Simple mathematical forms for the regression line
Although any shape of regression line can be drawn 'by eye' through a scatterplot, it is convenient to restrict attention to simple mathematical functions
y = ƒ ( x )
Using a model of this type, we can predict the response for any value, x, of the explanatory variable by inserting that value in the function — much easier than reading off a prediction from a curve that is drawn 'by eye'.
Linear equations
Although some relationships must be described by curves, a straight line is an adequate description of many bivariate data sets. A straight line corresponds to an equation of the form
y = b0 + b1 x
The constant b0 is called the intercept of the line and describes the y-value when the x-value is zero. The constant b1 is called the line's slope; it describes the change in y when x increases by one.
Intercept and slope
The following diagram illustrates the meaning of a line's slope and intercept.
Drag the two red arrows up and down to adust the values of the slope and intercept. Observe how the coloured arrows on the diagram are related to the parameter values.
When used to describe a relationship between two variables, such a straight line is called a linear model or a linear regression model.
Predictions are easily made from a linear model
A linear model can be used to predict the value of y that would be obtained at any value of x, using the equation
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= b0 + b1 x |
Predictions from the equation
The scatterplot below describes a relationship that can be described well by the linear model (drawn in grey in the diagram),
y = 2.0 + 0.4 x
Drag the vertical red line to display predictions of Y at different values of X.