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Multivariate Normal Distribution
Lecture 4
July 21, 2011
Advanced Multivariate Statistical Methods
ICPSR Summer Session #2
Lecture #4 - 7/21/2011
Slide 1 of 41
Last Time
■
Matrices and vectors
◆
Eigenvalues
◆
Eigenvectors
◆
Determinants
Overview
● Last Time
● Today’s Lecture
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
■
Basic descriptive statistics using matrices:
◆ Mean vectors
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
◆
Covariance Matrices
◆
Correlation Matrices
Slide 2 of 41
Today’s Lecture
Overview
● Last Time
■
Putting our new knowledge to use with a useful statistical
distribution: the Multivariate Normal Distribution
■
This roughly maps onto Chapter 4 of Johnson and Wichern
● Today’s Lecture
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 3 of 41
Multivariate Normal Distribution
■
The generalization of the univariate normal distribution to
multiple variables is called the multivariate normal
distribution (MVN)
■
Many multivariate techniques rely on this distribution in some
manner
■
Although real data may never come from a true MVN, the
MVN provides a robust approximation, and has many nice
mathematical properties
■
Furthermore, because of the central limit theorem, many
multivariate statistics converge to the MVN distribution as the
sample size increases
Overview
MVN
● Univariate Review
● MVN
● MVN Contours
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 4 of 41
Univariate Normal Distribution
■
The univariate normal distribution function is:
f (x) = √
Overview
MVN
1
2πσ 2
−[(x−µ)/σ]2 /2
e
● Univariate Review
● MVN
● MVN Contours
MVN Properties
■
The mean is µ
■
The variance is σ 2
■
The standard deviation is σ
■
Standard notation for normal distributions is N (µ, σ 2 ), which
will be extended for the MVN distribution
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 5 of 41
Univariate Normal Distribution
N (0, 1)
Univariate Normal Distribution
Overview
0.4
MVN
● Univariate Review
● MVN
● MVN Contours
0.3
MVN Properties
MVN Parameters
0.2
MVN in Common Methods
f(x)
MVN Likelihood Functions
Assessing Normality
0.0
0.1
Wrapping Up
−6
−4
−2
0
2
4
6
x
Lecture #4 - 7/21/2011
Slide 6 of 41
Univariate Normal Distribution
N (0, 2)
Univariate Normal Distribution
Overview
0.4
MVN
● Univariate Review
● MVN
● MVN Contours
0.3
MVN Properties
MVN Parameters
0.2
MVN in Common Methods
f(x)
MVN Likelihood Functions
Assessing Normality
0.0
0.1
Wrapping Up
−6
−4
−2
0
2
4
6
x
Lecture #4 - 7/21/2011
Slide 7 of 41
Univariate Normal Distribution
N (1.75, 1)
Univariate Normal Distribution
Overview
0.4
MVN
● Univariate Review
● MVN
● MVN Contours
0.3
MVN Properties
MVN Parameters
0.2
MVN in Common Methods
f(x)
MVN Likelihood Functions
Assessing Normality
0.0
0.1
Wrapping Up
−6
−4
−2
0
2
4
6
x
Lecture #4 - 7/21/2011
Slide 8 of 41
UVN - Notes
Overview
■
The area under the curve for the univariate normal
distribution is a function of the variance/standard deviation
■
In particular:
MVN
● Univariate Review
● MVN
P (µ − σ ≤ X ≤ µ + σ) = 0.683
● MVN Contours
MVN Properties
P (µ − 2σ ≤ X ≤ µ + 2σ) = 0.954
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
■
Also note the term in the exponent:
Assessing Normality
Wrapping Up
■
