Multivariate ANOVA - Jonathan Templin's Website

48 Pages·2011·0.46 MB·English

Multivariate ANOVA
Lecture 5
July 26, 2011
Advanced Multivariate Statistical Methods
ICPSR Summer Session #2
Lecture#5-7/26/2011 Slide1of47
Today’s Lecture
n Univariate ANOVA Example
n Multivariate ANOVA (MANOVA)
Overview
lToday’sLecture
u One-way MANOVA
ANOVAExample
MANOVA u Two-way MANOVA
One-WayMANOVAExample
TreatmentCIs
Two-WayMANOVA
Two-WayMANOVAExample
WrappingUp
Lecture#5-7/26/2011 Slide2of47
Setup
n Imagine you have a total of g populations, from which you
take a sample of n from each:
L
Overview
Population
ANOVAExample
lHungry? 1 2 . . . g
lCanYouGuess?
lSASCode X X . . . X
11 21 g1
MANOVA
X X . . . X
12 22 g2
One-WayMANOVAExample . . . .
. . . .
. . . .
TreatmentCIs
Two-WayMANOVA X X . . . X
1n1 2n2 gng
Two-WayMANOVAExample
WrappingUp n Each of these populations are N(µ , σ2) (note that each has
L
the same variance)
Lecture#5-7/26/2011 Slide3of47
Hypothesis Test
n Your goal is to test the null hypothesis that the means of all
populations are equal against the alternative hypothesis that
at least one mean is not equal to the others:
Overview
ANOVAExample
lHungry? H : µ = µ = . . . = µ
0 1 2 g
lCanYouGuess?
lSASCode
MANOVA
H : at least one µ not equal to the others
1
One-WayMANOVAExample
TreatmentCIs
Two-WayMANOVA
n The ANOVA model specifies that the mean for a given
Two-WayMANOVAExample
population, µ , is a function of an overall mean µ, and
L
WrappingUp
population specific effects, τ :
L
µ = µ + τ
L L
Lecture#5-7/26/2011 Slide4of47
Parameterization
n The ANOVA parameterization leads to an equivalent
hypothesis to be tested:
Overview
H : τ = τ = . . . = τ = 0
0 1 2 g
ANOVAExample
lHungry?
lCanYouGuess?
lSASCode
H : at least one τ not equal to zero
1
MANOVA
n All of this leads to the model for an individual observation,
One-WayMANOVAExample
X , the jth observation from population L:
TreatmentCIs Lj
Two-WayMANOVA
Two-WayMANOVAExample
X = µ + τ + e
Lj L Lj
WrappingUp
overall treatment random
mean effect error
! ! !
Lecture#5-7/26/2011 Slide5of47
How ANOVA Works
n The way the hypothesis of all τ parameters equal to zero is
tested is through a partitioning of the variability that is
present in the data set
Overview
ANOVAExample
n This partitioning is based on the sum of squares
lHungry?
lCanYouGuess?
lSASCode
n For instance, consider the following parameterization of the
MANOVA
ANOVA model:
One-WayMANOVAExample
TreatmentCIs
Two-WayMANOVA x = x¯ + (x¯ − x¯) + (x − x¯ )
Lj L Lj L
Two-WayMANOVAExample
estimated
overall random
WrappingUp
treatment
 
mean error
! !
effect
 
 
Lecture#5-7/26/2011 Slide6of47
How ANOVA Works
n If we then:
u Subtract the mean from both sides
u Square both sides (factoring out a zero cross product)
u Sum across all observations j
n We end up with the ANOVA decomposition:
g g g
n (x − x¯)2 = n (x¯ − x¯)2 + n (x − x¯ )2
L Lj L L L Lj L
l=1 j=1 L=1 L=1 j=1
SS
cor
P P P P P
SS SS
tr res
SS total
 
SS between SS within
! !
Corrected
 
 
Lecture#5-7/26/2011 Slide7of47
ANOVA Table
n Typically with ANOVA, we place all of our SS and df into a “convenient” table
g
(note n = N):
L
L=1
P
Source Degrees of
of variation Sum of squares freedom (d.f.)
Treatments SS g − 1
tr
Residual (Error) SS N − g
res
Total (corrected) SS N − 1
cor
n The null hypothesis of equal treatment means (or zero treatment effects) is
tested by comparing a ratio of sum of squares for between effects (divided by
it’s degrees of freedom) to the sum of squares for within effects (divided by
it’s degrees of freedom)
Lecture#5-7/26/2011 Slide8of47
H
Evaluating
0
n The test of H is given by:
0
SS /(g − 1)
tr
F =
SS /(N − g)
res
n F is compared with the critical value for significance level (α) of F (α),
g 1,N g
− −
which is obtained from a table (or Excel)
n If F is greater than the critical value, H is rejected, and you would conclude
0
that there is a difference in sample means
n The multivariate analog of this test is given by something very similar (only
now small values lead to rejection of H ):
0
SS
tr
SS + SS
tr res
Lecture#5-7/26/2011 Slide9of47
Example Data Set
Neter(1996,p. 676).
n “The Kenton Food Company wished to test four different
package designs for a new breakfast cereal
Overview
ANOVAExample n “Twenty stores, with approximately equal sales volumes,
lHungry?
lCanYouGuess? were selected as the experimental units
lSASCode
MANOVA n “Each store was randomly assigned one of the package
One-WayMANOVAExample designs, with each package design assigned to five stores
TreatmentCIs
Two-WayMANOVA n “The stores were chosen to be comparable in location and
sales volume
Two-WayMANOVAExample
WrappingUp
n “Other relevant conditions that could affect sales, such as
price, amount and location of shelf space, and special
promotional efforts, were kept the same for all of the stores
in the experiment”
Lecture#5-7/26/2011 Slide10of47