Berkson's Paradox: why handsome people are such a jerk?
It is not difficult to find a relationship between two variables when there is not. We may observe a negative correlation between variables, but the actual correlation between them is positive, or vice versa. Many times this phenomenon is explained by Berkson’s paradox.
When we are talking about correlation is simply refers to the measure of the relationship between two variables. A correlation measures the grade in which two variables are linearly associated. It can take a value that goes from -1 to 1, in which -1 means a perfect negative linear relationship: more \(X\) is associate with less \(Y\), or vice-versa. Conversely, a correlation equal to 1 means that more \(X\) is associated with more \(Y\). Finally, a 0 correlation coefficient between \(X\) and \(Y\) is interpreted as no relationship at all.
It is essential to acknowledge the differences between what we observed and the truth or what it is. In our daily lives, we are surrounded by plenty of scenarios in which we don’t think about the sample size or if groups are well represented within the sample. The thing is that we look at the world through our kaleidoscope, full of patterns, full of colours.
Imagine that you start thinking about the relationship between attractiveness and sympathy…and you begin inferring by your own dates experiences and stories of your fine selection of friends that handsome people are jerks; meanwhile, enjoyable people are not so good looking.
Why it’s so common to hear this? Are attractiveness and niceness put in conflict by our creators? It is widespread to hear from the popular wisdom library such a conundrum, but we don’t have many reasons to believe any relationship between these attributes. Indeed, both variables, in reality, correlate near to 0. But what could it explain this apparent contradiction between the observed and the “truth”? Berkson’s Paradox!
Let’s say that \(X\) = “Niceness” and \(Y\) = “Attractiveness”. Then we collect data from many people, measuring these attributes, which you can visualize in the below animation. Every attribute has a score from 0 to 100; the former means absent and the later fullness of the attribute, respectively. Note that in the animation title, you can see the correlation coefficient (\(\rho\)).
It’s likely and honest to think that, on average, one restricts the potential candidates in which we are interested in getting a date. You accept to date with someone who is not so good-looking if their eloquent and lovely behaviour compensates the absence of visual gracefulness. Otherwise, you could tolerate an idiot in the way in which their handsomeness diminished its jerkiness fragrance. As you can appreciate, from the above animation, your preferences remove from the map the group of candidates who don’t meet your minimum criteria. Let’s call this group “You would not date”.
Now the ugly truth: another group would use similar arguments to avoid our perfect combination of attractiveness and niceness, which we call “Would not date you”. This double mechanism of restrictions prevents us from “observed the world as it is” and explain how we go from a correlation close to 0 between attractiveness and niceness to a strong negative correlation of −0.7: the more handsome you are, the more jerkiness you spit out. An illusion of our kaleidoscope that gives us the impression of a false dichotomy between \(X\) and \(Y\).
Additional resources
- Read more about Berkson’s paradox
- This example was mention in the book “Calling Bullshit, The Art of Skepticism in a Data-Driven World” and elaborate in “How Not to Be Wrong: The Power of Mathematical Thinking”
- Avoiding the Piftalls of Selection Bias (Carl T. Bergstrom, 2021)
Code
The gif made by mounting three different plots created with ggplot2 (R) and using the tool Animated Gif Maker.