Lecture #4 - 7/21/2011
(x − µ)
σ
2
= (x − µ)(σ 2 )−1 (x − µ)
This is the square of the distance from x to µ in standard
deviation units, and will be generalized for the MVN
Slide 9 of 41
MVN
■
The multivariate normal distribution function is:
f (x) =
Overview
MVN
● Univariate Review
1
(2π)p/2 |Σ|
−1
−(x−µ)′ Σ
e
1/2
(x−µ)/2
● MVN
● MVN Contours
MVN Properties
■
The mean vector is µ
■
The covariance matrix is Σ
■
Standard notation for multivariate normal distributions is
Np (µ, Σ)
■
Visualizing the MVN is difficult for more than two dimensions,
so I will demonstrate some plots with two variables - the
bivariate normal distribution
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 10 of 41
Bivariate Normal Plot #1
µ=
Overview
"
0
0
#
,Σ =
"
1 0
0 1
#
MVN
● Univariate Review
● MVN
● MVN Contours
0.16
MVN Properties
0.14
0.12
MVN Parameters
0.1
MVN Likelihood Functions
0.08
MVN in Common Methods
0.06
Assessing Normality
0.04
0.02
Wrapping Up
0
4
4
2
2
0
0
−2
−2
−4
Lecture #4 - 7/21/2011
−4
Slide 11 of 41
Bivariate Normal Plot #1a
µ=
Overview
MVN
"
0
0
#
,Σ =
"
1 0
0 1
#
4
● Univariate Review
● MVN
3
● MVN Contours
MVN Properties
2
MVN Parameters
1
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
0
−1
Wrapping Up
−2
−3
−4
−4
Lecture #4 - 7/21/2011
−3
−2
−1
0
1
2
3
4
Slide 12 of 41
Bivariate Normal Plot #2
µ=
Overview
"
0
0
#
,Σ =
"
1 0.5
0.5 1
#
MVN
● Univariate Review
● MVN
● MVN Contours
MVN Properties
0.2
MVN Parameters
0.15
MVN Likelihood Functions
MVN in Common Methods
0.1
Assessing Normality
0.05
Wrapping Up
0
4
4
2
2
0
0
−2
−2
−4
Lecture #4 - 7/21/2011
−4
Slide 13 of 41
Bivariate Normal Plot #2
µ=
Overview
MVN
"
0
0
#
,Σ =
"
1 0.5
0.5 1
#
4
● Univariate Review
● MVN
3
● MVN Contours
MVN Properties
2
MVN Parameters
1
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
0
−1
Wrapping Up
−2
−3
−4
−4
Lecture #4 - 7/21/2011
−3
−2
−1
0
1
2
3
4
Slide 14 of 41
MVN Contours
■
Overview
The lines of the contour plots denote places of equal
probability mass for the MVN distribution
◆
MVN
● Univariate Review
The lines represent points of both variables that lead to
the same height on the z-axis (the height of the surface)
● MVN
● MVN Contours
■
MVN Properties
These contours can be constructed from the eigenvalues
and eigenvectors of the covariance matrix
MVN Parameters
MVN Likelihood Functions
◆
The direction of the ellipse axes are in the direction of the
eigenvalues
◆
The length of the ellipse axes are proportional to the
constant times the eigenvector
MVN in Common Methods
Assessing Normality
Wrapping Up
■
Specifically:
(x − µ)′ Σ−1 (x − µ) = c2
√
has ellipsoids centered at µ, and has axes ±c λi ei
Lecture #4 - 7/21/2011
Slide 15 of 41
MVN Contours, Continued
Overview
■
Contours are useful because they provide confidence
regions for data points from the MVN distribution
■
The multivariate analog of a confidence interval is given by
an ellipsoid, where c is from the Chi-Squared distribution
with p degrees of freedom
■
Specifically:
MVN
● Univariate Review
● MVN
● MVN Contours
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
(x − µ)′ Σ−1 (x − µ) = χ2p (α)
Assessing Normality
Wrapping Up
provides the confidence region containing 1 − α of the
probability mass of the MVN distribution
Lecture #4 - 7/21/2011
Slide 16 of 41
MVN Contour Example
■
Imagine we had a bivariate normal distribution with:
" #
"
#
0
1 0.5
µ=
,Σ =
0
0.5 1
■
The covariance matrix has eigenvalues and eigenvectors:
#
"
#
"
1.5
0.707 −0.707
,E =
λ=
0.5
0.707 0.707
■
We want to find a contour where 95% of the probability will
fall, corresponding to χ22 (0.05) = 5.99
Overview
MVN
● Univariate Review
● MVN
● MVN Contours
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 17 of 41
MVN Contour Example
■
This contour will be centered at µ
■
Axis 1:
Overview
MVN
√
µ ± 5.99 × 1.5
● Univariate Review
● MVN
● MVN Contours
"
0.707
0.707
#
−0.707
0.707
#
=
"
=
"
2.12
2.12
# "
,
MVN Properties
−2.12
−2.12
#
MVN Parameters
MVN Likelihood Functions
■
Axis 2:
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
√
µ ± 5.99 × 0.5
"
−1.22
1.22
# "
,
1.22
−1.22
#
Slide 18 of 41
MVN Properties
Overview
MVN
■
The MVN distribution has some convenient properties
■
If X has a multivariate normal distribution, then:
1. Linear combinations of X are normally distributed
MVN Properties
● MVN Properties
MVN Parameters
2. All subsets of the components of X have a MVN
distribution
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
3. Zero covariance implies that the corresponding
components are independently distributed
Wrapping Up
4. The conditional distributions of the components are MVN
Lecture #4 - 7/21/2011
Slide 19 of 41
Linear Combinations
■
If X ∼ Np (µ, Σ), then any set of q linear combinations of
variables A(q×p) are also normally distributed as
AX ∼ Nq (Aµ, AΣA′ )
■
For example, let p = 3 and Y be the difference between X1
and X2 . The combination matrix would be
h
i
A = )1 −1 0
Overview
MVN
MVN Properties
● MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
For X



µ1
σ11



µ =  µ2  , Σ =  σ21
µ3
σ31
σ12
σ22
σ32

σ13

σ23 
σ33
For Y = AX
µY =
Lecture #4 - 7/21/2011
h
µ1 − µ2
i
, ΣY =
h
σ11 + σ22 − 2σ12
i
Slide 20 of 41
MVN Properties
■
The MVN distribution is characterized by two parameters:
◆
The mean vector µ
◆
The covariance matrix Σ
Overview
MVN
MVN Properties
■
MVN Parameters
● MVN Properties
● UVN CLT
The maximum likelihood estimates for these parameters are
given by:
● Multi CLT
n
● Sufficient Stats
MVN Likelihood Functions
◆
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
◆
1X
1
′
The mean vector: x̄ =
xi = X’1
n i=1
n
The covariance matrix
n
1X
1
S=
(xi − x̄)2 = (X − 1x̄′ )′ (X − 1x̄′ )
n i=1
n
Slide 21 of 41
Distribution of x̄ and S
Recall back in Univariate statistics you discussed the Central
Limit Theorem (CLT)
Overview
MVN
It stated that, if the set of n observations x1 , x2 , . . . , xn were
normal or not...
MVN Properties
MVN Parameters
■
The distribution of x̄ would be normal with mean equal to µ
and variance σ 2 /n
■
We were also told that (n − 1)s2 /σ 2 had a Chi-Square
distribution with n − 1 degrees of freedom
■
Note: We ended up using these pieces of information for
hypothesis testing such as t-test and ANOVA.
● MVN Properties
● UVN CLT
● Multi CLT
● Sufficient Stats
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 22 of 41
Distribution of x̄ and S
We also have a Multivariate Central Limit Theorem (CLT)
Overview
It states that, if the set of n observations x1 , x2 , . . . , xn are
multivariate normal or not...
