Battlestar GalacticaGalactica Outline Outline BSG BSG • Statistics and Basics Basics Battlestar Galactica Estimation Human or Cylon? Estimation • The story so far… Identification Group testing on the Identification Covariates Group testing on the Covariates – Video NIH grant Battlestar Galactica NIH grant Christopher R. Bilder Department of Statistics University of Nebraska-Lincoln www.chrisbilder.com [email protected] Slide 1 of 37 Slide 2 of 37 www.chrisbilder.com www.chrisbilder.com Battlestar GalacticaGalactica Battlestar GalacticaGalactica Outline Outline BSG • Statistics and BSG •Dr. Gaius Baltar Basics Battlestar Galactica Basics – Asked to develop a Estimation • The story so far… Estimation Cylon detector Identification Identification Covariates –Video Covariates • Season 1’s Bastille NIH grant • Cylons 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? Slide 3 of 37 Slide 4 of 37 www.chrisbilder.com www.chrisbilder.com Battlestar GalacticaGalactica Battlestar GalacticaGalactica Outline Outline BSG • Dr. Gaius Baltar (continued) BSG • Dr. Gaius Baltar (continued) Basics – Season 1’s Tigh me up and Tigh me down Basics – Season 1’s Tigh me up and Tigh me down Estimation Estimation Identification Identification Covariates Covariates NIH grant 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 BSG • Individual testing BSG • Group testing Basics Basics Estimation Estimation Identification Identification Covariates Covariates NIH grant 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 BSG • Group testing (continued) BSG • Screening blood donations Basics – Time savings Basics – American Red Cross uses groups of size 16 Estimation – Save resources Estimation – HIV, Hepatitis B, Hepatitis C, … Identification Identification Covariates – Strategy works well when prevalence of a “trait” is Covariates – Screen about 6 million a year NIH grant small 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) Slide 9 of 37 Slide 10 of 37 www.chrisbilder.com www.chrisbilder.com Other examples NotationNotation Outline Outline BSG • Multiple vector transfer design experiments BSG • Individual responses Basics Basics th th – Estimate probability an insect vector – Yik = 1 if the i item in the k group has the “trait” Estimation transfers a pathogen to a plant Estimation (positive) and Identification Identification Yik = 0 otherwise (negative) for i=1, …, I and k=1, …, K Covariates – Swallow (Phytopathology, 1985, 1987) Covariates NIH grant • Veterinary 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) Slide 11 of 37 Slide 12 of 37 www.chrisbilder.com www.chrisbilder.com NotationNotation NotationNotation Outline Outline BSG • Group responses BSG • Example random variables Basics Basics – Zk = 1 denotes a positive response + or − + or − + or − + or − + or − + or − Estimation th Estimation Zk = 0 denotes a negative response for the k group Identification Identification Covariates – Zk are independent Bernoulli(θ) random variables Covariates NIH grant • θ = P(Zk = 1) 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 − Slide 13 of 37 Slide 14 of 37 www.chrisbilder.com www.chrisbilder.com NotationNotation NotationNotation Outline Outline BSG • Example observed values BSG • Example observed values Basics Basics Basics Basics y11 = 0 y21 = 0 y12 = 0 y22 = 1 y13 = 0 y23 = 0 Estimation Estimation Identification Identification Covariates Covariates NIH grant NIH grant - + + z1 = 0 z2 = 1 z3 = 1 y31 = 0 y41 = 0 y32 = 0 y42 = 0 y33 = 0 y43 = 1 Slide 15 of 37 Slide 16 of 37 www.chrisbilder.com www.chrisbilder.com PurposePurpose EstimateEstimate p Outline Outline BSG • Prevalence of a trait in a population (estimation problem) BSG • How can we estimate p = P(Yik = 1)? Basics • Which items are positive (identification problem) Basics – We observe information about the groups, not Estimation Estimation individuals! Identification Identification I Covariates Covariates – θ = 1 – P(Yik = 0, ∀i) = 1 – (1 – p) NIH grant 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 BSG • What if there is testing error? BSG • Dorfman (Annals of Mathematical Statistics, 1943) Basics Basics – Can incorporate sensitivity (η) and specificity (δ) – Retest all items in a positive group y12 = 0 y22 = 1 Estimation – Estimation – Often credited for the very first use Identification Identification Covariates Covariates of group testing NIH grant 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 BSG • U.S. national program funded by Centers for Disease BSG • Lindan et al. (J. Clinical Microbiology, 2005) Basics Control and Prevention Basics – Estimates that 12% of the laboratories in the U.S. are Estimation – Assess and reduce prevalence of chlamydia and Estimation already using group testing Identification gonorrhea Identification Covariates Covariates – Group testing has allowed “laboratories to achieve a NIH grant •Nebraska 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 Slide 21 of 37 Slide 22 of 37 www.chrisbilder.com www.chrisbilder.com • InfertilityInfertility PreventionPrevention ProgramProgram Heterogonous populations Outline Outline BSG • Quarter #1 of 2006 (continued) BSG • Individual responses Basics Basics – Individual data: 111/1,384 = 0.0802 – Yik are independent Bernoulli(pik) random variables Estimation – Group testing: Estimation – p = P(Y = 1) for item i in group k Identification Identification ik ik Covariates Covariates • Group responses NIH grant 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 BSG • Part of the data from Vansteelandt et al. (Biometrics, 2000) BSG • Model Basics Basics – logit(pik) = β0 + β1xik1 + … + βpxikp Estimation Estimation • Estimation of β , β , β , …, β Identification Identification 0 1 2 p Covariates Covariates – Note that Yik are not directly observed NIH grant 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 BSG •Alike BSG •Random Basics – Individuals with “similar” covariates are put into pools Basics – Individuals are assigned to pools at random Estimation – Smallest