The Likelihood Ratio Test Flashcards
What does the log-likelihood measure?
Lack of fit of a model to the data -> the worse the fit, the smaller the likelihood
What is the deviance?
A measure of lack of fit -> how poorly does the estimated model fit the data?
D= -2l+c (c is chosen such that D=0 for a saturated model)
We use it to compare two nested models
What is a saturated model?
As many parameters as we have observations
What is a nested model?
A subset of another model (less parameters)
What does the likelihood ratio test do?
It estimates both models and retain DR and DU
LR = Dr-Du
if the restrictions are unteasonable, then DR>DU
Is the difference in the deviance sufficiently large to reject the restricted model in favor of the unrestricted?
What does the likelihood ratio test not care about?
Parsimony -> The goal of modeling is to sinplify things, improving fit means increasing model conplexity. So Is the loss in parsimony really worth the improvement in fit?
What measures combine fit and parsimony?
The Akaike Information Criterion
The Bayesian Information Criterion
What is the AIC?
AIC = -2l + 2P = D + 2P
looks at how many parameters we add to the model and if thats worth it
our objective is to minimize AIC by reducing the deviance (model fit) or increasing the parsimony (number of parameters)
What is the BIC?
BIc = -2l + Pln n = D+ ln n
uses the natural logarithm of n instead of 2P
additional complexity is only worth it if the deviance is much smaller; it is more conservative and puts more weight into simplicity than the AIC
we are favorinh the model with the smallest BIC
What are Akaike Weights?
A transformation of AIC so that the transformed values add to 1 across models. Can be interpreted as model probabilities in a particular set: What is the probability that model Mj is the correct model in the set?
-> converting to relative AIC by conparibg the AIC to the minimum AIC
