multivariate analysis intro

Lecture #1 - 7/18/2011 Slide 1 of 28

Introduction to Multivariate Analysis

Lecture 1

July 18, 2011Advanced Multivariate Statistical Methods

ICPSR Summer Session #2

Overview

● Today’s Lecture

Course Overview

Data Organization

Descriptive Statistics

Graphical Techniques

Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 2 of 28

Today’s Lecture

■ Introductions

■ Syllabus and course overview

■ Chapter 1 (a brief review, really):

◆ Data organization/notation

◆ Graphical techniques

◆ Distance measures

■ Introduction to SAS

Overview

Course Overview

● Multivariate

● Course Structure

● Multivariate

Data Organization



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 3 of 28

Multivariate Statistics and Thinking

■ Although titled “Advanced Multivariate Statistical Methods”this course is an overview of thinking about data andmethods from a multivariate lens:

◆ Many methods fall under the label “multivariate statistics”(e.g., Multivariate ANOVA, Discriminant Analysis, PrincipalComponent Analysis)

◆ Many multivariate statistical distributions exist (e.g.,Multivariate Normal, Wishart)

◆ Many modern (univariate) statistical methods rely onthese multivariate distributions, especially the multivariatenormal distribution

■ This course will focus on multivariate thinking, not just aboutmethods, but also about the foundations of multivariatestatistical analysis

Overview

Course Overview

● Multivariate


● Multivariate

Data Organization



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 4 of 28

Course Structure

■ The course is organized around a central topic each week:

1. Foundations of Multivariate Thinking and TheMultivariate Normal Distribution◆ Matrix algebra◆ Multivariate normal distribution

2. Multivariate Normal and Linear Mixed Models◆ Multivariate ANOVA◆ Discrimination/classification◆ Linear models

3. Multivariate Data Reduction Procedures◆ Principal components analysis◆ Factor analysis and structural equation modeling

4. Generalized Multivariate Techniques◆ Distance methods◆ Finite mixture models◆ Categorical distributions

Overview

Course Overview

● Multivariate


● Multivariate

Data Organization



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 5 of 28

Multivariate Statistics

A taxonomy of multivariate statistical analyses shows that mosttechniques fall into one of the following categories:

1. Data reduction or structural simplification

2. Sorting and grouping

3. Investigation of the dependence among variables

4. Prediction

5. Hypothesis construction and testing

Overview

Course Overview

Data Organization

● Arrays



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 6 of 28

Data Organization

■ As a precursor of things to come, here is a preview of theways data are organized in this book/course

■ Multivariate data are a collection of observations (ormeasurements) of:

◆ p variables (k = 1, . . . , p)

◆ n “items” (j = 1, . . . , n)

■ “items” can also be though of assubjects/examinees/individuals or entities (when peopleare not under study)

■ In some disciplines (such as educational measurement),“items” are considered the variables collected perindividual

Lecture #1 - 7/18/2011 Slide 7 of 28

Data Organization

■ xjk = measurement of the kth variable on the jth entity

Variable 1 Variable 2 . . . Variable k . . . Variable p

Item 1: x11 x12 . . . x1k . . . x1p

Item 2: x21 x22 . . . x2k . . . x2p

......

......

...Item j: xj1 xj2 . . . xjk . . . xjp

......

......

...Item n: xn1 xn2 . . . xnk . . . xnp

Overview

Course Overview

Data Organization

● Arrays



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 8 of 28

Arrays

■ To represent the entire collection of items and entities, arectangular array can be constructed:

X =

x11 x12 . . . x1k . . . x1p

x21 x22 . . . x2k . . . x2p

......

......

xj1 xj2 . . . xjk . . . xjp

......

......

xn1 xn2 . . . xnk . . . xnp

■ In the next class, we will learn about how arrays like thishave an algebra that makes life somewhat easier

■ All arrays will be symbolized by boldfaced font

Overview

Course Overview

Data Organization

● Arrays



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 9 of 28

Array Example

■ So, putting things all together, envision standing outside ofthe bookstore, asking people for receipts

■ You are interested in looking at two variables:

◆ Variable 1: the total amount of the purchase

◆ Variable 2: the number of books purchased

■ You find four people, and here is what you see observe (withnotation:

x11 = 42 x21 = 52 x31 = 48 x41 = 58

x12 = 4 x22 = 5 x32 = 4 x42 = 3

Overview

Course Overview

Data Organization

● Arrays



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 10 of 28

Array Example (Continued)

■ The data array would the look like:

X =

x11 x12

x21 x22

x31 x32

x41 x42

=

42 4

52 5

48 4

58 3

■ Notice for any variable, xjk:

◆ The first subscript (j) represents the ROW location in thedata array

◆ The second subscript (k) represents the COLUMNlocation in the data array

Overview

Course Overview

Data Organization


● Sample Mean

● Sample Variance

● Sample Correlation


Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 11 of 28

Descriptive Statistics Review

■ When we have a large amount of data, it is often hard to geta manageable description of the nature of the variablesunder study

■ For this reason (and as a way of introducing a review topicsfrom previous courses), descriptive statistics are used

■ Such descriptive statistics include:

