Multivariate Data and Matrix Algebra Review

BMTRY 726Spring 2012

What is ‘Multivariate’ Data?• Data in which each sampling unit contributes to

more than one outcome. For example….

Sampling Unit

Cancer patients Serum concentrations on a panel of protein markers are collected in chemotherapy patients

Smoking cessation participants

Collect background information and smoking behavior at multiple visits

Post-operative patient outcome

Multiple measures of how a patient is doing post-operatively: patient self-reported pain, opioid consumption, ICU/Hospital length of stay

Diabetics Each subject assigned to different glucose control option (medication, diet, diet and medication). Fasting blood glucose is monitored at 0, 3, 6, 9, 12, and 15 months.

Goals of Multivariate Analysis• Data reduction and structural simplification – Say we collect 16 biological markers to examine patient

response to chemotherapy.– Ideally we might like to summarize patient response as

some simple combination of the markers.– How can variation in p=16 markers be summarized?

Goals of Multivariate Analysis• Sorting and grouping data– Participants are enrolled in a smoking cessation program

for several years– Information about the background of each subject and

smoking behavior at multiple visits– Some patients quit while others do not– Can we use the background and smoking behavior

information to classify those that quit and those that do not in order to screen future participants?

Goals of Multivariate Analysis• Investigating dependence among variables– Subjects take a standardized test with different categories

of questions• Sentence completion• Number sequences• Orientation of patterns• Arithmetic (etc.)

– Can correlation among scores be attributed to variation in one or more unobserved factors?• Intelligence• Mathematical ability

Goals of Multivariate Analysis• Prediction based on relationship between variables– We conduct a microarray experiment to compare tumor

and healthy tissue– We want to develop a reliable classification tool based on

the gene expression information from our experiment

Goals of Multivariate Analysis• Hypothesis testing– Participants in a diabetes study are placed into one of

three treatment groups– Fasting blood glucose is evaluated at 0, 3, 6, 9, 12, and 15

months– We want to test the hypothesis that treatment groups are

different.

Multivariate Data Properties• What property/ies of multivariate data make

commonly used statistical approached inappropriate?

Notation & Data Organization

• Consider an example where we have 15 tumor markers collected on 30 tissue samples

• The 15 markers are variables and our samples represent the subjects in the data.

• These data can most easily be expressed as an 15 by 30 array

Notation & Data Organization

• More generally, let j = 1,2,…,p represent a set of variables collected in a study

• And let i = 1,2,…,n represent the samples

11 12 1

21 22 2

n1 2

p

p

n p

n np

x x x

x x x

x x x

X

Random Vectors• Each experimental unit has multiple outcome

measures thus we can arrange the ith subject’s j = 1,2,…, p outcomes as a vector.

• is a random variable as are it’s individual elements

1

2

i

i

i

ip

X

X

X

X

iX

p denotes the number of outcomes for subject i

i = 1,2,…,n is the number subjects

Descriptive Statistics• We can calculate familiar descriptive statistics

for this array– Mean

– Variance

– Covariance (Correlation)

Arranged as Arrays

• Means

• Covariance

1

2

p

x

x

x

11 12 1

21 22 2

1 2

p

p

p p pp

s s s

s s s

s s s

Distance• Many multivariate statistics are based on the

idea of distance• For example, if we are comparing two groups

we might look at the difference in their means• Euclidean distance

Distance• But why is Euclidean distance inappropriate in

statistics?• This leads us to the idea of statistical distance• Consider a case where we have two measures

Distance• Consider a case where we have two measures

Distance• Consider a case where we have two measures

Distance• Our expression of statistical distance can be

generalized to p variables to any fixed set of points

1 2

1 2

22 2

1 1 2 2

11 22

, ...

p

p

p p

pp

P x x x

Q y y y

x yx y x yd Q P

s s s

Basic Matrix Operations

• Can I add A2x3 and B3x3?

• What is the product of matrix A and scalar c?

• When can I multiply the two matrices A and B?

Matrix Transposes

• The transpose of an n x m matrix A, denoted as A’, is an m x n matrix whose ijth element is the jith element of A

• Properties of a transpose:

''

'

'

( )

A

A B

AB

Types of Matrices• Square matrix:

• Idempotent:

• Symmetric:

• A square matrix is diagonal :

More Definitions

• An n x n matrix A is nonsingular if there exists an matrix Bn x n such that

• B is the multiplicative inverse of A and can be written as

• A square matrix with no multiplicative inverse is said to be….

• We can calculate the inverse of a matrix assuming one exists but it is tedious (let the computer do it).

