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Multivariate Data Analysis
Overview of Methods
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Motivation

• Sophisticated multivariate statistical methods
  are becoming standard practice in the physical,
  natural and social sciences, as well as in
  business
• Variations of existing methods are being
  developed, existing techniques are being applied
  to new applications, and new methods continue to
  be designed

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Motivation

• The accelerated use of advanced multivariate
  techniques is being driven by
• Growing complexity in the topics being addressed
• Ever-larger data sets
• Ability to apply computationally intensive
  methods through powerful computer tools
• Academic training

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Overview of Multivariate Data Analysis Methods
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Vocabulary Data types
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Vocabulary Variable Types

• Response vs explanatory
• Response or dependent variable
• Variable to be modeled or predicted
• Explanatory or independent variable
• Variables used to predict or model dependent
  variable
• Importance of identifying data and variable types
• Critical in determining analysis objectives and
  appropriate analysis method
• Avoid inappropriate variable operations

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Classification of Methods

• Dependence techniques
• One or a set of variables are regarded as
  dependent variables
• Objective is to predict or explain the value of
  the dependent variable(s) based on the values of
  a set of independent variables
• Examples
• What is the probability that a loan applicant
  will default?
• What factors best differentiate people whose
  primary news source is the Internet?

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Classification of Methods

• Dependence techniques
• Multiple regression
• Logistic regression
• Discriminant analysis
• Canonical correlation
• Structural equation modeling
• Analysis of variance
• Decision trees

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Classification of Methods

• Interdependence techniques
• No single group of variables defined as
  dependent or independent
• Objective is to identify and characterize
  underlying structure between the variables
• Examples
• What are the underlying factors that define a
  customers perception of a brand?
• Which signal returns arise from the same object
  and how many objects are present?

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Classification of Methods

• Interdependence techniques
• Factor analysis
• Multidimensional scaling
• Correspondence analysis
• Cluster analysis

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Classification of Methods

• Interdependence techniques are valuable data
  reduction methods
• Data reduction attempts to manage and interpret
  the large amounts of data gathered
• One goal is combine groups of cases measured over
  multiple variables into a relatively small number
  of understandable segments
• Or to group variables together into categories of
  latent traits and then characterize cases with
  respect to this smaller number of traits
• The reduced data variables are then often used as
  variables in dependence techniques

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Multiple Regression

• Multiple regression is a dependence technique
  used to model the relationship between the value
  of a single metric dependent variable and a set
  of metric independent variables
• Categorical variables can be included as dummy
  variables
• Model can be applied to predict changes in the
  dependent variables response to changes in the
  independent variables
• Regression also indicates the relative importance
  of independent variables on the response of the
  dependent variable

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Multiple Regression

• For example, a client may be interested in
  understanding the effect of price and promotional
  activity on a products market share among both
  loyal and not loyal customers
• Technical result is a linear model of the form
• Y a0 a1X1 a2X2 anXn
• Best visualizations of the results control all
  but one (or two) of the independent variables and
  examine how the value of dependent variable
  changes with respect to the free independent
  variables

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Multiple Regression
Market share for loyal customers
Market share for not-loyal customers
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Multiple Regression

• Properties
• Single interval scale dependent variable
• Multiple independent variables, preferably on
  interval scale
• Familiar and useful technique
• Issues
• Assumes linear relationship between dependent and
  independent variables
• Overused and often assumptions not fully checked
• Often misapplied to classification problems

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Logistic Regression

• Logistic Regression is a dependence techniques
  used to model the relationship between a single
  categorical dependent variable and a set of
  metric independent variables
• Typically dependent variable takes one of two
  values success/failure, buy/do not buy
• Multinomial formulations
• A logistic model gives the probability that the
  dependent variable takes a target value given the
  values of the independent variable

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Logistic Regression

• For example, which credit and demographic factors
  best predict whether a customer will keep a loan
  current
• Dependent variable taken as 60 days past due or
  worse
• Independent variables are credit and employment
  history, and demographic descriptors

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Logistic Regression

• Properties
• Powerful technique for predicting group
  membership and identifying important independent
  variables
• Becoming more widely used
• Procedures and results similar to linear
  regression
• Issues
• Adequate data
• Model validation
• Communicating probabilistic concepts

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Decision Trees

• Decision trees are a dependence technique used to
  develop a model to classify the value of a
  single dependent variable based on a set of
  independent variables
• Dependent and independent variables can be any
  data type
• The typical product of CART is a straightforward,
  easily interpretable set of segmentation rules
• For example, classify existing customers as high
  or low likelihood buyers of a new product based
  on demographics and historical purchasing
  behavior. Classification could be used to focus
  advertising campaign

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Decision Trees

• Decision trees can be also used to examine
  profiles of different market segments with
  respect to underlying demographic and
  psychographic variables
• For example, what are the most significant
  demographic variables determining whether the
  Internet is a persons most important information
  source?

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Decision Trees
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Decision Trees

• Properties
• Single dependent variable of any scale
• Multiple independent variables of any scale
• Free of model assumptions typical in other
  dependence techniques
• Powerful statistical learning algorithm able to
  identify complex variable interactions
• Issues
• Not as familiar
• Standard inferential statistics not applicable
• Often leads to asymmetric relationships

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Factor Analysis

• Factor analysis is an interdependence technique
  used to identify a set of underlying latent
  traits (factors) that explain the correlations
  between a large number of variables
• Data summarizing
• Derive a set of underlying concepts that
  summarize a larger set of variables
• Data reduction
• Develop a set of factor variables that serves as
  a more parsimonious description of the data

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Factor Analysis

• Interested in defining underlying dimensions
  influencing the perception of online destinations
• Survey respondents are asked to rate a set of
  destinations (including clients) with respect to
  a number of traits
• Factor analysis can be applied to develop a
  succinct set of perception dimensions
• This manageable set of dimensions can be used to
  characterize a clients site and to develop a
  focused plan to reposition it

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Factor Analysis

• Factors can then be used to provide visual
  summary of data

Competence Sophistication Trustworthy Exciting
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Factor Analysis

• Properties
• Very useful in identifying structure and
  relationships in data
• Provides tractable set of concepts for both
  managerial and analytical uses
• Provides opportunities for visualizations
• Issues
• Questionnaire design
• Variable selection
• Factor interpretation and validity

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Cluster Analysis

• Cluster analysis is an interdependence technique
  used to segment cases into homogeneous groups
  based on a specified set of variables
• Data reduction
• Develop a more parsimonious description of cases
  which can then be used in analytical
  classification methods
• Identify similarities between cases with respect
  to clustering variables
• Characterize clusters with respect to other sets
  of variables

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Cluster Analysis

• Want to identify and then characterize similar
  groups of TV pilot shows based on survey
  responses rating shows on various traits
• For one or two traits it may be possible to do
  this subjectively. Cluster analysis provides an
  objective method for multiple
  traits
• Clusters can be characterized with respect to
  variables not used in the analysis, such as show
  success, and cluster membership can be used as a
  dependent variable in classification method

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Cluster Analysis
Cluster 1 Low likelihood of success Cluster 2
Moderate likelihood of success Cluster 3 High
likelihood of success
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Cluster Analysis

• Properties
• Many cluster techniques are available for data of
  all scales
• Can identify structure in large data sets that
  may be difficult to discover in any other way
• Provides objective segmentation method
• Issues
• Selecting appropriate clustering method
• Determining appropriate number of clusters
• Validating clusters