To understand a log-linear model, let's look at a model that all of you should be familiar with already, and think about different parts of a model: objective, assumptions, parameters, interpretation, etc..

Two-way ANOVA

Does the amount of sunlight and watering affect the growth of geraniums? The plant growth of 16 plants is measured in centimeters. Each combinaton of sunlight and water has 4 plants; e.g., High water and high sunlight has 4 plants ranging in lenght from 21 to 30cm.

Objective: model the continuous response as function of two factors.

Model structure: Y_ijk = μ + α_i + β_j + γ_ij + e_ijk with e_ijk ∼ N(0, σ²), i = 1, ..., I, j = 1, ...., J, and k = 1, ..., n_ij

Model assumptions: At each combination of levels the outcome is normally distributed with the same variance: y_ijk ∼ N(μ_ij , σ²), where μ_ij = E(y_ijk) = μ + α_i + β_j + γ_ij

This model is over-parametrized because term γ_ij already has I × J parameters corresponding to the cell means μ_ij . The constant, μ, and the main effects, α_i and β_j give us additional 1 + I + J parameters.

We use constraints such as Σ_i α_i = Σ_j β_j = Σ_i Σ_j γ_ij = 0, to deal with this overparametrization.

The questions that we may ask and answer with the ANOVA model are:

(1) Does the amount (level) of watering affect the growth of potted geraniums? (Is there a significant main effect for factor B?, e.g. H₀ : α_i = 0 for all i)

(2) Does the amount (level) of sunlight affect the growth of potted geraniums? (Is there a significant main effect for factor A?)

(3) Does the effect of the level of sunlight depend on level of watering? (Is there a significant interaction between factors A and B?)

Based on the above output, we can conclude that there is a significant interaction effect (F=9.28, p=0.010), that is the effect of watering on the average plant growth depends on the amount of sunlight and vice versa. However, it seems that the significant interation effect is mostly driven by the amount of watering.