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Chapter 18Chapter 18
Measures of Measures of AssociationAssociation
McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All Rights Reserved.
18-2
Learning ObjectivesLearning 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 ObjectivesLearning 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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Invalid AssumptionsInvalid Assumptions
“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: PulsePoint: Research RevelationResearch Revelation
25 The percent of students using a credit card for college costs due to convenience.
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Measures of Association: Measures of Association: Interval/Ratio DataInterval/Ratio Data
Pearson correlation coefficientFor 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 correlationThree variables; relating two with the third’s effect taken out
Multiple correlationThree variables; relating one variable with two others
Bivariate linear regressionPredicting one variable from another’s scores
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Measures of Association: Measures of Association: Ordinal DataOrdinal Data
GammaBased on concordant-discordant pairs; proportional reduction in error (PRE) interpretation
Kendall’s tau bP-Q based; adjustment for tied ranks
Kendall’s tau cP-Q based; adjustment for table dimensions
Somers’s dP-Q based; asymmetrical extension of gamma
Spearman’s rhoProduct moment correlation for ranked data
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Measures of Association: Measures of Association: Nominal DataNominal Data
Phi Chi-square based for 2*2 tables
Cramer’s VCS based; adjustment when one table dimension >2
Contingency coefficient CCS based; flexible data and distribution assumptions
Lambda PRE based interpretation
Goodman & Kruskal’s tauPRE based with table marginals emphasis
Uncertainty coefficient Useful for multidimensional tables
Kappa Agreement measure
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Researchers Search for InsightsResearchers 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 Pearson’s Product Moment Correlation Correlation rr
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 Connections and DisconnectionsDisconnections
“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 RelationshipsScatterplots of Relationships
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ScatterplotsScatterplots
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Diagram of Common VarianceDiagram of Common Variance
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Interpretation of CorrelationsInterpretation of Correlations
X causes YX causes Y
Y causes XY causes X
X and Y are activated by one or more other variablesX and Y are activated by
one or more other variables
X and Y influence each other reciprocally
X and Y influence each other reciprocally
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Artifact CorrelationsArtifact Correlations
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Interpretation of CoefficientsInterpretation 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 Comparison of Bivariate Linear Correlation and RegressionCorrelation and Regression
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Examples of Different SlopesExamples of Different Slopes
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Concept ApplicationConcept Application
X
Average Temperature (Celsius)
Y
Price 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 Plot of Wine Price by Average TemperatureTemperature
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Distribution of Y for Distribution of Y for Observation of XObservation of X
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Wine Price Study ExampleWine Price Study Example
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Least Squares Line: Wine Price Least Squares Line: Wine Price StudyStudy
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Plot of Standardized ResidualsPlot of Standardized Residuals
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Prediction and Prediction and Confidence BandsConfidence Bands
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Testing Goodness of FitTesting 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 VariationComponents of Variation
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FF Ratio in Regression Ratio in Regression
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Coefficient of Determination: Coefficient of Determination: rr22
Total proportion of variance in Y explained by X
Desired r2: 80% or more
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Chi-Square Based MeasuresChi-Square Based Measures
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Proportional Reduction of Proportional Reduction of Error MeasuresError Measures
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Statistical Alternatives Statistical Alternatives for Ordinal Measuresfor Ordinal Measures
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Calculation of Concordant (Calculation of Concordant (PP), ), Discordant (Discordant (QQ), Tied (), Tied (Tx,TyTx,Ty), and ), and Total Paired Observations: Total Paired Observations: KeyDesign ExampleKeyDesign Example
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KDL Data for Spearman’s RhoKDL Data for Spearman’s Rho
_______ _____ Rank By_____ _____ _____
Applicant Panel x Psychologist y d d2
1
2
3
4
5
6
7
8
9
10
3.5
10.0
6.5
2.0
1.0
9.0
3.5
6.5
8.0
5.0
6.0
5.0
8.0
1.5
3.0
7.0
1.5
9.0
10.0
4.0
-2.5
5.0
-1.5
.05
-2
2.0
2.0
-2.5
-2
1.0
6.25
25.00
2.52
0.25
4.00
4.00
4.00
6.25
4.00
_1.00_57.00 .
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Key TermsKey Terms
• Artifact correlations
• Bivariate correlation analysis
• Bivariate normal distribution
• Chi-square-based measures
• Contingency coefficient C
• Cramer’s V
• Phi
• Coefficient of determination (r2)
• Concordant
• Correlation matrix
• Discordant
• Error term
• Goodness of fit
• lambda
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Key Terms 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 Key Terms
• Intercept
• Slope
• Residual
• Scatterplot
• Simple prediction
• tau
