This question came through email today, and I thought maybe some of you would also like to see the response.
“I feel like I keep getting mixed up with the theory behind these topics but can get the computational problems right on the exam. Can you please help me and list the differences in regards to testing and theory in bullet points for me that you think would be testable? I would really appreciate your help on this.”
The Pearson chi-square statistic is the same as the one that we use to test contingency tables, the sum of the ratios of the squared difference between the observed and the expected (aka model predicted) value divide by the expected/predicted value. When we think of the observed value as the prediction from the saturated model, we can see the Pearson statistic as a measurement of distance between the model in question and the saturated model.
The Pearson statistic is very similar to the deviance (D.3.3) in this way. In fact, the Pearson statistic is an approximation to the deviance. We want this one to be small if the model in question (which in this case we are taking as our null hypothesis) is to be judged to be good. The comparison to deviance makes this clear, since small deviance means that the likelihood of the model in question is close to the largest possible.
The change in the deviance is the statistic discussed in D.3.4, and it measures something that compares two different models of interest. Since the models are nested, the idea is that when you take the difference in the deviances the likelihood of the saturated model cancels out, leaving you with just the difference in the two log likelihoods of the models that you are interested in comparing.
The Likelihood ratio chi-square statistic is like the deviance in that it compares the model in question to some other standard model, but this time the model is the null model with no explanatory variables instead of the saturated model with bunches of variables. As a result, we want this one to be large in order to reject the null (which this time is the null model) and accept the model in question.
Something to keep in mind here is that the “smaller” model, the one with less parameters or less variables, is always taken as the null hypothesis model, and the “large” model is the alternative. This is why we sometimes want the statistic to be large and other times want it to be small, since our perspective on whether our model in question is the null hypothesis or the alternative changes.
Here’s a question for you: Given the likelihood ratio chi-square statistics for two models in question that are nested models, compute the change in the deviance statistic.