Akaike Information Criterion Vs Likelihood Ratio Test

[CGandC]

TL;DR Summary

In both Akaike Information Criterion and Likelihood Ratio Test we compare likelihoods to understand the better fit to the empirical data, but what's the difference between Akaike Information Criterion and Likelihood Ratio Test?

Hello,

I want to understand the difference between both goodness-of-fit tests, I would be glad if you could help me:

Akaike Information criterion is defined as:

$AIC_i = - 2log( L_i ) + 2K_i$

Where $L_i$ is the likelihood function defined for distribution model $i$ .
$K_i$ is the number of parameters of the distribution model. for example, for exponential distribution we have only lambda so $K_{exponential} = 1$

So if I want to know which distribution better fits the empirical data, I see which AIC is higher and choose the representative distribution for that high AIC.

Likelihood Ratio Test is defined as ( According to " Clauset , et al . Power law distributions in empirical data" ):

" The basic idea behind the likelihood ratio test is to compute the likelihood of
the data under two competing distributions. The one with the higher likelihood is
then the better fit. Alternatively, one can calculate the ratio of the two likelihoods,
or equivalently the logarithm R of the ratio, which is positive or negative depending
on which distribution is better, or zero in the event of a tie. "Bottom line:
We can see that both in the Akaike and Likelihood ratio test, essentially I compare the likelihood functions for different distribution models and I choose the bigger one which is representative of the better distribution fit. using both methods as described above yielded me very similar results in MATLAB.
So I don't really understand the difference between both methods, maybe I'm understanding these methods wrong?

 

[Stephen Tashi]

So if I want to know which distribution better fits the empirical data, I see which AIC is higher

Did you mean to say "lower"?

 

[CGandC]

Did you mean to say "lower"?

Yes, sorry.

 

[Stephen Tashi]

A practical distinction is that a larger number of free parameters in a model makes its AIC higher, thus making the model less preferable. The log liklihood ratio isn't directly affected by the numbers of free parameters in the models.

 

[CGandC]

I found another distinction:

Likelihood ratio test is defined to test for distributions that are nested.

For example: I can use for this test to determine whether power-law PDF or power-law with cutoff PDF fit the data better because the power-law PDF is a nested PDF model of the power-law with cutoff PDF.

However, Akaike Information Criterion is general and has no model limitations like the Likelihood ratio test has, but It indeed provides high values when computed.

Besides these main distinctions:
What are some advantages and disadvantages of using AIC over LRT ( likelihood ratio test )?

 

[Stephen Tashi]

I found another distinction:

Likelihood ratio test is defined to test for distributions that are nested.

Whether that's true is a matter of vocabulary. Under certain assumptions, the distribution of the liklihood ratio (as a statistic) is asymptotically chi-squared. Without the assumption that the distributions are nested, the statistic may not have a chi-squared distribution, but it may still be possible to estimate its distribution.

Some articles only use the term "likihood ratio test" in the case of nested models. Other articles use the term in a more general sense - e.g. the abstract of https://besjournals.onlinelibrary.wiley.com/doi/full/10.1111/j.2041-210X.2010.00063.x (I haven't read the article itself.)

What are some advantages and disadvantages of using AIC over LRT ( likelihood ratio test )?

Questions of about what is a "better" or "good" statistical method are not mathematical questions unless considerable quantitative context is specified - for example, a utility or penalty function that evaluates the cost or benefit of making statistical decisons (right or wrong) on the basis of the statistical tests involved.

So you are probably asking for an answer based on empirical experience. The answer will vary from person to person and from field of study to field of study. You need to get advice from people studying the same thing that you do. If the question involves publishing something in a jounal, you need to look at published articles to see which statistical methods are accepted for publication. Culture and tradition are also factors in choosing statistical methods.

I found these concise notes on model selection: https://www.stat.cmu.edu/~larry/=stat705/Lecture16.pdf They prefer using "cross-validation" to using the AIC.