◆ Means

◆ Variances

◆ Covariances

◆ Correlations

Overview

Course Overview

Data Organization


● Sample Mean

● Sample Variance



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 12 of 28

Sample Mean

■ For the kth variable, the sample mean is:

x̄k =1

n

n∑

j=1

xjk

■ An array of the means for all p variables then looks like this(which we will come to know as the mean vector):

x̄ =

x̄1

x̄2

x̄3

x̄4

Overview

Course Overview

Data Organization


● Sample Mean

● Sample Variance



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 13 of 28

Sample Variance

■ For the kth variable, the sample variance is:

s2

k = skk =1

n

n∑

j=1

(xjk − x̄k)2

■ Note the “kk” subscript, this will be important because theequation that produces the variance for a single variable is aderivation of the equation of the covariance for a pair ofvariables

■ Also note the division by n

◆ Reasons for this will become apparent in the near future(hint: it’s a type of estimate)

■ For a pair of variables, i and k, the sample covariance is:

sik =1

n

n∑

j=1

(xji − x̄i)(xjk − x̄k)

Overview

Course Overview

Data Organization


● Sample Mean

● Sample Variance



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 14 of 28

Sample Covariance Matrix

■ Making an array of all sample covariances give us:

Sn =

s11 s12 . . . s1p

s21 s22 . . . s2p

......

. . ....

sp1 sp2 . . . spp

Overview

Course Overview

Data Organization


● Sample Mean

● Sample Variance



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 15 of 28

Sample Correlation

■ Sample covariances are dependent upon the scale of thevariables under study

■ For this reason, the correlation is often used to describe theassociation between two variables

■ For a pair of variables, i and k, the sample correlation isfound by dividing the sample covariance by the product ofthe standard deviation of the variables:

rik =sik√

sii

√skk

■ The sample correlation:◆ Ranges from -1 to 1◆ Measures linear association◆ Is invariant under linear transformations of i and k◆ Is a biased statistic

Overview

Course Overview

Data Organization


● Sample Mean

● Sample Variance



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 16 of 28

Sample Correlation Matrix

■ Making an array of all sample correlations give us:

R =

1 r12 . . . r1p

r21 1 . . . r2p

......

. . ....

rp1 rp2 . . . 1

Overview

Course Overview

Data Organization



● Bivariate Scatterplots

● Trivariate Scatterplots

● Stars

● Chernoff Faces

● Dendrograms

● Variable Space

● Network Diagrams

Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 17 of 28


■ Displaying multivariate data can be difficult due to our naturallimitations of seeing the world in three dimensions

■ Several simple ways of displaying data include:

◆ Bivariate scatterplots

◆ Three-dimensional scatterplots

Lecture #1 - 7/18/2011 Slide 18 of 28

Bivariate Scatterplots

Lecture #1 - 7/18/2011 Slide 19 of 28

Trivariate Scatterplots

Overview

Course Overview

Data Organization



● Bivariate Scatterplots

● Trivariate Scatterplots

● Stars

● Chernoff Faces

● Dendrograms

● Variable Space

● Network Diagrams

Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 20 of 28


■ But you likely already have seen those plots

■ Some plots that can be achieved by multivariate methodsinclude:

◆ “Stars”

◆ Chernoff faces

◆ Dendrograms

◆ Bivariate plots, but of the variable space

◆ Network graphs

Lecture #1 - 7/18/2011 Slide 21 of 28

Stars

Lecture #1 - 7/18/2011 Slide 22 of 28

Chernoff Faces

Lecture #1 - 7/18/2011 Slide 23 of 28

Dendrograms

Lecture #1 - 7/18/2011 Slide 24 of 28

Variable Space Plots

Lecture #1 - 7/18/2011 Slide 25 of 28

Network Diagrams

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P000001

P000010

P000011

P000100

P000101

P000110

P000111

P001000

P001001

P001010

P001011

P001100

P001101

P001110

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P010000

P010001

P010010

P010011

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P010110

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P011011

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P101100P101101

P101110

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P110000

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P110101

P110110

P110111

P111000

P111001

P111010

P111011

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P111101

P111110

P111111

Pajek

Overview

Course Overview

Data Organization



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 26 of 28

Distance Measures

■ A great number of multivariate techniques revolve around thecomputation of distances:

◆ Distances between variables

◆ Distances between entities (people, objects, etc.)

■ The formula for the Euclidean distance formula between thecoordinate pair P = (x1, x2) and the origin O = (0, 0):

d(O, P ) =√

(x1 − 0)2 + (x2 − 0)2

Overview

Course Overview

Data Organization



Distance Measures

Wrapping Up

Lecture #1 - 7/18/2011 Slide 27 of 28

Distance Measures

■ Elaborate discussions of distance measures will be foundlater in the class

■ There are also some statistical analogs to distancemeasures, taking the variability of variables into account

■ Also be aware that there are literally an infinite number ofdistance measures!

■ To be considered an actual “distance”, a distance measuremust satisfy the following:

◆ d(P, Q) = d(Q, P )

◆ d(P, Q) > 0 if P 6= Q

◆ d(P, Q) = 0 if P = Q

◆ d(P, Q) ≤ d(P, R) + d(R, Q) (known as the triangleinequality)

Overview

Course Overview

Data Organization



Distance Measures

Wrapping Up

● Final Thoughts

Lecture #1 - 7/18/2011 Slide 28 of 28

Final Thoughts

■ We introduced what this course will be about - the wild worldof multivariate statistics

■ Things will become increasingly relevant as timeprogresses...but do not hesitate to ask “why?”

■ We will now head down to the lab for a SAS introductionsession

■ Tomorrow’s Class: Matrix algebra (Chapter 2 andSupplement 2A)

multivariate analysis intro

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