2D ANOVA Factors, Levels, and Cells

A factor is a basis for categorizing data. For example, if you count the number of sit-ups individuals can do, one basis of categorization is age. For age, you might have the following levels:

Level 06 years old to 10 years old
Level 111 years old to 15 years old

Another possible factor is weight, with the following levels:

Level 0less than 50 kg
Level 1between 50 and 75 kg
Level 2more than 75 kg

Now, suppose that you made a series of observations to see how many sit-ups people could do. If you took a random sampling of n people, you might find the following results:

Person 1 8 years old (level 0) 30 kg (level 0) 10 sit-ups
Person 2 12 years old (level 1) 40 kg (level 0) 15 sit-ups
Person 3 15 years old (level 1) 76 kg (level 2) 20 sit-ups
Person 4 14 years old (level 1) 60 kg (level 1) 25 sit-ups
Person 5 9 years old (level 0) 51 kg (level 1) 17 sit-ups
Person 6 10 years old (level 0) 80 kg (level 2) 4 sit ups

and so on.

If you plot observations as a function of factor A and factor B, they fall into cells of a matrix with factor A as rows and factor B as columns. Each cell must contain at least one observation, and each cell must contain the same number of observations.

To perform the analysis of variance, you specify an array X of observations, with values 10, 15, 20, 25, 17, and 4. The array Index A specifies the level (or category) of factor A to which each observation applies. In this case, the array would have the values 0, 1, 1, 1, 0, and 0.

The array Index B specifies the level (or category) of factor B to which each observation applies. In this case, the array would have the values 0, 0, 2, 1, 1, and 2. Finally, there are two possible levels for factor A and three possible levels for factor B, so you pass in a value of 2 for the A levels parameter and a value of 3 for the B levels parameter.

You can apply any one of the following models, where L is the specified observations per cell: