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Multivariate distributions

The Normal distribution

1.The Normal distribution parameters m and s

(or s2)

- Comment If m 0 and s 1 the distribution is

called the standard normal distribution

The probability density of the normal

distribution

If a random variable, X, has a normal

distribution with mean m and variance s2 then we

will write

The multivariate Normal distribution

- Let

a random vector

Let

a vector of constants (the mean vector)

- Let

a p p positive definite matrix

Definition

- The matrix A is positive semi definite if

Further the matrix A is positive definite if

Suppose that the joint density of the random

vector

The random vector, x1, x2, xp is said

to have a p-variate normal distribution with mean

vector and covariance matrix S We will

write

Example the Bivariate Normal distribution

with

and

Now

and

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Hence

where

Note

is constant when

is constant. This is true when x1, x2 lie on an

ellipse centered at m1, m2 .

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Surface Plots of the bivariate Normal distribution

Contour Plots of the bivariate Normal distribution

Scatter Plots of data from the bivariate Normal

distribution

Trivariate Normal distribution - Contour map

x3

mean vector

x2

x1

Trivariate Normal distribution

x3

x2

x1

Trivariate Normal distribution

x3

x2

x1

Trivariate Normal distribution

x3

x2

x1

example

- In the following study data was collected for a

sample of n 183 females on the variables

- Age,
- Height (Ht),
- Weight (Wt),
- Birth control pill use (Bpl - 1no pill, 2pill)
- and the following Blood Chemistry measurements
- Cholesterol (Chl),
- Albumin (Abl),
- Calcium (Ca) and
- Uric Acid (UA). The data are tabulated next page

The data

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Alb, Chl, Bp

Marginal and Conditional distributions

Theorem (Marginal distributions for the

Multivariate Normal distribution)

have p-variate Normal distribution

with mean vector

and Covariance matrix

Then the marginal distribution of is

qi-variate Normal distribution (q1 q, q2 p -

q)

with mean vector

and Covariance matrix

Theorem (Conditional distributions for the

Multivariate Normal distribution)

have p-variate Normal distribution

with mean vector

and Covariance matrix

Then the conditional distribution of given

is qi-variate Normal distribution

with mean vector

and Covariance matrix

Proof (of Previous two theorems)

is

The joint density of

,

and

where

where

,

and

also

and

,

,

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The marginal distribution of is

The conditional distribution of given

is

is called the matrix of partial variances and

covariances.

is called the partial covariance (variance if i

j) between xi and xj given x1, , xq.

is called the partial correlation between xi and

xj given x1, , xq.

is called the matrix of regression coefficients

for predicting xq1, xq2, , xp from x1, , xq.

Mean vector of xq1, xq2, , xp given x1, ,

xqis

Example

Suppose that

Is 4-variate normal with

The marginal distribution of

is bivariate normal with

The marginal distribution of

is trivariate normal with

Find the conditional distribution of

given

Now

and

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The matrix of regression coefficients for

predicting x3, x4 from x1, x2.

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Thus the conditional distribution of

given

is bivariate Normal with mean vector

And partial covariance matrix

Using SPSS

Note The use of another statistical package such

as Minitab is similar to using SPSS

- The first step is to input the data.

The data is usually contained in some type of

file.

- Text files
- Excel files
- Other types of files

After starting the SSPS program the following

dialogue box appears

If you select Opening an existing file and press

OK the following dialogue box appears

Once you selected the file and its type

The following dialogue box appears

If the variable names are in the file ask it to

read the names. If you do not specify the Range

the program will identify the Range

Once you click OK, two windows will appear

A window containing the output

The other containing the data

To perform any statistical Analysis select the

Analyze menu

To compute correlations select Correlate then

BivariateTo compute partial correlations select

Correlate then Partial

for Bivariate correlation the following dialogue

appears

the output for Bivariate correlation

for partial correlation the following dialogue

appears

the output for partial correlation

- - - P A R T I A L C O R R E L A T I O N C

O E F F I C I E N T S - - - Controlling for..

