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 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.

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

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.

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.

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.

Partial Correlation Coefficient Mathematical formula for Or

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:

Partial Correlation Coefficient Hypothesis: Test statistic:

Partial Correlation Coefficient Compare two models: y = 0 + 1x1 +  y = 0 + 1x1 + 2x2 +  H0: H0: 2=0

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.

Multiple Partial Correlation Mathematical formula for Hypothesis: Test statistic:

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:

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

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