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# Clinical Research Training Program 2021

Clinical Research Training Program 2021. CORRELATIONS. Fall 2004. www.edc.gsph.pitt.edu/faculty/dodge/clres2021.html. OUTLINE. Correlations Multiple Correlation Coefficients Partial Correlation Coefficients Multiple Partial Correlation. Multiple Correlation Coefficient.

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## Clinical Research Training Program 2021

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1. Clinical Research Training Program 2021 CORRELATIONS Fall 2004 www.edc.gsph.pitt.edu/faculty/dodge/clres2021.html

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 + 1x1 +  y = 0 + 1x1 + 2x2 +  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 + 1x1 +  y = 0 + 1x1 + 2x2 + 3x3 +  H0: H0: 2 = 3 =0

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