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The Multiple Correlation Coefficient. Definition. has ( p +1)-variate Normal distribution. with mean vector. and Covariance matrix. We are interested if the variable y is independent of the vector.

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slide2

Definition

has (p +1)-variate Normal distribution

with mean vector

and Covariance matrix

We are interested if the variable y is independent of the vector

The multiple correlation coefficient is the maximum correlation between y and a linear combination of the components of

slide3

Derivation

This vector has a bivariate Normal distribution

with mean vector

and Covariance matrix

We are interested if the variable y is independent of the vector

The multiple correlation coefficient is the maximum correlation between y and a linear combination of the components of

slide4

The multiple correlation coefficient is the maximum correlation between y and

The correlation between y and

Thus we want to choose to maximize

Equivalently

slide7

We are interested if the variable y is independent of the vector

The sample Multiple correlation coefficient

Then the sample Multiple correlation coefficient is

slide8

Testing for independence between y and

The test statistic

If independence is true then the test statistic F will have an F-distributions with n1 = p degrees of freedom in the numerator and n1 = n – p + 1degrees of freedom in the denominator

The test is to reject independence if:

slide9

Other tests for Independence

Test for zero correlation (Independence between a two variables)

The test statistic

If independence is true then the test statistic t will have a t -distributions with n = n –2degrees of freedom.

The test is to reject independence if:

slide10

Test for non-zero correlation (H0: r = r0 )

The test statistic

If H0 is true the test statistic z will have approximately a Standard Normal distribution

We then rejectH0 if:

slide11

Test for zero partial correlation correlation (Conditional independence between a two variables given a set of p Independent variables)

The test statistic

= the partial correlation between yi and yj given x1, …, xp.

If independence is true then the test statistic t will have a t -distributions with n = n – p - 2degrees of freedom.

The test is to reject independence if:

slide12

Test for non-zero partial correlation

The test statistic

If H0 is true the test statistic z will have approximately a Standard Normal distribution

We then rejectH0 if:

the problem
The problem

Quite often when one has collected data on several variables.

The variables are grouped into two (or more) sets of variables and the researcher is interested in whether one set of variables is independent of the other set.

In addition if it is found that the two sets of variates are dependent, it is then important to describe and understand the nature of this dependence.

The appropriate statistical procedure in this case is called Canonical Correlation Analysis.

canonical correlation an example
Canonical Correlation: An Example

In the following study the researcher was interested in whether specific instructions on how to relax when taking tests and how to increase Motivation , would affect performance on standardized achievement tests

  • Reading,
  • Language and
  • Mathematics
slide16
A group of 65 third- and fourth-grade students were rated after the instruction and immediately prior taking the Scholastic Achievement tests on:
  • how relaxed they were (X1) and
  • how motivated they were (X2).

In addition data was collected on the three achievement tests

  • Reading (Y1),
  • Language (Y2) and
  • Mathematics (Y3).

The data were tabulated on the next page

slide18

Definition: (Canonical variates and Canonical correlations)

have p-variate Normal distribution

and

with

Let

and

be such that U1 and V1 have achieved the maximum correlation f1.

Then U1 and V1 are called the first pair of canonical variates and f1 is called the first canonical correlation coefficient.

slide19

derivation: ( 1st pair of Canonical variates and Canonical correlation)

Now

Thus

has covariance matrix

slide20

derivation: ( 1st pair of Canonical variates and Canonical correlation)

Now

Thus

has covariance matrix

hence

slide21

Thus we want to choose

so that

is at a maximum

or

is at a maximum

Let

slide23

Thus

This shows that

is an eigenvector of

k is the largest eigenvalue of

and

is the eigenvector associated with the largest eigenvalue.

slide24

Also

and

slide25

Summary:

The first pair of canonical variates

, eigenvectors of the matrices

are found by finding

associated with the largest eigenvalue (same for both matrices)

The largest eigenvalue of the two matrices is the square of the first canonical correlation coefficientf1

slide26

The remaining canonical variates and canonical correlation coefficients

The second pair of canonical variates

, so that

are found by finding

1. (U2,V2) are independent of (U1,V1).

2. The correlation between U2 and V2 is maximized

The correlation, f2, between U2 and V2 is called the second canonical correlation coefficient.

slide27

The ith pair of canonical variates

, so that

are found by finding

1. (Ui,Vi) are independent of (U1,V1), …, (Ui-1,Vi-1).

2. The correlation between Ui and Vi is maximized

The correlation, f2, between U2 and V2 is called the second canonical correlation coefficient.

slide29

Now

and maximizing

Is equivalent to maximizing

subject to

Using the Lagrange multiplier technique

slide30

Now

and

also

gives the restrictions

slide31

These equations can used to show that

are eigenvectors of the matrices

associated with the 2nd largest eigenvalue (same for both matrices)

The 2ndlargest eigenvalue of the two matrices is the square of the 2nd canonical correlation coefficientf2

slide32

continuing

Coefficients for the ith pair of canonical variates,

are eigenvectors of the matrices

associated with the ith largest eigenvalue (same for both matrices)

The ithlargest eigenvalue of the two matrices is the square of the ith canonical correlation coefficientfi

example
Example

Variables

  • relaxation Score (X1)
  • motivation score (X2).
  • Reading (Y1),
  • Language (Y2) and
  • Mathematics (Y3).
summary
Summary

U1 = 0.197 Relax + 0.979 Mot

V1 = 0.504 Read + 0.900 Lang + 0.565 Math

f1 = .592

U2 = 0.980 Relax + 0.203 Mot

V2 = 0.391 Math - 0.361 Read - 0.354 Lang

f2 = .159