Effect sizes provide information about the magnitude of an effect. Unfortunately, they can be difficult to interpret or appear “small” to anyone unfamiliar with the typical effect sizes in a given research field. Rosenthal and Rubin (1992) provide an intuitive effect size, called the Binomial Effect Size Display, that captures the change in success rate due to a treatment.

The calculation is simple:

Treamtment BESD = 0.50 + (r / 2)
Control BESD = 0.50 - (r / 2)

where r is the correlation coefficient between treatment and survival (however defined). Many mathematical discussions exist, below is a simulation of one specific example by Randolph and Edmondson (2005). Please keep in mind the BESD is not without its critics (e.g., Thompson 1998).

The Example

Aziothymidine (AZT) is used to treat AIDS, and the correlation between AZT use and survival is 0.23. Using the equations above, we can calculate the BESD for the treatment and control groups.

# Survival

AZT_survive <- 0.50 + (0.23 / 2)
Placebo_survive <- 0.50 - (0.23 / 2)

So the survival percentages for each group are:

AZT_survive

[1] 0.615

Placebo_survive

[1] 0.385

Now we can simulate that process to see if our results match.

The Simulation

Preliminary set up:

k <- 1000
percent_treatment_survive <- numeric(k)
percent_control_survive <- numeric(k)

# The correlation between AZT and survival is 0.23

Sigma <- matrix(c(1.0, 0.23,
                    0.23, 1.0), 2, 2, byrow = T)

Running the process:

for(i in 1:k){
  
  # Draws from a binomial distribution with 0.50 base rate
  
  # The correlation between both vectors is 0.23
  
  # The first vector is treatment vs control assignment.
  # 1 = treatment ; 0 = control
  
  # The second vector is survive vs. not survive
  # 1 = survive ; 0 = not survive
  
  x <- rmvbin(5000, margprob = c(0.5, 0.5), bincorr = Sigma)
  x <- as.data.frame(x)
  
  # "Survive" is when column 2 is equal to 1
  
  total_survive <- x %>%
        filter(V2 == 1)
  
  # The amount of people in each group that survived
  
  treatment_survive <- sum(total_survive$V1 == 1) / nrow(total_survive)
  
  control_survive <- sum(total_survive$V1 == 0) / nrow(total_survive)
  
  # Save the results from each iteration
  
  percent_treatment_survive[i] <- treatment_survive
  percent_control_survive[i] <- control_survive
  
}

Comparison

Our original calculations were as follows:

AZT_survive

[1] 0.615

Placebo_survive

[1] 0.385

and here are the simulation results:

mean(percent_treatment_survive)

[1] 0.6159293

mean(percent_control_survive)

[1] 0.3840707

Keep in mind the BESD assumes a 50/50 base rate of success (however defined) with no treatment.

Bo\(^2\)m =)

The Binomial Effect Size Display

The Example

The Simulation

Comparison