# Same variable on both sides of a regression model

I am sketching a regression model for examining the effects of multiple variables on the difference between a perceived value $$B_i$$ and predicted value $$\hat{B}_i$$ at any given timepoint $$i$$. The $$\hat{B}_i$$ values are predicted by a non-statistical model.

$$Y_i = \beta_0 + \beta_1(A_i) + \beta_2(\hat{B}_i) +\ ...$$

where $$Y_i = B_i - \hat{B}_i$$.

In particular, I am interested in examining the effect of $$A_i$$ on $$Y_i$$, but I want to control for the non-statistical model predictions ($$\hat{B}_i$$), which I suspect deviate systematically from the perceived values ($$B_i$$) in my dataset. However, is it problematic to have the $$\hat{B}_i$$ variable on both the LHS and RHS in this way?

Alternatively, I could model $$B_i$$ as the response variable and continue to include $$\hat{B}_i$$ as a fixed effect, but modeling the difference between the two values seemed more interpretable for the particular theory I am evaluating. But, perhaps modeling $$B_i$$ is preferable?

• It might help to have more specific information on the situation / variables here. In particular, what is meant by "perceived value"? Are these the true values of $B$, or is this some kind of subjective judgment subject to random error? Are you concerned about something changing as a function of the value of $\hat{B}$ (eg, such that a curvilinear function of $B$ would be appropriate)? What is your reluctance to just modeling $\hat{B}$? Sep 21, 2022 at 18:57
• Thank you for your clarification questions. The perceived values $B$ are subjective judgements. In the full model (not specified above), I include a random effect for the subjects/participants who provided the judgements.
– nts
Sep 21, 2022 at 19:54
• Regarding the second question, the non-statical model is a model of expectation. Our hypothesis is that perception ($B$) deviates more or less from expectation ($\hat{B}$) based on $A$. However, the non-statistical model itself is likely error-prone in terms of predicting expectation, resulting in $\hat{B}$ being greater on average than $B$ (i.e., negative $Y$ values). I thus wanted to control for the error in the non-statistical model and tease it apart from the effect of $A$.
– nts
Sep 21, 2022 at 19:55

Your model reduces tor typical linear regression

\begin{align} B_i - \hat{B_i} &= \beta_0 + \beta_1(A_i) + \beta_2(\hat{B}_i) +\ ... = \\ B_i &= \beta_0 + \beta_1(A_i) + (1+\beta_2)(\hat{B}_i) +\ ... \end{align}

Having the difference on the left-hand side doesn't make it more interpretable, as you can always re-arrange the terms back to the previous form.

• Thank you for your answer. Just to clarify: If I am interested in exploring what modulates the differences between $B_i$ and $\hat{B}_i$ while having $B_i$ as the response variable, the model should be $B_i = \beta_0 + \beta_1(A_i) + (1 + \beta_2)(\hat{B}_i) + ...$, not $B_i = \beta_0 + \beta_1(A_i) + \beta_2(\hat{B}_i) + ...$?
– nts
Sep 21, 2022 at 19:57
• @nts I don't understand your comment. What I'm saying is that mathematically your model is just the linear regression above. $\beta_2$ would be just $+1$ higher, so you can subtract $1$ to get the parameter from your proposed model.
– Tim
Sep 21, 2022 at 20:10