1
$\begingroup$

Consider the following y-values ($(0,y_0),(1,y_1),...$):

$$580,382,854,193,128,901,283,294,854,490$$

Plotting the linear regression gives the following formula:

$$y = 4.5x + 475.8$$

However, switching around the order of the y-values, like so:

$$580,382,854,854,128,901,283,294,193,490$$

Gives the following line of best fit:

$$y = -35.6x + 656.1$$

Why does the order matter when it comes to a line of best fit? The elements are the same and the algorithm I am using has no interaction between x- and y-variables:

sx = 0; sy = 0; stt = 0; sts = 0;
yArray = {}; //ten numbers from above
for (i = 0; i < 10; ++i) {
    sx = sx + i;
    sy = sy + yArray[i];
}
for (i = 0; i < 10; ++i) {
    t = i - (sx / 10);
    stt = stt + (t * t);
    sts = sts + (t * yArray[i]);
}
slope = sts / stt;
intercept = (sy - (sx * slope)) / 10;

There's nothing like sx + sy or sx * sy, etc. I just don't see where the order matters here.

$\endgroup$
  • 4
    $\begingroup$ Because you're choosing different points. Consider: The line through (0,0) and (1,1) is $y=x$; the line through (0,1) and (1,0) is $y=1-x$. $\endgroup$ – Christopher Carl Heckman Feb 18 '16 at 6:39
  • 1
    $\begingroup$ As for your programming, changing the order of the $y$ coordinates changes the value of sts. $\endgroup$ – Christopher Carl Heckman Feb 18 '16 at 6:56
  • $\begingroup$ You can't switch the $y$ values without switching the corresponding $x$ values. $\endgroup$ – Giovanni Resta Feb 18 '16 at 8:30
1
$\begingroup$

The algorithm you are using must somehow assess the association between x and y. Otherwise, it can't give you a regression line. (I believe @CarlHeckman has put his finger on it in his second Comment.)

By changing the order of the y's without making a corresponding change in the order of the x's, you are destroying the bivariate nature of the data. Consider the following fake data:

 Subject:  1  2  3  4  5
 x:        2  4  6  8 10
 y1:       0  1  2  3  4

Here the x's and y's give points on an ascending line. Their correlation is 1.

But if I mix the y-values around, I destroy the 'pairing' that traces back to the subjects. Now the plotted (x, y) points no longer lie in a straight line. Their correlation is $r = 0.6$.

 Subject:  1  2  3  4  5
 x:        2  4  6  8 10
 y2:       0  3  2  1  4

And if I put the y's in reverse order, then I get a line that goes the other direction, and the correlation is -1.

 Subject:  1  2  3  4  5
 x:        2  4  6  8 10
 y3:       4  3  2  1  0

Changing the correlation changes the slope of the regression line. Here are plots of the original and changed data.

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.