On Thu, 23 Oct 2003, Rich Ulrich replied to Kevin Lausten, who wrote: > > I am having difficulty understanding how regressions perform. I > > have 2 vectors (x = [4, 4, 0], y = [3, 6, 3]) and want to map y to > > x. When this is done using a regression formula, I get a = 3 and b > > = .375. Now, when I map x to y I get a = 0 and b = 2/3. Why is the > > second result not the reflective inverse of the first?(about the > > line y = x)?
Because the criteria are different. In a regression analysis to predict Y from X, the errors of prediction are measured as (Y-observed minus Y-predicted) , and the sum of squares of these errors is minimized. When predicting X from Y, the errors of prediction are (X-observed minus X-predicted) , and the sum of squares of _these_ errors is minimized. Both regression lines pass through the center of gravity of the data (that is, through the point (mean of X, mean of Y) when the data are displayed in a scatterplot of Y vs. X), but that is the only point they have in common (unless the plot is linear with no error, that is, when r = +1 (or -1) exactly). For your data, the regression line y^ = a + b*x predicting Y from X passes through the points (X,Y) = (0,3) [which is a datum] and (4,4.5) [which is the mean of the data (4,3) and (4,6)]. The regression line x^ = a + b*y predicting X from Y passes through the (Y,X) points (3,2) [which is the mean of the data (3,0) and (3,4)] and (6,4) [which is a datum]. The two lines cross where X = 2 + 2/3 and Y = 3. > You are implying that a correlation of zero, which would > be graphed by a flat, horizontal line, should be reflected > to make a regression line that goes straight up. > > The regression slope, each time, is equal to the > correlation (r) times the ratio of the standard deviations. > As far as there is reflection... you can see that the > second is reflected about the line with slope r . ----------------------------------------------------------------------- Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
