Effect of Violations of Normality

1
Effect of Violations of Normality
On the Correlation Coefficient t-Test

• Edgell and Noon, 1984

2
Is the t-test for correlation coefficients robust
to violations of its assumptions?
3
Overview

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

4
Review Violation of Normality
Violation of Independence
t2 r2 / ((1-r2)/df)

• Bivariate normal assumption
• Both variables come from normal distributions
• OR
• One variable is from a normal distribution and
  the variables are independent
• Independence assumption
• Value of one variable is not influenced by the
  other

5
Review Violation of Normality
Violation of Independence
Method

• 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

6
Review Violation of Normality
Violation of Independence
Distributions
Exponential Distribution
7
Review Violation of Normality
Violation of Independence
Distributions
Uniform Distribution
8
Review Violation of Normality
Violation of Independence
Distributions
Cauchy Distribution
9
Review Violation of Normality
Violation of Independence
Results
10
Review Violation of Normality
Violation of Independence
Results
11
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

12
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

13
Review Violation of Normality Violation
of Independence
Mixed Bivariate Normal Distributions
P.5 ?1.3
P.5 ?2 -.3
? 0
14
Review Violation of Normality
Violation of Independence
Results
15
Conclusion
Is the t-test for correlation coefficients robust
to violations of normality?

• Violations of Normality
• Robust at .05
• At .01, only sensitive to extreme departures from
  normality

16
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
• t-Test could be considered a test of the
  hypothesis of independence