Linear Regression with lm()

Linear regression is the workhorse of statistical modelling in R, and lm() is the function that fits ordinary least-squares models. The first argument is a formula of the form response ~ predictors, where the tilde separates the dependent variable on the left from the independent variables on the right. Multiple predictors are added with +, interaction terms with * or :, and transformations can be applied inline (e.g., log(x), I(x^2)). The second argument is the data frame containing the variables. The fitted model object stores coefficients, residuals, fitted values, and the model matrix. The summary() method applied to an lm object produces the regression table with coefficients, standard errors, t-statistics, and p-values, as well as the R-squared, adjusted R-squared, and F-statistic for overall model significance. Individual components are extracted with coef(), residuals(), fitted(), vcov(), and confint(). The predict() function generates fitted values or forecasts for new data. Model diagnostics are crucial for validating that linear regression assumptions are met. plot(model) produces four standard diagnostic plots: residuals vs fitted (checks linearity and homoscedasticity), Q-Q plot of standardised residuals (checks normality), scale-location plot, and residuals vs leverage (identifies influential observations). The broom package provides tidy() and glance() for converting model output into data frames suitable for further manipulation and ggplot2 visualisation.

Example

data(mtcars)

# Simple linear regression: mpg ~ wt
model1 <- lm(mpg ~ wt, data = mtcars)
summary(model1)
# Coefficients:
#             Estimate Std. Error t value Pr(>|t|)
# (Intercept)  37.285      1.878  19.858  < 2e-16 ***
# wt           -5.344      0.559  -9.559  1.29e-10 ***

coef(model1)          # intercept and slope
confint(model1)       # 95% CI for each coefficient
residuals(model1)[1:5]
fitted(model1)[1:5]

# Multiple linear regression
model2 <- lm(mpg ~ wt + hp + cyl, data = mtcars)
summary(model2)

# Model with interaction
model3 <- lm(mpg ~ wt * cyl, data = mtcars)
# equivalent to: mpg ~ wt + cyl + wt:cyl
summary(model3)

# Polynomial term
model4 <- lm(mpg ~ wt + I(wt^2), data = mtcars)
summary(model4)

# Predict for new data
new_cars <- data.frame(wt = c(2.5, 3.0, 3.5),
                       hp = c(110, 130, 150),
                       cyl = c(4, 6, 8))
predict(model2, newdata = new_cars)
predict(model2, newdata = new_cars, interval = "confidence")

# ANOVA table for the model
anova(model2)

# Compare nested models
anova(model1, model2)    # F-test for added predictors

# Diagnostic plots (opens 4 plots)
# par(mfrow = c(2, 2))
# plot(model1)

# R-squared and F-statistic
s <- summary(model1)
s$r.squared          # 0.7528
s$adj.r.squared      # 0.7446
s$fstatistic         # F-value and df

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