F04-Correlation

F04-Correlation - PubH 7405: REGRESSION ANALYSIS...

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PubH 7405: REGRESSION ANALYSIS Correlation Analysis
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CORRELATION & REGRESSION We have 2 continuous measurements made on each subject, one is the response variable Y, the other predictor X. There are two types of analyses: Correlation : is concerned with the association between them, measuring the strength of the relationship; the aim is to determine if they are correlated – the roles are exchangeable. Regression : To predict response from predictor .
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ROLES OF VAIRABLES In Regression Analysis , each has a well- defined role; we’ll predict “response Y” from a given value of “predictor X” In Correlation Analysis , the roles of “X” and “Y” are exchangeable; in the coefficient of correlation “r” is symmetric with respect to X and Y : we get the same result regardless of which one is X – no special “label”.
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SCATTER DIAGRAM In quarters I and III, For positive association, For stronger relationship most of the dots, being closely clustered around the line, are in these two quarters; the above sum is large. 0 ) )( ( y y x x 0 ) )( ( y y x x ) , ( y x Quarter (I) Quarter (III) (II) (IV)
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SCATTER DIAGRAM In quarters II and IV, For negative association, For stronger relationship most of the dots, being closely clustered around the line, are in these two quarters; the sum is a large negative number. 0 ) )( ( y y x x 0 ) )( ( y y x x ) , ( y x Quarter (II) Quarter (IV) (I) (III)
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SUMMARY The “ sum of products summarizes the “ evidence of the relationship under investigation; It is zero or near zero for weak associations and is large, negative or positive, for stronger associations. The sum of products can be used as a measure of the strength of the association itself. However, it is “ unbounded making it hard to use because we cannot tell if we have a strong association (how large is “large”?). We need to “ standardize it. ) )( ( y y x x
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COEFFICIENT OF CORRELATION With a standardization, we obtain a statistic r is called the Correlation Coefficient measuring the strength of the relationship:  ] ) ( ][ ) ( [ ) )( ( 2 2 y y x x y y x x r WE can “explain” the denominator as necessary for standardization: to obtain a statistic in [-1,1].
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There are many different ways to express the coefficient of correlation n; one of which is often mentioned as the “short-cut” formula : y x xy s s s r  ] ) ( ][ ) ( [ ) )( ( ] ) ( ][ ) ( [ ) )( ( 2 2 2 2 2 2 n y y n x x n y x xy y y x x y y x x r s xy is the “sample covariance” of X and Y
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Another very useful formula is to express the coefficient of correlation r as the “ Average Product” in “standard units where s x and s y are the (sample) standard deviations of X and Y, respectively: y _ x _ s y y s x x 1 n 1 r
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The following is an important and very interesting characteristic: Y) r(X, b) aY d, r(cX
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This note was uploaded on 11/21/2011 for the course PUBH 7405 taught by Professor Staff during the Fall '08 term at Minnesota.

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F04-Correlation - PubH 7405: REGRESSION ANALYSIS...

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