Multivariate Data and Matrix Algebra Review BMTRY 726 Spring 2012

What is ‘Multivariate’ Data? Data in which each sampling unit contributes to more than one outcome. For example…. Sampling Unit Cancer patientsSerum 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 DiabeticsEach 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

Random Vectors Each experimental unit has multiple outcome measures thus we can arrange the i th subject’s j = 1,2,…, p outcomes as a vector. is a random variable as are it’s individual elements 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

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

Basic Matrix Operations Can I add A 2x3 and B 3x3 ? 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 ij th element is the ji th element of A Properties of a transpose:

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 B n 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).

Matrix Determinant The determinant of a square matrix A is a scalar given by What is the determinant of

Matrix Determinant The determinant of a square matrix A is a scalar given by What is the determinant of

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

Matrix Determinant Using this result what is the determinant of

Orthogonal an Orthonormal vectors A collection of m-dimensional vectors, x 1, x 2, …, x p 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, c 1,c 2, …,c p not all zero for which

Linear Dependence The p of m-dimensional vectors,, are linearly dependent if there is a set of constants, c 1,c 2, …,c p not all zero for which Conversely, if no such set of non-zero constants exists, the vectors are linearly independent.

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?

Orthogonal Matrices A square matrix A n x n is said to be orthogonal if its columns form an orthonormal set. This can be easily be determined by showing that

Eigenvalues and Eigenvectors The eigenvalues of an A n x n matrix are the solutions to for a set of eigenvectors,. We typically normalize so that

Quadratic Forms Given a symmetric matrix A n x n and an n-dimensional vector x, we can write the quadratic form as. For example, find the quadratic form where

Trace Let A be an n x n matrix, the trace of A is given by Properties of the trace:

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

Positive Definite Matrices A real symmetric matrix is:

Back to Random Vectors Define Y as a random vector Then the population mean vector is:

Random Vectors Cont’d So Y i is a random variable whose mean and variance can be expressed by:

Covariance of Random Vectors We then define the covariance between the i th and j th trait in Y as Yielding the covariance matrix

Correlation Matrix of Y The correlation matrix for Y is

Properties of a Covariance Matrix is symmetric (i.e.  ij =  ji for all i,j ) is positive semi-definite for any vector of constants

Linear Combinations Consider linear combinations of the elements of Y If Y has mean  and covariance , then

Linear Combinations Cont’d If  is not positive definite then for at least one