Consider a two-armed study comparing a placebo and treatment. In general, the probabilistic index (PI) is defined as,

and is interpreted as **the probability that a subject in the treatment group will have an increased response compared to a subject in the placebo group**. The probabilistic index is a particularly useful effect measure for ordinal data, where effects can be difficult to define and interpret owing to absence of a meaningful difference. However, it can also be used for continuous data, noting that when the outcome is continuous, and the PI reduces to .

suggests an increased outcome is equally likely for subjects in the placebo and treatment group, while suggests an increased outcome is more likely for subjects in the treatment group compared to the placebo group, and the opposite is true when .

## Simulation

Suppose and represent the independent outcomes in the placebo and treatment groups, respectively and an increased value of the outcome is the desired response.

We simulate observations from each group such that treatment *truly* increases the outcome and the variances within each group are equal such that .

```
# Loading required libraries
library(tidyverse)
library(gridExtra)
# Setting seed for reproducibility
set.seed(12345)
```

```
# Simulating data
n_X = n_Y = 50
sigma_X = sigma_Y = 1
mu_X = 5; mu_Y = 7
outcome_X = rnorm(n = n_X, mean = mu_X, sd = sigma_X)
outcome_Y = rnorm(n = n_Y, mean = mu_Y, sd = sigma_Y)
df <- data.frame(Group = c(rep('Placebo', n_X), rep('Treatment', n_Y)),
Outcome = c(outcome_X, outcome_Y))
```

Examining side-by-side histograms and boxplots of the outcomes within each group, there appears to be strong evidence that treatment increases the outcome as desired. Thus, we would expect a probabilistic index close to 1 as most outcomes in the treatment group appear larger than those of the placebo group.

```
# Histogram by group
hist_p <- df %>%
ggplot(aes(x = Outcome, fill = Group)) +
geom_histogram(position = 'identity', alpha = 0.75, bins = 10) +
theme_bw() +
labs(y = 'Frequency')
# Boxplot by group
box_p <- df %>%
ggplot(aes(x = Outcome, fill = Group)) +
geom_boxplot() +
theme_bw() +
labs(y = 'Frequency')
# Combine plots
grid.arrange(hist_p, box_p, nrow = 2)
```

Continue reading The Probabilistic Index for Two Normally Distributed Outcomes