Nested GLM Models

Question

I know that GLM models are nested when one can obtain the smaller model by removing $\beta$'s from the larger model. However, when doing some exercises I found on the internet, I stumbled about doing a partial deviance test for the following two models:

Here, let covariate A be a continuous covariate and B be a factor covariate with 2 levels

Model 1: Intercept, A

Model 2: No Intercept, A, B coded as [0, 1] and [1, 0] for the two levels.

In the exercise, it has been claimed that Model 1 is nested within Model 2.

Here, obtaining Model 1 by setting parameters of Model 2 to zero is not really possible. However, Model 1 is kind of included when we take $\beta_{intercept} = \beta_{factor1} + \beta_{factor2}$. Does this then imply that the two models are nested? I always thought that the same covariates must be included in the model to have nested models.

Thank you in advance for your help and clarification! :)

Answer 1

Model 1 is nested within Model 2 ( so model 2 is the larger model) means that every fit by Model 1 can be matched exactly by Model 2 (but parameter values can well be different). I explained this in another context here: Why is sum of squared residuals non-increasing when adding explanatory variable?

Such a definition is necessary to take account of different parametrizations, your example case is the same linear model parametrized in two different ways. In the case of linear models $Y_1=X\beta_1 + \epsilon_1, Y_2=Z\beta_2 + \epsilon_2$, this can be paraphrased using the concept (from linear algebra formulation of linear models) of model space, which is the rank space of the design matrix, that is, $$ R(X)=\{ x\beta \colon \beta\in\mathbb{R}^p \} $$ that is, the space of all possible expected values the model can produce. Model 1 is then nested in model 2 if $R(X) \subseteq R(Z)$. With this definition nesting is independent of parametrization.

Your understanding is related to Thomas Lumley's concept of symbolic nesting: What is a "symbolically nested" model?