Consider a factorial design with two treatments A and B. Four possible cases are: administer none of them, administer A, administer B and administer both.
By introducing indicator variables $I_A$ and $I_B$ specifying whether treatment A or B, respectively has been administered, we define following linear model for the response variable:
$Y_i = \beta_0+\beta_1 I_{Ai}+\beta_2 I_{Bi}+\beta_3 I_{Ai}I_{Bi}+\epsilon_i$
where $i=1,...,N$, $\epsilon_i \sim N(0,\sigma^2)$, and $Cov(\epsilon_i,\epsilon^*_i)$. For simplicity, let's assume that each option was sampled $N/4$ times.
If $\beta_3=0$, meaning that there is no interaction term: define design matrix $X$. The maximum likelihood estimator can be found as $\boldsymbol{\hat{\beta}} = (X^T X)^{-1}X^TY$, and the variance matrix of the estimates is $Cov(\boldsymbol{\hat{\beta}}) = \sigma^2 (X^T X)^{-1}$. The treatment effect of none is $\beta_0$, A is $\beta_0+\beta_1$, B is $\beta_0+\beta_2$ and A&B is $\beta_0+\beta_1+\beta_2$. Using $Cov(\boldsymbol{\hat{\beta}})$, we can calculate the covariance between these linear combinations of the coefficients.
Can we make a similar analysis when $\beta_3 \neq 0$?