## Role of vaccine

Problem Set 2

C. Durso

Introduction

These questions were rendered in R markdown through RStudio (https://www.rstudio.com/wp-content/uploads/2015/02/rmarkdown-cheatsheet.pdf, http://rmarkdown.rstudio.com ).

Please generate your solutions in R markdown and upload both a knitted doc, docx, or pdf document in addition to the Rmd file.

The questions in this problem set use material from the slides on discrete probability spaces and the Rmd “Discrete_Probability_Distributions_2_3_3.Rmd”.

Load Data

data(“PolioTrials”)

dat<-PolioTrials

Question 1

Please carry out the analysis below and answer the questions that follow. For this assignment, please do all calculations in R and show the code and the results in the knit document.

Frame Question

The basic question “did the vaccine work?” was addressed in problem set 1 using the “rbinom” function to implement the idea that populations in the “Vaccinated” and “Placebo” groups in the “Randomized Control” experiment were the same in regards to paralytic polio cases by using the “rbinom” function to generate a count of polio cases in a population of the size “Vaccinated” population with a probability of paralytic polio estimated from the combined groups. The “rbinom” function with arguments “rbinom(n,size,prob)” draws n random samples from the binomial distribution Binom(size,prob)used to model the number of successes in “size” independent Bernoulli trials with probability of success equal to “prob”.

Note that the function “dbinom(x,size,prob)” gives the value of the density function for Binom(size,prob) at x. Thus “dbinom(x,size,prob)” give the probability of exactly x successes in “size” independent Bernoulli trials in which the probability of success is “prob”. Likewise,the function “pbinom(x,size,prob)” returns the probability of the event that the number of successes is in the set {0,1,…x}.

Q1, part 1

(5 points)

Consider the null model that the number of paralytic polio cases in the “Vaccinated” group follows the binomial distribution with “size” equal to the number of participants in the “Vaccinated” group of the “RandomizedControl” experiment and “prob” equal to the proportion of paralytic polio cases in the pooled “Vaccinated” and “Placebo” groups of the “RandomizedControl” experiment. What is the probability under this model of the event of that the number of cases is less than or equal to the observed number of cases? Please calculate this directly rather than simulating it.

Q1, part 2

(5 points)

Is the value computed in part 1 strong evidence against the null model?

Question 2

Please carry out the analysis below and answer the questions that follow.

Frame Question

In this section, you will address the question of whether the “NotInoculated” and “Placebo” groups in the “Randomized Control” experiment had significantly different rates of paralytic polio using a binomial probability space directly.

Let the null model for the number of paralytic polio cases in the “Placebo” group of the “RandomizedControl” be the binomial probability space with “size” number of participants in the “Placebo” group of the “RandomizedControl” experiment and “prob” equal to the proportion of paralytic polio cases in the pooled “Placebo” and “NotInnoculated” groups of the “RandomizedControl” experiment, capturing the idea that populations in the “NotInnoculated” and “Placebo” groups in the “RandomizedControl” experiment were the same in regards to paralytic polio cases.

Q2, part 1