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PSY 1950 Correlation 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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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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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 r 2 –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

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