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I have the following model: $$ |X| = B_0 +B_1 \cdot y + B_2 \cdot z , $$ where $z$ and $y$ are normally distributed random variables, and $B_1$ and $B_2$ denote the coefficients.

My dependent variable contains either the positive or negative values of $X$. The positive and negative values of $X$ together are normally distributed, with a mean and median of around zero. For statistical reasons I would like to separately test the positive and negative values. This splits the left and right side of the otherwise normally distributed variable $X$. This clearly violates the normality assumption used for linear regression models. If I want to compare groups I would use a nonparametric type of test, but in this case I want to run the equivalent of a linear regression. Is this possible?

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  • $\begingroup$ The normal distribution is symmetric, so you only need a solution for one case. $\endgroup$ – emcor Jun 25 '14 at 22:56
  • $\begingroup$ I presume you mean with one case it is either positive or negative. $\endgroup$ – Mr. Radical Jun 25 '14 at 23:00
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    $\begingroup$ It is difficult, if not impossible, to formulate a legitimate statistical model based on this information. Could you please explain the context of this question and the basis for your suppositions about $X$? $\endgroup$ – whuber Jun 26 '14 at 1:42
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    $\begingroup$ "For statistical reasons I would like to separately test the positive and negative values" - what are these reasons? I'm afraid you are trying to formulate the problem that you dont need to solve $\endgroup$ – Aksakal Oct 13 '15 at 15:10
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    $\begingroup$ Is it possible to restate the model? As it stands it doesn't make a lot of sense since the left side is nonnegative but the right side can be negative. $\endgroup$ – dsaxton Mar 1 '16 at 13:57
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I think you can run a regression with $X_+$ and $X_-$ as independents, long as you assume a linear relationship to $Y$.

The "normality assumption" for linear regression is only regarding the error term assumed normally distributed around the true value.

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  • $\begingroup$ I have tried this, but unfortunately I got results that were insignificant. $\endgroup$ – Mr. Radical Jun 25 '14 at 23:21
  • $\begingroup$ It may be the case, that linear regression is the wrong model then. Linear regression assumes a purely linear relationship. You may think of just plotting X vs. Y, to see if their relationship is linear. $\endgroup$ – emcor Jun 25 '14 at 23:27
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    $\begingroup$ That doesn't necessarily mean the results are wrong or that the wrong technique was used. $\endgroup$ – Peter Flom - Reinstate Monica Jun 25 '14 at 23:27
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    $\begingroup$ By $X_+$ you mean a dummy variable for when $X$ is positive correct? Given the OP's comments, I would suggest a paramterization that includes the interactions with one dummy (you can't include both as the are collinear unless you do not include the intercept). $\endgroup$ – Andy W Jun 25 '14 at 23:31
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    $\begingroup$ The relationship between X and y has been established before in prior research and is a linear relation. With regard to z there is also evidence of a linear relationship. I was wondering if the problem was with the separation of positive and negative values. $\endgroup$ – Mr. Radical Jun 25 '14 at 23:35

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