Biostatistics: Life in Probabilities Cecilia Cotton Department of - PowerPoint PPT Presentation

Biostatistics: Life in Probabilities Cecilia Cotton Department of Statistics and Actuarial Science For the Love of Math and Computer Science October 14, 2017 1/30

Good Morning! What is Biostatistics? Using the tools of statistics, biostatisticians help answer pressing research questions in medicine, biology and public health, such as whether a new drug works, what causes cancer and other diseases, and how long a person with a certain illness is likely to survive. 1 Today’s Goal: Explore a specific topic in public health using probability theory and statistics to help us gain a deeper understanding of the burden of disease in individuals and the population 1 https://www.biostat.washington.edu/about/biostatististics 2/30

World Health Organization - Measles Fact Sheet 2 ◮ Measles is a highly contagious virus is spread by coughing and sneezing, close personal contact or direct contact with infected nasal or throat secretions ◮ In 1980, before widespread vaccination, measles caused an estimated 2.6 million deaths each year ◮ Measles is one of the leading causes of death among young children even though a safe and cost-effective vaccine is available ◮ In 2015, there were 134 200 measles deaths globally ◮ From 2000-2015, measles vaccination prevented an estimated 20.3 million deaths ◮ In 2016, about 85% of the world’s children received one dose of measles vaccine by their first birthday through routine health services ◮ No specific antiviral treatment exists for the measles virus 2 http://www.who.int/mediacentre/factsheets/fs286/en/ 3/30

Measles in Canada ◮ Measles was eliminated from Canada in 1998 ◮ “The absence of endemic measles transmission in a defined geographic area for 12 months or more, in the presence of a well-performing surveillance system” 3 ◮ Number of measles cases in Canada (Public Health Agency of Canada) 4 ◮ 2014: 127 cases ◮ 2015: 196 cases ◮ 2016: 11 cases ◮ 2017: 45 cases (up to September 23, 2017) 5 3 http://www.wpro.who.int/immunization/documents/measles elimination verification guidelines 2013/en/ 4 https://www.canada.ca/en/public-health/services/diseases/measles/surveillance-measles.html 5 http://www.cbc.ca/news/canada/toronto/toronto-measles-exposure-1.4263354 4/30

Measles surveillance in Canada: 2015 6 ◮ 196 cases in four provinces, 5.5 cases per 1,000,000 population ◮ 172 cases were never immunized or immunization was not up-to-date for age ◮ 9 cases were age-ineligible for measles-containing vaccine ◮ 7 cases were up-to-date with measles vaccination ◮ Remaining cases had unknown vaccination status ◮ 9 imported cases originating from China (2), India (2), Ethiopia, Pakistan, South Africa, Tunisia, and United States ◮ 4 outbreaks (190 cases) and 6 spontaneous sporadic cases ◮ Most cases (81.1%, n=159) were in a non-immunizing religious community. Index case was exposed to measles during travel to “a popular theme park in California” 6 Sherrard L, Hiebert J, Cunliffe J, Mendoza L, Cutler J. Measles surveillance in Canada: 2015. Can Comm Dis Rep 2016;42(7):139-145 5/30

Vaccination and Herd Immunity ◮ Measles can be prevented with a 2 dose vaccine (Ontario: 12 months & 4 years) 7 ◮ The direct effect of vaccination is to protect the vaccinated individual from contracting measles ◮ Herd Immunity is the indirect protection of unvaccinated persons, whereby an increase in the prevalence of vaccine-immunity prevents circulation of infectious agents in unvaccinated susceptible populations. 8 ◮ The vaccination rate necessary to achieve herd immunity depends on the vaccine efficacy, and the infectiousness of the disease ◮ Let’s explore the effect of vaccination rate using some demonstrations 7 http://www.health.gov.on.ca/en/pro/programs/immunization/docs/immunization schedule.pdf 8 Kim TH, Johnstone J, Loeb M. Vaccine herd effect. Scandinavian Journal of Infectious Diseases. 2011;43(9):683-689. 6/30

Herd Immunity Demonstration ◮ If your handout has an empty box (Round 1 and/or Round 2) answer the following question: ◮ Does your birthday occur on a day greater than or equal to the 4 th of the month? ◮ Birthday: February 3 - write NO in the box ◮ Birthday: May 4 - write YES in the box ◮ Birthday: August 30 - write YES in the box ◮ Now our population is about to receive 5 visitors infected with measles ◮ If the visitor infects their initial contact, everyone is eventually exposed ◮ If the visitor does not infect their initial contact, there are no further exposures ◮ Round 1: 75% vaccination rate; Round 2: 95% vaccination rate ◮ Online simulation from The Guardian 9 9 https://www.theguardian.com/society/ng-interactive/2015/feb/05/-sp-watch-how-measles-outbreak-spreads-when-kids-get-vaccinated 7/30

https://www.theguardian.com/society/ng-interactive/2015/feb/05/-sp-watch-how-measles-outbreak-spreads-when-kids-get-vaccinated 8/30

