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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?

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  • $\begingroup$ 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 $\hat{B}$? $\endgroup$ Sep 21 at 18:57
  • $\begingroup$ 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. $\endgroup$
    – nts
    Sep 21 at 19:54
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    $\begingroup$ 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 $A$. $\endgroup$
    – nts
    Sep 21 at 19:55

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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.

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  • $\begingroup$ 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 $B_i = \beta_0 + \beta_1(A_i) + \beta_2(\hat{B}_i) + ...$? $\endgroup$
    – nts
    Sep 21 at 19:57
  • $\begingroup$ @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. $\endgroup$
    – Tim
    Sep 21 at 20:10

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