# Is GLM with one continuous variable of 4 levels a nested model of GLM with 3 dummy variables?

Y is just a binary variable 0 and 1.

X is a variable with 4 levels 0, 1, 2, 3.

We fit a logistic model A regarding X as continuous variable. Then we fit a logistic model B regarding X as categorical variable, so we have 3 dummy variables.

We want to know if B improves the model fit significantly, so we use likelihood ratio test.

However, I assume one model has to be nested of the other in order to use likelihood ratio test.

So is A a nested model of B? It seems to me that A's term is not a subset of B's terms.

Therefore, I want to know if following situation are also nested models?

1. Model A: $$y = a + b +c$$, Model B: $$y = ab$$
2. Model A: $$y = a^2 + a^3$$, Model B: $$y = a$$

To see if model A is nested in model B, it is not enough to compare the symbolic model structure, but see What is a "symbolically nested" model?. What matters is that, for every set of values of the parameters in A, we can find parameters for B that gives the same predicted values. And that is clearly the case for your first example, so A is nested in B, although not symbolically nested.

1. neither model is nested in the other

2. neither model is nested in the other

EDIT to clarify:

Let A be given by the model function $$f_A(y;x, \theta_A)$$ and B by the model function $$f_B(y;x, \theta_B)$$. (A model function for a random variable $$Y$$ means a density/probability mass function for $$Y$$, parametrized by some parameter varying over some parameter space, left implicit above.) Then A is nested in B if any predictions given by A can be matched by B, that is, if given $$y,x$$ and some $$\theta_A$$ there is some $$\theta_B$$ such that $$f_B(y;x, \theta_B)=f_A(y;x,\theta_A)$$.

Applying this to your question: The only difference between the glm's is in the linear predictor (so we assume the same model form, the same link function, ...). The linear predictor for A is $$\eta_A(x)= \alpha_0 + \alpha_1 x$$, for model B is $$\eta_B(x)= \beta_0+ \beta_{11}I(x=1)+\beta_{12}I(x=2)+\beta_{13}I(x=3)$$ (we have used $$x=0$$ as reference level, this choice does not matter.)

Given some value for $$\eta_A$$, say $$\eta_A(x)=1+0.5 \cdot x$$, finding a match for $$\eta_B$$ is only a matter of finding one solution (don't matter if there are more, we just need one) of the following equation system: \begin{align} 1+0.5 \cdot 0 &= \beta_0 \\ 1+0.5 \cdot 1 &= \beta_0 + \beta_{11} \\ 1+0.5 \cdot 2 &= \beta_0 + \beta_{12} \\ 1+0.5 \cdot 3 &= \beta_0 + \beta_{13} \end{align} as this is a linear system in four unknowns and four equations, it does have a solution.

• I'm sorry I still don't get "gives the same predicted values"? They are two different models and have different numbers of parameters/coefficients. How are their predicted values the same? Even for the "symbolical nested model", they should give different predicted values when we fit the same data. Commented Feb 19, 2020 at 18:15
• Ir Is not Will give, IT is can give, AS functions. Commented Feb 19, 2020 at 18:20
• Commented Feb 19, 2020 at 19:06
• terrific answer! Thanks! It makes much more sense now! Commented Feb 19, 2020 at 21:47
• According to your definition, why are not the two additional example nested models? Commented Feb 20, 2020 at 0:46