Cohen's d Archives • Statisticelle

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

$\text{PI} = P(X < Y) + \frac{1}{2} P(X = Y)$

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, $P(X = Y) = 0$ and the PI reduces to $P(X < Y)$ .

$PI = 0.5$ suggests an increased outcome is equally likely for subjects in the placebo and treatment group, while $PI > 0.5$ suggests an increased outcome is more likely for subjects in the treatment group compared to the placebo group, and the opposite is true when $PI < 0.5$ .

Simulation

Suppose $X \sim N(\mu_X, \sigma^{2}_{X})$ and $Y \sim N(\mu_Y, \sigma^{2}_{Y})$ represent the independent outcomes in the placebo and treatment groups, respectively and an increased value of the outcome is the desired response.

We simulate $n_X = n_Y = 50$ observations from each group such that treatment truly increases the outcome and the variances within each group are equal such that $\sigma^{2}_{X} = \sigma^{2}_{Y}$ .

# 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)

plot of chunk unnamed-chunk-3

Continue reading The Probabilistic Index for Two Normally Distributed Outcomes