What are some ways of generating three-way tables of counts? We essentially have the same sampling schemes as we saw for two-way tables:
- Poisson unrestricted sampling – nothing is fixed, each cell is a Poisson random variable with a rate μijk
- Multinomial sampling with fixed total sample size n
And now with these more complex tables since we have more "total" sample sizes to fix, we have two additional ways...
- Multinomial sampling with fixed sample size for each partial table, e.g. n++k
- Product-Multinomial sampling within each partial table, e.g. fix ni+k, that is fix the rows within each partial table. You can review the properties of product-multinomial sampling in two-way tables or in the supplemental sections.
Next, let's see more on different independence and association models that can capture relationships between three categorical variables.