General Linear Model

\(Y = X\beta + \epsilon\) \(\beta = (X'V^{-1}X)^{-1}X'V^{-1}Y\)

Why

The GLM is a powerfultool, since many different types of analyses can be carried out using it including: one-sample t-tests, two-sample t-tests, paired t-tests, ANOVA, and ANCOVA. The first section illustrated simple linear regression in the example where processing speed was modeled as a function of age. Figure A.2 shows some common analyses that are carried out using the GLM where the top example is the simplest model, a one-sample t-test.

Figure A.2. Examples of GLM models for popular study designs including: One-sample t-test, two-sample t-test, paired t-test, and two-way ANOVA. The first column describes the model, the second column describes how the data are ordered in the outcome vector, the third column shows the design matrix, and the last column illustrates the hypothesis tests and corresponding contrasts. Note, in the ANOVA example F-tests are used for all contrasts, whereas t-tests are used for the other examples.