Relationship between dependent and independent variables

Question

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?

Answer 1

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.