MVN
MVN Properties
■
The distribution of x̄ would be normal with mean equal to µ
and variance/covariance matrix Σ/n
■
We are also told that (n − 1)S will have a Wishart
distribution, Wp (n − 1, Σ), with n − 1 degrees of freedom
MVN Parameters
● MVN Properties
● UVN CLT
● Multi CLT
● Sufficient Stats
MVN Likelihood Functions
MVN in Common Methods
◆
Assessing Normality
Wrapping Up
■
Lecture #4 - 7/21/2011
This is the multivariate analogue to a Chi-Square
distribution
Note: We will end up using some of this information for
multivariate hypothesis testing
Slide 23 of 41
Distribution of x̄ and S
Overview
MVN
MVN Properties
MVN Parameters
● MVN Properties
● UVN CLT
● Multi CLT
■
Therefore, let x1 , x2 , . . . , xn be independent observations
from a population with mean µ and covariance Σ
■
The following are true:
√
◆
n X̄ − µ is approximately Np (0, Σ)
◆
n (X − µ)′ S−1 (X − µ) is approximately χ2p
● Sufficient Stats
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 24 of 41
Sufficient Statistics
■
The sample estimates X̄ and S) are sufficient statistics
■
This means that all of the information contained in the data
can be summarized by these two statistics alone
■
This is only true if the data follow a multivariate normal
distribution - if they do not, other terms are needed (i.e.,
skewness array, kurtosis array, etc...)
■
Some statistical methods only use one or both of these
matrices in their analysis procedures and not the actual data
Overview
MVN
MVN Properties
MVN Parameters
● MVN Properties
● UVN CLT
● Multi CLT
● Sufficient Stats
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 25 of 41
Density and Likelihood Functions
Overview
■
The MVN distribution is often the core statistical distribution
for a uni- or multivariate statistical technique
■
Maximum likelihood estimates are preferable in statistics due
to a set of desirable asymptotic properties, including:
MVN
MVN Properties
◆
MVN Parameters
MVN Likelihood Functions
● Intro to MLE
◆
● MVN Likelihood
MVN in Common Methods
◆
Assessing Normality
Wrapping Up
■
Lecture #4 - 7/21/2011
Consistency: the estimator converges in probability to
the value being estimated
Asymptotic Normality: the estimator has a normal
distribution with a functionally known variance
Efficiency: no asymptotically unbiased estimator has
lower asymptotic mean squared error than the MLE
The form of the MVN ML function frequently appears in
statistics, so we will briefly discuss MLE using normal
distributions
Slide 26 of 41
An Introduction to Maximum Likelihood
■
Maximum likelihood estimation seeks to find parameters of a
statistical model (mapping onto the mean vector and/or
covariance matrix) such that the statistical likelihood function
is maximized
■
The method assumes data follow a statistical distribution, in
our case the MVN
■
More frequently, the log-likelihood function is used instead of
the likelihood function
Overview
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
● Intro to MLE
● MVN Likelihood
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
◆
The “logged” and “un-logged” version of the function have
a maximum at the same point
◆
The “logged” version is easier mathematically
Slide 27 of 41
Maximum Likelihood for Univariate Normal
Overview
■
We will start with the univariate normal case and then
generalize
■
Imagine we have a sample of data, X, which we will assume
is normally distributed with an unknown mean but a known
variance (say the variance is 1)
■
We will build the maximum likelihood function for the mean
■
Our function rests on two assumptions:
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
● Intro to MLE
● MVN Likelihood
