Analysis of purely categorical data set Total statistics noob here. Imagine a data set like this, in which two categorical pieces of information ("SLOT"/"ITEM") are mapped:
PARTICIPANT 1
SLOT1: ITEM_A
SLOT2: ITEM_A
SLOT3: ITEM_C
SLOT4: ITEM_B
SLOT5: ITEM_F

There are hundreds of these records. All the "SLOT"s will always be mapped exactly once per participant, and all possible values are known. Now, when I visualize the data, it is super clear to see that there is a preference for certain ITEMs in certain SLOTs. But visual proof is not really enough here as it's very vague. Is there any statistical method I can employ to do so? Better yet, is there a way to tell how the slot influences the probability of a given ITEM?
Thanks a lot in advance!
 A: If you want some kind of causal inference, the problem gets (much) harder. However, if you just want to know the probability of variable $Y$ being a particular category given that variable $X$ is a particular category, the problem is easy: calculate it.
Let $X\in\{\text{dog},\text{cat},\text{bird}\}$ and $Y\in\{\text{bone},\text{yarn},\text{worm}\}$. If you want to know $P(Y=\text{worm}\vert X=\text{bird})$, calculate it as usual.
$P(Y=\text{worm}\vert X=\text{bird}) = \dfrac{
P(Y=\text{worm}\text{ AND }X=\text{bird})
}{
P(X=\text{bird})
}$
That is, calculate the probability of being a bird, and then calculate the probability of being both a bird and a worm. Equivalently, you can calculate the number of birds in your sample and then, out of those, consider how many have $Y=\text{worm}$. Then divide the number of birds with worms by the number of birds. The other eight pairings work analogously, as do all nine reverse pairings (e.g., $P(X=\text{bird}\vert Y=\text{worm})$ instead of $P(Y=\text{worm}\vert X=\text{bird})$, as these might be equal but need not be).
If you need explicit hypothesis testing, the $\chi^2$ test is a reasonable option. The Wikipedia article has a good example using neighborhoods and blue collar/white collar. We have questions and answers on here that discuss the $\chi^2$ test, too, though none come to mind.
A more complicated approach is to think of this as a multinomial logistic regression analogue to the usual ANOVA, where we have a different multinomial distribution, depending on the category, much as the usual ANOVA has a different Gaussian distribution, depending on the category, though you wind up with the same conditional probabilities. Such a view of the probem, however, might help you in creating confidence intervals for differences in probabilities, much as you can do group-by-group comparisons in the usual ANOVA after an overall test of equality.
