic Investigation, 2004) • p can be thought of as the “individual probability” or • Public health studies “prevalence in a population”

– Prevalence of HCV (Liu et al., Transfusion, 1997) – Yik’s are not directly observed (at least initially) – Prevalence of HIV (Verstraeten et al., Trop. Med. & International Health, 2000)

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Outline Outline ƒ BSG • Group responses ƒ BSG • Example random variables ƒ Basics ƒ Basics – Zk = 1 denotes a positive response + or − + or − + or − + or − + or − + or − ƒ Estimation th ƒ Estimation Zk = 0 denotes a negative response for the k group ƒ Identification ƒ Identification ƒ Covariates – Zk are independent Bernoulli(θ) random variables ƒ Covariates

ƒ NIH grant • θ = P(Zk = 1) ƒ NIH grant • Individual and group response relationship + or - + or - + or -

– Zk = 1 if and only if Zk = 0 if and only if

+ or − + or − + or − + or − + or − + or −

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NotationNotation NotationNotation

Outline Outline ƒ BSG • Example observed values ƒ BSG • Example observed values ƒ Basics ƒ Basics Basics Basics y11 = 0 y21 = 0 y12 = 0 y22 = 1 y13 = 0 y23 = 0 ƒ Estimation ƒ Estimation ƒ Identification ƒ Identification ƒ Covariates ƒ Covariates ƒ NIH grant ƒ NIH grant

- + + z1 = 0 z2 = 1 z3 = 1

y31 = 0 y41 = 0 y32 = 0 y42 = 0 y33 = 0 y43 = 1

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Outline Outline ƒ BSG • Prevalence of a trait in a population (estimation problem) ƒ BSG • How can we estimate p = P(Yik = 1)? ƒ Basics • Which items are positive (identification problem) ƒ Basics – We observe information about the groups, not ƒ Estimation ƒ Estimation individuals! ƒ Identification ƒ Identification I ƒ Covariates ƒ Covariates – θ = 1 – P(Yik = 0, ∀i) = 1 – (1 – p) ƒ NIH grant ƒ NIH grant – Then p = 1 – (1 – θ)1/I – MLE for p: • Unequal group sizes – Likelihood function

where

θk = positive probability for group k Slide 17 of 37 Slide 18 of 37 www.chrisbilder.com www.chrisbilder.com Ik = size of group k

Testing error Identification

Outline Outline ƒ BSG • What if there is testing error? ƒ BSG • Dorfman (Annals of Mathematical Statistics, 1943) ƒ Basics ƒ Basics – Can incorporate sensitivity (η) and specificity (δ) – Retest all items in a positive group y12 = 0 y22 = 1 ƒ Estimation – ƒ Estimation – Often credited for the very first use ƒ Identification ƒ Identification ƒ Covariates ƒ Covariates of group testing ƒ NIH grant ƒ NIH grant •Sterrett(Annals of Mathematical Statistics, 1957) z2 = 1 – Individually retest until first positive is found – Re-group remaining items • If group is negative, STOP y32 = 0 y42 = 0 • If group is positive, repeat – Expected number retests is smaller than Dorfman

Slide 19 of 37 Slide 20 of 37 www.chrisbilder.com www.chrisbilder.com • Gupta and Malina (Statistics in Medicine, 1999) provides a summary InfertilityInfertility PreventionPrevention ProgramProgram InfertilityInfertility PreventionPrevention ProgramProgram

Outline Outline ƒ BSG • U.S. national program funded by Centers for Disease ƒ BSG • Lindan et al. (J. Clinical Microbiology, 2005) ƒ Basics Control and Prevention ƒ Basics – Estimates that 12% of the laboratories in the U.S. are ƒ Estimation – Assess and reduce prevalence of chlamydia and ƒ Estimation already using group testing ƒ Identification gonorrhea ƒ Identification ƒ Covariates ƒ Covariates – Group testing has allowed “laboratories to achieve a ƒ NIH grant •Nebraska ƒ NIH grant significant increase in specimen loads.” – Swab or urine specimens are sent to the Nebraska Public • Quarter #1 of 2006, chlamydia testing Health Laboratory at U. of Nebraska Medical Center – Urine specimens – 1,384 total –NATs – Ignore sensitivity and specificity here – About 30,000 individual tests done per year – Individual data: 111/1,384 = 0.0802 • Group testing! – Group testing: • Randomly put known individual responses into groups of size I = 2

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InfertilityInfertility PreventionPrevention ProgramProgram Heterogonous populations

Outline Outline ƒ BSG • Quarter #1 of 2006 (continued) ƒ BSG • Individual responses ƒ Basics ƒ Basics – Individual data: 111/1,384 = 0.0802 – Yik are independent Bernoulli(pik) random variables ƒ Estimation – Group testing: ƒ Estimation – p = P(Y = 1) for item i in group k ƒ Identification ƒ Identification ik ik ƒ Covariates ƒ Covariates • Group responses ƒ NIH grant ƒ NIH grant – Zk are independent Bernoulli(θk) random variables

