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(gx)>n. Division of both sides of the first normal equation in (4) y # a bx # y # (7) bx. This is equivalent to writing #
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If desired, we can first find b from (5) or (6) and then use (7) to find a the least-squares line as y y # b(x x) # or y y # a (x a (x
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The result (8) shows that the constant b, which is the slope of the line (3), is the fundamental constant in determining the line. From (8) it is also seen that the least-squares line passes through the point (x, y), which is called # # the centroid or center of gravity of the data. The slope b of the regression line is independent of the origin of coordinates. This means that if we make the transformation (often called a translation of axes) given by x xr h y yr k (9)
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where h and k are any constants, then b is also given by n a xryr b n a xr2 Q a xrR Q a yrR Q a xrR
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(10)
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CHAPTER 8 Curve Fitting, Regression, and Correlation
where x, y have simply been replaced by xr, yr [for this reason we say that b is invariant under the transformation (9)]. It should be noted, however, that a, which determines the intercept on the x axis, does depend on the origin (and so is not invariant). In the particular case where h x, k y, (10) simplifies to # # b a xryr 2 a xr (11)
The results (10) or (11) are often useful in simplifying the labor involved in obtaining the least-squares line. The above remarks also hold for the regression line of x on y. The results are formally obtained by simply interchanging x and y. For example, the least-squares regression line of x on y is x x # a (x a( y x)( y # y)2 # y) # (y y) # (12)
It should be noted that in general (12) is not the same line as (8).
The Least-Squares Line in Terms of Sample Variances and Covariance
The sample variances and covariance of x and y are given by s2 x a (x n x)2 # , s2 y a( y n y)2 # , sxy a (x x)( y # n y) # (13)
In terms of these, the least-squares regression lines of y on x and of x on y can be written, respectively, as y y # sxy s2 x (x x) # and x x # sxy s2 y (y y) # (14)
if we formally define the sample correlation coefficient by [compare (54), page 82] r then (14) can be written y sy y # x x ra s #b x and x sx x # y y # ra s b y (16) sxy sxsy (15)
In view of the fact that (x x)>sx and (y y)>sy are standardized sample values or standard scores, the # # results in (16) provide a very simple way of remembering the regression lines. It is clear that the two lines in (16) 1, in which case all sample points lie on a line [this will be shown in (26)] and there are different unless r is perfect linear correlation and regression. It is also of interest to note that if the two regression lines (16) are written as y a bx, x c dy, respectively, then bd r2 (17)
Up to now we have not considered the precise significance of the correlation coefficient but have only defined it formally in terms of the variances and covariance. On page 270, the significance will be given.
The Least-Squares Parabola
The above ideas are easily extended. For example, the least-squares parabola that fits a set of sample points is given by y a bx cx2 (18)
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