Are these factors considered random or fixed factors? I am in a little bit of a mix right now regarding this scenario right here. Assuming I have a 2x3 design, looking at the effects of:


*

*Two mood-inducing conditions: 


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*Happy

*Neutral
2.Three levels of task difficulty:


*

*Easy

*Moderate

*Difficult



I wish to look at whether task performance will be affected by mood a certain mood induced. A total sample of 100 participants will be distributed 50-50 in factor 1. Factor 2 is within-subjects.
Factor 1 is a fixed factor here, since I am only specifically interested in two mood types. However, is factor 2 considered fixed right here since I am looking at three designated difficulty levels, or is factor 2 considered random, as difficulty is subjective, lying on a continuum? Or, is factor 2 considered random, because the 100 samples used are just random samples from the much bigger population of people on earth? 
 A: I think the random/fixed effect terminology can be a little confusing: Andrew Gelman has  blogged multiple times on this point. I believe that the best definition is that concerning pooling: as stated in a great answer on this site, random effects are estimated with partial pooling, while fixed effects are not. 
In your case, use fixed effects for Factor 1 and 2, and add random effects for Factor 2, but only if if you have multiple observations with different values of Factor 2, for at least some of the 100 subjects. As a matter of fact, if you still had the same 100 subjects, but you didn't have multiple observation per subject, then you wouldn't pool estimates among different subjects.
The reason why you use random effects is that you expect different individuals to react differently to the same levels of Factor 1 and Factor 2. Now, estimating a different set of parameters $\hat{\boldsymbol{\beta}}_i=(\hat{\beta}_{0i},\hat{\beta}_{1i},\hat{\beta}_{2i},\hat{\beta}_{3i})$ for each subject $i$ isn't a good idea (this would be the no pooling model in the multilevel modeling terminology). This would be an exceedingly flexible model (too many parameters), and also an useless model, because how would you make predictions for a new individual not included in the 100 ones? Also, unless you have the same number of replications for each subject, estimates for individuals with few replicates will have higher variance than estimates for individuals with more replicates.
At the same time, you don't want to pool the data from all individuals together and just estimate 4 parameters $\hat{\boldsymbol{\beta}}=(\hat{\beta}_0,\hat{\beta}_1,\hat{\beta}_2, \hat{\beta}_3)$ (complete pooling model), because in this way you're neglecting the info that you had multiple measurements, with varying levels of Factor 2 (but not of Factor 1) for the each individual. What you can do is then to estimate fixed effects $\hat{\boldsymbol{\beta}}=(\hat{\beta}_0,\hat{\beta}_1,\hat{\beta}_2,\hat{\beta}_3)$ , plus a random intercept and two random coefficients for Factor 2 for each individual, which however you assume to come from a common multivariate normal distribution. This way you shrink the estimates for individuals with few replicates, towards the fixed effect estimates. This is the partial pooling or mixed effects model.
A: Irrespective of how you measure Factor 2, it should be included as a fixed effect. You are only interested in these three levels, not in the "population" of possible difficulties they were drawn from.
You do need to a add a random effect for subject though, to model the dependence between measurements on the same subject, albeit under different difficulties. The subjects are drawn from a large population, and you want to do inference on this population, not on the 100 subjects you have randomly drawn.
