ANOVA - Modeling

Oneway ANOVA or Completely Randomized Design:

We have the following notation:
Y_ij = μ + α_i + e_ij, i=1,..,n_j, j=1,..,k

where:
k is the number of groups
n_j is the number of observations in the j^th group
Y_ij is the i^th observation in the j^th group
μ is the overall mean.
α_i is the deviation from the overall mean of the j^th group
e_ij is the residual of the i^th observation in the j^th group

Example (Mothers and Cocaine Use):
k =3
n₁=39, n₂=19, n₃=36
Y₁₁=44.3, Y₁₂=45.1, Y₂₁=45.3, and so on

How can we interpret this model?
Say you are asked to use the data set to find an estimate for the mean length of a newborn baby without knowing anything about the mother. Then your best guess is the sample mean of all the babies lengths: 49.5 cm is the estimate of μ.

Now somebody tells you that actually the babies mother belongs to the first trimester group. How is this going to change your estimate? Well, the mean length of babies in this group is 49.3 cm, so you would change your guess by 49.3-49.5= -0.2 cm. This is an estimate of α₂. So α₂ is the adjustment to the overall mean if you know the observation belongs to group 2.
Similarly we find an estimate of α₁ to be 51.1-49.5= 1.6 cm and α₃ to be 48.0-49.5= -1.5 cm.
So the α's incorporate the change to the overall mean if we know which group an observation belongs to. But if there is no difference between the group means, then we would not need any adjustments, and so the null hypothesis of no difference between the group means becomes:

H₀: α₁=α₂=...=α_k=0 (means are the same for all groups)
H_a: α_i≠0 for some i

Previously we have written the hypotheses as follows:

H₀: μ₁ = .. = μ_k
H_a: μ_i ≠ μ_j for some i≠j

These two are absolutely equivalent, but the first one is easier to generalize to more than one factor so we will use it.

Twoway ANOVA

a) Completely randomized block design: One factor is the focus of the study, the other is included because it might effect the outcome
b) Factorial design: both factors are of equal interest.
but for our purposes the difference does not matter.

Twoway Additive Model:

Twoway Model with Interaction:

which leads to the three hypotheses tests:
H₀: α₁=α₂=...=α_I=0 (no difference of the means for first factor)
H₀: β₁=β₂=...=β_J=0 (no difference of the means for second factor)
H₀: g₁₁=g₁₂=...=g_IJ=0 (no interaction)
were the third test is for the presence of interaction. Note: this third test requires repeated measurments!

Note: Stat>ANOVA>Twoway by default fits a model with interaction, but Stat>ANOVA>General Linear Model fits an additive model.