STAT424 Spring 2010
Homework #6 Apr 27, 2010
Homework 6
Due: Tuesday, May 4, 2010 1)Although tea is the worlds most widely consumed beverage after water, little is known about its nutritional value. Folacin is the only B vitamin present in any signicant a
STAT 424 Course Notes
The Gauss-Markov Theorem
Spring 2015
The linear model is
Y
= X +
where Y (n 1) is random and observable, X (n k) is xed
and known, (k 1) is xed and unknown, and (n 1) is
random and unobservable.
Throughout, we make the usual second-
STAT 424 Course Notes
The ANOVA F -Test
Spring 2015
The linear model is
Y
= X +
where Y (n 1) is random and observable, X (n k) is xed
and known, (k 1) is xed and unknown, and (n 1) is
random and unobservable.
Throughout, we make the usual normal theory
STAT 424 Course Notes
Expected Quadratic Forms
Spring 2015
A quadratic form in n 1 random vector Y is of the form
Y AY
where A is a constant symmetric n n matrix.
Recall that ANOVA sums of squares are quadratic forms of Y ,
with a projection matrix for A.
STAT 424 Course Notes
Normal-Theory Distributions
Spring 2015
Noncentral Chi-Square
Denition
If Z N (, I ) then the distribution of
W = ZZ =
Z
2
is noncentral chi-square with
degrees of freedom and
noncentrality parameter = 2 ,
and we write
W 2 ()
(Proof
STAT 424 Course Notes
Summary of Estimation
for the Linear Model
Spring 2015
The linear model is
Y
= X +
where Y (n 1) is random and observable, X (n k) is xed
and known, (k 1) is xed and unknown, and (n 1) is
random and unobservable.
Alternative perspec
STAT 424 Course Notes
Linear Combinations
and Estimability
Spring 2015
Linear Combinations using
A linear combination of the elements of Rk is
= c = c1 1 + + ck k
where c = c1 ck
is a vector of known constants.
For example, any j is a linear combination
STAT 424 Course Notes
The Multivariate Normal
Distribution
Spring 2015
Let Z be a n 1 vector whose elements are independent
standard normal random variables.
Denition
The multivariate normal distribution with parameters
(n 1) and (n n non-negative denite
STAT 424 Course Notes
Random Vectors and Matrices
Spring 2015
A random matrix (or random vector) is a matrix (vector)
whose elements are random variables. Its distribution is
characterized by the joint distribution of its elements.
Two random matrices Y1
STAT 424 Course Notes
Some Matrix Algebra
Spring 2015
Quadratic Forms & Symmetric Matrices
A quadratic form in x Rn is
Q(x) = x Ax
where A is a symmetric n n matrix.
A is non-negative denite if
x Ax 0
for all x Rn
A is positive denite if
x Ax > 0
for all
STAT 424 Course Notes
The Analysis of Variance
Spring 2015
Orthogonal Decomposition
Consider again the linear model
Y = X +
Let V = col(X).
Then the vector of least squares tted values is
Y
= PV Y
and the least squares residual vector is
e = Y Y
= (I PV
STAT 424 Course Notes
Least Squares
Spring 2015
Recall: The projection of y Rn onto subspace V Rn is
the unique vector y V such that
yy V
Intuitively, y should be the vector in V that is closest to y.
(Geometry .)
1
Indeed, y solves a certain least square
STAT 424 Course Notes
Projection Matrices
Spring 2015
The Projection Matrix
Theorem
For every subspace V Rn there exists a unique n n matrix
PV such that, for all y Rn ,
y = PV y
is the projection of y onto V .
PV is called the projection matrix onto V .
STAT 424 Course Notes
Projections
Spring 2015
Projections of Vectors
Theorem
For any subspace V Rn and any y Rn , there exists a
unique vector y V such that
yy V
We call y the projection of y onto V .
(Textbook notation: p(y|V )
1
Projections of Vectors
T
STAT 424 Course Notes
Introduction
Spring 2015
We wish to statistically model or analyze random variables
Y1 , Y2 , . . . , Yn
which
may or may not be independent
usually are not identically distributed: they can have
dierent means and possibly dierent va
STAT 424 Course Notes
Sequential and Partial
Sums of Squares
Spring 2015
Y
= X +
Assume the model starts with an explicit intercept, and
partition the remaining columns of X into p groups:
X =
1 n X1 X2 Xp
0
1
= 2
.
.
.
p
where Xj is n kj and, co
STAT 424 Course Notes
Generalized Least Squares
Spring 2015
Y
= X +
but now, instead of the usual second-order conditions, assume
E() = 0
Cov() = 2 A
( 2 > 0)
or, equivalently,
E(Y ) = X
Cov(Y ) = 2 A
We will assume A is positive denite and known.
1
( 2
STAT424 Spring 2010
Homework #2 Feb 9, 2010
Homework 2
Due: Tuesday, Feb 16, 2010 1) Consider the linear model y1 y2 y3 = y4
+ + 1 + 1 + , + + 2 + 2
E( ) = 0,
Cov( ) = 2 I.
(a) Write down the design matrix X . Find a basis for C (X ). (b) Find the projec
STAT424 Spring 2010
Homework #3 Feb 23, 2010
Homework 3
Due: Tuesday, March 2, 2010 1) Another example for marginally normally distributed and uncorrelated do not imply independent . Suppose X N (0, 1). Then generate the other random variable Y by tossing
STAT424 Spring 2010
Homework #4 Mar 17, 2010
Homework 4
Due: Tuesday, March 30, 2010 1) Circle the best answer. If you choose None of the above, please provide your correct answer. 1. A one-way ANOVA with three levels was conducted and the null hypothesis
STAT424 Spring 2010
Homework #5 Apr 1, 2010
Homework 5
Due: Thursday, Apr 8, 2010 1) A child whose mother drank too much alcohol while pregnant with the child can show symptoms of fetal alcohol syndrome, which includes being more impulsive than normal, th
STAT 424 Course Notes
Analysis of Covariance
Spring 2015
Background
Consider a designed experiment to assess the relative eects of
two or more treatments.
Experimental units are randomly divided into treatment
groups, and each receives the treatment that
STAT 424 Course Notes
Two-Way ANOVA for
Balanced Data
Spring 2015
Consider two-way ANOVA for which the data are balanced:
nij = n0
i = 1, . . . , I
j = 1, . . . , J
This is just a special case, but it is worth separate
consideration because
I
computations
STAT 424 Course Notes
Two-Way ANOVA
Spring 2015
Suppose all observations are unambiguously cross-classied
according to the values of two categorical variables,
designated A and B.
We will refer to these two variables as factors, and their
possible values
STAT 424 Course Notes
Three-Way ANOVA
Spring 2015
We consider a linear model for observations that are
unambiguously cross-classied according to the levels of three
factors (categorical variables): A, B, and C.
Let the levels of A, B, and C be coded as
A