MVN in Common Methods
Assessing Normality
1. All data follow a normal distribution
Wrapping Up
2. All observations are independent
■
Lecture #4 - 7/21/2011
Put into statistical terms: X is independent and identically
distributed (iid) as N1 (µ, 1)
Slide 28 of 41
Building the Likelihood Function
Overview
■
Each observation, then, follows a normal distribution with the
same mean (unknown) and variance (1)
■
The distribution function begins with the density – the
function that provides the normal curve (with (1) in place of
σ 2 ):
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
1
(Xi − µ)
f (Xi |µ) = p
exp −
2(1)2
2π(1)
● Intro to MLE
● MVN Likelihood
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
2
■
The density provides the “likelihood” of observing an
observation Xi for a given value of µ (and a known value of
σ2 = 1
■
The “likelihood” is the height of the normal curve
Slide 29 of 41
The One-Observation Likelihood Function
The graph shows f (Xi |µ = 1) for a range of X
0.4
Overview
MVN
f(X|mu)
MVN Parameters
MVN Likelihood Functions
● Intro to MLE
0.1
● MVN Likelihood
0.2
0.3
MVN Properties
MVN in Common Methods
0.0
Assessing Normality
Wrapping Up
−4
−2
0
2
4
X
The vertical lines indicate:
Lecture #4 - 7/21/2011
■
f (Xi = 0|µ = 1) = .241
■
f (Xi = 1|µ = 1) = .399
Slide 30 of 41
The Overall Likelihood Function
Overview
■
Because we have a sample of N observations, our likelihood
function is taken across all observations, not just one
■
The “joint” likelihood function uses the assumption that
observations are independent to be expressed as a product
of likelihood functions across all observations:
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
● Intro to MLE
● MVN Likelihood
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
L(x|µ) = f (X1 |µ) × f (X2 |µ) × . . . × f (XN |µ)
!
P
N
N/2
N
2
Y
1
i=1 (Xi − µ)
L(x|µ) =
exp −
f (Xi |µ) =
2
2π(1)
2(1)
i=1
■
The value of µ that maximizes f (x|µ) is the MLE (in this
case, it’s the sample mean)
■
In more complicated models, the MLE does not have a
closed form and therefore must be found using numeric
methods
Slide 31 of 41
The Overall Log-Likelihood Function
■
For an unknown mean µ and variance σ 2 , the likelihood
function is:
!
PN
N/2
2
1
2
i=1 (Xi − µ)
L(x|µ, σ ) =
exp −
2πσ 2
2σ 2
■
More commonly, the log-likelihood function is used:
Overview
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
● Intro to MLE
● MVN Likelihood
MVN in Common Methods
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
2
L(x|µ, σ ) = −
N
2
log 2πσ
2
−
PN
i=1 (Xi −
2σ 2
µ)
2
!
Slide 32 of 41
The Multivariate Normal Likelihood Function
■
For a set of N independent observations on p variables,
X(N ×p) , the multivariate normal likelihood function is formed
by using a similar approach
■
For an unknown mean vector µ and covariance Σ, the joint
likelihood is:
Overview
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
● Intro to MLE
● MVN Likelihood
MVN in Common Methods
L(X|µ, Σ) =
Assessing Normality
N
Y
1
′
p/2 |Σ|1/2
(2π)
i=1
−1
exp − (xi − µ) Σ
(xi − µ) /2
Wrapping Up
=
Lecture #4 - 7/21/2011
1
N
X
!
1
′ −1
exp
−
(x
−
µ)
Σ (xi − µ) /2
i
np/2 |Σ|n/2
(2π)
i=1
Slide 33 of 41
The Multivariate Normal Likelihood Function
Overview
MVN
■
Occasionally, a more intricate form of the MVN likelihood
function shows up
■
Although mathematically identical to the function on the last
page, this version typically appears without explanation:
MVN Properties
MVN Parameters
L(X|µ, Σ) = (2π)
MVN Likelihood Functions
● Intro to MLE
● MVN Likelihood
MVN in Common Methods
Assessing Normality
"
exp −tr Σ−1
N
X
i=1
−np/2
|Σ|−n/2
(xi − x̄) (xi − x̄)′ + n (x̄ − µ) (x̄ − µ)′
!#
!
