How to compute a confidence interval for the expected value of a response variable (in R)

Task

If we have a simple linear regression model, $y = \beta_0 + \beta_1x + \epsilon$, where $\epsilon$ is some random error, then given any $x$ input, $y$ can be veiwed as a random variable because of $\epsilon$. Let’s consider its expected value. How do we construct a confidence interval for that expected value, given a value for the predictor $x$?

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Solution

Let’s assume that you already have a linear model. We construct an example one here from some fabricated data.

# Make the linear model
x <- c(34, 9, 78, 60, 22, 45, 83, 59, 25)
y <- c(126, 347, 298, 309, 450, 187, 266, 385, 400)
model <- lm(y ~ x)

Construct a data frame containing just one entry, the value of the independent variable for which you want to compute the confidence interval. That data frame can then be passed to R’s predict function to get a confidence interval for the expected value of $y$.

# Use your chosen value of x below:
data <- data.frame(x=40)
# Compute the confidence interval for y:
predict(model, data, interval="confidence", level=0.95) # or choose a different confidence level; here we use 0.95

  fit      lwr     upr     
1 313.7217 226.648 400.7954

Our 95% confidence interval is $[226.648, 400.7954]$. We can be 95% confident that the true average value of $y$, given that $x$ is 40, is between 226.648 and 400.7954.

Content last modified on 24 July 2023.

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Contributed by Elizabeth Czarniak (CZARNIA_ELIZ@bentley.edu)