Between-subjects, factorial, crossed, cross-classified: all the same thing? Suppose I have a test with $j$ items taken by $i$ persons. I wish to obtain the mean item score (y) taking into account that both items and persons are a sample of what I expect. As such, I take both as having random-effects.
I know persons and items, as random-effects, are crossed with one another. I also wish to know if item_type has any effect on the mean item score. Thus, if I'm an R user, I would probably use (suppose y is normally distributed):
library(lme4)
lmer(y ~ item_type + (1|persons_id) + (1|items_id), data = data) ## DON'T RUN


Question: Which of the following terminologies applies to the design described above: $(1)$ Between-subjects, $(2)$ factorial, $(3)$ crossed, $(4)$ cross-classified?


 A: A few points to note:

*

*y does not need to be normally distributed for an LMM. It is the conditional distribution (ie the residuals) that should be approximately normal, in order for certain inferences to be made.


*crossed and cross-classified are the same.


*a factorial design looks very much like a crossed design. For example where each level of two or more factors is applied to the subject; hpwver not the context - we usually say that the factor was "applied" to a subject: in factorial designs the factors are often variables that are controlled by the experimenter and so are naturally thought of as fixed effects, and there are usually few levels of the factors involved, whereas in crossed designs where the factors are random we often think of them as "belonging" to the factor. Of course this is not universal. As in crossed designs, factorial designs can be incomplete - that is not all combinations of the factor levels are applied - this happens with incomplete lock designs and split block designs. With crossed models we say "partially crossed" (or equaivalantly "partially nested")


*Between-subjects is a term I don't like much and don't use often at all. Basically it just refers to a variable which only varies between subjects - that is, it does not vary within subjects. Sex assigned at birth would be an exmaple. Sometimes a variable can be both in different contexts - Age is often a between subjects variable in a cross-sectional study, but can be a within variable foor a longitudinal study. Note here that it is not necessary for the variable to be a factor in order for it to be "between" or "within" subjects.
