Title: Error and coefficient
Course: UGQuants (SSPS) > Section: Multiple Regression

Error and coefficient

We know that this summary line will not fit all the points perfectly: so another way of thinking about our data is to say that this regression line (a+bX) needs to take into account some error (ε) in order to be equal to the observed values (Y):

 Ŷ =  (a + bX) + ε

 DATA = FIT + RESIDUAL

With interval or ratio levels of measurement we can use Pearson’s r - the correlation coefficient - to measure how tightly the data points cluster around the regression line.  If the model were perfect (i.e. if there were no error), then there would be perfect correlation (positive or negative, depending on the sign of the slope), meaning that all the data points lie exactly on the line.  Any deviations from the straight line would represent errors in the model, such that greater differences would lead to correlation coefficients getting closer to zero (with r = 0 indicating a complete lack of fit).