/2
Wrapping Up
Lecture #4 - 7/21/2011
Slide 34 of 41
The MVN Log-Likelihood Function
Overview
MVN
■
As with the univariate case, the MVN likelihood function is
typically converted into a log-likelihood function for simplicity
■
The MVN log-likelihood function is given by:
MVN Properties
MVN Parameters
l(X|µ, Σ) = −
MVN Likelihood Functions
● Intro to MLE
● MVN Likelihood
MVN in Common Methods
Assessing Normality
1
2
N
X
i=1
np
n
log (2π) − log (|Σ|) −
2
2
!
′
(xi − µ) Σ−1 (xi − µ)
Wrapping Up
Lecture #4 - 7/21/2011
Slide 35 of 41
But...Why?
Overview
■
The MVN distribution, likelihood, and log-likelihood functions
show up frequently in statistical methods
■
Commonly used methods rely on versions of the distribution,
methods such as:
MVN
MVN Properties
MVN Parameters
◆
MVN Likelihood Functions
◆
MVN in Common Methods
● Motivation for MVN
◆
● MVN in Mixed Models
● MVN in SEM
◆
Assessing Normality
Wrapping Up
◆
■
Lecture #4 - 7/21/2011
Linear models (ANOVA, Regression)
Mixed models (i.e., hierarchical linear models, random
effects models, multilevel models)
Path models/simultaneous equation models
Structural equation models (and confirmatory factor
models)
Many versions of finite mixture models
Understanding the form of the MVN distribution will help to
understand the commonalities between each of these
models
Slide 36 of 41
MVN in Mixed Models
■
From SAS’ manual for proc mixed :
Overview
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
● Motivation for MVN
● MVN in Mixed Models
● MVN in SEM
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 37 of 41
MVN in Structural Equation Models
■
From SAS’ manual for proc calis:
■
This package uses only the covariance matrix, so the form of
the likelihood function is phrased using only the Wishart
Distribution:
Overview
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
● Motivation for MVN
● MVN in Mixed Models
● MVN in SEM
Assessing Normality
Wrapping Up
Lecture #4 - 7/21/2011
w (S|Σ) =
−1 (n−p−2)
|S|
exp −tr SΣ
/2
Qp
1
p(n−1)/2
p(p−1)/4
(n−1)/2
2
π
kΣ|
i=1 Γ 2 (n − i)
Slide 38 of 41
Assessing Normality
■
Recall from earlier that IF the data have a Multivariate
normal distribution then all of the previously discussed
properties will hold
■
There are a host of methods that have been developed to
assess multivariate normality - just look in Johnson &
Wichern
■
Given the relative robustness of the MVN distribution, I will
skip this topic, acknowledging that extreme deviations from
normality will result in poorly performing statistics
■
More often than not, assessing MV normality is fraught with
difficulty due to sample-estimated parameters of the
distribution
Overview
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
● Assessing Normality
● Transformations to Near
Normality
Wrapping Up
Lecture #4 - 7/21/2011
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Transformations to Near Normality
■
Historically, people have gone on an expedition to find a
transformation to near-normality when learning their data
may not be MVN
■
Modern statistical methods, however, make that a very bad
idea
■
More often than not, transformations end up changing the
nature of the statistics you are interested in forming
■
Furthermore, not all data need to be MVN (think conditional
distributions)
Overview
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
● Assessing Normality
● Transformations to Near
Normality
Wrapping Up
Lecture #4 - 7/21/2011
Slide 40 of 41
Final Thoughts
Overview
■
The multivariate normal distribution is an analog to the
univariate normal distribution
■
The MVN distribution will play a large role in the upcoming
weeks
■
We can finally put the background material to rest, and begin
learning some statistics methods
■
Tomorrow: lab with SAS - the “fun” of proc iml
■
Up next week: Inferences about Mean Vectors and
Multivariate ANOVA
MVN
MVN Properties
MVN Parameters
MVN Likelihood Functions
MVN in Common Methods
Assessing Normality
Wrapping Up
● Final Thoughts
Lecture #4 - 7/21/2011
Slide 41 of 41