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which for linear regression are equivalent. The formula (29) is often referred to as the product-moment formula for linear correlation. Formulas equivalent to those above, which are often used in practice, are n a xy r B and Sn a r x2 Q a xR Q a yR
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CHAPTER 8 Curve Fitting, Regression, and Correlation
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If we use the transformation (9), page 267, we find n a xryr r B Sn a xr2 Q a xrR Q a yrR
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Q a yrR T x, k # y, (33) becomes # (34)
(33)
which shows that r is invariant under a translation of axes. In particular, if h r B a xryr Qa xr2 R Q a yr2 R
which is often useful in computation. The linear correlation coefficient may be positive or negative. If r is positive, y tends to increase with x (the slope of the least-squares line is positive) while if r is negative, y tends to decrease with x (the slope is negative). The sign is automatically taken into account if we use the result (29), (31), (32), (33), or (34). However, if we use (30) to obtain r, we must apply the proper sign.
Generalized Correlation Coefficient
The definition (29) [or any of the equivalent forms (31) through (34)] for the correlation coefficient involves only sample values x, y. Consequently, it yields the same number for all forms of regression curves and is useless as a measure of fit, except in the case of linear regression, where it happens to coincide with (30). However, the latter definition, i.e., r2 explained variation total variation a ( yest a( y y)2 # y)2 # (35)
does reflect the form of the regression curve (via the yest) and so is suitable as the definition of a generalized correlation coefficient r. We use (35) to obtain nonlinear correlation coefficients (which measure how well a nonlinear regression curve fits the data) or, by appropriate generalization, multiple correlation coefficients. The connection (25) between the correlation coefficient and the standard error of estimate holds as well for nonlinear correlation. Since a correlation coefficient merely measures how well a given regression curve (or surface) fits sample data, it is clearly senseless to use a linear correlation coefficient where the data are nonlinear. Suppose, however, that one does apply (29) to nonlinear data and obtains a value that is numerically considerably less than 1. Then the conclusion to be drawn is not that there is little correlation (a conclusion sometimes reached by those unfamiliar with the fundamentals of correlation theory) but that there is little linear correlation. There may in fact be a large nonlinear correlation.
Rank Correlation
Instead of using precise sample values, or when precision is unattainable, the data may be ranked in order of size, importance, etc., using the numbers 1, 2, . . . , n. If two corresponding sets of values x and y are ranked in such manner, the coefficient of rank correlation, denoted by rrank, or briefly r, is given by (see Problem 8.36) rrank where d n 1 6 a d2 n(n2 1) (36)
differences between ranks of corresponding x and y number of pairs of values (x, y) in the data
The quantity rrank in (36) is known as Spearman s rank correlation coefficient.
CHAPTER 8 Curve Fitting, Regression, and Correlation
Probability Interpretation of Regression
A scatter diagram, such as that in Fig. 8-1, is a graphical representation of data points for a particular sample. By choosing a different sample, or enlarging the original one, a somewhat different scatter diagram would in general be obtained. Each scatter diagram would then result in a different regression line or curve, although we would expect that these would not differ significantly from each other if the samples are drawn from the same population. From the concept of curve fitting in samples, we are led to curve fitting for the population from which samples are drawn. The scatter of points around a regression line or curve indicates that for a particular value of x, there are actually various values of y distributed about the line or curve. This idea of distribution leads us naturally to the realization that there is a connection between curve fitting and probability. The connection is supplied by introducing the random variables X and Y, which can take on the various sample values x and y, respectively. For example, X and Y may represent heights and weights of adult males in a population from which samples are drawn. It is then assumed that X and Y have a joint probability function or density function, f(x, y), according to whether they are considered discrete or continuous. Given the joint density function or probability function, f (x, y), of two random variables X and Y, it is natural from the above remarks to ask whether there is a function g(X) such that E5[Y A curve with equation y the following theorem: g(X)]26 a minimum (37)
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