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Model Specification and Assumptions ◮ n = 100 individuals in each population ◮ All in close contact and mix regularly ◮ Each equally likely to come into to contact with an outsider ◮ Vaccination Rate = Prob[vaccination] = p (unspecified, for now) ◮ Vaccine Efficacy = Prob[protected | vaccinated] = 0.99 ◮ Each population gets 5 visits from an infectious outsider ◮ If they contact a protected individual: measles does not enter the community ◮ If they contact a susceptible individual (unvaccinated or vaccinated but susceptible): ◮ Risk of Infection = Prob[infection | susceptible] = 0.90 ◮ If measles enters community, all susceptible individuals will eventually be exposed 10/30

Susceptible p (100 − p ) Unvaccinated Vaccinated 100% 99% 1% Susceptible Susceptible Protected 100% 10% 90% 10% 90% Uninfected Infected Uninfected Uninfected Infected 11/30

Some Probability Definitions and Rules ◮ The Probability than event A occurs is denoted P ( A ) where 0 ≤ P ( A ) ≤ 1 ◮ The probability of an event not occurring is one minus the probability of it occurring: P (¯ A ) = 1 − P ( A ) ◮ If A and B are two events in the sample space S , then the Conditional Probability of A given B is: P ( A | B ) = P ( A ∩ B ) when P ( B ) > 0 , P ( B ) ◮ Law of Total Probability : If B 1 , B 2 , . . . is a partition of the sample space S , then for any event A : � � P ( A ) = P ( A ∩ B i ) = P ( A | B i ) P ( B i ) i i ◮ This simplifies to P ( A ) = P ( A | B ) P ( B ) + P ( A | ¯ B ) P (¯ B ) 12/30

Probability of Infection ◮ Define the following events for a random individual ◮ V = vaccination ◮ P = protection from measles ◮ I = infection ◮ First we condition on whether or not individual was vaccinated P ( I ) = P ( I | V ) P ( V ) + P ( I | ¯ V ) P ( ¯ V ) ◮ If the individual is vaccinated we have: P ( I | V ∩ P ) P ( P | V ) P ( V ) + P ( I | V ∩ ¯ P ) P (¯ P ( I | V ) P ( V ) = P | V ) P ( V ) = 0 . 90(0 . 01) p ◮ For the unvaccinated: P ( I | ¯ V ) P ( ¯ P ( I | ¯ V ∩ P ) P ( P | ¯ V ) P ( ¯ V ) + P ( I | ¯ V ∩ ¯ P ) P (¯ P | ¯ V ) P ( ¯ V ) = V ) = 0 . 90(1 − p ) 13/30

Susceptible p (1 − p ) Unvaccinated Vaccinated 1 0 . 01 0 . 99 Susceptible Susceptible Protected 1 0 . 10 0 . 90 0 . 10 0 . 90 Uninfected Infected Uninfected Uninfected Infected 0 . 90(1 − p ) 0 . 90(0 . 01) p 14/30

Probability of an “Outbreak” in the Population ◮ We just showed: P ( Infection ) = 0 . 90(1 − p ) + 0 . 90(0 . 01) p = 0 . 90(1 − 0 . 99 p ) ◮ Recall: Each population gets 5 visits from an infectious outsider ◮ What is the probability that an “outbreak” occurs in a given population? P ( Outbreak ) = 1 − P ( No Outbreak ) 5 P ( i th visit does not lead to an infection ) � = 1 − i =1 1 − [ P ( No Infection )] 5 = 1 − [1 − 0 . 90(1 − 0 . 99 p )] 5 = 15/30

Probability of an “Outbreak” in the Population Vaccination Theoretical Rate Probability 10.00 99.98 30.00 99.33 50.00 95.17 58.50 90.75 68.90 81.46 74.40 74.16 83.80 56.49 86.00 51.22 90.00 40.33 95.00 24.06 99.70 5.70 16/30

Simulation Study ◮ We will further explore the population dynamics using Monte Carlo Simulation ◮ Frequently used in biostatistics when exact analytical derivations are not possible ◮ A typical Monte Carlo Simulation involves the following: 10 1. General N independent data sets under the conditions of interest 2. Compute the numerical values of the estimator of the statistic of interest T (data) for each data set: T 1 , . . . , T n 3. If N is large enough, summary statistics across T 1 , . . . , T n should be good approximations of the true properties of the estimator under the conditions of interest 10 http://www4.stat.ncsu.edu/ davidian/st810a/simulation handout.pdf 17/30

Simulation Study ◮ For this study, simulate N = 1000 different populations of size n = 100 ◮ Each individual’s vaccination and protection/susceptibility status were independently generated from Binomial random variables ◮ Up to 5 visits from an infectious outsider, Infection status also randomly generated ◮ Statistics of interest: ◮ Does an “Outbreak” occur? (i.e. does the infection enter the population?) ◮ If the infection enters the population, what proportion of individuals are: ◮ Vaccinated, Protected, (Uninfected) ◮ Unvaccinated, Uninfected ◮ Vaccinated, Susceptible, Uninfected ◮ Unvaccinated, Infected ◮ Vaccinated, Susceptible, Infected ◮ Repeat the simulation study for 11 different vaccination rates ◮ Coding was done using R statistical programming language 18/30