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Math Help - Inverting a Regression doesn't yield symmetric results( !)

  1. #1
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    Inverting a Regression doesn't yield symmetric results( !)

    I have two vectors X and Y
    I produce one regression line y = mx + c regressing Y on X and another x = ny + d regressing X on Y
    Intuitively I would expect n = 1/m and d = -c/m but this is not the case
    The R^2s are exactly the same
    Without rigorous math can someone explain to me intuitively why this is so?
    Thanks, HB
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  2. #2
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    Re: Inverting a Regression doesn't yield symmetric results( !)

    I believe the problem is that (unless R^2 = 1) the lines are not y = mx + c and x = ny +d but y = mx + c + e1 and x = ny + d + e2, where the e are errors.
    If you have R you can see this this way
    x <- c(1,3,6,7,9)
    y <- 3*x + 2
    m1 <- lm(y~x)
    m2 <- lm(x~y)
    coef(m1)
    coef(m2)

    where a strict relationship holds. But add some error
    y2 <- 3*x + 2 + rnorm(5)
    and re-run
    m3 <- lm(y2~x)
    m4 <- lm(x~y2)

    and the relationship fails
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  3. #3
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    Re: Inverting a Regression doesn't yield symmetric results( !)

    Thanks a lot! - I did some work on this one and it turns out that the two necessary conditions are R^2 = 1 and sigma_x = sigma_y.
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