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] 0.6162115`

`mean(percent_control_survive)`

`## [1] 0.3837885`

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

Bo\(^2\)m =)