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Battlestar Galactica Battlestar Galactica Galactica Battlestar Outline Outline Outline Statistics and BSG BSG BSG BSG Battlestar Galactica Basics Basics Basics Human or Cylon Cylon? ? Human or Estimation

  1. Battlestar Galactica Battlestar Galactica Galactica Battlestar Outline Outline Outline • Statistics and � BSG � BSG � BSG BSG Battlestar Galactica � Basics � Basics � Basics Human or Cylon Cylon? ? Human or � Estimation � Estimation � Estimation • The story so far… � Identification � Identification � Identification Group testing on the Group testing on the – Video � Covariates � Covariates � Covariates Battlestar Galactica Battlestar Galactica � NIH grant � NIH grant � NIH grant Christopher R. Bilder Department of Statistics University of Nebraska-Lincoln www.chrisbilder.com chris@chrisbilder.com Slide 1 of 37 Slide 2 of 37 www.chrisbilder.com www.chrisbilder.com Battlestar Galactica Battlestar Galactica Galactica Battlestar Galactica Battlestar Galactica Galactica Battlestar Battlestar Outline Outline • Statistics and • Dr. Gaius Baltar � BSG � BSG BSG BSG Battlestar Galactica – Asked to develop a � Basics � Basics � Estimation � Estimation • The story so far… Cylon detector � Identification � Identification – Video • Season 1’s Bastille � Covariates � Covariates Day episode • Cylons � NIH grant � NIH grant – # of Cylons in fleet is – Centurion expected to be small – Humanoid form – 47,905 individuals to (new) test! • How can you distinguish a human from a Cylon? Slide 3 of 37 Slide 4 of 37 www.chrisbilder.com www.chrisbilder.com

  2. Battlestar Battlestar Galactica Galactica Galactica Battlestar Galactica Battlestar Galactica Galactica Battlestar Battlestar Outline Outline • Dr. Gaius Baltar (continued) • Dr. Gaius Baltar (continued) � BSG � BSG BSG BSG – Season 1’s Tigh me up and Tigh me down – Season 1’s Tigh me up and Tigh me down � Basics � Basics � Estimation � Estimation � Identification � Identification � Covariates � Covariates � NIH grant � NIH grant – Video – Video – (47,905 blood tests) ∗ (11 hours each) = 21,956 days Slide 5 of 37 Slide 6 of 37 www.chrisbilder.com www.chrisbilder.com Battlestar Galactica Battlestar Galactica Galactica Battlestar Galactica Battlestar Galactica Galactica Battlestar Battlestar Outline Outline • Individual testing • Group testing � BSG � BSG BSG BSG � 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 - Cylons • Problems: • If the GROUP is positive, then at least ONE of the 4 – Time individuals is a Cylon – Limited resources Slide 7 of 37 Slide 8 of 37 – “Retesting” can be done to determine who is a Cylon www.chrisbilder.com www.chrisbilder.com

  3. Battlestar Galactica Battlestar Galactica Galactica Other examples Other examples Battlestar Other examples Outline Outline • Group testing (continued) • Screening blood donations � BSG � BSG BSG – Time savings – American Red Cross uses groups of size 16 � Basics � Basics Basics � Estimation � Estimation – Save resources – HIV, Hepatitis B, Hepatitis C, … � Identification � Identification – Strategy works well when prevalence of a “trait” is – Screen about 6 million a year � Covariates � Covariates small � NIH grant � 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 Other examples Notation Notation Other examples Notation Outline Outline • Multiple vector transfer design experiments • Individual responses � BSG � BSG – Y ik = 1 if the i th item in the k th group has the “trait” – Estimate probability an insect vector � Basics � Basics Basics Basics � Estimation � Estimation transfers a pathogen to a plant (positive) and � Identification � Identification Y ik = 0 otherwise (negative) for i =1, …, I and k =1, …, K – Swallow ( Phytopathology , 1985, 1987) � Covariates � Covariates – Y ik are independent Bernoulli( p ) random variables • Veterinary � NIH grant � NIH grant • p = P ( Y ik = 1) – Bovine viral diarrhea in cattle (Peck, Beef , 2006) • Homogenous population – Avian pneumovirus (APV) in turkeys (Maherchandani et al., J. Veterinary Diagnostic Investigation , 2004) • p can be thought of as the “individual probability” or “prevalence in a population” • Public health studies – Y ik ’s are not directly observed (at least initially) – Prevalence of HCV (Liu et al., Transfusion , 1997) – 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

  4. Notation Notation Notation Notation Notation Notation Outline Outline • Group responses • Example random variables � BSG � BSG – Z k = 1 denotes a positive response � Basics � Basics + or − + or − + or − + or − + or − + or − Basics Basics Z k = 0 denotes a negative response for the k th group � Estimation � Estimation � Identification � Identification – Z k are independent Bernoulli( θ ) random variables � Covariates � Covariates • θ = P ( Z k = 1) � NIH grant � NIH grant • Individual and group response relationship + or - + or - + or - – Z k = 1 if and only if Z k = 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 Notation Notation Notation Notation Notation Notation Outline Outline • Example observed values • Example observed values � BSG � BSG � Basics � Basics Basics Basics y 11 = 0 y 21 = 0 y 12 = 0 y 22 = 1 y 13 = 0 y 23 = 0 � Estimation � Estimation � Identification � Identification � Covariates � Covariates � NIH grant � NIH grant - + + z 1 = 0 z 2 = 1 z 3 = 1 y 31 = 0 y 41 = 0 y 32 = 0 y 42 = 0 y 33 = 0 y 43 = 1 Slide 15 of 37 Slide 16 of 37 www.chrisbilder.com www.chrisbilder.com

  5. Purpose Purpose Estimate Estimate p Purpose Estimate p p Outline Outline • Prevalence of a trait in a population (estimation problem) • How can we estimate p = P( Y ik = 1)? � BSG � BSG • Which items are positive (identification problem) – We observe information about the groups, not � Basics � Basics Basics � Estimation � Estimation Estimation individuals! � Identification � Identification – θ = 1 – P ( Y ik = 0, ∀ i ) = 1 – (1 – p ) I � Covariates � Covariates – Then p = 1 – (1 – θ ) 1/ I � NIH grant � NIH grant – MLE for p : • Unequal group sizes – Likelihood function where θ k = positive probability for group k Slide 17 of 37 Slide 18 of 37 I k = size of group k www.chrisbilder.com www.chrisbilder.com Testing error Testing error Identification Identification Testing error Identification Outline Outline • What if there is testing error? • Dorfman ( Annals of Mathematical Statistics , 1943) � BSG � BSG – Can incorporate sensitivity ( η ) and specificity ( δ ) – Retest all items in a positive group � Basics � Basics y 12 = 0 y 22 = 1 � Estimation � Estimation Estimation – – Often credited for the very first use � Identification � Identification Identification of group testing � Covariates � Covariates • Sterrett ( Annals of Mathematical � NIH grant � NIH grant Statistics , 1957) z 2 = 1 – Individually retest until first positive is found – Re-group remaining items • If group is negative, STOP y 32 = 0 y 42 = 0 • If group is positive, repeat – Expected number retests is smaller than Dorfman • Gupta and Malina ( Statistics in Medicine , 1999) provides Slide 19 of 37 Slide 20 of 37 www.chrisbilder.com www.chrisbilder.com a summary

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