CORRELATION

Overview of Correlation u What is a Correlation? u Correlation Coefficients u Coefficient of Determination u Test for Significance u Correlation and Causality u Partial and Part Correlations

What is a Correlation? u Degree of linear relationship between variables u Each individual is measured on both variables

What is a Correlation? u Comparison of the way scores deviate from their means on the two variables u Standardized covariance

Cross-Product Deviation u Find the difference between each person’s scores and the mean of the variable (deviation). u For each person, multiply the two deviations together. u Do the deviations tend to go in the same direction?

Covariance u Add up all the cross-product deviations and average them. u The more covariance, the more the two variables go together, or co-vary. u Covariance is not standardized, so it’s hard to interpret.

Pearson r u Standardized covariance u Used for two interval/ratio variables u Varies from -1 to +1

Pearson r u Absolute value indicates strength of relationship u.1 - small u.3 - medium u.5 - large

Pearson r u Sign indicates direction of correlation u Positive: increases on one variable correspond to increases on the other variable u Negative: increases on one variable correspond to decreases on the other variable

Other Correlation Coefficients u Ordinal variables u Spearman rho or Kendall’s tau u Dichotomous variable with interval/ratio variable u Point biserial r (discrete dichotomy) u Biserial r (continuous dichotomy)

Other Correlation Coefficients u Two dichotomous variables u Phi coefficient

About Dichotomous Variables u Dichotomous variables are usually at the nominal level. u Numbers are assigned to the two categories in an arbitrary way. u The way the numbers are assigned determines the sign of the correlation coefficient.

Coefficient of Determination u Measures proportion of explained variance in Y based on X u r 2

Testing r for Significance u Null hypothesis is usually that r is zero in the population. u One tailed vs. two-tailed

Assumptions u Appropriate types of data u Independent observations u Normal distributions u Linear relationship

Correlation and Causality u A correlation by itself does not show that one variable causes the other. u A correlation may be consistent with a causal relationship.

Example APA format Example APA format The Pearson r was computed between rated enjoyment of frog legs and level of neuroticism. The correlation was statistically significant, r (58) =.28, p =.03.

The Third Variable Problem u A correlation between X and Y could be caused by a third variable influencing both X and Y.

The Directionality Problem u A correlation between X and Y could be a result of X causing Y or Y causing X.

Partial Correlation u Used to “partial out” the effects of a third variable (X2) on the relationship between X1 and Y u Correlation between X1 and Y with the influence of X2 removed from both X1 and Y

Y X1 X2 Partial r 2

Interpreting Partial Correlations u Compare the simple bivariate correlation to the partial correlation. u If the partial correlation is lower, it suggests that X2 is mediating the relationship between X1 and Y.

Part Correlation u Also called: semi-partial correlation u Correlation between X1 and Y with the influence of X2 (and other predictor variables) removed from just X1 u Indicates amount of unique variance in Y explained by X1 u Used in Multiple Regression Analysis

X2 X1 Part r 2 Y

Y X1 X2 Partial r 2

Take-Home Points u A correlation coefficient tells you how much and in what direction two variables are related. u Correlation alone does not indicate causation, nor does it indicate lack of causation.