Joint Linear Regression hypothesis test

Suppose I have the following joint model:

$$y_{f_i} = \beta_1 + \beta_2 t_i + \beta_3 s_i +\epsilon_i$$

$$y_{m_i} = \beta_4 + \beta_5t_i + \beta_6 s_i +\epsilon_i$$

where $$y_{f_i}$$ corresponds to the female portion of the response variable, and $$y_{m_i}$$ corresponds to the male portion, and $$t_i,s_i$$ are some predictors.

I want to do a hypothesis test such as

$$H_0: \beta_2 = \beta_5 =0 \quad \text{and}\quad\beta_3 = \beta_6,\quad \text{against} \quad \text{not}\quad H_0$$

I would like to use the F test, but I don't understand how I would deal with the fact that we are using a joint model. Any help is appreciated.

• Why you dont consider one model with dummy variables? Commented Mar 11, 2019 at 20:20
• Since no one replied for a while, that's what I did in the end - was just wondering if that's the only way Commented Mar 11, 2019 at 20:31
• IMHO, it is the correct way. You can improvise something, but I wouldn't recommend it Commented Mar 11, 2019 at 20:35
• In order to compare between two different models they have to be on the same data set and preferably nested. Commented Mar 12, 2019 at 8:03

The proper way to do it is by estimating $$y_i = \beta_0 + \beta_1t_i + \beta_2D_i + \beta_3D_is_i + \beta_4s_i + \epsilon_i$$ where $$D_i=1$$ if $$i$$th patient is a male. This is your full model. The restricted model will be $$y_i = \beta_0 + \beta_2D_i + \beta_3s_i + \epsilon.$$ Now you can take the $$R^2$$ of each model and compute the partial $$F$$-test.

In order to combine both regression equations, you can introduce a dummy variable $$g_i$$ into your data set. If the $$i^\text{th}$$ observation is belonging to a male person the dummy variable will be equal to $$1$$. The dummy variable will vanish if the $$i^\text{th}$$ observation is belonging to a female person.

The regression will be

$$y_i = \beta_1 + \alpha_1g_i + \beta_2t_i + \alpha_2g_it_i + \beta_3s_i+\alpha_3g_is_i+\varepsilon_i$$

The relationship $$\beta_1+\alpha_1=\beta_4$$, $$\beta_2+\alpha_2=\beta_5$$ and $$\beta_3+\alpha_3=\beta_6$$, can be derived if you consider $$g_i=1$$.

You wanted to test $$\beta_2=\beta_5=0$$ and $$\beta_3=\beta_6$$. Looking at the previous equations you should see that, we need to show the following conditions for the combined regression.

$$\beta_2=0 \quad \wedge \quad \alpha_2 = 0$$ $$\alpha_3=0$$

When I write that something needs to equal to $$0$$, I mean that the confidence interval must include zero.