Session 13: Correlation (Zar, Chapter 19)

(1)Regression vs. correlation Regression: R 2 is the proportion that the model explains of the variability of y.

Correlation: Relationship (x & y both random) Linear Relationship measured as the correlation coefficient: “Pearson Correlation Coefficient” “Pearson Linear Correlation Coefficient”

The correlation coefficient can be negative depending upon the numerator. So Perfect Negative Perfect Positive 0 -- No Linear Relationship

No Linear Relationship

(2) Hypotheses about r Can use table B.17 directly to perform the test.

Example 19.1a

Cannot just replace 0 with Fisher developed the following transformation:

For the 1-sided Tests:

F.N. David showed that Fisher’s z is good if n ≥ 20. Note K   K  are from Table B.2. Table B.3 can be used (with  ). Note: Table B.18 calculates the transform.

(3) Spearman rank correlation coefficient If there are ties, where

The Spearman rank correlation coefficient, computed for the data of Example 19.1.

(4) Correlation in Tables: Cohen’s kappa Recall the coded agreement tables & preference tables of Chapter 9: Define P ij =f ij /n

Define sum of observed probability on main diagonal sum of main diagonal estimates

Test 2-sided: Test 1-sided:

where and Or we can use the following if testing versus 0:

Remember the Dental Mildness Study: THETA THETA THETA KAPPA= THETA4 = A= B= C= THETA5= VAR1= (Large Sample) VAR2= (Null Model of Independence)