Statistical Analysis Professor Lynne Stokes

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Presentation transcript:

Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 6QF Multivariate Normal Distribution, Chi-square Distribution of Quadratic Forms, Testing the Significance of Factor Effectrs

Quadratic Forms Distributional properties of q depend on both the properties of the known matrix A and the distribution of the random vector x.

Multivariate Normal Distribution

Properties of the Covariance Matrix Nonsingular Symmetric Positive Definite Positive (Semi-) Definite Matrices Similar Definitions: Negative (Semi-) Definite, Indefinite

Distribution of Quadratic Forms in Normal Random Variables

Trace of a Square Matrix Definition Properties Cyclic Permutations li = eigenvalues of A Symmetric Idempotent Matrix li = 1 or 0

Sample Variance Probability Distribution

Total Sum of Squares Quadratic Form Show Degrees of Freedom n -1 = ar - 1 = rank(AT) = tr(AT)

Main Effect Sum of Squares Quadratic Form Show Degrees of Freedom a -1 = rank(AA) = tr(AA)

Main Effect Sum of Squares Probability Distribution Show

Main Effect Noncentrality Parameter

Error Sum of Squares Quadratic Form Degrees of Freedom Show n - a = rank(AE) = tr(AE)

Error Sum of Squares Probability Distribution Show

Pairwise Independence of Quadratic forms Independence of the Main Effect and Error Sums of Squares

Statistical Tests for (Fixed) Main Effects and Interactions : Balanced Complete Factorials Single-Factor Experiment Response Distribution y ~ N(m1 + XAa , s2I)

Statistical Tests for (Fixed) Main Effects and Interactions : Balanced Complete Factorials Distributional Properties SSA & SSE are statistically independent

Statistical Tests for (Fixed) Main Effects and Interactions : Balanced Complete Factorials iff a1 = ... = aa = 0

Testing Factor Effects Single-Factor Model yij = m + ai + eij i = 1, ..., a; j = 1, ..., r Equivalent Simultaneous Test for Main Effects H0: a1 = a2 = ... = aa vs. Ha: ai aj for some (i,j)

Test Statistic

Assignment Verify the ‘Show’ Results