Multiple regression
Objective. Fit, interpret and diagnose multivariable linear models.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Multiple regression models a response variable y as a linear combination of several predictors, y = β0 + β1x1 + ... + βpxp + ε, with coefficients estimated by least squares to minimise the sum of squared residuals. Interpreting β_j requires holding other predictors fixed, and diagnostics such as residual plots check for non-linearity or non-constant variance, while variance inflation factors detect multicollinearity among predictors. For a model with two correlated predictors, high multicollinearity can inflate standard errors even if the overall fit remains good.
Worked example
In a multiple regression model, ŷ = 5 + 2x1 − 3x2, predict ŷ when x1 = 4 and x2 = 1.
- 1Substitute x1 = 4 into the term 2x1 to get 8
- 2Substitute x2 = 1 into the term −3x2 to get −3
- 3Sum with the intercept: 5 + 8 − 3
- 4Simplify to the predicted value
Answer: ŷ = 10
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. A common error is interpreting a regression coefficient as the total effect of a predictor while ignoring that it only represents the effect holding all other predictors constant. Always state coefficient interpretations conditional on the other variables in the model.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.