Plot Simultaneous and Pointwise Confidence Bands for Linear Regression

When performing linear regression analysis it can be useful to plot the confidence bands around the regression line to indicate the range of possible values for the response variable. 

There are two types of confidence bands that can be plotted: pointwise and simultaneous.

• Pointwise confidence bands indicate the uncertainty in the predicted response variable at each value of the predictor variable. 
• Simultaneous confidence bands indicate the range of the possible values for the entire regression line.

 We will use the ggplot2 package in R Programming Language to plot both pointwise and simultaneous confidence bands for linear regression.

Simultaneous Confidence Bands

In linear regression, simultaneous confidence bands are used to provide a visual representation of uncertainty associated with the estimated regression line. These bands show the range within which the true regression line is likely to fall with a certain level of confidence.

R

library(ggplot2)
data(mtcars)

#regression model
fit <- lm(mpg ~ wt, data = mtcars)

ci <- predict(fit, interval = "confidence",
              level = 0.95)

ggplot(data = mtcars, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  geom_ribbon(aes(ymin = ci[, 2], ymax = ci[, 3]),
              alpha = 0.2)

Output:

Simultaneous Confidence Bands

Pointwise Confidence Bands

The Pointwise confidence bands, on the other hand, provide a visual representation of uncertainty associated with a single regression coefficient. These bands show the range within which the true value of the coefficient is likely to fall with a certain level of confidence.

R

library(ggplot2)
data(mtcars)

#regression model
fit <- lm(mpg ~ wt, data = mtcars)

ggplot(data = mtcars, aes(x = wt, y = mpg)) +
  geom_point() +
  stat_smooth(method = "lm", se = TRUE)

Output:

Pointwise Confidence Bands

Now let’s plot the simultaneous as well as pointwise confidence bands on the same graph and compare the two visually.

R

library(dplyr)
# sample datasets
setosa <- iris %>% filter(Species == "setosa")
                %>% select(Sepal.Length,
                           Sepal.Width,
                           Species)
virginica <- iris %>% filter(Species == "virginica")
                    %>% select(Sepal.Length,
                               Sepal.Width,
                               Species)

# compute simultaneous confidence bands
setosa <- setosa %>% arrange(Sepal.Length)
virginica <- virginica %>% arrange(Sepal.Length)

 compute linear models
Model <- as.formula(Sepal.Width ~ poly(Sepal.Length,2))
fit1  <- lm(Model, data = setosa)
fit2  <- lm(Model, data = virginica)
# compute design matrices
X1  <- model.matrix(fit1)
X2  <- model.matrix(fit2)
#  general linear hypotheses
cht1 <- multcomp::glht(fit1, linfct = X1)
cht2 <- multcomp::glht(fit2, linfct = X2)
#  simultaneous confidence bands (finally!)
cc1 <- confint(cht1); cc1 <- as.data.frame(cc1$confint)
cc2 <- confint(cht2); cc2 <- as.data.frame(cc2$confint)
setosa$LowerBound <- cc1$lwr
setosa$UpperBound <- cc1$upr
virginica$LowerBound <- cc2$lwr
virginica$UpperBound <- cc1$upr

# combine datasets
mydata <- rbind(setosa, virginica)

# plot both simultaneous confidence
# bands and pointwise confidence
# bands, to show the difference
library(ggplot2)
# prepare a plot using dataframe mydata,
# mapping sepal Length to x,
# sepal width to y, and grouping the data by species
ggplot(data = mydata, aes(x = Sepal.Length,
                          y = Sepal.Width,
                          color = Species)) +
  # add data points
  geom_point() +
  # add quadratic regression with orthogonal polynomials and 95% pointwise
  # confidence intervals
  geom_smooth(method ="lm", formula = y ~ poly(x,2)) +
  # # add 95% simultaneous confidence bands
  geom_ribbon(aes(ymin = LowerBound,
                  ymax = UpperBound),
              alpha = 0.5,
              fill = "grey70")

Output:

Simultaneous Confidence Bands and Pointwise Confidence Bands