Nimrod Hertz-Palmor Yes, it is statistically possible to observe a significant three-way interaction while not observing 3 significant two-way interactions. Similar to how in a two-way ANOVA, one can observe a significant interaction effect with only one significant main effect or no significant main effects.

In a balanced full factorial design all effects are independent of each other (orthogonal). So the sums of squares attributed to two-way interaction can in theory and practice be 0 and the three-way interaction have non-zero SS and be significant.

Its unusual in practice because in general higher order effects account for less variation than lower order effects. This is the hierarchical ordering principle. Note this is a general tendency that arises for a mix of different reasons - it isn't inevitable:

http://jakewestfall.org/blog/index.php/2015/05/11/the-hierarchical-ordering-principle/

Blaine Tomkins Thom Baguley Thank you both very much!