13.1 Contour plot

A contour plot is a graphical representation where contour lines connect points of equal value. It is particularly useful for visualizing interactions between two continuous variables.

Let’s consider the risk of cardiovascular disease (CVD) as a function of three predictors: age, cholesterol level, and smoking.

$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3$$

where $y$ is the response variable (CVD), $x_1$ and $x_2$ are age and cholesterol, respectively. While $x_3$ is the interaction term given by the product of the two predictors, $x_1 \times x_2$.

$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 (x_1 \times x_2)$$

We can visualize the interaction effect between age and cholesterol on CVD risk using a contour plot. The contour lines will show how the risk of CVD changes with different combinations of age and cholesterol levels.

To visualize if within this data there is a interaction effect between variables, and what type of interaction effect it is, we can use a contour plot.

Imagine these are the coefficients for the model, we can create a contour plot to visualize the interaction effect between two predictors, $x_1$ and $x_2$.

library(tidyverse)
set.seed(123)

beta0<- rep(0,200)
beta1<- rep(1,200)
beta2<- rep(1,200)

x1<- runif(200, min = 0, max = 1)
x2 <- runif(200, min = 0, max = 1)
e <- rnorm(200)

Then we simulate the interaction effect using three different scenarios: antagonism, additive (no interaction), and synergism.

• $\beta_3 \approx 0$, it indicates no interaction effect, meaning that the risk of CVD is simply the sum of the individual effects of age and cholesterol without any additional interaction.

• $\beta_3 > 0$, it indicates that the interaction effect is positive, meaning that as both age and cholesterol increase, the risk of CVD increases more than if they were considered independently. Conversely,

• $\beta_3 < 0$, it suggests a negative interaction effect, meaning that the risk of CVD does not increase as much with increasing age and cholesterol.

beta3_antagonism <- rep(-10,200) # Antagonism
beta3_additive <- rep(0,200) # Additive (no interaction)
beta3_synergism <- rep(10,200) # Synergism

We build three models to represent the three different interaction effects: antagonism, additive, and synergism.

y1 = beta0 + beta1*x1 + beta2*x2 + beta3_antagonism*(x1*x2) + e
y2 = beta0 + beta1*x1 + beta2*x2 + beta3_additive*(x1*x2) + e
y3 = beta0 + beta1*x1 + beta2*x2 + beta3_synergism*(x1*x2) + e

observed1 <- tibble(y1, x1, x2)
observed2 <- tibble(y2, x1, x2)
observed3 <- tibble(y3, x1, x2)

mod1 <- lm(y1 ~ x1*x2, data = observed1)
mod2 <- lm(y2 ~ x1*x2, data = observed2)
mod3 <- lm(y3 ~ x1*x2, data = observed3)

observed1$z <- predict(mod1, observed1)
observed2$z <- predict(mod2, observed2)
observed3$z <- predict(mod3, observed3)

We need to create a grid of points to plot the contour lines. This grid will cover the range of values for $x_1$ and $x_2$.

grid1 <- with(observed1, 
             interp::interp(x = x1, y = x2, z))
griddf1 <- subset(data.frame(x = rep(grid1$x, nrow(grid1$z)),
                            y = rep(grid1$y, 
                                    each = ncol(grid1$z)),
                            z = as.numeric(grid1$z)),!is.na(z))

grid2 <- with(observed2, 
             interp::interp(x = x1, y = x2, z))
griddf2 <- subset(data.frame(x = rep(grid2$x, nrow(grid2$z)),
                            y = rep(grid2$y, 
                                    each = ncol(grid2$z)),
                            z = as.numeric(grid2$z)),!is.na(z))

grid3 <- with(observed3, 
             interp::interp(x = x1, y = x2, z))
griddf3 <- subset(data.frame(x = rep(grid3$x, nrow(grid3$z)),
                            y = rep(grid3$y, 
                                    each = ncol(grid3$z)),
                            z = as.numeric(grid3$z)),!is.na(z))

We can now create the contour plots for each of the three scenarios using ggplot2. Each plot will show the contour lines representing the predicted values of $y$ based on the interaction between $x_1$ and $x_2$.

p1 <- ggplot(griddf1, aes(x, y, z = z)) +
  geom_contour(aes(colour = after_stat(level),
                   linetype = factor(after_stat(level))),
                   linewidth = 2) +
  scale_color_viridis_c() +
  guides(linetype = "none") +
  labs(title="Antagonism",
       color="Prediction", x = "x1", y = "x2") +
  theme(legend.position = "top")

p2 <- ggplot(griddf2, aes(x, y, z = z)) +
  geom_contour(aes(colour = after_stat(level),
                   linetype = factor(after_stat(level))),
                   linewidth = 2) +
  scale_color_viridis_c()+
  guides(linetype = "none")+
  labs(title="Additive (no interaction)",
       color="Prediction", x = "x1", y = "x2")+
  theme(legend.position = "top")

p3 <- ggplot(griddf3, aes(x, y, z = z)) +
  geom_contour(aes(colour = after_stat(level),
                   linetype = factor(after_stat(level))),
                   linewidth = 2) +
  scale_color_viridis_c()+
  guides(linetype = "none")+
  labs(title="Synergism",
       color="Prediction", x = "x1", y = "x2")+
  theme(legend.position = "top")

Combine the three contour plots using patchwork:

library(patchwork)
p1 + p2 + p3