Given some real life dataset where the attributes of the dataset take on categorical values, with corresponding r(1) × r(2) × … × r(m)... Show moreGiven some real life dataset where the attributes of the dataset take on categorical values, with corresponding r(1) × r(2) × … × r(m) contingency table with nonzero rows or nonzero columns, we will be testing the goodness-of-fit of various independence models to the dataset using a variation of Metropolis-Hastings that uses Markov bases as a tool to get a Monte Carlo estimate of the p-value. This variation of Metropolis-Hastings can be found in Algorithm 3.1.1. Next we will consider the problem: ``out of all possible undirected graphical models each associated to some graph with m vertices that we test to fit on our dataset, which one best fits the dataset?" Here, the m attributes are labeled as vertices for the graph. We would have to conduct 2^(mC2) goodness-of-fit tests since there are 2^(mC2) possible undirected graphs on m vertices. Instead, we consider a backwards selection method likelihood-ratio test algorithm. We first start with the complete graph G = K(m), and call the corresponding undirected graphical model ℳ(G) as the parent model. Then for each edge e in E(G), we repeatedly apply the likelihood-ratio test to test the relative fit of the model ℳ(G-e), the child model, vs. ℳ(G), the parent model, where ℳ(G-e) ⊆ℳ(G). More details on this iterative process can be found in Algorithm 4.1.3. For our dataset, we will be using the alcohol dataset found in https://www.kaggle.com/datasets/sooyoungher/smoking-drinking-dataset, where the four attributes of the dataset we will use are ``Gender" (male, female), ``Age", ``Total cholesterol (mg/dL)", and ``Drinks alcohol or not?". After testing the goodness-of-fit of three independence models corresponding to the independence statements ``Gender vs Drink or not?", ``Age vs Drink or not?", and "Total cholesterol vs Drink or not?", we found that the data came from a distribution from the two independence models corresponding to``Age vs Drink or not?" and "Total cholesterol vs Drink or not?" And after applying the backwards selection likelihood-ratio method on the alcohol dataset, we found that the data came from a distribution from the undirected graphical model associated to the complete graph minus the edge {``Total cholesterol”, ``Drink or not?”}. Show less
