I don't believe that likelihood ratio tests work in the context of regression because the likelihood functions for model A versus B aren't the same thing so we're using different standards to judge different things and thinking the standards don't matter when they should.
They must be used to compare nested models.
In statistics, the likelihood function (often simply called the likelihood) is formed from the joint probability of a sample of data given a set of model parameter values; it is viewed and used as a function of the parameters given the data sample.[a]
The likelihood function describes a hypersurface whose peak, if it exists, represents the combination of model parameter values that maximize the probability of drawing the sample obtained.
A likelihood ratio tests basically says if model A has a likelihood greater than model B by a certain number, than model A is better than model B where A is nested in B. What determines the likelihood function is the set of parameter values. A nested model will always include at most the number of parameter values in the full model.
My concern is that the loglikelihood function for model A is not the same thing as the loglikelihood function for model B.
We can compare log likelihoods to say some set of model parameter values is better than another set assuming we're talking only about a very particular likelihood function, but the nested and full model have different likelihood functions so we're comparing apples to oranges when we're comparing nested models. Under standard A, the full model has a loglikelihood of 500 but under standard B the nested model has a loglikelihood of 700 but what if standrd A is less strict than standard B? The judges aren't the same in both cases. Under the likelihood ratio test 500 is significantly less than 700 so choose the null model which doesn't make sense.