GLM Hypothesis Testing - Partitioning Sums of Squares

A fundamental principle of least squares methods is that variation on a dependent variable can be partitioned, or divided into parts, according to the sources of the variation. Suppose that a dependent variable is regressed on one or more predictor variables, and that for convenience the dependent variable is scaled so that its mean is 0. Then a basic least squares identity is that the total sum of squared values on the dependent variable equals the sum of squared predicted values plus the sum of squared residual values. Stated more generally,

where the term on the left is the total sum of squared deviations of the observed values on the dependent variable from the dependent variable mean, and the respective terms on the right are (1) the sum of squared deviations of the predicted values for the dependent variable from the dependent variable mean and (2) the sum of the squared deviations of the observed values on the dependent variable from the predicted values, that is, the sum of the squared residuals. Stated yet another way,

Total SS = Model SS + Error SS

Note: the Total SS is always the same for any particular data set, but that the Model SS and the Error SS depend on the regression equation. Assuming again that the dependent variable is scaled so that its mean is 0, the Model SS and the Error SS can be computed using

Model SS = b'X'Y

Error SS = Y'Y - b'X'Y

Whole Model Tests

Six Types of Sums of Squares

Error Terms for Tests

Testing Specific Hypotheses

Testing Hypotheses for Repeated Measures and Dependent Variables