7. Suppose the data consist of repeated observations (y;it, X), t = 1, -.. ,T, for each in- dividual i = 1,... ,n. Here...
Suppose the data consist of repeated observations (yt,X), t = 1,-.. ,T, for each in- dividual i 1,.. , n. Here yit is the response and xjt is a covariate vector. A linear mixed-effects model for analysing the population-averaged and subject-specific effects of Xit is of the following form ZiBWibi Ei, Уi (yil ,yiT)T; Zi is a T x p design matrix built from {xjt} for the fixed are i.i.d. MVN(0, Q) > 0 where y effects B; Wi is...
Question 1 (10 marks) Suppose the data consist of repeated observations (Yit, xT), t = 1, ... ,T, for each individual i = 1,..., n. Here Yit € {0, 1, 2,... } is a count response and Xit is a k x 1 covariate vector. Consider a log-linear random intercept model Yit|bi ~ Poisson(1it), with lit E(Yit|bi) = elit and Nit = zB+bi. Here Zit is a p x 1 design vector built from Xit, and {bi} are i.i.d. N(0,02)...
1) Consider n data points with 3 covariates and observations {xil, Гіг, xī,3, yi); i-1,.,n, and you fit the following model, y Bo+B+B32+Br+e that is yi-An + ßiXiut Ali,2 + Asri,3 + Ei where є,'s are independent normal distribution with mean zero and variance ơ2 For a observed covariate vector-(1, ri, ^2, r3) (with the intercept and three regressor variables) and observed yg at that point a) write the expression for estimated variance for the fit zs at z. (Let...
Problem 4.12 & Problem 4.14 ?
4.12 Suppose y is N4(u, 2), where 8 9 -3 6 3-3 23 μ= PROBLEMS 107 (a) Find the distribution ofz-: 4y1-2y2 + y3-3y4 (b) Find the joint distribution of zy y2y3y4 and z22yi + (c) Find the joint distribution of zı = 3y1 +N2-4y3-N4, z2--yı-3y2+ (d) What is the distribution of y3? (e) What is the joint distribution of y2 and y4? (f) Find the joint distribution of yi, 1(yi + y2), yit...
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
linear stat modeling & regression
1) Consider n data points with 3 covariates and observations {xn, ^i2, xi3,yid; i,,n, and you fit the following model, y Bi+Br2+Br+e that is yi A) +Ari,1 +Ari,2 +Buri,3 + єї where є,'s are independent normal distribution with mean zero and variance ơ2 . H the vectors of (Y1, . . . ,Yn). Assume the covariates are centered: Σίχί,,-0, k = 1,2,3. ere, n = 50, Let L are Assume, X'X is a diagonal matrix...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we have a linear model, = SXY/SXX and A,-ㄚ-Ax be the least-square estimates so that E(X) = β0 +ATp Let given in lecture. (a) Show that E(5xx)-A5xx and E(Y)-Ao +A2. (b) Use (a) to show that E(A)-A and E(A)-A. În other words, these are unbiased estimators (c) The fitted values Yi = ArtAz; are used as estimates of E(K), and the residuals ei = Y-...
1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or pmf f(y; A;) = i= 1, . .. ,n. Yi = 0, 1,2, -..; ^; > 0; Y;! Note that the pdf from an exponential family has the following general form b(0) + c(y, a(o) y0 exp f(y; 0, 6) = Suppose the linear predictor of the GLM is n = a+Bxi, with (a,B) being the...
How can I get the (a) 3*2 matrix A?
x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0 y (a) Find a 3x2 matrix A whose column space is V and the entries a a1 0 = (b) Find an orthonormal basis for V by applying the Gram-Schmidt procedure (c) Find the projection matrix P projecting onto the left nullspace (not the column space) of A (d) Find an SVD (A...
Linear Algebra! Practice exam #1 question 1 Thanks for sloving!
1- Transformations (3 points each) a) Given a linear transformation T :N" N" T(x,y)-(x-y,x+y) and B= {< l, 0>.< 1,1 >} , B = {< l, l>,< 0, l>} V,-< 2, l> Find V,T,and TVg) b) Given a linear transformation T:n'->n2 T(x,y,2)-(x-z,x +2y)and V =< 2,-I, I> B= {<l, 0, 1>.< 1, 1, 0 >, < 0, l, 0 >}, B' = {<l, l >, < 0, 1 >} Find...