Linear Regression in R: lm(), summary() & Diagnostics

To run a linear regression in R, fit a model with the built-in lm() function — for example model <- lm(y ~ x, data = mydata) — then read the results with summary(model). The summary gives you the intercept and slope coefficients, their p-values, and the R² value that tells you how much of the variation in the outcome the model explains. This guide walks through every step: fitting the model, interpreting the summary() output, plotting the fit, checking assumptions and making predictions.

What Is Regression Analysis?

In statistics, regression analysis is used to study the relationship between an independent and a dependent variable. The method ‘regresses’ the value of y, the dependent variable, on x, the independent variable. In plain terms, regression measures how y changes as x changes, and lets you predict y for values of x you have not directly observed.

Knowing which of your variables is the predictor and which is the response is the first decision you make, because it determines the order in which they go into the lm() formula.

What Is Linear Regression?

In linear regression the relationship between x and y is modelled as a straight line: as x increases, y tends to increase or decrease at a constant rate. A single-predictor (simple) linear regression is described by the equation:

y = b₀ + b₁x + ε — The simple linear regression model

Where:

• y is the dependent or ‘response’ variable;
• x is the independent or ‘predictor’ variable;
• b₀ is the intercept (the value of y when x = 0);
• b₁ is the slope (the change in y for a one-unit increase in x);
• ε is the random error term.

When you have one predictor it is simple linear regression; when you add two or more predictors it becomes multiple linear regression, which uses the same lm() function with extra terms on the right of the formula.

What Does ‘R’ Mean in Linear Regression?

There are two senses of ‘R’ that often get confused, so it is worth separating them clearly:

• R (lower-case r in many texts) is the correlation coefficient — a value between −1 and +1 measuring the strength and direction of the linear relationship between the observed and predicted values of y.
• R² (the coefficient of determination) is the square of that value. It ranges from 0 to 1 and gives the proportion of the variation in y that the model explains. An R² of 0.85 means the model accounts for 85% of the variability in the response.

R² is reported automatically by summary() in R, so you rarely calculate it by hand. To find it manually you would compute R² = 1 − (SS_residual / SS_total), where SS_residual is the sum of squared residuals and SS_total is the total sum of squares. From a scatter plot, R itself is the correlation, obtained in R with cor(x, y); squaring it gives R².

The Two Functions You Need

Two base-R functions do most of the work. You do not need to install any package for them.

1. lm() — fits the model. The basic syntax is:

lm(formula, data)

• formula — the relationship, written as response ~ predictor (read the ~ as “is modelled by”);
• data — the data frame that contains those variables.

2. predict() — generates predictions. Once the model is fitted you can predict y for new values of x:

predict(object, newdata)

• object — the model created with lm();
• newdata — a data frame containing the new predictor values (the column name must match the predictor used in the model).

Reading the summary() Output

Calling summary(model) is where you actually interpret the regression. The annotated table below maps each part of a typical output to what it tells you.

The original example below uses base R to model a person’s weight (the dependent variable) from their height (the independent variable).

Linear Regression in R Step by Step

Here is the full workflow, from a fresh RStudio session to an interpreted result.

Step 1 — Install R and RStudio. Download base R from CRAN and RStudio Desktop. In RStudio, click File > New File > R Script so you can save and re-run your code.

Step 2 — Load helper packages (optional). Base R can fit and plot a regression with no add-ons, but tidyverse (data manipulation and ggplot2 visualisation) and ggpubr (publication-ready plots) make life easier. Install once with install.packages(), then load them each session:

install.packages(c("tidyverse", "ggpubr"))   # run once
library(tidyverse)
library(ggpubr)
theme_set(theme_pubr())

Step 3 — Load and inspect your data. Import a CSV and check it before modelling:

mydata <- read.csv("mydata.csv")
head(mydata)      # first six rows
summary(mydata)   # min, max, mean of each column

Step 4 — Fit the model with lm(). Put the response on the left of the ~ and the predictor(s) on the right:

# Simple linear regression (one predictor)
model <- lm(weight ~ height, data = mydata)

# Multiple linear regression (several predictors)
model2 <- lm(weight ~ height + age, data = mydata)

Step 5 — Read the results. Inspect the coefficients, R² and p-values:

summary(model)
confint(model)    # 95% confidence intervals for the coefficients

Step 6 — Plot the data and the fitted line. In base R:

plot(weight ~ height, data = mydata,
     main = "Weight vs Height")
abline(model, col = "red")   # adds the regression line

Or, with ggplot2 for a cleaner figure:

ggplot(mydata, aes(x = height, y = weight)) +
  geom_point() +
  geom_smooth(method = "lm", se = TRUE)

Step 7 — Predict new values. Estimate weight for new heights:

newdata <- data.frame(height = c(170, 182))
predict(model, newdata)

Step 8 — Report in words. Translate the slope into plain language, e.g. “For every 1 cm increase in height, predicted weight rose by 0.67 kg (p < 0.001), and the model explained 86% of the variation in weight (R² = 0.86).”

Assumptions and Diagnostic Plots

A linear regression is only trustworthy if its assumptions hold. There are four to check:

1. Linearity — the relationship between predictor and response is genuinely a straight line.
2. Independence — the residuals (errors) are independent of one another.
3. Homoscedasticity — the residuals have roughly constant variance across all fitted values (no funnel shape).
4. Normality — the residuals are approximately normally distributed.

R checks all four with a single command. Calling plot() on a model produces four diagnostic charts:

par(mfrow = c(2, 2))   # show all four plots together
plot(model)

Worked Example: Predicting Weight from Height

# 1. Enter the data
height <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
weight <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)
mydata <- data.frame(height, weight)

# 2. Fit the model
model <- lm(weight ~ height, data = mydata)

# 3. Read the output
summary(model)

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept) -38.4551     8.0440   -4.78  0.0014 **
height        0.6746     0.0511   13.20  1.0e-06 ***

Multiple R-squared:  0.956,  Adjusted R-squared: 0.951
F-statistic: 174.2 on 1 and 8 DF,  p-value: 1.0e-06

predict(model, data.frame(height = 170))
# -38.4551 + 0.6746 * 170 ≈ 76.2 kg

Plotting the points with the fitted line shows the positive linear trend at a glance:

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