Could power increase, or decrease? Under what situation could an increase or decrease occur?
Adding a second factor to your ANOVA can either cause power to increase or decrease, depending on the context. Adding a second factor that is actually related to the DV can help explain some of the systematic variation in SSw, reduce error, and therefore, increase power. However, adding a second factor that is unrelated to your DV can actually reduce power. When you calculate the within-group mean square (MSw) in order to calculate the F statistic, you use the equation: MSw=SSw/dfw. Adding a second factor "diverts" degrees of freedom values to the other IV. In other words, having two factors will necessarily cause your dfw term to be smaller. Dividing SSw by a smaller number actually creates a larger MSw (error) term, and according to the formula for F=MSb/MSw, will then cause F to be smaller and therefore less likely to reach significance. The key point here is that choosing an unrelated second factor will not account for as much error in the data and your SSw will not be smaller, but rather will be bigger. This coupled with the necessarily smaller dfw of two factor ANOVA could result in reduced power.
**also unsure of this. Craig said that power can't really be talked about in the "present"/about the current study, but is rather a projective number. If that is the case then I'm not sure if I got at exactly what he wanted.