Pivot Points in Bivariate Linear Regression

There are little-noticed points in the plane, which are artifacts of linear regression. The points, which are called pivot points, are the intersections of sets of regression lines. We derive the coordinates of the pivot point and ex-plain its sources. We show how a pivot point arises in a certain notable data set, which has been analyzed often for points of high leverage. We obtain the application of pivot points that shortens calculations when updating a set of bivariate observations by adding a new point.


Introduction
It is common to produce many lines to fit bivariate data as the observations are being altered in some way. For example, in order to determine a particular data point's influence on the best fit, the point may be moved by changing its ycoordinate and a new line created. Some diagnostic tests are based on this. A point, which is called the pivot point, is the intersection of certain lines that are often used for examining influence.
An example of a pivot point is presented in Section 2. In Section 3, we derive the coordinates of the pivot point. We show that a pivot point can be created in two ways. One way is augmenting an original set of bivariate observations with an additional point, which can have arbitrary multiplicity. Another way is altering an existing observation's y-coordinate as described above. Section 4 presents the benefit of the pivot point in that it can be useful to shorten calculations when adding a new observation.

Illustrative Example
linear regression, it is common to generate a sequence of regression lines. These lines fit the same set of observations, except that the y-coordinate is made to vary while its x-coordinate is unchanged for the specified data point of interest. The influence of Case 18 on the least-squares regression line is examined by keeping its x-coordinate of 42 and giving its y-coordinate the values 57, 77, 97, 117, and 137.
This produces the five regression lines in Figure 1. Clearly, Case 18 could have a large influence on the regression line. Some authors have illustrated and evaluated leverage in this way [8] [9] [10] [11]. All these regression lines pass though a common point, called the pivot point [12]. In Figure 1, the pivot point (12.3, 96.1) is shared by the five lines, and its location is indicated by the symbol ∆.

Derivation of the Pivot Point
We derive the formula for the coordinates of the pivot point. The pivot point can be created by augmenting an original set of bivariate observations with an additional point, which can have arbitrary multiplicity, which is another method to diagnose influence on the line [5] [13] [14] [15]. We show that formulation to be equivalent to varying the location of a single point, while keeping the same first coordinate, as is done in Figure 1.
which is on line (1) for S m . When m > 0 and u ≠ 0, the pivot point  (3) is called the pivot point of R with respect to S 0 , because P is on all regression lines for S m , which have different slopes. Because the y-coordinate v of R is absent from the coordinates of P, it is also called the pivot point of u with respect to S 0 . The set of regression lines that is created by adding copies of R, is called a pencil of lines or fan of lines through P.

Computational Shortcuts When Augmenting a Bivariate Set
The pivot point offers two shortcuts for computing equations for regression lines.
This is analogous to adding the n + 1 st value a to the data set { } whose mean is x . The new mean can be calculated using ( ) ( ) 1 nx a n + + , which requires considerably less computation than not using x [11].
One shortcut is, given set S 0 , the regression line for S m can be computed as the line containing the point of means (2) and the pivot point (3). Recall that in (4) Equations (5) are seen by substituting a 0 and b 0 from (1), a ∞ and b ∞ from (4), and w from (6) into the right-hand sides of (5), which yields a m and b m in (1).

Conclusion
Pivot points are omnipresent in applications of bivariate linear regression. In particular, they are points through which new lines pass when a data point is altered. One important purpose of altering a point is to determine its influence.
We have displayed this phenomenon with the well-known data set of ages at first word versus Gesell scores, which has been analyzed by many authors from many points of view. A pivot point is a handy and efficient tool for shortening calculations when new data arises.