When it comes to model selection between two non-nest models, the information criteria, e.g. AIC or BIC, is often used and the model with a lower information criteria is preferred.
However, even with AIC or BIC, we are still unable to answer the question of whether the model A is significantly better than the model B probabilistically. Proposed by Quang Vuong (1989), Vuong test considers a better model with the individual log likelihoods significantly higher than the ones of its rival. A demonstration of Vuong test is given below.
First of all, two models for proportional outcomes, namely TOBIT regression and NLS (Nonlinear Least Squares) regression, are estimated below with information criteria, e.g. AIC and BIC, calculated and the likelihood of each individual record computed. As shown in the following output, NLS regression has a lower BIC and therefore might be considered a “better” model.
Next, with the likelihood of each individual record from both models, Vuong test is calculated with the formulation given below
Vuong statistic = [LR(model1, model2) – C] / sqrt(N * V) ~ N(0, 1)
LR(…) is the summation of individual log likelihood ratio between 2 models. “C” is a correction term for the difference of DF (Degrees of Freedom) between 2 models. “N” is the number of records. “V” is the variance of individual log likelihood ratio between 2 models. Vuong demonstrated that Vuong statistic is distributed as a standard normal N(0, 1). As a result, the model 1 is better with Vuong statistic > 1.96 and the model 2 is better with Vuong statistic < -1.96.
As shown in the output, although the model2, e.g. NLS regression, is preferred by a lower BIC, Vuong statistic doesn’t show the evidence that NLS regression is significantly better than TOBIT regression but indicates instead that both models are equally close to the true model.