A very useful data visualisation tool in science, particularly in medical and sports settings, is the Bland-Altman/Tukey Mean-Difference plot. When comparing two sets of measurements for the same variable made by different instruments, it is often required to determine whether the instruments are in agreement or not.

Correlation and linear regression can tell us something about the bivariate relationship which exists between two sets of measurements. We can identify the strength, form and direction of a relationship but this approach is not recommended for comparative analyses.

The Bland-Altman plot’s first use was in 1983 by J.M Bland and D.G Altman who applied it to medical statistics. The Tukey Mean-Difference Plot was one of many exploratory data visualisation tools created by John Tukey who, interestingly, also created the beloved boxplot.

A simple implementation of the Bland-Altman/Tukey Mean-Difference plot in R is described below. The dataset used for this example is one of R’s built-in datasets, Loblolly, which contains data on the growth of Loblolly pine trees.

Prepare the workspace and have a look at the structure of the data.

library(dplyr)
library(ggplot2)
pine_df <- Loblolly
glimpse(pine_df)

Randomly split the data into two samples of equal length. These samples will be used as our two sets of pine tree height measurements. Arrange in order of descending height to mimic realistic dual measurements of single trees and combine into a dataframe. Note that these data are used simply to demonstrate and the measures in reality are NOT of the same tree using different instruments.

sample_df <- data.frame(sample_n(pine_df, size = nrow(pine_df) * 0.5) %>%
 select(height) %>% 
 arrange(desc(height)), 
 sample_n(pine_df, size = nrow(pine_df) * 0.5) %>%
 select(height) %>%
 arrange(desc(height)))

Assign sensible names to the two pine tree height samples in the dataframe.

names(sample_df) <- c("Sample_1", "Sample_2")

Each row in the dataframe consists of a pair of measurements. The Bland-Altman plot has the average of the two measures in a pair on the x-axis. The y-axis contains the difference between the two measures in each pair. Add the averages and differences data to the dataframe.

sample_df$Avg <- (sample_df$Sample_1 + sample_df$Sample_2) / 2
sample_df$Dif <- sample_df$Sample_1 - sample_df$Sample_2

Finally, code the plot and add the mean difference (blue line) and a 95% confidence interval (red lines) for predictions of a mean difference. This prediction interval gives the level of agreement (1.96 * SD).

ggplot(sample_df, aes(x = Avg, y = Dif)) +
  geom_point(alpha = 0.5) +
  geom_hline(yintercept = mean(sample_df$Dif), colour = "blue", size = 0.5) +
  geom_hline(yintercept = mean(sample_df$Dif) - (1.96 * sd(sample_df$Dif)), colour = "red", size = 0.5) +
  geom_hline(yintercept = mean(sample_df$Dif) + (1.96 * sd(sample_df$Dif)), colour = "red", size = 0.5) +
  ylab("Diff. Between Measures") +
  xlab("Average Measure")