# Comparing two groups of proportions through the logistic normal likelihood ratio test: A bias analysis using the Laplace 6 approximation

Many cluster-randomized designs aim at comparing the proportions of individuals in different intervention groups who have a specified characteristic, such as developing a disease by a fixed point in time. This experimental design requires a methodology to compare two groups of proportions in the presence of clustering. Among the methodologies to analyze this experimental design, the Logistic-Normal Likelihood Ratio test (LNLR test) stands out in the literature for two reasons: (a) its potential to provide optimal inferences for the difference between two groups of proportions since it is a parametric test, and (b) its lack of popularity due to the drawbacks of the commonly used techniques to estimate its parameters A recently proposed estimation technique for hierarchical logistic models, the Laplace 6 approximation, has been shown in some studies to produce statistically more satisfying estimates for the parameters of these types of models than those obtained from the commonly used penalized quasi-likelihood (PQL) technique. The random-intercept logistic model whose parameter estimates are used to construct the LNLR test belongs to the family of hierarchical logistic models; but the Laplace 6 estimates of the parameters of this model have not yet been compared to those obtained from the PQL technique. Large biases or large mean square errors produced by the PQL estimators of the variance of the random components of hierarchical logistic models have prevented the exploration of the statistical properties of these types of models, and consequently the analysis of the statistical properties of the LNLR test to date In this study two statistical properties of this model are compared when estimating its parameters with the Laplace 6 approximation and the PQL technique. The Laplace 6 and the PQL estimated biases and mean square errors of the model whose parameter estimates are used to construct the LNLR test are compared in this study through Monte Carlo simulations for several scenarios of interest. These scenarios model pairs of groups of proportions typical of those of rare diseases since many prevention trials involve small proportions The considerably smaller biases obtained in this study with the Laplace 6 approximation in comparison with those obtained with the PQL technique have laid the cornerstone to further explore other statistical properties of the LNLR test such as its power and the complementary computation of sample sizes. These statistical properties and other future research projects are suggested in the conclusions of this dissertation