Interaction terms

You can introduce dummy variable is not just intercept shifter, but as interacted with X variables.

For example, We have program that are randomly assigned to participant. We have female and male participants. You might be interested in knowing whether treatment affects men differently than women. To do that you need to run regression with interaction terms. Esentially interaction terms, are just multiplication of two dummy variables.

\[Y_i = \alpha + \beta D_i + \gamma M_i + \delta M_i*D_i + \epsilon_i (1)\]

Let assume that D is the dummy variable 1 if the patient recived vaccination shot, 0 if not. Y is the probability of the person catching coronovirus after one month. M is the gender dummy variable 1 if person is male, and 0 if the person is female.

Gender	Vaccination	Formula
Male	Not vaccinated	$\alpha + \gamma$
Male	Vaccinated	$\alpha + \beta + \gamma + \delta$
Female	Not vaccinated	$\alpha$
Female	Vaccinated	$\alpha + \beta$

Coefficient	Interpretation
$\alpha$	Probability of not vaccinated average female to contract coronovirus after one month
$\gamma$	Difference in probabilities of contracting coronovirus after one month for not vaccinated males versus females
$\beta$	Difference in probabilities of contracting coronovirus after one month for vacinated females versus unvaccinated females
$\delta$	Difference in difference between vaccinated males and unvaccinated males versus difference between vaccinated females and unvaccinated females.

tip $$\delta$$ is here esentially is Difference-in-Difference estimator

Gender	Vaccination	Formula
Male	Not vaccinated	\(\alpha + \gamma\)
Male	Vaccinated	\(\alpha + \beta + \gamma + \delta\)
Female	Not vaccinated	\(\alpha\)
Female	Vaccinated	\(\alpha + \beta\)

Coefficient	Interpretation
\(\alpha\)	Probability of not vaccinated average female to contract coronovirus after one month
\(\gamma\)	Difference in probabilities of contracting coronovirus after one month for not vaccinated males versus females
\(\beta\)	Difference in probabilities of contracting coronovirus after one month for vacinated females versus unvaccinated females
\(\delta\)	Difference in difference between vaccinated males and unvaccinated males versus difference between vaccinated females and unvaccinated females.