How to find the intersecting point in the regression line

Question

In the above graph two vertical and horizontal line segments are drawn which will intersect the regression line. How to write code to draw those lines to find the intersecting point?

Answer 1

The formula for each regression line is printed on the plot, so we can get the information by simple algebra.

First we will plot each regression line:

x   <- c(0, 22)
y0  <- 27.46 + 0.31 * x
y5  <- 40.18 + 0.49 * x
y10 <- 55.54 + 0.67 * x
y15 <- 71.63 + 0.84 * x

plot(x, y0, type = "l", ylim = c(0, 105), xlim = c(0, 25),
     ylab = "Percentage of VO2max", xlab = "Load (kg)")
lines(x, y5)
lines(x, y10)
lines(x, y15)

Now we just rearrange the appropriate regression line formulas with y = 50, y = 60, and y = 75:

x5  <- (50 - 40.18) / 0.49
x10 <- (60 - 55.54) / 0.67
x15 <- (75 - 71.63) / 0.84

So we can add these to our plot to show that we have the intersections:

abline(h = 50, lty = 2, col = "red")
abline(h = 60, lty = 2, col = "blue")
abline(h = 75, lty = 2, col = "green")
lines(c(x5, x5), c(50, 0), lty = 2, col = "red")
lines(c(x10, x10), c(60, 0), lty = 2, col = "blue")
lines(c(x15, x15), c(75, 0), lty = 2, col = "green")
points(c(x5, x10, x15), c(50, 60, 75))

This looks good. So our three intersections are:

data.frame(x = c(x5, x10, x15), y = c(50, 60, 75))
          x  y
1 20.040816 50
2  6.656716 60
3  4.011905 75

---

EDIT

With some data added in the comments:

df <- data.frame(load = rep(c(0,4.4,10.7,17,21.4), each = 4), 
                 Gradient = c(0,5,10,15), 
                 VO2max= c(28.0,41.0,56.3,71.3,28.2,41.1,57.0,
                           75.0,31.0,45.4,63.6,82.1, 32.0,48.8,
                           66.8,85.5,34.6,50.5,69.9,89.3))

df$Gradient <- as.factor(df$Gradient)

It is possible to do this in ggplot2:

library(ggplot2)

ggplot(df, aes(load, VO2max, group = Gradient)) + 
  geom_point(aes(shape = Gradient), size = 3) + 
  geom_abline(aes(slope = 0.31, intercept = 27.46)) +
  geom_abline(aes(slope = 0.49, intercept = 40.18)) +
  geom_abline(aes(slope = 0.67, intercept = 55.54)) +
  geom_abline(aes(slope = 0.84, intercept = 71.63)) +
  geom_segment(data = data.frame(x = c(x5, x10, x15),
                                 y = c(50, 60, 75),
                                 Gradient = factor(c(50, 60, 75))),
               aes(x, y, xend = x, yend = 0, colour = Gradient),
               linetype = 2) +
  geom_point(data = data.frame(load = c(x5, x10, x15), 
                               VO2max = c(50, 60, 75),
                               Gradient = 1)) +
  coord_cartesian(ylim = c(0, 105), xlim = c(0, 25),
                  expand = 0) +
  geom_hline(data = data.frame(y = c(50, 60, 75), 
                               Gradient = factor(c(50, 60, 75))),
             aes(yintercept = y, colour = Gradient), linetype = 2) +
  theme_minimal() +
  theme(axis.line = element_line()) +
  guides(colour = "none")