Adding extra variables
The simplest way to model how a single explanatory variable affects the response is with a simple linear model,
y = b0 + b1 x
This can be easily extended for a second explanatory variable, Z, by adding a further linear term,
y = b0 + b1 x + b2 z
and so on with more explanatory variables,
y = b0 + b1 x + b2 z + b3 w + ...
This is not the only way to use several explanatory variables to predict a response, but it is the simplest.
This type of model is called a multiple regression model.
Least squares estimates
The method used to get estimates of the coefficients of a multiple regression model is an extension of the method that was used to fit a simple linear model — least squares. Least squares will be described later in this section.
Body fat
The diagram below shows the least squares equation that predicts body fat from weight.
Use the checkboxes below the terms in the equation to add other explanatory variables to the model. Observe that the existing coefficients change when you add a new variable.
We have shown how extra variables can be added to a least squares equation to help predict the response.
However we have not investigated whether adding more variables actually improves predictions.
This type of question cannot be addressed until we introduce methodology to use data to ask questions about the process from which the data arose — a collection of methods called inference.