Strength of Association

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Strength of Association

• What is the difference between a significance
  test statistic and a measure of association
• What does each one tell us about a bivariate
  relationship?
• How are they related?
• How can we measure the Strength of Association
  between two variables?
• Coefficient should range between .0 1
• Coefficient should not be affected by N
• Coefficient should not be affected by a
  variables metric or scale of measurement
• Coefficient values should be interpretable or
  meaningful

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Strength of Association (cont.)

• What does bivariate association mean?
  Generally reflects two distinct ideas
• Covariance or co-occurrence between variables
  (versus independence)
• Measures of agreement or covariance
• Predictability of one variable from the other
  (versus randomness)
• Proportional Reduction of Error (PRE) Measures
• A number of different measures of association
• Based on different levels of measurement
• Based on different logical models criteria

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Strength of Association (continued)

• Association between 2 numerical variables ?
  Covariance Correlation
• Coefficient of Association Pearsons r
• r-squared proportion of variance in common
• May use Spearmans r if data are ranked
• Association between 1 categoric variable and 1
  numeric variable ? ANOVA mean differences
  (F-tests and t-tests)
• Coefficient of Association eta (?)
• eta-squared proportion of variance between
  groups

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Strength of Association (continued)

• Association between 2 categoric variables ?
  Ordinal or Nominal
• Approaches to nonparametric measures of
  association ? (a) Chi-square-based (b) PRE (c)
  Concordance/agreement
• Nominal (unordered) variables
• Chi-square-derived
• Contingency coefficient, C
• Cramers V coefficient ? use this for 2x3 or
  larger
• Phi coefficient, F ? use this for 2x2 tables
• PRE-derived
• Lambda (asymmetric)

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Strength of Association (continued)

• Association between 2 categoric variables
  (continued)
• Ordinal (ordered) variables
• Concordance-based statistics
• Gamma, ? ? most commonly used
• Others available Kendalls tau (t) Somers d
  (less frequently used)
• Rank-order statistics
• Use only if many categories few ties (i.e.,
  cases with the same value on a variable)
• Can also use Chi-square-based measures

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Strength of Association (continued)

• Categoric Measures A Summary
• Nominal variables
• Phi, F for 2x2 tables
• Kramers V for larger than 2x2 tables
• Ordinal variables
• Gamma, ? ? most commonly used
• Yules Q ? same statistic in a 2x2 table
• Use rank-order statistics (Spearmans r) only if
  many values few ties
• Can also use Phi and Kramers V

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Strength of Association (continued)

• Categoric measures of association
• Different coefficients will generally not yield
  the same values on the same crosstab
• Gamma ( Yules Q) will almost always compute
  higher values than Kramers V ( Phi) on the same
  tables

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Strength of Association (continued)

• Categoric measures of association
• How to Compute them?
• By Hand see formulas in the textbook
• Chi-square-based easiest to compute
• Note special computing formulas for 2x2 tables
• Note X Y variables in crosstab must be
  formatted in the same direction for ordinal-level
  statistics (e.g., Gamma)
• In SPSS Click statistics box in Crosstabs
  pop-up menu, then select appropriate coefficients
  (do not select them all)

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Strength of Association (continued)

• Going beyond bivariate analysis to multivariate
  analyses
• We often wish to consider more than two variables
  at a time because other variables may be involved
  in more complex patterns
• Termed Partialling or controlling ?
  statistically consider the confounding influence
  of additional variables (3rd variable problem)
• This invokes the concept of Ceteris paribus
  (holding everything else equal)

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Strength of Association (continued)

• Going beyond bivariate analysis to multivariate
  analyses
• Partialling or controlling ? statistically
  consider the confounding influence of additional
  variables (3rd variable problem)
• In cross-tabulations, this means nesting the
  crosstabs within the 3rd variable
• i.e., compute separate sub-crosstabs within
  levels of the 3rd variable
• See the example on the handout

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Strength of Association (continued)

• Going beyond bivariate analysis to multivariate
  analyses
• Partialling or controlling ? statistically
  consider the confounding influence of additional
  variables (3rd variable problem)
• In cross-tabulations, this means nesting the
  crosstabs within the 3rd variable
• i.e., compute separate sub-crosstabs within
  levels of the 3rd variable
• See the example on the handout
• Only useful when 3rd variable is associated with
  both X and Y variables.