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Multivariate Statistics: Distributional Properties and Testing of Factor Effects

Explore the properties of multivariate normal distribution, chi-square distribution, and factor effects in statistical analysis. Understand the significance of quadratic forms and covariance matrices. Learn about testing factor effects and the distribution of quadratic forms. Dive into the main effect sum of squares and error sum of squares, examining their independence and statistical tests for fixed main effects and interactions.

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Multivariate Statistics: Distributional Properties and Testing of Factor Effects

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  1. 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

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

  3. Multivariate Normal Distribution

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

  5. Distribution of Quadratic Forms in Normal Random Variables

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

  7. Probability Distribution Sample Variance

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

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

  10. Main Effect Sum of Squares Probability Distribution Show

  11. Main Effect Noncentrality Parameter

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

  13. Error Sum of Squares Probability Distribution Show

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

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

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

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

  18. 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: aiaj for some (i,j)

  19. Test Statistic

  20. Assignment • Verify the ‘Show’ Results

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