Antecedentes y consecuentes de la cohesión grupal

The purpose of Chapter 3 is to show that our methods can be extended to more complicated models than were demonstrated in Chapter 2. In Chapter 2, the model studied was basic; it featured a binomial response with a random effect. We had two ways to view the model: one as the sum of independent Bernoulli random variables where the only explanatory infor- mation was at the group level, and not at the individual Bernoulli unit level. The other way was to view a binomial random variable with a random effect as being the level of the obser- vational unit. We recognized that to expand the model depends on which interpretation is initially used. If we used the sum of Bernoulli random variables version, the extension could include an explanatory variable at the level of the observational unit. If, instead, we were to view the response as binomial random variable with a random effect for each observational unit then extension can only occur by adding a higher level to the model’s hierarchy. While both extensions are considered in the thesis, in Chapter 3 we focused on adding an additional layer. We extend using the alternative approach in Chapter 4.

The additional layer increased the number of variables in our model, which necessitated the need for additional prior specifications. Posterior distributions, approximate and exact, were easily sampled using Gibbs sampling. All the conditional distributions, save one, were recognizable. Instead of using an advanced sampling technique such as adaptive-rejection sampling or slice sampling to sample this unknown distribution we proposed tw normal approximations.

We tested this method of using a Gibbs sampler with the normal approximation on the Hospital, Doctor, and Patient (HDP) Dataset. The results showed that our method is not unreasonable compared with JAGS. We further compared the two methods using simulated data where our approximate methods preformed as well as JAGS.

mensional distributions. In Chapter 2, when we encountered this issue, we developed the SR method. The SR method is obtaining what we called sufficient reduction components that greatly reduce the dimension of what needs to be sampled. The SR method, like the normal approximation method, produced results similar to those obtained using JAGS. The results from the HDP data were reasonable, but they were much improved in the simulated data they were much improved, provided the number of observational units per cluster was not appreciably larger than the number of clusters.

Even though we expanded the model, both of our approximate methods were successful. The results each method produced are encouraging. However, our methods are still limited by the virtue of the assumed model being constrained in that we could not include patient level covariates in our analyses in this chaptaer. In Chapter 4, we demonstrate the final extension of our model so that it can potentially be used in any GLMM setting.

Chapter 4

Logistic Regression with Fixed Effects

at Base Level

4.1

Introduction

Chapter 4’s purpose, like Chapter 3’s, is to demonstrate how our method can be used in a GLMM setting. In Chapter 2, we initially tested our method on one of the most basic GLMM: logistic regression with a random effect per observational unit. In Chapter 3, we extended the model and showed how our method can accomodate such an extension. While our ultimate goal is to extend our method to accommodate the entire class of GLMMs in this chapter we focus on one aspect of this.

The model we focused on in Chapter 2 featured a logistic regression problem. In Chapter 3 we referred to the model in Chapter 2 in one of two equivalent ways: First as a Bernoulli response without any main effects but we did include cluster level ones. Then we modeled the sum of the Bernoulli responses as binomial with a random effect on each binomial. Secondly as binomial response with a random effect at each observation. Extending the model with

higher levels was accomplished in Chapter 3. But if we view the model in the hierarchical, the first way, there is another way to extend the model- include fixed effects at the level of the observational unit.

This extension may seem trivial, but it is not. Previously, we were able to merge the fixed effects at each level into the random effect at that level (assuming normal random effects with zero mean implies that the addition of fixed effects makes the random effect normal with mean at the fixed effect with the variance equal to the random effect’s variance). This interpretation was critical to our method it allowed us to use an asymptotic approximation within the Gibbs sampler. With explanatory data at the observational unit level the cluster level’s random effect cannot just absorb the information. Instead, we will need to reconsider the form of the model and adjust our methods accordingly.