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I have trouble understanding the following statement on Wikipedia about mixed design ANOVAs:

Thus, in a mixed-design ANOVA model, one factor (a fixed effects factor) is a between-subjects variable and the other (a random effects factor) is a within-subjects variable. Thus, overall, the model is a type of mixed effect model.

The page gives an example from Andy Field’s book: Participants in an experiment are each presented nine individuals of the opposite sex and asked to indicate how likely they are to go on a date with that person. The nine individuals that are presented are chosen according to their attractiveness (three levels) and their personality (three levels). Gender is mentioned as a "between-subjects" variable, while attractiveness and personality are mentioned as "repeated measurements".

My understanding is that gender differs between the subjects, while attractiveness and personality differ within the measurements for a single subject. The sentence in the opening paragraph that I have highlighted in bold suggests to me that gender (the between-subjects variable) is a fixed effects factor, while attractiveness and personality (the within-subjects variables) are random effects factors.

I have found many different definitions of random effects but many seem to have in common that they are "categorical variables whose levels are chosen at random from a larger population" (see here). Following this definition, it seems to me that "subject" (participant) is a random effect (more specifically, a random intercept), since the participants are assumed to be drawn from a larger population, while "gender", "attractiveness" and "personality" are all fixed effects (since the individuals that were presented to the participant were carefully chosen for their characteristics).

The example in Wikipedia comes from Andy Field’s book, Chapter 14. This chapter is about repeated measures ANOVA, and Field does not use the terminology of mixed models in this chapter, nor does he does call the within-subjects variables "random effects" like the Wikipedia page does. Instead, when he does introduce multilevel models in Chapter 19, he writes that "All of the effects in this book so far we have treated as fixed effects.", and he defines random effects as those where "the experiment contains only a random sample of possible treatment conditions". So I would say that Field’s definition is consistent with the second interpretation.

So… are attractiveness and personality random effects because they are within-subjects variables, or are they fixed effects because their levels are known and all possible levels occur in the data?

I know that there are a lot of contradicting definitions of random effects... To phrase the question more practically, how would you analyse Field’s repeated measures ANOVA data in a tool that asks you to specify random and fixed effects, such as SPSS’s procedures for linear mixed models?

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    $\begingroup$ +1 Good question, pity it went unnoticed. Your understanding is all correct, and the Wikipedia quote is wrong (or at best sloppy). $\endgroup$ – amoeba Apr 22 '18 at 22:09

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