What these models mean

• Grand mean:
  
  $$F_{ij}=\tau_0$$
  
  
  • All cells in the table have the same expected value
  • e.g., in a 1-D table (frequency distribution) we might expect the same amount in each category
  • An example would be suicides by day of the week: we might first try a base model that says day of the week has no effect on numbers of suicides
  • A 2-D example could be suicides by day-of-week and season: a reasonable base model might say neither affects the number of suicides, and we should expect approximately the same number on a Sunday in winter as a Tuesday in autumn.

• Equiprobability on one dimension:
  
  $$F_{ij}=\tau_0\tau_i$$
  
  
  • For instance, we might find that the suicide rate varies by day-of-week, but not by time of year.

• Independence:
  
  $$F_{ij}=\tau_0\tau_i\tau_j$$
  
  
  • We might find that there is a seasonal effect as well as a day-of-week effect, but that these don't interact: e.g., the seasonal effect is approximately the same for all days, or the weekly effect is the same all year round.

• Association:
  
  $$F_{ij}=\tau_0\tau_i\tau_j\tau_{ij}$$
  
  
  • Where independence doesn't hold we have association.
  • For at least some combinations of categories of the two variables there is an extra effect: another way of saying this is that there is an `interaction' between the variables.
  • In the day-of-week/season example, there could be a weekly effect that is different in different seasons, or equivalently a seasonal effect that is different for different days of the week (e.g., winter Fridays may be exceptionally high compared with other Fridays or other winter days, summer Tuesdays exceptionally low, etc. ).