Modelling interaction How does adding interaction term in the model adjust for it or why do we need to add interaction?  I am working on logistic regression model with treatment and race as predictors.  I have added interaction of treatment and race in the model. Are we accounting for variability caused by the interaction term here?  I will be grateful to intuitive explanation.
 A: Suppose you have two treatment levels (Treatment and Control) and five race levels (A, B, C, D, and E), without interaction, the model would look like:
\begin{equation}
y = \beta_0 + \beta_1 Trt + \beta_2 B+ \beta_3 C+ \beta_4 D+ \beta_5 E
\end{equation}
This model implies that treatment and race are independently related to the outcome, y. This means race A on average would have a mean difference of $\beta_1$ between the treatment and control groups, and same for B, C, D, and E. Alternately, regardless if you're in treatment or not, the difference between the outcome is always $\beta_2$ for race B when compared to race A, $\beta_3$ for race C when compared to race A, so on so forth.
Graphically, if you connect to tips of a bar plot showing y by treatment and race, the 2 lines by treatment status will be parallel across races, and the 5 lines by race will be parallel across treatment statuses.
This is not often the case. In fact, sometimes some races may respond to the treatment better/worse than the others. In those cases, we suspect an interaction, and will change the model to:
\begin{equation}
\begin{aligned}
y = & \beta_0 + \beta_1 Trt + \beta_2 B+ \beta_3 C+ \beta_4 D+ \beta_5 E+ \\
    & \beta_6 B\times Trt+ \beta_7C\times Trt+ \beta_8D\times Trt+ \beta_9E\times Trt
\end{aligned}
\end{equation}
If you count the $\beta$s we now have 10 estimates, corresponding to 2 treatments by 5 races = 10 group means. By fitting interaction we lift the limitation of being an independent predictor, and now the effect of treatment can vary depending on race, and vice versa.

As for why do we need to add interaction, it is to investigate if the independent variables are modifying the effect of each other. Practically, this investigation can help isolating different subgroup behaviors. For example, suppose a hypertensive drug will help people who are not overweight but will harm people who are overweight. If the trial consists of a sample that is 50% overweight, the overall results will say that the drug is largely non-efficacious (positive results from the non-overweight got washed out by the negative results from the overweight). However, once a treatment by weight status interaction is accounted for, the drug will actually appear beneficial to some of the people.
