# Need help interpreting significant two-way interaction and insignificant three-way interaction

So typically I would have no problems with such interpretations. But my design is a little complicated so would appreciate any insight into this.

I am interested in looking at the effect of transitioning across experimental wealth levels on generosity (how much money they gave to a recipient after transitioning). In my study, I have four different between-subject groups. People who transition from poor to rich via merit, people who transition from poor to rich via luck, people who transition from rich to rich via merit, and people who transition from rich to rich via luck.

The problem here is - there are already baseline differences between groups at Time 1, which makes effects at Time 2 a little hard to interpret. To deal with this, I modelled Block (block 1 is before transitioning and block 2 is after) as a factor in my model as follows: ResponseModel1 <- lme4::lmer(response ~ f_transition*f_manner*f_block + (1 | id), data = aggrdata, control = lmerControl(optimizer = "bobyqa", optCtrl=list(maxfun=200000000)))

The two-way interaction shown below is significant, but the three-way is not. The two-way seems to suggest that the difference between the poor to rich, and rich to rich groups is only significant in the luck condition. However, is this two-way interaction misleading because there are already baseline differences in groups at Block 1? What would be the best way to interpret this? Should I only look at the three-way?

• You would need to provide more context about your experimental design. For example, what is baseline Time1? If you randomized, why are there (meaningful) differences at baseline? [There is disagreement about how to deal with so-called "failure to randomize"]. Further, there is no three-way interaction in your model specification, so it unclear what you are referring to – stefgehrig Jun 10 at 16:00
• Sorry the model specification was incorrect. I've corrected it now. – user3635550 Jun 10 at 16:05
• @stefgehrig I'm not sure why there are differences at baseline. At time 1, subjects were randomly assigned to be at the 'Poor' or 'Rich' wealth level, then they went through a luck vs merit manipulation, and based on that were told that they became richer (either via merit or luck). The differences between the Poor and Rich groups at Time 1 are understandable. However, there are also differences between the P-Luck and P-Merit groups at Time 1.These however, don't seem to be significant. – user3635550 Jun 10 at 16:06
• "The differences between the Poor and Rich groups at Time 1 are understandable." Differences in what? Was there a generosity elicitation already before the transition? – stefgehrig Jun 10 at 16:31
• Ok I think I got it. You should ignore the difference between P-Luck and P-Merit, and also between R-Luck and R-Merit at Time 1, because they proceed(!) the experimental manipulation and are thus noise/sampling variation (given you experimental implementation is sound in all other aspects). – stefgehrig Jun 10 at 16:33