# Does a non-hierarchical linear model while doing ANOVA make sense?

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?