# Change Score or Regressor Variable Method - Should I regress $Y_1$ over $X$ and $Y_0$ or $(Y_1-Y_0)$ over $X$

I have data about investment preferences 1 year before the Covid and during the Covid lockdown.

Some changes appear using simple T-Test. I want to be able to assess if these changes are particularly strong for some specific demographics (e.g., older individuals ($$X_1$$), individuals with lower income ($$X_2$$), etc...).

Should I use the initial level of my dependant variable in the regressions? Basically, if I want to use OLS regressions to investigate which independant variable correlate with the change in my dependant variable, which model is preferrable?

Model 1 (apparently called Change Score Method): $$(Y_2-Y_1)= \beta_1 . X_1+ \beta_2 . X_2$$

Model 2 (apparently called Regressor Variable Method) Score Method): $$Y_2= \beta_1 . X_1+ \beta_2 . X_2 + \beta_3 . Y_1$$

Thank you so much for your help - Any reference would also be much appreciated!

Both methods have been used. See here for example. It depends what question you want to answer. If you want to talk mostly about "change" you can use

(Y2-Y1) ~ X1 + X2            # (1)


Basal (Y1) should not be added to above equation as it will always be correlated with difference (Y2-Y1) - see comments below by @EdM and here.

On the other hand, if you want to discuss factors affecting "final value", you can use

Y2 ~ X1 + X2 + Y1            # (2)


However, since repeated measurements (Y1,Y2 at 2 times) have been done on same subject, hence mixed model is also often used. (including interactions as commented by @dbwilson below):

Y ~ X1 + X2 + time + X1*time + X2*time + (1|subject)


Following simplified version of formula is effectively same as above:

Y ~ X1*time + X2*time + (1|subject)            # (3)


There is another method commonly used, especially in biomedical literature: "Percent change", i.e.

(100*(Y2-Y1)/Y1) ~ X1 + X2            # (4)


It is not correct to keep Y1 as a predictor variable in this last method as there will be strong correlation between baseline and percent change.

I think this last method (percent change) is most understandable.