Understanding Partial Solutions in PROC GLM for Statistical Hypothesis Testing - Prof. Bri, Exams of Statistics

Download Understanding Partial Solutions in PROC GLM for Statistical Hypothesis Testing - Prof. Bri and more Exams Statistics in PDF only on Docsity! PROC GLM PROC GLM is set up mainly for testing of statistical hypotheses. It uses a less than full rank coding for the indicator variables in the design matrix. For instance, its design matrix for two factors (3 levels & 2 levels, 2 obs. per cell) is X œ Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ Ù Õ Ø 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 Interactions would be coded as six extra columns (products of cols 2-4 with 5-6). The matrix is singular, and theX Xw normal equations do not have a solution. GLM uses a which allows a partialgeneralized inverse solution to the normal equations. The ordinary inverse of A produces A A I#" œ A property of the ordinary inverse is A AA A#" #" #"œ A of the matrix , denoted , is anygeneralized inverse A A# matrix such that A AA A# # #œ One such matix is found by finding a smaller matrix A which can be inverted. Write A A AA Aœ ” •"" "##" ## where is an invertible matrix (say, ). ThenA "" 7 ‚7 A A 00 0 # "" #" "# #" ## œ ” • is a generalized inverse, where the ' s are matrices of0 zeros. For example, recall the matrix A œ ” •3 16 2 A generalized inverse is A# œ ” •13 00 0