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We are examining our understanding of interaction terms in categorical variables. We have a three category variable, and the model is:

$y=\alpha+\beta x+\beta_0x_0+\beta_1x_1+\beta_2xx_0+\beta_3xx_1$

Here $x$ is a continuous (almost) variable, and $x_i$ indicates true (1) if the variable belongs to category $C_i$, and false (0) otherwise. Note $C_{0,1,2}$ are exclusive, such that for each data point $x_1\neq x_2$. In this model $C_2$ is a reference category.

We compare this to three regressions:

$y=A_i+B_i x$

For each of our categories $C_i$ - each regression is run for $\{x|x\in C_i\}$.

It seemed to us, given the nature of our categories, that we should get exactly:

  1. $A_0=\alpha+\beta_0$, $B_0=\beta+\beta_2$
  2. $A_1=\alpha+\beta_1$, $B_1=\beta+\beta_4$
  3. $A_2=\alpha$,$B_2=\beta$

We are using Stata if that matters, but it seems results are not the same. They are consistent within the uncertainties, but we expected an exact match as for each set $\{x|x\in C_i\}$, the interaction model is the same as the corresponding indivdual regression. I have neglected uncertainty terms in the model, but we are not sure they are comparable.

Why do the two differ?

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  • $\begingroup$ This notation is unconventional and very confusing, at least to me. $\endgroup$ – Peter Flom Oct 4 '18 at 18:15
  • $\begingroup$ @PeterFlom No problem - what notation do you prefer? $\endgroup$ – kabanus Oct 4 '18 at 18:19
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    $\begingroup$ @PeterFlom Do you prefer I use $\beta_0$...$\beta_4$? I am not sure what is confusing you, I just wanted different names for the slope coefficients, and for the individual regression slope coefficients. Do you prefer $m$ and $n$ for the regular regression coefficients? $\endgroup$ – kabanus Oct 4 '18 at 18:22
  • $\begingroup$ Yes. The usual way of writing it is $y = \beta_0 + \beta_1x_1 + \dots \beta_px_p$ When you have interactions you have to add e.g. $\beta_{p+1}x_1x_2$ Also, all the x's should have subscripts. $\endgroup$ – Peter Flom Oct 4 '18 at 18:53
  • $\begingroup$ @PeterFlom Just edited - do not hesitate to let me know if this is still unclear. Note there are interaction terms of the form $xx_i$ as well, I thought indexing the slopes with both interacting variables was clearer, but I stand corrected. $\endgroup$ – kabanus Oct 4 '18 at 18:54

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