Effect of Violations of Normality Edgell and Noon, 1984 On the Correlation Coefficient t-Test

Is the t-test for correlation coefficients robust to violations of its assumptions?

Overview   Review t-test of the Correlation Coefficient   Violations Bivariate Normal Assumption Independence Assumption

Review Violation of Normality Violation of Independence  Bivariate normal assumption Both variables come from normal distributions Both variables come from normal distributions OROR One variable is from a normal distribution and the variables are independent One variable is from a normal distribution and the variables are independent  Independence assumption Value of one variable is not influenced by the other Value of one variable is not influenced by the other t 2= r 2 / ((1-r 2 )/df)

Review Violation of Normality Violation of Independence  Run 10,000 samples  Very Non-normal distributions  Range of sample sizes  Determine the proportion of samples that were significant at the.05 and.01 level Method

Review Violation of Normality Violation of Independence Distributions Exponential Distribution

Review Violation of Normality Violation of Independence Distributions Uniform Distribution

Review Violation of Normality Violation of Independence Distributions Cauchy Distribution

Review Violation of Normality Violation of Independence Results

Review Violation of Normality Violation of Independence Results

Review Violation of Normality Violation of Independence Method  Run 10,000 samples  Range of sample sizes  Zero correlations with dependency  Determine the proportion of samples that were significant at the.05 and.01 level

Review Violation of Normality Violation of Independence Method Zero-Correlations with dependency 1) Second variable is the square of the First Variable 2) Mixed Bivariate Normal Distributions - Population is aggregate of smaller subpopulations

Review Violation of Normality Violation of Independence P=.5ρ1=.3P=.5 ρ2= -.3 ρ =0 Mixed Bivariate Normal Distributions

Review Violation of Normality Violation of Independence Results

 Violations of Normality Robust at.05 Robust at.05 At.01, only sensitive to extreme departures from normality At.01, only sensitive to extreme departures from normality Conclusion Is the t-test for correlation coefficients robust to violations of normality?

Conclusion Is the t-test for correlation coefficients robust to violations of independence?  Not Robust  But Non independent variables are not likely to have a correlation of zero Non independent variables are not likely to have a correlation of zero t-Test could be considered a test of the hypothesis of independence t-Test could be considered a test of the hypothesis of independence