Interactions: radon and smoking

Source: Interactions/interactions.Rmd

The original page is a small plotting example from Chapter 1: two straight-line risk curves whose slopes differ by smoking status. The point is the visual meaning of an interaction: the effect of home radon exposure is larger for smokers than for nonsmokers.

Plot the two interaction lines

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

radon = np.linspace(0, 12.5, 200)
curves = pd.DataFrame({
    "radon": np.tile(radon, 2),
    "smoking": np.repeat(["Smokers", "Nonsmokers"], len(radon)),
})
curves["prob_lung_cancer"] = np.where(
    curves["smoking"].eq("Smokers"),
    0.07409 + 0.0134 * curves["radon"],
    0.00579 + 0.0026 * curves["radon"],
)

fig, ax = plt.subplots(figsize=(6, 4))
for label, grp in curves.groupby("smoking"):
    ax.plot(grp["radon"], grp["prob_lung_cancer"], color="black", lw=1)

ax.text(10, 0.07409 + 10 * 0.0134 - 0.02, "Smokers")
ax.text(10, 0.00579 + 10 * 0.0026 + 0.01, "Nonsmokers")
ax.set(
    xlim=(0, 12.5), ylim=(0, 0.25),
    xlabel="Home radon exposure (pCi/L)",
    ylabel="Probability of lung cancer",
    title="Example of an interaction",
)
ax.spines[["top", "right"]].set_visible(False)

Regression form

A linear interaction model for these two lines can be written as

[ () = + x + S + xS, ]

where x is radon exposure and S=1 for smokers. Taking nonsmokers as the baseline gives:

alpha = 0.00579
beta = 0.0026
gamma = 0.07409 - 0.00579
delta = 0.0134 - 0.0026
pd.Series({"Intercept": alpha, "radon": beta, "smoker": gamma, "radon:smoker": delta})

Intercept       0.00579
radon           0.00260
smoker          0.06830
radon:smoker    0.01080
dtype: float64

The interaction coefficient is the difference in slopes. In this stylized example, each additional pCi/L of radon changes the probability by 0.0026 for nonsmokers and by 0.0134 for smokers.