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1 Clinical Research Training Program 2021 CORRELATIONS Fall 2004

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

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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.

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

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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.

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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.

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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.

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8 Partial Correlation Coefficient Mathematical formula for Or

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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 :

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10 Partial Correlation Coefficient Hypothesis: Test statistic:

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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

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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.

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

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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 :

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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

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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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