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I have a dependent variable $Y$, an independent variable $X$, and a categorical independent variable T (which can take 2 levels). Now I have 4 models:

MI: different slopes, different intercepts $$y_{i1}=\alpha_1+\beta_1x_{i1}, y_{i2}=\alpha_2+\beta_2x_{i2}; \text{or}\; y_{ij}=\beta_0+\beta_1X_{ij}+\beta_2T_i+\beta_3X_{ij}T_j$$

MII: same slope, different intercepts $$y_{i1}=\alpha_1+\beta x_{i1}, y_{i2}=\alpha_2+\beta x_{i2}; \text{or}\; y_{ij}=\beta_0+\beta_1X_{ij}+\beta_2T_j$$

MIII: same intercept, different slopes $$y_{i1}=\alpha+\beta_1x_{i1}, y_{i2}=\alpha+\beta_2x_{i2}$$

MIV: one regression line for both $$y_{ij}=\alpha+\beta x_{ij}; \text{or}\; y_{ij}=\beta_0+\beta_1X_i$$

Now I am trying to compare these models one against the other. I am not sure, if I am right with what I think:
Comparing MI to MII tests $H_0:\, \beta_1=\beta_2$ (this I do know), which means that $H_0$: same slope against $H_a$: different slopes (which I am not sure of). If I compare these in R, I get a p-value. Now what I think is that, if the p-value < alpha level, I can reject $H_0$, which would mean I should stick to the first model. If it is a large value, I cannot reject the null hypotheses, which means I should stick with the second model (as there I have the same slope). I am not sure if I am right with that.
Plus I cannot write down the second equation for the model third model, can someone help out?

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MIII could be written $y_{ij} = \beta_0 + \beta_1X_{ij} + \beta_2X_{ij}T_{ij} $

And then comparing MI to MIII tests whether $\beta_2$ in MI is significantly different from 0.

However, I wouldn't compare models against each other this way. I would use AIC or BIC (I don't have a strong preference between them). Better yet, I'd use substantive knowledge to figure out which model is most sensible.

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