Bivariate Analysis: Measures of Association

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Bivariate AnalysisMeasures of Association

• Chapter 17

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Correlation Analysis

• With a simple two variable correlation, you need
  to know the strength and direction of the
  correlation

Scatterplots help illustrate the relationships
between variables
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CorrelationLinear Association

• The purpose of correlation analysis is to find
  the measure of the degree of linear association
  (correlation) between two variables
• The Pearson correlation coefficient is commonly
  used for this purpose

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Interpreting Correlation

• Values range from -1 to 1
• Correlations of 0 mean that there is no
  relationship between variables
• Extreme values (-1 and 1) show that there is a
  strong linear relationship
• Correlations of -0.7 and 0.7 are equal in
  strength, but in opposite directions.
• A score of 0.5 would mean that there is linear
  trend with some deviation from the line
• Positive and negative
• Positive correlations mean that as one variable
  increases, the other increases
• Negative correlations mean that as one variable
  increases, the other decreases

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Regression Analysis

• Using one variable to predict another variable
• Example Predicting fast food purchases based on
  gender, age, income, and education
• Bivariate (simple) regression
• Single predictor
• Multivariate regression
• Multiple predictors

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Components of Regression

• Independent variables
• Variables that drive other variables
• Dependent variables
• Outcome or predicted variable

Independent variables Highway conditions Average
temperature of previous three days Local weather
forecast Newspaper space devoted to
advertising Average of previous three weeks
attendance
Dependent variable Ski resort weekend attendance
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Regression Formula

• Y a ßx1 e
• a mean of population when x1 0
• ß change in Y population mean per unit change
  in x1
• e error drawn independently from a normally
  distributed universe error is independent of x1
• Since e is impossible to predict and averages to
  0, it is typically left out of the equation

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Strength of Association

• Variation in Y can be explained by two things
• Variation explained by the regression (sum of
  squares due to regression)
• Variation not explained by the regression (sum of
  squares of deviation from the regression)
• The coefficient of determination (r2) measures
  the proportion of variation in Y explained by
  changes in X
• Value is between 0 and 1
• Is a measure of significance

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Strength of Association

• The equation for r2 is given bywhere the top
  half of the equation is deviance from the
  regression and the bottom half is total variation
• The closer r2 gets to 1, the better the
  regression is at predicting Y. If r2 1, then
  the regression perfectly predicts the actual
  values in virtually every instance

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Rank Correlations

• Associations between variables can be found when
  data is given as rankings rather than interval
  scales through rank correlations

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Spearman Rank Correlation

• The best-known and easiest technique is the
  Spearman rank correlation (rs)
• rs is given by the equationwhere d is the
  difference between rankings in two ranking
  methods
• When N ? 10, rs can be used to calculate a
  t-score with the equation and the
  resulting t-score is used in a two-tailed test of
  significance

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Kendall Rank Correlation Coefficient (?)

• More complicated than the Spearman rank
• Should be used when three or more sets of
  rankings are compared
• Calculated by the proportion of concordant pairs
  minus the proportion of discordant pairs
• There exist two bivariate observations, (xi,yi)
  and (xj,yj)
• Concordant pairs are when (xi-xj)(yi-yj) are
  positive
• Discordant pairs are when (xi-xj)(yi-yj) is
  negative
• Scores range from -1 to 1

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Goodman and Kruskals Lambda (?)

• ? is used when nominal scales are used
• Spearman rank and ? wont work because the
  ordering element is missing with nominal scales
• ? can be calculated by statistical packages

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Final Note

• Because these calculations can be run simply with
  statistical software packages like SPSS and SAS,
  and even Excel, it is more important to
  understand the significance of results than to
  memorize equations