1. Clinical Research Training Program 2021
CORRELATIONS
Fall 2004

2. OUTLINE Correlations Multiple Correlation Coefficients
Partial Correlation Coefficients
Multiple Partial Correlation

3. Multiple Correlation Coefficient
The multiple correlation coefficient, denoted as Ry|x1,…, xp, is a measure of the overall linear association of one dependent variable y with p independent variables x1,…, xp.
The least-squares solution
has the largest value of Ry|x1,…, xp.
When p = 1, R = ? r.

4. Multiple Correlation Coefficient
Mathematical formula for Ry|x1,…, xp
Its square:
absolute
value

5. Multiple Correlation Coefficient
The quantity , measures the proportionate reduction in the total sum of squares to due to the multiple linear regression of y on x1,…, xp.
The quantity Ry|x1,…, xp, is the correlation
of the observed value y with the predicted value , and this correlation is always nonnegative.

6. Partial Correlation Coefficient
The partial correlation coefficient, denoted as ryx*|x1,…, xp is a measure of the strength of the linear relationship between the dependent variable y and one independent variables, say x*, after we control for the effects of other p independent variables x1,…, xp.

7. Partial Correlation Coefficient
The order of the partial correlation coefficient depends on the number of variables that are being controlled for.
First-order partials: ryx*|x1
Second-order partials: ryx*|x1, x2
Third-order partials: ryx*|x1, x2, x3
…etc.

8. Partial Correlation Coefficient
Mathematical formula for Or

9. Partial Correlation Coefficient
The quantity measures the proportion of the residual sum of squares that is accounted for by the addition of x* to a regression model already involving x1,…, xp.
The partial F statistic F(x*| x1,…, xp) is used to test H0:

10. Partial Correlation Coefficient
Hypothesis:
Test statistic:

11. Partial Correlation Coefficient
Compare two models:
y = 0 + 1x 
y = 0 + 1x1 + 2x 
H0:
H0: 2=0

12. Multiple Partial Correlation
The multiple partial correlation coefficient, denoted as ry(z1, …, zq)|x1,…, xp is a measure of the strength of the linear relationship between the dependent variable y and a set of independent variables, say z1,…, zq, after we control for the effects of other p independent variables x1,…, xp.

13. Multiple Partial Correlation
Mathematical formula for
Hypothesis:
Test statistic:

14. Multiple Partial Correlation
The quantity measures the proportion of the residual sum of squares that is accounted for by the addition of z1,…, zq to a regression model already involving x1,…, xp.
The partial F statistic F(z1,…, zq| x1,…, xp) is used to test H0:

15. Multiple Partial Correlation
Compare two models:
y = 0 + 1x 
y = 0 + 1x1 + 2x2 + 3x3 + 
H0:
H0: 2 = 3 =0

16. Partial Correlation Coefficient
y = SBP
Variable
Type I
Partial Corr.
Type II
Intercept -
Weight
rSBP,WGT
rSBP,WGT|AGE,HGT
Age
rSBP,AGE|WGT
rSBP,AGE|WGT,HGT
Height
rSBP,HGT|WGT,AGE