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We have a situation where we want to test the association between X and Y, but the change in X from baseline is more interpretable.

There are several possibilities I see for setting up the model. But I would appreciate some help in understanding the implications of each choice, like how the interpretations change.

In these 3 choices, we are modeling our outcome at all time points:

A1. $Y_{ij} = \beta_0 + \beta_1*X_{ij}$

Here we model Y and X, without any baseline adjustment

A2. $Y_{ij} = \beta_0 + \beta_1*(X_{ij}-X_{i0})$

Here we adjust for the change score of $(X_{ij}-X_{i0})$, without any baseline adjustment. At time 0, this change score will equal 0.

A3. $Y_{ij} = \beta_0 + \beta_1*X_{i0}+\beta_2*(X_{ij}-X_{i0})$

Here we adjust for the change score, but also for the baseline level of the covariate, X0.

Looking at the results from A1 and A3 models: $\beta_1 (A1) \approx \beta_2 (A3) \approx 2*\beta_1 (A2).$

But also, $\beta_1 (A3) \approx \beta_2 (A3)$, which I think corresponds to: $\beta_0 + \beta_1*X_{i0}$ at baseline, and $\beta_0 + (\beta_1-\beta_2)*X_{i0} + \beta_2*X_{ij} = \beta_0 + \beta_2*X_{ij}$ after baseline, so essentially, we are just in the A1 case.

It seems that there might be little difference in models A1 and A3, and that it would not actually be correct to interpret $\beta_2(A3)$ as the effect of the change in X on Y, right?

But would it be useful/correct to use an interaction with $(X_{ij}-X_{i0})$ (e.g. $(X_{ij}-X_{i0})*Z$) to test if post-baseline level of X differs across levels of Z? This seems analogous to a using an interaction with a spline, where we might expect things to change more or less after baseline.

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    $\begingroup$ I have a few relevant answers, stats.stackexchange.com/a/51587/1036 and stats.stackexchange.com/a/21781/1036. The second link I explain why I do not like the A3 specification and the first I explain how A2 is a restrictive model. Although I will need to think about the interaction question a bit more. $\endgroup$
    – Andy W
    Commented Oct 28, 2014 at 11:58
  • $\begingroup$ Thanks for your reply. I read through your second link, and I agree with your illustration/example. But when I actually ran this on some data, it looked like it forced the relationship to be such that $\beta_1 \approx \beta_2$. This seems that though we tried estimate some linear function of a covariate, we are essentially not changing anything. Thus, maybe the interpretation doesn't really change, as if you are at time 0, the effect is $\beta_1*X + 0$, and if you are at anytime after time 0, the effect is $0 + \beta_2*X$. $\endgroup$
    – rjweyant
    Commented Oct 29, 2014 at 13:15
  • $\begingroup$ But if we consider interactions, if you have interactions with both the baseline and the change score, I think you remain in a similar situation to what you listed in your 2nd link. But if you only have an interaction with the change score, you now have a term that is 0 at baseline, and not after. I'm struggling with how to interpret this, and if there is even a reason this should be done. $\endgroup$
    – rjweyant
    Commented Oct 29, 2014 at 13:25
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    $\begingroup$ Here is an illustration using R code of the point I make in the second link (in my original comment). If you write the equations in terms of expectations (e.g. for OLS) the relationships between the coefficients is exactly as I stated. Sorry I don't have the time (or mental energy!) to think about this problem at the moment. What I would do is start from writing the equations out in the levels (and include your interactions), and then see where your theory doesn't fit those equations in levels. $\endgroup$
    – Andy W
    Commented Oct 31, 2014 at 13:18
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    $\begingroup$ Then try to write out how would the change score better fits the relationship you are trying to model. The point in my prior questions was that the change scores on the right hand side are frequently equivalent to some specification of the levels. $\endgroup$
    – Andy W
    Commented Oct 31, 2014 at 13:21

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