The Hypotheses behind Regression Analysis

The method of least squares explained above makes at least 4 assumptions, the adherence to which may be checked a posteriori. These assumptions concern with the error e_i = $\hat{y}_{i}^{}$ - y_i, i.e., the difference between the best fit prediction and the observation. The assumptions are that these errors (a). are mutually independent (b). have zero mean (c). have a constant variance across all the values of the statistical variables and (d). are normally distributed. Violation of these assumptions can be identified in many cases by simply examining a plot of e_i vs. x_i. Note that the first of Eq. 4 guarantees that the mean value of e_i is 0 within the precision of the computation (To see this better rewrite that equation as $\sum$(b + ax_i - y_i) = 0). However, this is not necessarily true of non-linear regression analysis.