The regression form of the model for a factor $\tau$ with three levels and factor $\alpha$ with two levels is $Y_i = \beta_0 + \beta_1 \tau_{i,2} + \beta_2 \tau_{i,3} + \beta_3 \alpha_{i,2} + \epsilon_i$ Note that the first level for each factor has been dropped to create a dummy variable. Each $\tau_{i}$ and $\alpha_i$ is an indicator variable with value $1$ if observation $i$ is a member of the factor level. We can interpret the results of the regression as follows: | | $\tau_1$ | $\tau_2$ | $\tau_3$ | | ---------- | ------------------------------- | ----------------------------------------- | ----------------------------------------- | | $\alpha_1$ | $\mu_{1,1} = \beta_0$ | $\mu_{1,2} = \beta_0 + \beta_1$ | $\mu_{1,3} = \beta_0 + \beta_2$ | | $\alpha_2$ | $\mu_{2,1} = \beta_0 + \beta_3$ | $\mu_{2,2} = \beta_0 + \beta_1 + \beta_3$ | $\mu_{2,3} = \beta_0 + \beta_2 + \beta_3$ | This is derived directly from setting the indicator variable for each level of each factor to either $1$ or $0$ in the regression form of the model. We can interpret the intercept term $\beta_0$ as the average value of the response at the first level of each factor. Moving from level to level in each factor, the mean is given by the table above. # two-way ANOVA as linear regression model with interaction term The two-way ANOVA with interaction term has the form $Y_{ijk} = \mu + \tau_{j} + \alpha_{k} + (\tau \alpha)_{jk} + \epsilon_ijk$ where $(\tau \alpha)_{jk}$ is the interaction term. The model has no clear interpretation for either factor, but rather can only be interpreted within each level of the alternate factor.