In the formula above the interaction effect is, of course, dosegendertype. The ANOVA results can be seen below (we have only kept the line presenting the third-order interaction effect).
Df Sum Sq Mean Sq F value Pr(>F) dose:gender:type 2 187 93.4 22.367 3.81e-10The interaction effect is statistically significant: F(2)=22.367, p<0.01. In other words, we do have a third-order interaction effect. In this situation, it is not advisable to report and interpret the second-order interaction effects (they could be misleading). Therefore, we are going to compute the simple second-order interaction effects.
This is definitely not a trivial article, but there are useful techniques in it.