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I tried to develop an empirical equation by using multiple regression analysis. In my case I use aerosol as dependent variables and relative humidity and winds components ($U$ and $V$) as independent variables. Before developing an equation, I have converted aerosol to dry form by: $$ {\rm aerosol}_{dry}= {\rm aerosol}(1-RH) $$ The final equation is something like: $$ {\rm aerosol}_{dry} = {\rm intercept} - x_1 RH + x_2 U + x_3 V $$

where, $x_1$, $x_2$, and $x_3$ are regression coefficients. Now the problem is I have used relative humidity in independent and dependent variable (indirectly). Are there any ways to justify this situation where we have used RH in both equations?

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  • $\begingroup$ What is RH? What is U? Both of them are relative humidity? $\endgroup$ – user158565 Oct 16 '18 at 16:42
  • $\begingroup$ @a_statistican sorry for not making question clear...actually RH is the relative humidity, U & V are the wind vector components representing for horizontal and vertical movement. In the equation wind components act as source factor and RH act as removing factor. $\endgroup$ – muna sapkota Oct 16 '18 at 17:01
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    $\begingroup$ Possible duplicate of Regressing a ratio on a component of the ratio $\endgroup$ – Firebug Oct 16 '18 at 17:10
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    $\begingroup$ I don't think this is a close enough duplicate to merge. $\endgroup$ – Peter Flom Oct 17 '18 at 12:51
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Sometimes I met this situation: $$ Y-X_1 = \alpha + \beta_1X_1 + ...+ \epsilon$$ It is OK because it is equivalent to $$ Y = \alpha + (1+\beta_1)X_1 + ...+ \epsilon$$ But your situation is not so simple. Let $Y$ be aerosol_dry, $X_1$ be RH, and $Z$ be aerosol. Then we have $$Y = Z + ZX_1$$ Then your model is: $$ Z + ZX_1 = \alpha +\beta_1 X_1 + ... $$ It is hard to explain this model. So my suggestion is: 1) If aerosol_dry is important, then fit a model without RH. 2) If the relationship between aerosol and RH is important, use aerosol as dependent variable directly.

Of course, after model fitting, you need to check if there are obvious evidence of violating the assumptions.

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  • $\begingroup$ dear @a_statistician thank you for giving this detail explanation. As you suggested, I have tried without RH to model dry aerosol but without it, performance is very low (i.e around r2 of 0.2). After including r2 is something like 0.65. simillarly i also tried to model aerosol with RH, U and V, in this case also it did not work. Yes its really hard to give justification :( $\endgroup$ – muna sapkota Oct 16 '18 at 17:38
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    $\begingroup$ @muna Clearly $R^2 = 0.65$ is created by RH appearing in the both sides of the equation. So I think you need to accept the fact that relative humidity and winds components have no meaningful linear relationship with aerosol. Maybe you can draw some scatter plots to see is there any nonlinear relationship. $\endgroup$ – user158565 Oct 16 '18 at 17:44
  • $\begingroup$ yes you are right that highR2 is because of RH in both sides. I have already checked and relationship with different independent variables and found that aerosol has a very poor relationship with all independent variables. But interestingly after converting into dry forms correlation with U and V also increased. So I was thine king if we could justify physically ..but I am not sure...maybe I have to accept the fact that as you said :( $\endgroup$ – muna sapkota Oct 16 '18 at 17:53

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