Chapter 1 of 7

Adding a Second Variable

Multiple Linear Regression

In simple regression, we used one variable — salary — to predict market value. R² = 70.8%. Can we do better?

Estimated regression equation

\hat{y} = -0.9 + 3.90 \times \text{salary}

But these players have different positions. It may help us improve the model.

The single line underpredicts outfield players and overpredicts goalkeepers. It seems that position matters.

Add position as a second variable. Each group gets its own intercept — but the same slope.

Estimated regression equation

\hat{y} = -1.4 + 3.04 \times \text{salary} + 10.8 \times \text{outfield}

R² 0.0%

The binary variable is a switch. It shifts the line up for outfield players by a fixed amount.

Full estimated equation

\hat{y} = -1.4 + 3.04 \times \text{salary} + 10.8 \times \text{outfield}

Goalkeeper (outfield = 0)

\hat{y} = -1.4 + 3.04 \times \text{salary} + 10.8 \times \textcolor{#b45309}{0}

\hat{y} = -1.4 + 3.04 \times \text{salary}

Outfield (outfield = 1)

\hat{y} = -1.4 + 3.04 \times \text{salary} + 10.8 \times \textcolor{#2563eb}{1}

\hat{y} = -1.4 + 3.04 \times \text{salary} + 10.8

\hat{y} = 9.4 + 3.04 \times \text{salary}

Both lines have the same slope (3.04). The only difference is the starting point — the vertical shift of 10.8.