Selecting on a Collider

1 Introduction

Selecting on a collider may introduce bias.

2 Call Libraries

library(ggplot2) # graphics

library(patchwork) # graphics

library(dplyr) # data wrangling

library(sjPlot) # nice tables

set.seed(1234) # set random seed

3 Simulated Data

N <- 1000 # sample size

x <- rnorm(N, 100, 10) # randomly distributed x

e <- rnorm(N, 0, 5) # normal error

y <- 1 * x + e # y is a function of x + random error (e)

df <- data.frame(x, y) # data frame

head(df) # replay

          x         y
1  87.92934  81.90268
2 102.77429 104.28163
3 110.84441 103.14869
4  76.54302  79.71988
5 104.29125 107.80601
6 105.06056  95.53114

4 Select Data

We generate a collider: \(z = x + y\) and then select observations if \(z > 215\) (an admittedly arbitrary, but illustrative value).

df$z <- df$x + df$y # z is a collider

df$selected <- df$z > 215

df_selected <- df %>% # data selected on collider
  filter(selected)  # selection criterion

5 Graphs

We graph the data to get an idea of the full dataset, with selected and non-selected observations, as well as a graph of the selected data only.

Relationships Differ in the Overall and Selected Data

We note that relationships, as typified by the regression line, are quite different in the overall and non-selected data.

p1 <- ggplot(df,
             aes(x = x,
                 y = y)) + 
  geom_point(aes(color = selected)) +
  geom_smooth(method = "lm") +
  labs(title = "y ~ x",
       subtitle = "Entire Sample") +
  scale_color_manual(values = c('TRUE' = 'blue', 
                                'FALSE' = 'red')) +
  ylim(50, 150) + # specify y scale
  theme_minimal()

p2 <- ggplot(df_selected,
             aes(x = x,
                 y = y)) + 
  geom_point(aes(color = selected)) +
  geom_smooth(method = "lm") +
  labs(title = "y ~ x",
       subtitle = "Selected On Collider") +
  scale_color_manual(values = c('TRUE' = 'blue', 
                                'FALSE' = 'red')) +
  ylim(50, 150) + # specify y scale
  theme_minimal()

p1 + p2

6 Regressions

Similarly, regressions find quite different coefficients for x in the two datasets.

fit1 <- lm(y ~ x, data = df)

fit2 <- lm(y ~ x, data = df_selected)

tab_model(fit1, fit2,
          dv.labels = c("Entire Sample", "Selected on Collider"))

	Entire Sample			Selected on Collider
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	-2.71	-5.76 – 0.35	0.083	25.68	13.09 – 38.27	<0.001
x	1.03	1.00 – 1.06	<0.001	0.79	0.68 – 0.90	<0.001
Observations	1000			228
R2 / R2 adjusted	0.814 / 0.814			0.462 / 0.460