Multiple linear regression is a method we can use to quantify the relationship between two or more predictor variables and a response variable.

This tutorial explains how to perform multiple linear regression by hand.

**Example: Multiple Linear Regression by Hand**

Suppose we have the following dataset with one response variable *y* and two predictor variables X_{1} and X_{2}:

Use the following steps to fit a multiple linear regression model to this dataset.

**Step 1: Calculate X _{1}^{2}, X_{2}^{2}, X_{1}y, X_{2}y and X_{1}X_{2}.**

**Step 2: Calculate Regression Sums.**

Next, make the following regression sum calculations:

- Σx
_{1}^{2 }= ΣX_{1}^{2 }– (ΣX_{1})^{2}/ n = 38,767 – (555)^{2}/ 8 =**263.875** - Σx
_{2}^{2 }= ΣX_{2}^{2 }– (ΣX_{2})^{2}/ n = 2,823 – (145)^{2}/ 8 =**194.875** - Σx
_{1}y = ΣX_{1}y – (ΣX_{1}Σy) / n = 101,895 – (555*1,452) / 8 =**1,162.5** - Σx
_{2}y = ΣX_{2}y – (ΣX_{2}Σy) / n = 25,364 – (145*1,452) / 8 =**-953.5** - Σx
_{1}x_{2}= ΣX_{1}X_{2}– (ΣX_{1}ΣX_{2}) / n = 9,859 – (555*145) / 8 =**-200.375**

**Step 3: Calculate b _{0}, b_{1}, and b_{2}.**

The formula to calculate b_{1 }is: [(Σx_{2}^{2})(Σx_{1}y) – (Σx_{1}x_{2})(Σx_{2}y)] / [(Σx_{1}^{2}) (Σx_{2}^{2}) – (Σx_{1}x_{2})^{2}]

Thus, **b _{1 }**= [(194.875)(1162.5) – (-200.375)(-953.5)] / [(263.875) (194.875) – (-200.375)

^{2}] =

**3.148**

The formula to calculate b_{2 }is: [(Σx_{1}^{2})(Σx_{2}y) – (Σx_{1}x_{2})(Σx_{1}y)] / [(Σx_{1}^{2}) (Σx_{2}^{2}) – (Σx_{1}x_{2})^{2}]

Thus, **b _{2 }**= [(263.875)(-953.5) – (-200.375)(1152.5)] / [(263.875) (194.875) – (-200.375)

^{2}] =

**-1.656**

The formula to calculate b_{0 }is: y – b_{1}X_{1} – b_{2}X_{2}

Thus, **b _{0 }**= 181.5 – 3.148(69.375) – (-1.656)(18.125) =

**-6.867**

**Step 5: Place b _{0}, b_{1}, and b_{2} in the estimated linear regression equation.**

The estimated linear regression equation is: ŷ = b_{0} + b_{1}*x_{1} + b_{2}*x_{2}

In our example, it is **ŷ = -6.867 + 3.148x _{1} – 1.656x_{2}**

**How to Interpret a Multiple Linear Regression Equation**

Here is how to interpret this estimated linear regression equation: ŷ = -6.867 + 3.148x_{1} – 1.656x_{2}

**b _{0} = -6.867**. When both predictor variables are equal to zero, the mean value for y is -6.867.

**b _{1 }= 3.148**. A one unit increase in x

_{1 }is associated with a 3.148 unit increase in y, on average, assuming x

_{2 }is held constant.

**b _{2 }= -1.656**. A one unit increase in x

_{2 }is associated with a 1.656 unit decrease in y, on average, assuming x

_{1 }is held constant.

**Additional Resources**

The following tutorials provide additional information about linear regression:

An Introduction to Multiple Linear Regression

How to Perform Multiple Linear Regression in Excel

How to Perform Simple Linear Regression by Hand