Multivariate Statistical Analysis: Analyzing Criterionpredictor Association

1
Multivariate Statistical AnalysisAnalyzing
Criterion-predictor Association

• Chapter 18

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Knowing Which Techniques to Use

• The type of analysis to be performed depends on
  several things
• The number of variables
• The existence of dependencies
• Number of dependencies
• Measurement of data (scalingnominal, interval)

3
Covariation

• In many marketing research studies, the goal is
  to explain variation in one set of variables
  (i.e., sales) with matching variation in other
  sets of variables (advertising)
• Identifying criterion and predictor variables
  (dependent and independent variables)
• Creating a function or equation that estimates
  the criterion from predictor variables
• Establishing confidence in the relationship
  between variables

4
Multiple Regression

• Same as simple regressions, but with more
  predictive terms
• In this new case, ß1 is a partial regression
  coefficient with respect to x1
• Because x1 is usually correlated to x2, the
  question remains how to tell whether multiple
  regression is better than simple regression

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Regression Output Measures of Interest

• Regression equation
• R2both sample-determined and population adjusted
• F-test for the overall equation
• Individual t-tests and standard error for testing
  each partial regression coefficient
• Partial correlation coeffecients
• Accounted-for variance explained by each
  predictor (overlap goes to preceding predictor)

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Contribution to Explained Variance

• Example
• Bivariate regression
• Y 0.491 0.886 x1
• Same data, multiple regression
• Y 0.247 0.493 x1 0.484 x2
• Independent variables may be correlated, so
  looking at R2 values is critical
• R2 of the first equation is 0.723, meaning 72 of
  the variation in Y is explained by variation in
  x1
• R2 of the second equation is 0.783, meaning that
  adding the x2 to the equation only explains 6
  more of the variation

7
Multicollinearity

• Problem in regressions when predictor variables
  are highly correlated
• Distorts coefficients, making it tough to tell
  which predictor variables are most relevant
• No standard as to how much collinearity is too
  much
• 3 solutions for multicollinearity
• Ignore it
• Delete one or more correlated predictors
• As a rule of thumb, some researchers toss one or
  more correlated variables when the correlation is
  .9 or more
• Find predictor variable combinations that are
  uncorrelated

8
Cross-validation

• Randomly split cases into groups
• Compute separate regressions for each group
• Use equations to predict Y-values in other groups
• Check coefficients to check for
  agreementalgebraic sign and magnitude
• Compute a regression for the entire sample, using
  only the variables that were similar between
  groups

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Stepwise Regression

• Run a regression with many predictor variables
• Reviewing results and removing the least
  important variables
• Keep re-running the regression and removing
  variables until a solid model emerges
• Can also start with a single variable and adding
  more, keeping those that seem promising

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Downsides of Stepwise Regression

• Several supposedly optimal equations emerge,
  depending on the variables started with and the
  method (eliminating or adding variables)
• Multicollinearity clouds which variables might
  make the most predictive model

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Discriminant Analysis

• Discriminant analysis is used when the predictor
  variables are interval-scaled data but the
  criterion is categorical
• Example A B2B company has used its database to
  segment its customers into groups that prefer
  ordering over the internet, through a catalog, or
  from a salesperson.With discriminant analysis,
  the company can use data on new customers (annual
  sales, number of employees) to predict which
  segment they might fall into.

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Two Types of Discriminant Analysis

• Discriminant predictive (or explanatory) analysis
• Used to optimize predictive functions
• Discriminant classification analysis
• Like other grouping techniques, this analysis
  seeks to minimize differences within a single
  group, and then maximize differences between
  several groups

13
Chi-square Automatic Interaction Detection (CHAID)

• CHAID analysis is used to find dependencies
  between variables
• Typically used in data mining
• Useful when
• No desire to make assumptions about relationships
• Many potential explanatory variables
• Large number of observations

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CHAID Process

• Computer takes a large data set and tests
  different ways data can be split up, selecting
  the best available split
• Groups can be further split
• Because there are so many splits, CHAID requires
  very large samples

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CHAID Example

• A bank is planning a direct-mail campaign to sign
  up new customers for a credit card.
• CHAID analysis can be run on a customer database
  to find variables that will lead to a greater
  response.
• CHAID analysis might show that households earning
  50,000 per year or more are more likely to sign
  up. Within that group women are more likely to
  respond, and women with household size of 5 or
  more are more likely to respond.
• Purchasing a mailing list from a credit reporting
  agency for women in large households where
  reported income is over 50,000 results in a
  greater response rate over past campaigns4 for
  a targeted list compared to 1 for a broad
  mailingat reduced cost.

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Canonical Correlation

• Generalization of multiple correlation to two or
  more criterion variables
• Answers the question
• How well does one group of variables predict
  another?

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Correspondence Analysis

• Specialized set of canonical correlation
  techniques
• Useful for studying brands and attributes
• Includes a visual display of results to help
  interpret data

18
Probit and Logit

• Deals with the same type of problem as a
  regression, but is designed for nominal- or
  ordinal-scaled data
• Probit Response is assumed to be normally
  distributed
• Logit Logistic distribution is assumed

19
Path Analysis / Causal Modeling

• Hybrid of factor analysis and simultaneous
  equation regression
• The goal is measuring links between observed
  measures and unobservable constructs
• These advanced techniques require the right
  software and expertise on the part of the
  researcher