TU-L0022 - Statistical Research Methods D, Lecture, 2.11.2021-6.4.2022

Maximum likelihood estimates of two different models that are nested can be compared using the likelihood ratio test.

The idea is the same as in regression analysis and the F test. So in an F test, in regression analysis, we have two models. One is the constraint model and another one is the unconstrained model. Here, Model 2 is the more general, unconstraint model. And Model 1 is a special case of Model 2, because we get Model 1 from Model 2 by saying that these regression coefficients that are estimated here are actually zeros in Model 1, because we don't include these variables.

Then we do some math, we calculate the sum of squares or R-squared of these models, we compare that to degrees of freedom, we get the statistic that follows the F distribution.

In maximum likelihood estimates, we don't have the R-squared, we don't have the sum of squares, instead, we use the deviance statistic. So here in Kraimer's paper, we have two models.

So Model 1 is the constraint model, Model 2 is the unconstrained model, because we have this coefficient here that's estimated, here it is constrained to be zero. So, we have one degree of freedom difference between these two models.

Then we can calculate the likelihood ratio test of whether adding this one more parameter increases the model fit more than what can be expected by chance only, by comparing these deviances or -2 times log-likelihood. So, Model 2 is the unrestricted model, Model 1 is the restricted model.

We calculate the difference between the deviances, which is 3.79. And that difference follows the chi-square distribution with one degree of freedom, because there is only one parameter difference here. And the p-value for that would be 0.05, which says that there is no statistically
significant difference between the models that's on the border of 0.05 level. And it's also shown here that this is not very, very significant here.

In contrast to the F test, which always, if you have one parameter, gives you the same exact p-value as t test, the significance test statistic here from z test, and the likelihood ratio, p-value, they don't necessarily have the exact same value, because these are based on large sample approximations that may not work as exactly as intended in a small samples.

So there can be some scenarios like here, we have a significant coefficient, p is less than 0.05. But here we don't, we have p that is more than 0.05. But it's on the boundary. So we could just say that there is weak evidence for the existence of this relationship.