# Is likelihood ratio test comparing apples to oranges?

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.

I'm going to hopefully clarify some things about the likelihood ratio test. If you still disagree, comment below.

First, a nested model is (pragmatically) a model for which we assume the value(s) of the parameter(s). Consider a vector of parameters partitioned into two disjoint sets $$\mathbf{\Theta} = [\theta; \theta_0]$$ . In this case, we are free to estimate $$\theta$$ but we assert that the value of the other parameter(s) is $$\theta_0$$.

When we do this, the likelihood is exactly the same, we are just interested in the manifold where the other set of parameters is $$\theta_0$$.

We then compare this nested model where we are free to estimate every parameter from the data. If the likelihoods are drastically different (read, more different than we would expect by chance given that the parameter values are actually $$\theta_0$$) then we conclude that the model for which we can estimate all the parameters better fits the data.

So the likelihoods are in fact the same. We are just comparing different parts to one another.

The key thing for a statistical test is that the distribution of the test statistic can be computed under the null hypothesis. The null hypothesis for a likelihood ratio test is typically the less complex model (meaning fewer free parameters). If the less complex model holds (let's say that's model A), the likelihood from model B should normally be very similar to the likelihood from model A, because all the additional parameters should be estimated close to 0 (or in general close to the values assumed for them in model A). We're comparing "apples and oranges" only in the sense that the likelihood from model B will still be bigger because of additional degrees of freedom. So saying "model B is better because likelihood B is bigger" would indeed be unfair.

However, this is not what the likelihood ratio test is doing. The likelihood ratio tests asks whether the likelihood from model B is so much larger than that from model A, that it would be unrealistic if indeed model A were true. The corresponding probability can be computed from the distribution of the likelihood ratio under model A, and this is what the test is based on.

Note that I'm assuming that model A is "nested" within model B, meaning that model A corresponds to model B with some fixed parameter choices, which is the standard setup for the likelihood ratio test.