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PSY 1950 Correlation November 5, 2008. Definition Correlation quantifies the strength and direction...
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PSY 1950Correlation
November 5, 2008
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Definition• Correlation quantifies the strength and direction of a linear relationship between two variables
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History
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The First Scatterplot (Galton, 1885)
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Importance• Prior to correlation, “there was no way to discuss -- let alone measure -- the association between variables that lacked a cause-effect relationship”
• Correlation underlies many advanced statistical techniques– Factor analysis– Structural equation modeling
• Correlation informs– Prediction of a unkown variable– Validity of a measure– Reliability of a measure– Validity of a theory
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Covariance• Covariance measures how much two variables change together– The more they change together, the higher the covariance
– Variance is a special case of covariance
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X Y X Y Product1 4 -2 -2 42 5 -1 -1 13 6 0 0 04 7 1 1 15 8 2 2 4
Score DeviationX Y X Y Product1 8 -2 2 -42 5 -1 -1 13 4 0 -2 04 5 1 -1 -15 8 2 2 4
DeviationScore
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The Problem with Covariation
• It reflects not only the degree of a bivariate relationship, but also the variation of each variables
• In other words, its units depends on the variables
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Pearson Product-Moment Correlation (r)
• Special case of covariance– Standardized covariance– Covariance of standardized variables
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Example
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Interpreting r• Things to consider carefully
– Correlation versus causation– Restricted Range– Group sampling– Outliers– Linearity– Size– Homoscedasticity– Significance
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Correlation versus Causation
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Correlation versus Causation
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Restriction of Range• When the bivariate range is artificially limited– In the case of linear relationship, the correlation is almost spuriously attenuated
– In the case of curvilinear relationship, can result in a spuriously large correlation
• Possibly a grouping/selection effect– The correlation between height and basketball ability among NBA players
• http://www.ruf.rice.edu/~lane/stat_sim/restricted_range/index.html
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Grouping• Grouping of heterogeneous groups (either a priori via sampling or a posteriori via data segregation) can inflate correlation– e.g., the correlation between height and basketball ability among small people and tall people
– e.g., the correlation between height and weight in men and women•For men, r = .60, for women r = .49•Together, r = .78
• http://www.ruf.rice.edu/~lane/stat_sim/restricted_range/index.html
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Outliers• Correlation is very sensitive to outliers– For all three plots, r, means, and SD are equal
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Linearity
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Size• The magnitude of r
• The magnitude of r2
– The coefficient of determination– The proportion of variability in one variable accounted for by variability in the other variable
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Homoscedasticity• Same as homogeneity of variance assumption
• Variance for Y does not depend on value of Y and vice-versa
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Significance• To test the null hypothesis that the population correlation, (“rho”) = 0, use:
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Other measures of correlation
• Computationally identical to r– Point-biserial
•One dichotomous variable
– Phi•Two dichotomous variables
– Spearman•Both variables on ordinal scale•Tests monotonicity of relationship•As X increases, so does Y•No accurate significance test
• Computationally novel techniques– e.g., Kendll’s Tau