<p>Sample of Test Questions:</p><p>1. Q is a p×p orthogonal matrix if |Qx|=|x|. Prove that Q’Q=QQ’=I.</p><p>2. Define Householder transformations. What is it used for?</p><p>3. Show that if H is a composition of Householder transformations then H is orthogonal.</p><p>4. Let X be a n×p data matrix with n observations and p variables. Define the QR decomposition.</p><p>5. How do you use Houlsholder transformations to calculate the QR decomposition of a data matrix X.</p><p>6. How do you apply Given’s rotations to calculate the QR decomposition of a data matrix X.</p><p>7. Show how to compute the Least Squares regression estimate using the QR decomposition.</p><p>8. Define the singular value decomposition of a matrix M.</p><p>9. Let X be a n×p data matrix with n observations and p variables. Show how to use the SVD to calculate the principal components of the data. 10. What do you need to assume about n and p in prob. 9? Suppose that n < p, can you figure out how to calculate the principal components of X.</p><p>11. Suppose that A is a nonsingular covariance matrix. Show that the sequence </p><p> xi+1= Axi/|xi| converges to the largest eigenvalue of A.</p><p>12. Write the logistic regression model. What is the model with link function? What is the variance function?</p><p>13. Which algorithm should be used to calculate the MLE estimator of the logistic model?</p><p>14. What is the difference between fix effects and random effects? </p><p>15. What is a mix effects model? What do we estimate?</p><p>16. Define the 3 and 4 parameters Emax model. Find a relationship to the logistic model.</p><p>17. Draw a picture of the Emax dose response model and show the parameters on the graph. </p>