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A horizontal regression is defined as the following:

$$m=\frac{\sum_{i=1}^n (x_i-\operatorname{average(x)})(y_i-\operatorname{average(y))}}{\sum_{i=1}^n (x_i-\operatorname{average(x)})^2}$$

whereas a vertical regression is defined as

$$m=\frac{\sum_{i=1}^n (y_i-\operatorname{average(y)})^2}{\sum_{i=1}^n (x_i-\operatorname{average(x))}(y_i-\operatorname{average(y))}}$$

In several math-books it says that you use the horizontal regression if you want to calculate the x values to given y values and the vertical regression formula if you want to find the corresponding y values to given x values.

But how can it be that the function I get with the vertical regression formula is more accurate than the horizontal one for x on y values?

For example:

$ x := \{1,2,3,4,5,6,7,8\}$

$ y := \{0.3,0.5,0.7,1,1,1.5,1.6,2.1\}$

That gives me the vertical function: $f(x)=0.24404*x-0.0107$ and the horizontal function: $f(x)=0.25256*x-0.04902$

If I calculate the least-squares-sum (x on y):

$$\sum_{i=0}^7 (x_i-f(y_i))^2$$

I get 181.33... for the vertical one but 183.14... for the horizontal one.

Why is in general the horizontal regression associated with "x on y" if the vertical one obviously be more accurate?

Thanks a lot in advance!

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I think it must be minimized $\sum_{i=0}^7 (y_i-f(x_i))^2$, not $\sum_{i=0}^7 (x_i-f(x_i))^2$, doesn't it? –  H. Kabayakawa Jun 2 '12 at 15:12
$\sum_{i=0}^7 (y_i-f(x_i))^2$ tells me how good the y-values correspond; but I want to know how good the x-values correspond. So I think my formula should be alright... –  libjup Jun 2 '12 at 15:16
Now I see what you meant. You are (half) right... it should be $\sum_{i=0}^7 (x_i-f(y_i))^2$ since I get x-values from my function... –  libjup Jun 2 '12 at 15:24
still that doesn't solve my problem... it was just a typo –  libjup Jun 2 '12 at 15:26
The whole point of the horizontal regression is that it minimizes the $x$-on-$y$ residual, so I'd recommend checking your work and seeing if you got all your numbers right. –  Gerry Myerson Jun 3 '12 at 11:02
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2 Answers

I see the problem. You've computed linear fits of the form $y=f(x)$, and then you're comparing $x$ with $f(y)$! But of course, if $y=f(x)$, then $x$ is $f^{-1}(y)$, not $f(y)$. This is why your sum of squared residuals are so large for such small inputs. If you want to compare residuals in $x$, you need to compute $\sum \left(x_i - f^{-1}(y_i)\right)^2$ instead.

The inverse functions of the "vertical" and "horizontal" regressions are $$x \approx f^{-1}(y)=4.09756y+0.0439024$$ and $$x \approx f^{-1}(y)=3.95944y+0.194109$$ respectively. The respective sums of squared residuals in $x$ are $1.46513$ and $1.41574$. As you can see, the horizontal regression does better.

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The idea is that one variable (usually $x$) is a value you input-maybe the temperature or concentration of something-which you know accurately. The other one (usually $y$) is less well known as it is something you measure with errors. The fit then assumes the $x$ values are exact and minimizes the sum of the squares of the errors in $y$. If your data fit a straight line well, the difference will be small. If not, it will be large. This is the origin of the correlation coefficient equation.

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Sure, but if you get the variables mixed up and try to do a regression of $x$ on $y$ instead, the line you get should still give you a smaller $x$ on $y$ residual than the line you would have gotten if you had done it the right way, right? And I think OP is claiming to have a counterexample, and that shouldn't happen. –  Gerry Myerson Jun 3 '12 at 11:20
@Ross: It seems like you didn't get me right: I minimized the residual for the x-values by using the horizontal regression formula. Still my vertical-regression formula gives me a more accurate line than I get with the horionztal one. ( $\sum_{i=0}^7(x_i-f(y_i))^2$ is smaller with the vertical than with the horizontal function) –  libjup Jun 3 '12 at 14:50
@libjup: you say the error is always smaller when minimizing the horizontal residuals, but I don't see why that should be the case. For your data the variation in $y$ is much smaller than the variation in $x$-that may have something to do with it. Also the difference is pretty small. –  Ross Millikan Jun 3 '12 at 19:54
Then there is the regression that minimizes the sum of the squares of the distances to the regression line itself. This is independent of the coordinate system, and can be done with a few additional statistics. –  marty cohen Jul 3 '12 at 16:02
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