Can a random slope in a linear mixed model mask the effect of my intervention? I want to assess the impact of my intervention in a repeated-measures design. I have subject as a random intercept in order to account for the dependence of measurements within subjects:
Outcome ~ Condition*Time + (1|Subject), where Condition is treatment or placebo. I expected the effect of my intervention to increase over time, and I indeed see a significant Condition*Time interaction, which supports my hypothesis.
However, in reality, I guess that subjects will have variable effects of Time on Outcome. Indeed a model comparison confirms that the model Outcome ~ Condition*Time + (1+Time|Subject) fits the data significantly better, but my Condition*Time interaction is no longer significant in the presence of this random slope.  
Now, I am wondering if the random slope may have masked the effect of my intervention, by falsely attributing the variation in the effect of Time to Subject, rather than to Condition. Is this possible?
 A: The problem with the idea that the random slope in the mixed model is  "falsely attributing the variation in the effect of Time to Subject, rather than to Condition" is that you are then assuming that Time:Condition interaction is ground truth--but this is the hypothesis you are trying to confirm. You could just as easily surmise that the former model is mistaking subject level variability as a Time/Condition interaction. Either way, you'd be cherry picking the model that gives the effects you want to see rather than interpreting possible effects from a model based in the design of your experiment and/or the nature of your hypothesis. 
Neither model is true, but the significantly better fit (log-likelihood?, AIC?) suggests the mixed model might be better for analysis. Which model is best for testing your hypothesis and does the best job of accounting for sources of variability, error and/or bias? Are there other modes of analysis that might be better suited for your experiment based on its design? This is why it is often best to create an analysis plan before data collection and only deviate when necessary to deal with messy data (missing values leading to lack of balance, etc.)
