Are the following GLM models nested?

Question

I am new to GLM modeling but committed to learning as much as possible...

I have the following situation. Data can fall into one of 4 buckets.

It is a GLM model with a poisson distribution and a log link.

The three models can be described as:

1) A  A+B  A+B+B   A+B+B+B

The design matrix for this would be with lets say one observation for each bucket:

1 0
1 1
1 2
1 3

or

2) A  A+B  A+B+C   A+B+C+C

1 0 0
1 1 0
1 1 1
1 1 2

or

3) A  A+B  A+B+C   A+B+C+D   (not realistic. for ex)

1 0 0 0
1 1 0 0
1 1 1 0
1 1 1 1

It may not be obvious, but we are saying we need a new parameter to describe the last two buckets in model 2, and we need two new parameters for 3.

Would these be considered nested models?

How could I compare them to determine which one is best?

Thank you for your help.

Answer 1

Let $X_{i} \in \{1,2,3,4\}$ be which "bucket" you're in and $Y_{i}$ be whatever your dependent variable is. Define $A_{i} = \mathcal{I}\{X_{i} \geq 2\}$, $B_{i} = \mathcal{I}\{X_{i} \geq 3\}$, and $C_{i} = \mathcal{I}\{X_{i} = 4\}$ and $\mathcal{I}\{\cdot\}$ denotes the indicator function. Your description of Model 3 seems to indicate the model

$$ E(Y_{i}|A_{i}, B_i, C_i) = \beta_{0} + \beta_{1} A_{i} + \beta_{2} B_{i} + \beta_{3} C_{i} $$

Model 2 is the special case where $\beta_{2}=\beta_{3}$.

Model 1 is the special case where $\beta_{1}=\beta_{2}=\beta_{3}$.

So, yes, Model 1 is nested within Model 2, which is nested within Model 3. Nested models like this can be compared using the likelihood ratio test if the models are fit by maximum likelihood.