Title: Regression line
Course: UGQuants (SSPS) > Section: Multiple Regression

Regression line

We model data in order to simplify what is inevitably a more complex situation, whilst recognising that our model can never be a perfect representation of reality.  It is useful to think here of the equation D = F + R, which describes our ‘DATA’ (observations) in relation to two components: the model, which we call the ‘FIT’, and those elements of each observation which do not get portrayed exactly by the model, which are known as the ‘RESIDUALS’, or the error in the model (not in the observations).  It should be said at this stage that the observations themselves are not likely to be a perfect reflection of what they are supposed to be measuring for a variety of reasons, including the fact that typically samples are drawn to reflect populations and are themselves subject to bias.  Models are usually developed to try to say something valuable about populations, not to be a perfect fit to a sample.

At the interval or ratio levels of measurement we can draw a straight line, known as the regression line, to model a summary of the relationship between a dependent variable (Y) and an independent variable (X) in terms of the equation:

Ŷ = a+bX

The constants, a and b, in this equation can be calculated using ordinary least squares regression (OLS).  We can interpret the equation by saying that an estimated value of Y (Y-hat: Ŷ) can be produced from two components: the slope of the straight line (b), which measures the change in Y for a unit change in X, and the point at which this line intercepts the Y axis when X equals zero (a).