Lack-of-fit Sum of Squares - Probability Distributions - Sums of Squares

Sums of Squares

Suppose the error terms ε _{i j} are independent and normally distributed with expected value 0 and variance σ2. We treat x _i as constant rather than random. Then the response variables Y _{i j} are random only because the errors ε _{i j} are random.

It can be shown to follow that if the straight-line model is correct, then the sum of squares due to error divided by the error variance,

has a chi-squared distribution with N − 2 degrees of freedom.

Moreover, given a total number of observations N, a number of observables n and a number of parameters in the model p:

• The sum of squares due to pure error, divided by the error variance σ2, has a chi-squared distribution with N − n degrees of freedom;
• The sum of squares due to lack of fit, divided by the error variance σ2, has a chi-squared distribution with n − p degrees of freedom (here p = 2 as there are two parameters in the straight-line model);
• The two sums of squares are probabilistically independent.