Measures Of Association

MEASURES OF ASSOCIATION

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Bivariate Correlation vs. Nonparametric Measures of Association  Parametric

correlation requires two continuous variables measured on an interval or ratio scale  The coefficient does not distinguish between independent and dependent variables

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Bivariate Correlation Analysis Pearson correlation coefficient

• r symbolized the coefficient's estimate of • •

linear association based on sampling data Correlation coefficients reveal the magnitude and direction of relationships Coefficient’s sign (+ or -) signifies the direction of the relationship

 Assumptions

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of r

Linearity Bivariate normal distribution

Bivariate Correlation Analysis Scatterplots

• Provide a means for visual inspection of data • the direction of a relationship • the shape of a relationship • the magnitude of a relationship (with practice)

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Interpretation of Coefficients  Relationship

does not imply causation  Statistical significance does not imply a relationship is practically meaningful

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Interpretation of Coefficients  Suggests

alternate explanations for correlation results

• X causes Y. . . or • Y causes X . . . or • X & Y are activated by one or more other •

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variables . . . or X & Y influence each other reciprocally

Interpretation of Coefficients  Artifact

Correlations  Goodness of fit

• F test • Coefficient of determination • Correlation matrix • used to display coefficients for more than two variables

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Bivariate Linear Regression  Used

to make simple and multiple predictions  Regression coefficients

• Slope • Intercept

 Error

term  Method of least squares 8

Interpreting Linear Regression •

Residuals

• what remains after the line is fit or (Y -Y ) i

 Prediction

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and confidence bands

i

Interpreting Linear Regression 

Goodness of fit

• Zero slope

• Y completely unrelated to X and no systematic pattern is evident • constant values of Y for every value of X • data are related, but represented by a nonlinear function

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Nonparametric Measures of Association  Measures

for nominal data

• When there is no relationship at all, coefficient •

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is 0 When there is complete dependency, the coefficient displays unity or 1

Nonparametric Measures of Association  Chi-square

based measure

• Phi • Cramer’s V • Contingency coefficient of C

 Proportional

• Lambda • Tau

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reduction in error (PRE)

Characteristics of Ordinal Data  Concordant-

subject who ranks higher on one variable also ranks higher on the other variable  Discordant- subject who ranks higher on one variable ranks lower on the other variable

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Measures for Ordinal Data  No

assumption of bivariate normal distribution  Most based on concordant/discordant pairs  Values range from +1.0 to -1.0

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Measures for Ordinal Data  Tests

• • • • •

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Gamma Somer’s d Spearman’s rho Kendall’s tau b Kendall’s tau c