The Difference between Two Beta Distributions

The Probability Density Function of the difference of two Beta Distributions.

Density, distribution function and quantile function for the distribution of the difference of two Beta distributions with parameters parX and parY. We denote X and Y as two random variables representing the response rate of Control and Treatment group respectively. The assignment of Control and Treatment is practically interchangeable. We denote Z as the difference between two groups such that Z = Y-X.

Usage

dbetadiff(
  z,
  parY,
  parX,
  eps = .Machine$double.eps,
  rel.tol = .Machine$double.eps^0.1
)

Arguments

z

(numeric):
vector of differences between Control and Treatment arms such that Z = Y-X

parY

(numeric):
two parameters of Y's Beta distribution (Treatment)

parX

(numeric):
two parameters of X's Beta distribution (Control)

eps

(number):
lowest floating point number as lower bound of integration

rel.tol

(number):
used in stats::integrate()

Value

The density values

Note

X and Y can be either Control or Treatment and Z = X-Y, subject to assumptions.

Examples

# The following examples use these parameters:
parX <- c(1, 52) # Control group parameters
parY <- c(5.5, 20.5) # Treatment group parameters

# The difference between Control and Treatment is denoted as z.
z <- seq(from = -1, to = 1, length = 100)
plot(z, dbetadiff(z, parY = parY, parX = parX),
  type = "l"
)


# Calculate probability of Go, if difference was at least 15%.
# Explanation: given density of the difference Y - X between two beta distributions X (from control response rate) and Y (from experimental response rate), then we can calculate
# P(Y - X > 0.15), and when this probability is high then we could go.
test <- integrate(
  f = dbetadiff,
  parY = parY,
  parX = parX,
  lower = 0.15,
  upper = 1
)
str(test)
#> List of 5
#>  $ value       : num 0.677
#>  $ abs.error   : num 2.41e-07
#>  $ subdivisions: int 1
#>  $ message     : chr "OK"
#>  $ call        : language integrate(f = dbetadiff, lower = 0.15, upper = 1, parY = parY, parX = parX)
#>  - attr(*, "class")= chr "integrate"
test$value
#> [1] 0.677447

# Calculate probability of Stop, if difference was at most 50%.
# Explanation: given density of the difference Y - X between two beta distributions X (from control response rate) and Y (from experimental response rate), then we can calculate
# P(Y - X < 0.5), and when this probability is high then we could stop.
integrate(
  f = dbetadiff,
  parY = parY,
  parX = parX,
  lower = -1,
  upper = 0.5
)
#> 0.9993974 with absolute error < 8.6e-06