Here is the code if you want to reproduce the example:
library(ggplot2)
library(dplyr)
library(latex2exp)
# Multivariate normal dist parameters:
mu <- c(50, 50)
Sigma <- matrix(c(200, 4, 8, 200), 2)
# Generate 10.000 data points
set.seed(323)
df <- as.data.frame(MASS::mvrnorm(5e3, mu, Sigma))
# Compute correlation between V1 and V2
cor0 <- round(cor(df$V1, df$V2), 2)
# Initial plot
p0 <- df %>%
ggplot() +
geom_point(aes(V1, V2), alpha = .3, size = .2, color = "steelblue") +
scale_x_continuous(breaks = seq(0, 100, 20), n.breaks = seq(0, 100, 5)) +
scale_y_continuous(breaks = seq(0, 100, 20)) +
coord_cartesian(xlim = c(0, 100), ylim = c(0, 100), clip = "off") +
labs(x = "Niceness",
y = "Attractiveness",
title = TeX(paste("$\\rho$ =", as.character(cor0)))) +
annotate(geom = "text", x = c(10, 90), y = -15,
label = c("JERK", "NICE"), size = 4) +
annotate(geom = "text", x = -10, y = c(10, 90),
label = c("NOT", "HOT"), size = 4) +
theme_bw(base_size = 8) +
theme(plot.margin = margin(1.5, 1.5, 1.5, 1.5, "cm"),
plot.title = element_text(hjust = 0.5),
panel.grid.major = element_blank(),
panel.grid.minor = element_blank())
# Lower line:
# ------------------------------------------------------------------------------
# Create a lower diagonal line and assign an identifier to each group (2 labels)
# C: {LA=Lower Area, MA = Middle Area}
lower_line <- function(x, m=-1, b = 85) m * x + b
df <- df %>%
mutate(C = case_when(lower_line(V1) < V2 ~ "LA",
TRUE ~ "MA"))
# Compute cor1
cor1 <- round(cor(df[df$C == "MA", "V1"], df[df$C == "MA", "V2"]), 2)
# Create p1
p1 <- df %>%
ggplot() +
geom_point(aes(V1, V2, colour = C), alpha = .27, size = .2) +
scale_x_continuous(breaks = seq(0, 100, 20)) +
scale_y_continuous(breaks = seq(0, 100, 20)) +
coord_cartesian(xlim = c(0, 100), ylim = c(0, 100), clip = "off") +
scale_colour_manual(guide = FALSE, values = c("steelblue", "grey45")) +
labs(x = "Niceness",
y = "Attractiveness",
colour = NULL,
title = TeX(paste("$\\rho$ =", as.character(cor1)))) +
annotate(geom = "text", x = c(10, 90), y = -15,
label = c("JERK", "NICE"), size = 4) +
annotate(geom = "text", x = -10, y = c(10, 90),
label = c("NOT", "HOT"), size = 4) +
annotate("text", label = "YOU WOULD\nNOT DATE", x = 20, y = 20, size = 3.5) +
geom_segment(aes(x=-5, y=lower_line(-5), xend=90, yend=lower_line(90))) +
theme_bw(base_size = 8) +
theme(plot.margin = margin(1.5, 1.5, 1.5, 1.5, "cm"),
plot.title = element_text(hjust = 0.5),
panel.grid.major = element_blank(),
panel.grid.minor = element_blank())
# Upper line:
# ------------------------------------------------------------------------------
# Create a upper diagonal line and assign an identifier (3 labels)
upper_line <- function(x, m=-1, b=115) b + x*m
df <- df %>%
mutate(C = case_when((lower_line(V1) <= V2) & (V2 < upper_line(V1)) ~ "MA",
upper_line(V1) >= V2 ~ "UA",
lower_line(V1) < V2 ~ "LA"))
# Compute cor2
cor2 <- round(cor(df[df$C == "MA", "V1"], df[df$C == "MA", "V2"]), 2)
# Create p2
p2 <- df %>%
ggplot() +
geom_point(aes(V1, V2, colour = C), alpha = .27, size = .2) +
scale_x_continuous(breaks = seq(0, 100, 20)) +
scale_y_continuous(breaks = seq(0, 100, 20)) +
coord_cartesian(xlim = c(0, 100), ylim = c(0, 100), clip = "off") +
scale_colour_manual(guide = FALSE, values = c("grey45", "steelblue", "grey45")) +
labs(x = "Niceness",
y = "Attractiveness",
colour = NULL,
title = TeX(paste("$\\rho$ =", as.character(cor2)))) +
annotate(geom = "text", x = c(10, 90), y = -15,
label = c("JERK", "NICE"), size = 4) +
annotate(geom = "text", x = -10, y = c(10, 90),
label = c("NOT", "HOT"), size = 4) +
annotate("text", label = "YOU WOULD\nNOT DATE", x = 20, y = 20, size = 3.5) +
geom_segment(aes(x=-5, y=lower_line(-5), xend=90, yend=lower_line(90))) +
annotate("text", label = "WOULD NOT\nDATE YOU", x = 80, y = 80, size = 3.5) +
geom_segment(aes(x=10, y=upper_line(10), xend=105, yend=upper_line(105))) +
theme_bw(base_size = 8) +
theme(plot.margin = margin(1.5, 1.5, 1.5, 1.5, "cm"),
plot.title = element_text(hjust = 0.5),
panel.grid.major = element_blank(),
panel.grid.minor = element_blank())
# Save plots
ggsave("/Users/YourUser/Desktop/bp0.png", p0, width = 7.5, height = 7.5, dpi = "retina")
ggsave("/Users/YourUser/Desktop/bp1.png", p1, width = 7.5, height = 7.5, dpi = "retina")
ggsave("/Users/YourUser/Desktop/bp2.png", p2, width = 7.5, height = 7.5, dpi = "retina")