# Hypotheses check with mixed ANOVA, ANCOVA, regression or multilevel linear model?

I conducted a repeated-measures experiment and obtained for each subject the following data:

IV - continous (personality trait)
DV - continous, under condition 0 (default condition)
DV - same variable as above, but under condition 1
DV - same variable as above, but under condition 2


From theory, I have multiple hypotheses:

• The higher IV in a subject is, the higher is the change of DV between condition 1 and 0 (DV.1 - DV.0).
• The higher IV in a subject is, the higher is the change of DV between condition 2 and 0 (DV.2 - DV.0).
• DV under condition 2 is higher than under condition one for subjects with a high IV.

What is the recommended way to check those hypotheses?

1. First I thought about using a median split on IV to separate it in a low/high group so that IV is categorial, and hence a mixed ANOVA would work (dependent variable = DV, whitin = condition, between = IV). But doing a median split loses a lot of information and might lead to wrong conclusions, so I am not happy with this. Also, I get unequal group sizes which are problematic for mixed ANOVAs. Other than that ANOVA would be good, I could just use planned contrast to check my hypotheses.

2. Then I thought about using an ANCOVA, because an ANCOVA is an ANOVA with an added continous covariate (in this case IV), meaning IV does not have to be split artificially and can stay metric. The problem here is that if I understood it correctly the ANCOVA would partial out the effect of IV - which would not be acceptable, because I am mainly interested in the effect of IV on the DV under different conditions.

3. I then thought I just use a simple linear regression: lm(DV.1 ~ IV + DV.0), or lm(DV.1 - DV.0 ~ IV). With that I can check if I get a significant regression coefficient. But how do I know whether the difference between DV.1 and DV.0 is significant? Because if they are not, I have to reject my first hypothesis.

4. I also thought about using a multilevel linear models to model my problem and circumvent the problems that a mixed ANOVA can only have categorial independent variables. But I am not sure if that is even possible? I thought about using lme in R with subject as a random argument to account for the repeated measures design. Would that work?

As you probably can tell, I am new to the world of statistics, especially to hypothesis testing. I would appreciate it if you could help me choose the right kind of analysis method for my hypotheses!

Your explanation of your design is incorrect. You have just 1 DV (not 3) and 2 IVs (not 1). Your IVs are trait and condition. You also appear to have measured the same individual under multiple conditions, which suggests that you need a linear mixed-effects model with a random intercept (and possibly slope) for individual.
The simplest model that would be appropriate for this problem is (in lmer syntax ):
lmer(DV ~ trait + condition + (1|individual))

Of course, you may want to consider whether there could be interactions between trait and condition, and differences in how individuals respond to changes in condition