In a repeated measures design with multiple trials, what would be some appropriate statistical techniques used to analyze data when the variable of interest is a continuous variable? For example, let's say participants level of extraversion is measured on a continuous scale, where higher scores reflect extraversion and lower scores reflect intraversion. The same participants are given 15 puzzles to solve. The puzzles are categorized by difficulty (easy, moderate, hard); each participant solves 5 easy, 5 moderate and 5 hard puzzles to solve. The time to solve each puzzle is recorded. What statistical techniques should I consider if I want to assess whether level of extraversion has any bearing on the time to solve puzzles?
A multilevel regression or one of its extensions sounds suitable. If you are interested in whether Extraversion is related to the time to solve a puzzle differently for different difficulty levels, you'd want to include an interaction between extraversion and the difficulty variable (which would probably be reasonable to include as categorical).
You'd first want to arrange your data in long format with a 3-level difficulty level variable and then you could use a multilevel regression. E.g. in R
model<-lmer(time_to_solve ~ (1|participant) + Extraversion * difficulty, data=data) #if you just want to know whether extraversion has an effect in general, you'd change the * into a + in the code to get the extraversion main effect on time.
Then you can further explore the simple slopes for Extraversion for different difficulty levels. The (1|participant) term is a random intercept estimating the variance in outcome that is attributable to the participant, and takes care of the non-independence of your observations (i.e., the fact that you have several observations from the same participants). Check out some multilevel regression tutorials for an overview.
However, your outcome variable - time to solve the puzzles - may have a distribution for which the Gaussian (=regular linear multilevel model as above) is not the best, so you may need to consider using a linear model extension (glmer models in R) and some other distribution family.
Edit. so for this analysis your data structure should be like
ID Diff Extr. time 1 Easy 3.2 20 1 Easy 3.2 25 1 Easy 3.2 26 1 Easy 3.2 19 1 Easy 3.2 17 1 Moderat 3.2 26 1 Moderat 3.2 33 etc. #i.e. each participant gets 15 rows