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

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Presentation on theme: "1 Clinical Research Training Program 2021 CORRELATIONS Fall 2004 www.edc.gsph.pitt.edu/faculty/dodge/clres2021.html."— Presentation transcript:

1 1 Clinical Research Training Program 2021 CORRELATIONS Fall 2004

2 2 OUTLINE  Correlations  Multiple Correlation Coefficients  Partial Correlation Coefficients  Multiple Partial Correlation

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

4 4 Multiple Correlation Coefficient  Mathematical formula for R y|x 1,…, x p  Its square: absolute value

5 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 x 1,…, x p.  The quantity R y|x 1,…, x p, is the correlation of the observed value y with the predicted value, and this correlation is always nonnegative.

6 6 Partial Correlation Coefficient  The partial correlation coefficient, denoted as r yx * |x 1,…, x p 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 x 1,…, x p.

7 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: r yx * |x 1  Second-order partials: r yx * |x 1, x 2  Third-order partials: r yx * |x 1, x 2, x 3 …etc.

8 8 Partial Correlation Coefficient  Mathematical formula for Or

9 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 x 1,…, x p.  The partial F statistic F(x*| x 1,…, x p ) is used to test H 0 :

10 10 Partial Correlation Coefficient Hypothesis: Test statistic:

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

12 12 Multiple Partial Correlation  The multiple partial correlation coefficient, denoted as r y(z 1, …, z q )|x 1,…, x p is a measure of the strength of the linear relationship between the dependent variable y and a set of independent variables, say z 1,…, z q, after we control for the effects of other p independent variables x 1,…, x p.

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

14 14 Multiple Partial Correlation  The quantity measures the proportion of the residual sum of squares that is accounted for by the addition of z 1,…, z q to a regression model already involving x 1,…, x p.  The partial F statistic F(z 1,…, z q | x 1,…, x p ) is used to test H 0 :

15 15 Multiple Partial Correlation Compare two models: y =  0 +  1 x 1 +  y =  0 +  1 x 1 +  2 x 2 +  3 x 3 +  H0:H0: H 0 :  2 =  3 =0

16 16 Partial Correlation Coefficient VariableType I Partial Corr. Type II Partial Corr. Intercept-- Weightr SBP,WGT r SBP,WGT|AGE,HGT Ager SBP,AGE|WGT r SBP,AGE|WGT,HGT Heightr SBP,HGT|WGT,AGE y = SBP


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