I did some analysis that showed that one of my two factors was significant when I modeled it as a dummy coded variable (each of my two predictor variables has just two levels, and so the "on" in comparison to the "off" was shown to be significant). However, when I use a factor effects model, IE something like $\mu_{ij} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij}$, I no longer find either of my predictors significant. They are shown to have higher p-values in a factor effects model.
I'm mostly just curious, why is this?
Edit: note, I am not asking how introducing a variable changes my result; I am asking how these two separate versions of building a model with the same variables can produce different results.