If I understood it correctly, while doing ANOVA to determine which factors (and interactions between factors) have effects on the output variable ($Y$), we can express their influence by a linear model (e.g. explained here). Then if we have two factors $A$ and $B$, this model is given by: $$ Y_{ijk} = \alpha_i + \beta_j + (\alpha\beta)_{ij} + \varepsilon_{ijk}$$
$$\text{where }\begin{cases} \alpha_i &= \text{effect of level } i \text{ of factor } A\\ \beta_i &= \text{effect of level } j \text{ of factor } B\\ (\alpha\beta)_{ij} &= \text{effect of the interaction between } A_i \text{ and } B_j\\ \varepsilon_{ijk}&=\text{residual of this model for observation } k \end{cases}$$
Starting from this model we can analyse by using the $p_{values}$ which factors are not significant and thus removing them from the model. However, if by computing the $p_{values}$ we determine that the factor $A$ is not significant but the interaction betwen $A$ and $B$ is significant, Would it make sense to remove $A$ from our model?
Working with Minitab it seems to be possible because it offers this possibility, but I can't understand why. If we remove $A$ from our model, aren't we assuming that the factor $A$ doesn't have an influence on $Y$? If then, using the interaction wouldn't it mean that $A$ has still an influence in our model?
Thanks in advance!
