LINEAR STATISTICL MODELS
LINEAR STATISTICL MODELS STAT 714
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This 3 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 714 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/229671/stat-714-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.
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Date Created: 10/26/15
STAT 7147 FALL 2008 MIDTERM 1 TAKE HOME DIRECTIONS There are 3 questions Each question is worth 10 points Show all of your work Explain all of your reasoning This part is open book open notes If you use software eg Maple R SAS etc attach all output You may not talk with anyone besides the instructor 1 Consider two linear models for the same data Model 1 yXbe Model 2 y Wc e Here X is N gtlt p W is N gtlt q b is p gtlt 1 and c is q gtlt 1 Suppose that CW C For Model 1 let yX EX and PX denote the vector of least squares tted values the vector of least squares residuals and the perpendicular projection matrix onto The quantities 37W EW and PW are de ned analogously For each part you may use the previous parts in your answer eg to prove c you may use the results from a and b etc a Show that W XC for some p gtlt q matrix C Show that Pxpw c Show that yX 7 WWW 0 CD Show that yy YiVYW YX YWYQA X YW XEX 2 The effectiveness of three skin creams was studied in an experiment on N subjects On the forearm of each subject three locations were speci ed The three creams were randomly allocated to locations on each subject that is each subject received a complete set of three treatments It is assumed that the three observations on the same individual are correlated and that observations on different subjects are uncorrelated A statistical model for this experiment is yij M 5m where o yij denotes the 2th measurement on subject j and o M denotes the mean response for the 2th skin cream Assume that 61739 for 239 1 23 andj 12 N are random variables with Ee 0 varej 02 and correjej p for 239 31 2quot Note that corre 6177 0 whenj 31 j regardless of 239 because 67 and 677 correspond to different subjects a Assuming that M is xed not random express this model as y Xb e De ne all vectors and matrices Your design matrix X should be full rank b Compute covy c Find b the least squares estimator of b Your answer should be a vector I dont want to see matrices in your nal answer d Compute covb PAGE 1 STAT 7147 FALL 2008 MIDTERM 1 TAKE HOME 3 Let PX denote the perpendicular projection matrix onto a Give a detailed argument showing that I 7 PX is the perpendicular projection matrix onto b Let 1 l 0 l l 0 X l 0 l l 0 l l 0 1 Compute PX and I 7 PX c Express 1 2 y 3 4 5 as the sum of two vectors7 one in CX and one in d For the X matrix in part b7 describe in words what CX and NX are PAGE 2 STAT 7147 FALL 2008 MIDTERM 1 IN CLASS DIRECTIONS There are 4 questions Each question is worth 10 points Show all of your work Explain all of your reasoning This part is closed book7 closed notes No calculators are allowed 1 A researcher studies the effects of 3 different diets on weight gains in rats She chooses 12 rats at random from a large population and assigns the rats to exactly one of the 3 diets She also records the covariate z initial weight7 because she knows the response y weight gain will depend on x Assume that 4 rats are assigned to each diet a Write out a linear model7 in non matrix notation7 that relates the response y to the covariate z and the diets Clearly de ne all of your notation Use appropriate subscripts to denote the different diets and rats b Take your model in a and write it in the form y Xb e De ne all vectors and matrices c What assumptions must be true for your model to be a Guass Markov model 2 De ne the matrix A by 2 1 1 A 1 1 0 1 0 1 a Find a basis for CA b Find a vector c 31 0 such that Ac 0 c Find a generalized inverse of A A d If y is random vector with mean u 17 71 0 and covariance matrix 13 nd EAy and covAy 3 Suppose the system Ax c is consistent and that G is a generalized inverse of A a What is a particular solution to the system the general solution b If A is symmetric7 prove that G G is a generalized inverse of A c Prove that the generalized inverse in b is symmetric This shows that there does exist a generalized inverse of A7 A symmetric7 that is symmetric itself 4 True or False A true statement is one that is always true A false statement may be true some of the time7 but not always Each question is worth 2 points No partial credit will be given7 so no explanation is necessary a True or False If the columns of A are linearly dependent7 then lAl 0 b True or False If A is idempotent7 then rA trA c True or False A linear system Ax c is consistent if c E CA d True or False For the general linear model y Xb e7 the normal equations have a unique solution when X is of full column rank e True or False Suppose that x and y are random vectors lf z x y7 then covz covx covy PAGE 1