– θk = P(Zk = 1) for group k • Covariates th th – xik1, xik2, …, xikp for the i item in the k group – Approximate cost per test – Incorporate factors which influence trait status • $16 for urine – Not really done until recently in group testing! • $11 for swab

Slide 23 of 37 Slide 24 of 37 www.chrisbilder.com www.chrisbilder.com Kenyan pregnant women study Heterogonous populations

Outline Outline ƒ BSG • Part of the data from Vansteelandt et al. (Biometrics, 2000) ƒ BSG • Model ƒ Basics ƒ Basics – logit(pik) = β0 + β1xik1 + … + βpxikp ƒ Estimation ƒ Estimation • Estimation of β , β , β , …, β ƒ Identification ƒ Identification 0 1 2 p ƒ Covariates ƒ Covariates – Note that Yik are not directly observed ƒ NIH grant ƒ NIH grant z1 = 1 – Vansteelandt et al. (Biometrics, 2000)

• Likelihood function is written in terms of the Zk

z2 = 1 –Xie(Statistics in Medicine, 2001)

• Likelihood function is written in terms of the Yik • EM algorithm used Slide 25 of 37 Slide 26 of 37 www.chrisbilder.com www.chrisbilder.com

Forming groups Forming groups

Outline Outline ƒ BSG •Alike ƒ BSG •Random ƒ Basics – Individuals with “similar” covariates are put into pools ƒ Basics – Individuals are assigned to pools at random ƒ Estimation – Smallest variability in parameter estimates ƒ Estimation – Emulates chronological if no dependence over time ƒ Identification ƒ Identification ƒ Covariates – How implement? ƒ Covariates •Different ƒ NIH grant • One covariate: Sort by covariate, then assign ƒ NIH grant – Pool individuals with covariates as different as possible successive individuals to pools – Emulates “worse case scenario” (?) • Multiple covariates: ? – Usually requires one to have all individual testing specimens up front and available for testing at the same time

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Outline Outline • Simulate data from model fitted to the individual • One simulated data set ƒ BSG ƒ BSG True Individual estimated ƒ Basics observations in Vansteelandt et al. (Biometrics, 2000) ƒ Basics Group estimated (alike) • Relative efficiency Group estimated (random) Group estimated (different) ƒ Estimation ƒ Estimation – logit(pik) = β0 + β1xik = –1.97 – 0.024xik ƒ Identification ƒ Identification ƒ Covariates – Simulate the individual and group responses ƒ Covariates ƒ NIH grant • I = 7 subjects per group ƒ NIH grant

• K = 100 groups Estimated probability • Overall sample size is I∗K = 700 0.00 0.05 0.10 0.15 0.20 0.25

15 20 25 30 35 40

Covariate

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Forming groups Forming groups

Outline Outline ƒ BSG • 100 simulated data sets ƒ BSG • Last slide examined a fixed I∗K ƒ Basics ƒ Basics • What if we fix the number of groups (tests), K, instead? ƒ Estimation ƒ Estimation – Settings ƒ Identification ƒ Identification ƒ Covariates ƒ Covariates • logit(pik) = –2 + 0.6931xik ƒ NIH grant ƒ NIH grant • xik ~ Uniform(–70.079, 1.663)

• 0.001 < pik < 0.3

• Average value of pik is 0.02 • 500 simulated data sets for each simulation – Relative efficiency: I K = 500 2 5 10 Alike 2.20 4.62 6.72 Random 1.61 1.79 1.50 Slide 31 of 37 • Pearson Slide 32 of 37 www.chrisbilder.com correlations: www.chrisbilder.com Different 1.16 0.51 0.22 NIH Grant NIH Grant

Outline Outline ƒ BSG • Content removed ƒ BSG • Content removed ƒ Basics ƒ Basics ƒ Estimation ƒ Estimation ƒ Identification ƒ Identification ƒ Covariates ƒ Covariates ƒ NIH grant ƒ NIH grant

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NIH Grant NIH Grant

Outline Outline ƒ BSG • Content removed ƒ BSG • Content removed ƒ Basics ƒ Basics ƒ Estimation ƒ Estimation ƒ Identification ƒ Identification ƒ Covariates ƒ Covariates ƒ NIH grant ƒ NIH grant

Slide 35 of 37 Slide 36 of 37 www.chrisbilder.com www.chrisbilder.com Outline ƒ BSG ƒ Basics ƒ Estimation Human or Cylon? ƒ Identification Group testing on the ƒ Covariates Group testing on the ƒ NIHProposed grant Battlestar Galactica

Christopher R. Bilder Department of Statistics University of Nebraska-Lincoln www.chrisbilder.com [email protected]

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