Discussion-Binomial and Poisson

Discussion-Binomial and Poisson

Let’s assume that a hospital’s neurosurgical team performed N procedures for in-brain bleeding last year. x of these procedures resulted in death within 30 days. If the national proportion for death in these cases is π, then is there evidence to suggest that your hospital’s proportion of deaths is more extreme than the national proportion?

Binomial model

# x is a random variable of the numbers of deaths

#what is N?
n <- 100

#probability of success 
pi <- 0.15

# Number of deaths within 30 days  
x <- 20

#Check for conditions 

## Outcomes are mutually exclusive 
## Each trial is independent 
## result of a count 
## there are only two possible outcomes 

#probability statement 

## P(x >= 20 | n = 100, pi=0.15)

?dbinom

p_dbinom <- 1 - pbinom(q = 19, size = 100, prob = 0.15, lower.tail = TRUE)
p_dbinom

## [1] 0.1065443

print(paste0("the result is ", round(x = p_dbinom, digits = 7)))

## [1] "the result is 0.1065443"

Double checking result

sum(dbinom(x = 20:100, size = 100, prob = 0.15))

## [1] 0.1065443

pbinom(q = 19, size = 100, prob = 0.15, lower.tail = FALSE)

## [1] 0.1065443

Poisson example

lambda <- (n * pi)

?ppois
result2 <- ppois( q = 19, lambda = 15, lower.tail = FALSE) 
print(paste0("the result is ", round(x = result2, digits = 7)))

## [1] "the result is 0.1247812"

sum(dpois(x = 20:100, lambda= 15) ) 

## [1] 0.1247812

Both asnwer are a close due to the fact that the poisson distribution is actually a liminting case of the binomial distribution. The Poisson distribution is a good aproxiamtion to the binomial distribution when the numbers of trials is large and the probability of success is small, such tha np is close the lambda. In my example we can see that np = 15 and lambda is also = 15.