Comparing Mean Vectors for Several
Populations
Compare mean vectors for g treatments (or populations).
Randomly assign n units to the -th treatment (or take
independent random samples from g populat
Other hypotheses of interest (contd)
In addition to the simple null hypothesis of no treatment
eects, we might wish to test other hypothesis of the general
form (examples follow):
H0 : Ckg gp = 0, Co
Introduction
In many observational or designed studies, observations are
collected simultaneously on more than one variable on each
experimental unit.
Multivariate analysis is the collection of meth
Comparing Mean Vectors for Two Populations
Independent random samples, one sample from each of two
populations
Randomized experiment: n1 units are randomly allocated to
treatment 1 and n2 units are
Condence Regions
Condence regions are multivariate extensions of univariate
condence intervals.
Recall the denition of a 100(1 )% CI for a parameter
: for X f (x|), , the interval (t1(X ), t2(X ) is
Inferences about a Mean Vector
In the following lectures, we test hypotheses about a
p 1 population mean vector = (1, 2, . . . , p)
We could test p disjoint hypothesis (one for each j in ) but
that
Paired Comparisons and Repeated Measures
Hotellings T-squared tests for a single mean vector have
useful applications for studies with paired comparisons or
repeated measurements.
1. Paired compariso
Two-way MANOVA
We now consider designs with two factors. Factor 1 has g
levels and factor 2 has b levels.
If Xikr is the p 1 vector of measurements on the rth unit
in the ith level of factor 1 and t
Basic Concepts in Matrix Algebra
An column array of p elements is called a vector of dimension p and is
written as
x1
x
xp1 = .2 .
.
.
xp
The transpose of the column vector xp1 is row vector
x = [
Carapace Measurements for Female Turtles
Data on three dimensions of female turtle carapaces (shells):
X1=log(carapace length)
X2=log(carapace width)
X3=log(carapace height)
Since the measurement
Graphical Representation of Multivariate Data
One diculty with multivariate data is their visualization, in
particular when p > 3.
At the very least, we can construct pairwise scatter plots of
varia
Principal Components I
When a very large number p of variables is measured on each
sample unit, interpreting results of analyses might be dicult.
It is often possible to reduce the dimensionality of
Multivariate Linear Regression Models
Regression analysis is used to predict the value of one or
more responses from a set of predictors.
It can also be used to estimate the linear association betwe
Multivariate Normal Distribution I
We will almost always assume that the joint distribution of
the p 1 vectors of measurements on each sample unit is the
p-dimensional multivariate normal distributio
Moment-generating Function of the Multivariate
Normal Distribution
If X Np(, ), then the moment-generating function is
given by mX(t) I cfw_exp (t X) = exp (t + 1 t t).
E
2
134
More Features of the
M
Characterization of the Multivariate Normal
Distribution
Cramer (1946) showed that the following characterizes a
multivariate normal distribution:
X Np(, ) if and only if a X N (0, 2) for every pvar
STAT 501
Spring 2001
Instructions:
1.
FINAL EXAM
Name _
Write your answers in the space provided on this exam. If you need more space,
use the back of a page or attach extra sheets of paper, but clear
Assessing Normality The Univariate Case
In general, most multivariate methods will depend on the
distribution of X or on distances of the form
n(X ) S 1(X )
.
Large sample theory tells us that if th