Multivariate distributions

The Normal distribution

1.The Normal distribution – parameters  and  (or  2 ) Comment: If  = 0 and  = 1 the distribution is called the standard normal distribution Normal distribution with  = 50 and  =15 Normal distribution with  = 70 and  =20

The probability density of the normal distribution If a random variable, X, has a normal distribution with mean  and variance  2 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, [x 1, x 2, … x p ] is said to have a p-variate normal distribution with mean vector and covariance matrix  We will write:

Example: the Bivariate Normal distribution withand

Now and

Hence where

Note: is constant when is constant. This is true when x 1, x 2 lie on an ellipse centered at  1,  2.

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 x1x1 x2x2 x3x3 mean vector

Trivariate Normal distribution x1x1 x2x2 x3x3

x1x1 x2x2 x3x3

x1x1 x2x2 x3x3

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 - 1=no pill, 2=pill) and the following Blood Chemistry measurements Cholesterol (Chl), Albumin (Abl), Calcium (Ca) and Uric Acid (UA). The data are tabulated next page:

The data :

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 q i -variate Normal distribution (q 1 = q, q 2 = 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 q i -variate Normal distribution with mean vector and Covariance matrix

Proof: (of Previous two theorems) is where, The joint density of and

where, and

also, and

,

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 x i and x j given x 1, …, x q. is called the partial correlation between x i and x j given x 1, …, x q.

is called the matrix of regression coefficients for predicting x q+1, x q+2, …, x p from x 1, …, x q. Mean vector of x q+1, x q+2, …, x p given x 1, …, x q is:

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

The matrix of regression coefficients for predicting x 3, x 4 from x 1, x 2.

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. 1.Text files 2.Excel files 3.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 Bivariate To 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

- - - 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 ( 0) ( 178) ( 178) ( 178) P=. P=.082 P=.000 P=.002 ALB ( 178) ( 0) ( 178) ( 178) P=.082 P=. P=.000 P=.101 CA ( 178) ( 178) ( 0) ( 178) P=.000 P=.000 P=. P=.020 UA ( 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 the output for partial correlation:

Compare these with the bivariate correlation:

CHL ALB CA UA CHL ALB CA UA Partial Correlations 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: 1.Age 2.Calcium Intake in diet (CAI) 3.Bone Mass density (BMI)

The data

Bivariate correlations

Partial correlations

Scatter plot CAI vs BMI (r = )

3D Plot Age, CAI and BMI

Independence

Note: two vectors,, are independent if If is multivariate Normal with mean vector and Covariance matrix Then the conditional distribution of given is equal to the marginal distribution of Then the two vectors,, are independent if

The components of the vector,, are independent if  ij = 0 for all i and j (i ≠ j ) i. e.  is a diagonal matrix

Transformations

Theorem Let x 1, x 2,…, x n denote random variables with joint probability density function f(x 1, x 2,…, x n ) Let u 1 = h 1 (x 1, x 2,…, x n ). u 2 = h 2 (x 1, x 2,…, x n ). u n = h n (x 1, x 2,…, x n ). define an invertible transformation from the x’s to the u’s

Then the joint probability density function of u 1, u 2,…, u n is given by: where Jacobian of the transformation

Example Suppose that u 1, u 2 are independent with a uniform distribution from 0 to 1 Find the distribution of Solving for u 1 and u 2 we get the inverse transformation

also Hence and

The Jacobian of the transformation

The joint density of u 1, u 2 is f(u 1, u 2 ) = f 1 (u 1 ) f 2 (u 2 ) Hence the joint density of z 1 and z 2 is:

Thus z 1 and z 2 are independent Standard normal. The transformation is useful for converting uniform RV’s into independent standard normal RV’s

Example Suppose that x 1, x 2 are independent with density functions f 1 (x 1 ) and f 2 (x 2 ) Find the distribution of u 1 = x 1 + x 2 u 2 = x 1 - x 2 Solving for x 1 and x 2 we get the inverse transformation

The Jacobian of the transformation

The joint density of x 1, x 2 is f(x 1, x 2 ) = f 1 (x 1 ) f 2 (x 2 ) Hence the joint density of u 1 and u 2 is:

Theorem Let x 1, x 2,…, x n denote random variables with joint probability density function f(x 1, x 2,…, x n ) Let u 1 = a 11 x 1 + a 12 x 2 +…+ a 1n x n + c 1 u 2 = a 21 x 1 + a 22 x 2 +…+ a 2n x n + c 2 u n = a n1 x 1 + a n2 x 2 +…+ a nn x n + c n define an invertible linear transformation from the x’s to the u’s

Then the joint probability density function of u 1, u 2,…, u n is given by: where

Theorem Suppose that The random vector, [x 1, x 2, … x p ] 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

Theorem Suppose that The random vector, [x 1, x 2, … x p ] 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 Also and hence QED

Theorem (Linear transformations of Normal RV’s) Suppose that The random vector, has a p-variate normal distribution with mean vector and covariance matrix  with mean vector and covariance matrix then has a p-variate normal distribution Let A be a q × p matrix of rank q ≤ p

proof then is invertible. and covariance matrix Let B be a (p - q) × p matrix so that is p–variate normal with mean vector

Thus the marginal distribution of and covariance matrix is q–variate normal with mean vector

Summary – Distribution Theory for Multivariate Normal Marginal distribution Conditional distribution

(Linear transformations of Normal RV’s) Suppose that The random vector, has a p-variate normal distribution with mean vector and covariance matrix  with mean vector and covariance matrix then has a p-variate normal distribution Let A be a q × p matrix of rank q ≤ p

Recall: Definition of eigenvector, eigenvalue Let A be an n × n matrix Let then 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, 1, …, n, with corresponding eigenvectors Assume

Applications of these results to Statistics Suppose that The random vector, [x 1, x 2, … x p ] has a p-variate normal distribution with mean vector and covariance matrix  Then and covariance matrix  is positive definite. Suppose 1, …, p are the eigenvalues of  corresponding eigenvectors of unit length Note 1 > 0, …, p > 0

Let

Suppose that the random vector, [x 1, x 2, … x p ] 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

Thus the components of are independent normal with mean 0 and variance 1. and Has a  2 distribution with p degrees of freedom