AB BA I

1B A

Matrix Determinant• The determinant of a square matrix A is a scalar

given by

• What is the determinant of

11

1

1 11

1

11n j

j jj

a n

naA

A

11 12

21 22

a a

a aA

Matrix Determinant• The determinant of a square matrix A is a scalar

given by

• What is the determinant of

11

1

1 11

1

11n j

j jj

a n

naA

A

11 12

21 22

11 22 12 21

a a

a a

a a a a

A

A

Matrix Determinant• What about the determinant of the 3x3 matrix?

11 12 13

21 22 23

31 32 33

22 23 21 23 21 22

11 12 1332 33 31 33 31 32

11 22 33 12 23 31 13 21 32 13 22 31 12 21 33 11 23 32

a a a

a a a

a a a

a a a a a aa a aa a a a a a

a a a a a a a a a a a a a a a a a a

A

A

A

Matrix Determinant• Using this result what is the determinant of

?1 4 0

2 2 1

1 3 0

A A

Orthogonal an Orthonormal vectors

• A collection of m-dimensional vectors, x1, x2,…, xp are orthogonal if…

• The collection of vectors is said to be orthonormal if what 2 conditions are met?

Linear Dependence• The p of m-dimensional vectors, , are

linearly dependent if there is a set of constants, c1,c2,…,cp not all zero for which

1 2, ,..., px x x

Linear Dependence• The p of m-dimensional vectors, , are

linearly dependent if there is a set of constants, c1,c2,…,cp not all zero for which

• Conversely, if no such set of non-zero constants exists, the vectors are linearly independent.

1 2, ,..., px x x

Rank of a Matrix• Row rank is the number of rows• Column rank is the number of cols• Find the column rank of

• How are row and column rank related?• What does rank tell us about linear dependence of

the vectors that make up the matrix?

1 2 4

3 0 6

5 3 13

A

Orthogonal Matrices

• A square matrix Anxn is said to be orthogonal if its columns form an orthonormal set.

• This can be easily be determined by showing that

' A A I

Eigenvalues and Eigenvectors

• The eigenvalues of an Anxn matrix

are the solutions to

for a set of eigenvectors, . We typically normalize so that

1 2 ... 0p

i i ie eA

1 2, ,..., pe e e

'

'

1

0 for all

i i

i j

e e

e e i j

Quadratic Forms

• Given a symmetric matrix Anxn and an n-dimensional vector x, we can write the quadratic form as .

• For example, find the quadratic form where '1 2x xx

'x Ax

2 21 1 2 23 5x x x x

2 21 1 2 23 5x x x x

Trace• Let A be an nxn matrix, the trace of A is given by

• Properties of the trace:

1.

2.

3.

4. ' ' ' '

tr c c tr

tr tr tr

tr tr

tr tr tr tr

A A

A B A B

AB BA

AB B A AB A B

Positive Definite Matrices• A symmetric matrix A is said to be positive definite if

this implies

a. ' 0 for all except ' 0 0 0p x Ax x R x

1

1. rank

2. 0

3. exists

p

A

A

A

Positive Definite Matrices

• A real symmetric matrix is:

b. negative definite if ' 0 for all non-zero

c. positive semi-definite if ' 0 for all non-zero

d. negative semi-definite if ' 0 for all non-zero

x Ax x

x Ax x

x Ax x

Back to Random Vectors• Define Y as a random vector

• Then the population mean vector is:

1

2

p

Y

Y

Y

Y

1

2

p

E Y

E YE

E Y

Y

Random Vectors Cont’d

• So Yi is a random variable whose mean and variance can be expressed by:

2 2 2

mean:

variance:

i i i i i i

i i i i i i i i i ii

E Y y f y dy

V Y y f y dy E Y

Covariance of Random Vectors• We then define the covariance between the ith and jth

trait in Y as

• Yielding the covariance matrix

Cov , i j i i j j ijY Y E Y Y

11 12 1

21 22 2

1 2

p

p

p p pp

V

Y Σ

Correlation Matrix of Y• The correlation matrix for Y is

12 1

21 2

1 2

2 2

1 11 1

1 12 2

1 1

1

1

1

where

Note

p

p

p p

ijij ji

i j

p p

P

P

Properties of a Covariance Matrix

• is symmetric (i.e. sij = sji for all i,j)

• is positive semi-definite for any vector of constants

Σ

Σ

1

2 yielding ' 0

p

c

c

c

c c c

Linear Combinations• Consider linear combinations of the elements of Y

• If Y has mean m and covariance S, then

1 1 2 2

1

2

1 2

...

'

p p

p

p

Z c Y c Y c Y

Y

Yc c c

Y

c Y

1 1 2 2

1 1

' ...

0

p p

p p

i j iji j

E Z c c c

V Z c c

c μ

c'Σc

Linear Combinations Cont’d• If S is not positive definite then

for at least one

' ' 0V c Y c Σc

1

2 0

p

c

c

c

c