AGE HT WT CHL

ALB CA UA CHL 1.0000

.1299 .2957 .2338 (

0) ( 178) ( 178) ( 178)

P . P .082 P .000 P .002 ALB

.1299 1.0000 .4778 .1226

( 178) ( 0) ( 178) ( 178)

P .082 P . P .000 P

.101 CA .2957 .4778 1.0000

.1737 ( 178) ( 178) (

0) ( 178) P .000 P .000

P . P .020 UA .2338

.1226 .1737 1.0000 ( 178)

( 178) ( 178) ( 0) P

.002 P .101 P .020 P . (Coefficient /

(D.F.) / 2-tailed Significance) " . " is printed

if a coefficient cannot be computed

Compare these with the bivariate correlation

Partial Correlations

CHL ALB CA

UA CHL 1.0000 .1299 .2957

.2338 ALB .1299 1.0000

.4778 .1226 CA .2957

.4778 1.0000 .1737 UA

.2338 .1226 .1737 1.0000

Bivariate Correlations

- In the last example the bivariate and partial

correlations were roughly in agreement. - This is not necessarily the case in all stuations
- An Example
- The following data was collected on the following

three variables

- Age
- Calcium Intake in diet (CAI)
- Bone Mass density (BMI)

The data

Bivariate correlations

Partial correlations

Scatter plot CAI vs BMI (r -0.447)

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3D Plot

Age, CAI and BMI

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Independence

Note two vectors, , are

independent if

Then the conditional distribution of given

is equal to the marginal distribution of

If is multivariate Normal with

mean vector

and Covariance matrix

Then the two vectors, , are

independent if

The components of the vector, , are

independent if

s ij 0 for all i and j (i ? j )

i. e. S is a diagonal matrix

Transformations

Transformations

Theorem Let x1, x2,, xn denote random variables

with joint probability density function f(x1,

x2,, xn ) Let u1 h1(x1, x2,, xn).

u2 h2(x1, x2,, xn).

?

un hn(x1, x2,, xn).

define an invertible transformation from the xs

to the us

Then the joint probability density function of

u1, u2,, un is given by

where

Jacobian of the transformation

Example

Suppose that u1, u2 are independent with a

uniform distribution from 0 to 1 Find the

distribution of

Solving for u1 and u2 we get the inverse

transformation

also

and

Hence

The Jacobian of the transformation

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The joint density of u1, u2 is f(u1, u2) f1

(u1) f2(u2)

Hence the joint density of z1 and z2 is

Thus z1 and z2 are independent Standard normal.

The transformation

is useful for converting uniform RVs into

independent standard normal RVs

Example

Suppose that x1, x2 are independent with density

functions f1 (x1) and f2(x2) Find the

distribution of u1 x1 x2

u2 x1 - x2

Solving for x1 and x2 we get the inverse

transformation

The Jacobian of the transformation

The joint density of x1, x2 is f(x1, x2) f1

(x1) f2(x2) Hence the joint density of u1 and u2

is

Theorem Let x1, x2,, xn denote random variables

with joint probability density function f(x1,

x2,, xn ) Let u1 a11x1 a12x2 a1nxn c1

u2 a21x1 a22x2 a2nxn c2

?

un an1 x1 an2 x2 annxn cn

define an invertible linear transformation from

the xs to the us

Then the joint probability density function of

u1, u2,, un is given by

where

then

has a p-variate normal distribution

with mean vector

and covariance matrix

then

has a p-variate normal distribution

with mean vector

and covariance matrix

Proof

then

since

and

Also

and

hence

QED

Theorem (Linear transformations of Normal RVs)

Suppose that The random vector, has a p-variate

normal distribution with mean vector and

covariance matrix S

with mean vector

and covariance matrix

proof

Let B be a (p - q) p matrix so that

is invertible.

then

is pvariate normal with mean vector

and covariance matrix

Thus the marginal distribution of

is qvariate normal with mean vector

and covariance matrix

Recall Definition of eigenvector, eigenvalue

- Let A be an n n matrix
- Let

then l is called an eigenvalue of A and

and is called an eigenvector of A and

Thereom If the matrix A is symmetric with

distinct eigenvalues, l1, , ln, with

corresponding eigenvectors

Assume

Then and covariance matrix S is positive

definite.

Suppose l1, , lp are the eigenvalues of S

corresponding eigenvectors of unit length

Note l1 gt 0, , lp gt 0

Let

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then

has a p-variate normal distribution

with mean vector

and covariance matrix

Thus the components of

are independent normal with mean 0 and variance 1.

and

Has a c2 distribution with p degrees of freedom