- Write and describe a multiple linear regression model equation
- Calculate and describe the unadjusted coefficient of determination
- Assess the model assumptions with a residual or a predicted values plot
- An explanatory variable is an independent variable, one that may explain or cause a change in another variable.
- A response variable is a dependent variable, one that changes in response to the explanatory variable.
However, what if we found that there are more than one explanatory variable that explain the response variable?
multiple linear regression model
A linear regression model with two or more explanatory variables is called a multiple linear regression model. Since there is more than one explanatory variable, the model is no longer a line. In fact, we can include [latex]p[/latex] explanatory variables in our model.
The equation for the estimated model that uses [latex]p[/latex] variables is
[latex]\hat{y} = a + b_1 \cdot x_1 + b_2 \cdot x_2 + ... + b_p \cdot x_p[/latex]
where [latex]b_1, b_2, ... ,b_p[/latex] are the regression coefficients for explanatory variables [latex]x_1, x_2, ... ,x_p[/latex], respectively.
In multiple linear regression, [latex]b_1, b_2, ... , b_p[/latex] are called partial slopes.
We can interpret the regression coefficients for each explanatory variable in the model in terms of the relationship with the response variable. The explanation is very similar to what we have seen in simple linear regression models. However, since it is a partial slope, we have to make sure that we hold any other explanatory variables constant in our interpretation.
[latex]\hat{y} = a + b_1 \cdot x_1 + b_2 \cdot x_2 + ... + b_p \cdot x_p[/latex]
the partial slope, [latex]b_1[/latex], represents the expected change in the response variable, [latex]y[/latex], for every one unit increase in [latex]x_1[/latex], holding explanatory variables [latex]x_1, x_2, ... , x_p[/latex] constant.