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McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved. MEASURES OF ASSOCIATION Chapter 18
McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
MEASURES OF ASSOCIATION
Chapter 18
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Learning Objectives
Understand . . .How correlation analysis may be applied
to study relationships between two or more variables
The uses, requirements, and interpretation of the product moment correlation coefficient.
How predictions are made with regression analysis using the method of least squares to minimize errors in drawing a line of best fit.
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Learning Objectives
Understand . . .How to test regression models for
linearity and whether the equation is effective in fitting the data.
Nonparametric measures of association and the alternatives they offer when key assumptions and requirements for parametric techniques cannot be met.
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Pull Quote
“Consumer behavior with digital editions of magazines is very much like their behavior with print editions of magazines, and very much unlike their behavior with websites. Readers typically swipe through tablet editions from front to back, for example, the same way they work their way through print editions. They browse—taking in ads as they go—instead of jumping directly to specific articles the way web surfers do.”
Scott McDonald, senior vice-president for research and insights,
Conde Nast
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Measures of Association: Interval/Ratio Data
Pearson correlation coefficient
For continuous linearly related variables
Correlation ratio (eta)For nonlinear data or relating a main effect to a continuous dependent variable
BiserialOne continuous and one dichotomous variable with an underlying normal distribution
Partial correlation Three variables; relating two with the third’s effect taken out
Multiple correlation Three variables; relating one variable with two others
Bivariate linear regression Predicting one variable from another’s scores
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Measures of Association: Ordinal Data
GammaBased on concordant-discordant pairs; proportional reduction in error (PRE) interpretation
Kendall’s tau b P-Q based; adjustment for tied ranks
Kendall’s tau c P-Q based; adjustment for table dimensions
Somers’s d P-Q based; asymmetrical extension of gamma
Spearman’s rho Product moment correlation for ranked data
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Measures of Association: Nominal Data
Phi Chi-square based for 2*2 tables
Cramer’s V CS based; adjustment when one table dimension >2
Contingency coefficient C CS based; flexible data and distribution assumptions
Lambda PRE based interpretation
Goodman & Kruskal’s tau PRE based with table marginals emphasis
Uncertainty coefficient Useful for multidimensional tables
Kappa Agreement measure
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Researchers Search for Insights
Burke, one of the world’s leading research companies, claims researchers add the most value to a project when they look beyond the raw numbers to the shades of gray…what the data really mean.
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Pearson’s Product Moment Correlation r
Is there a relationship between X and Y?
What is the magnitude of the relationship?
What is the direction of the relationship?
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Connections and Disconnections
“To truly understand consumers’ motives and actions, you must determine relationships between what they think and feel and what they actually do.”
David Singleton, vp of insightsZyman Marketing Group
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Scatterplots of Relationships
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Scatterplots
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Plot of Forbes 500 Net Profits with Cash Flow
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Diagram of Common Variance
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Interpretation of Correlations
X causes Y
Y causes X
X and Y are activated by one or more other variables
X and Y influence each other reciprocally
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Artifact Correlations
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Interpretation of Coefficients
A coefficient is not remarkable simply because it is statistically significant!
It must be practically meaningful.
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Comparison of Bivariate Linear Correlation and Regression
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Examples of Different Slopes
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Concept Application
XAverage Temperature (Celsius)
YPrice per Case
(FF)12 2,000
16 3,000
20 4,000
24 5,000
Mean =18 Mean = 3,500
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Plot of Wine Price by Average Temperature
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Distribution of Y for Observation of X
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Wine Price Study Example
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Least Squares Line: Wine Price Study
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Plot of Standardized Residuals
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Prediction and Confidence Bands
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Testing Goodness of Fit
Y is completely unrelated to X and no systematic pattern is evident
There are constant values ofY for every value of X
The data are related but represented by a nonlinear function
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Components of Variation
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F Ratio in Regression
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F Ratio in Regression
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Coefficient of Determination: r2
Total proportion of variance in Y explained by X
Desired r2: 80% or more
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Chi-Square Based Measures
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Proportional Reduction of Error Measures
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Statistical Alternatives for Ordinal Measures
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Calculation of Concordant (P), Discordant (Q), Tied (Tx,Ty), and Total Paired Observations: KeyDesign Example
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KDL Data for Spearman’s Rho
_______ _____ Rank By_____ _____ _____
Applicant Panel x Psychologist y d d2
123456789
10
3.510.06.52.01.09.03.56.58.05.0
6.05.08.01.53.07.01.59.0
10.04.0
-2.55.0-1.5.05-22.02.0-2.5-21.0
6.2525.002.520.254.004.004.006.254.00
_1.00_57.00 .
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Key Terms
Artifact correlationsBivariate correlation
analysisBivariate normal
distributionChi-square-based
measuresContingency
coefficient CCramer’s V
PhiCoefficient of
determination (r2)ConcordantCorrelation matrixDiscordantError termGoodness of fitlambda
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Key Terms
• Linearity• Method of least
squares• Ordinal measures• Gamma• Somers’s d• Spearman’s rho• tau b• tau c
• Pearson correlation coefficient
• Prediction and confidence bands
• Proportional reduction in error (PRE)
• Regression analysis• Regression
coefficients
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Key Terms
• Intercept• Slope• Residual
• Scatterplot• Simple
prediction• tau
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McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
ADDITIONAL DISCUSSION OPPORTUNITIESChapter 18
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Snapshot: Oscars
Does the Oscar have any measurable effect on movie viewership?
Brief online survey via OmniPulse.
Event hype only a small influence.
Do women respond differently than men?
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PicProfile: Constellation Wines
Recession affected wine behavior.
Word sorts to reveal how Blackstone compared to other brands.
‘Masculine’ wasn’t threatening but a strength.
Positioning research using focus groups.
Three ads by Amazon Advertising were tested; “Count on it” strongest.
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Snapshot: Environsell
Does how you drive affect how you shop?
Observation studies.
Brits and Aussies shop as they drive.
Envirosell tracked shoppers’ behaviors.
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Snapshot: Advanced Statistics
“A risk score model was embedded in the daily transactions processing system to automatically determine how much cash each member can withdraw from an ATM or receive when making deposits.”
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Pull Quote
“The invalid assumption that correlationimplies cause is probably among the twoor three most serious and common errorsof human reasoning.”
Stephen Jay Gouldpaleontologist and science writer
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PulsePoint: Research Revelation
25 The percent of students using a credit card for college costs due to convenience.
McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
MEASURES OF ASSOCIATION
Chapter 18
18-48
Photo AttributionsSlide Source
8 Courtesy of Burke Research Inc.41 ©Image Source, all rights reserved42 Purestock/SuperStock43 Purestock/SuperStock44 Comstock Images/Getty Images
