Multiple linear regression
Predicting a quantity with a linear model on several variables: reading a coefficient, residuals, R² and adjusted R², the residual plot, polynomial terms, overfitting and the limits of extrapolation.
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Regression: what is it?
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A prediction task whose result is a numerical quantity (a time, a price, a mark) computed from other variables.
It is distinguished from classification, whose result is a category. Predicting a response time in milliseconds is a regression.
A model predicts a home's price: price (€) = 20,000 + 3,000 × area (m²) − 5,000 × distance to the centre (km). What does the −5,000 mean?
- At equal area, each kilometre further from the centre lowers the predicted price by €5,000
- At equal area, each kilometre further from the centre lowers the predicted price by 5%
- A home far from the centre costs €5,000 less, whatever the distance
- Each kilometre lowers the predicted price by €5,000, but for small areas only
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At equal area, each kilometre further from the centre lowers the predicted price by €5,000
A coefficient is the change in the predicted quantity per unit of its variable, the others held fixed. It is expressed in the quantity's unit per unit of the variable: here euros per kilometre, not a percentage. With no interaction term, this effect is the same for every area.
In a multiple regression, a coefficient reads “…”: the other variables are assumed ….
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In a multiple regression, a coefficient reads “all else being equal”: the other variables are assumed held fixed.
A variable's coefficient often changes when a correlated one is added: it does not measure an isolated effect, but the effect once the other is accounted for.
Multiple linear regression: what is it?
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A model y = b0 + b1·x1 + b2·x2 + … that predicts a quantity as a weighted sum of several explanatory variables.
Each coefficient reads “all else being equal”: b1 is the change in y when x1 goes up by one unit while the other variables stay fixed.