Generalized Linear Mixed Model In Longitudinal Data With Binary Response

Longitudinal data with binary outcomes are widely used in medicine, psychology, social science and other fields, and particularly are commonly seen in clinical studies of experimental medicines. This type of data have two characteristics:first, the data do not follow normal distribution; second, the data from the same individual at different time points are correlated. Because of these, the traditional statistical methods cannot be used to analyze longitudinal data with binary outcomes. Currently, there are two popular methods to analyze longitudinal binary data, namely the method of generalized estimating equations (GEEs) and the method of generalized linear mixed models (GLMMs). There has been relatively mature research as evidenced in literature on the method of generalized estimating equation, but there is not much research conducted on generalized linear mixed models. Particularly, there is only limited literature on parameter estimation methods under generalized linear mixed models and the research on this topic is not complete..Objective:To compare the pros and cons of various parameter estimation methods under generalized linear mixed models and to explore the impact of sample size, covariance structures, and mechanisms and proportion of missing on various parameter estimation methods.Method:Using Monte Carlo simulation, we employ the following Ivaluation indexes to compare the pros and the cons of various parameter estimation methods in GLMMs:bias, mean square error(MSE), average (over time) mean square error(AMSE), maximum (over time) of mean square error(MMSE), coverage probability of 95% confidence interval; and we compare various parameter estimation methods, considering different sample sizes, different covariance structures, or different mechanisms and proportion of missing. In the end, we apply the findings from simulation to solve a real-world problem, which comes from a clinical trial with longitudinal binary data.Result:When sample sizes are large and there are no missing data, numerical integration approximation methods, when compared to other estimation methods, lead to smaller bias and higher coverage probability, as well as smaller MSE, AMSE and MMSE, regardless whether covariance structure of the data is compound symmetry or unstructured correlation. That is, numerical integration approximation method is more accurate and stable in the analysis of longitudinal binary data with large sample size. The advantage of numerical integration approximation method is also more obvious in the analysis of longitudinal binary data when the variance of random effect is relatively large (≥1). When the variance is relatively small (<1), different methods lead to very similar bias, and coverage probability of 95% confidence interval is also roughly the same.The findings from large sample size do not apply to the case of small sample size. In the analysis of longitudinal with binary outcomes of small sample size, RSPL and MSPL methods are more stable, and RMPL and MMPL methods lead to higher coverage probability of 95% confidence interval, when compared to other methods. Estimation for the variance of a random variable obtained by linearization methods seems to be more accurate. All these indicate that linearization methods are more suitable in the analysis of longitudinal binary data with small sample sizes.The robustness of various parameter estimation methods to covariance structure may depend on sample size too. In the case of small sample sizes, MSE, AMSE and MMSE obtained by RSPL and MSPL methods are smaller than those obtained by other methods. The proportion of positive definite of G matrix obtained by RSPL and MSPL is higher than that obtained by other methods. Therefore, the robustness of RSPL and MSPL methods is better in the analysis of longitudinal binary data with small sample sizes. When sample sizes are relatively large, numerical integration approximation methods are better than linearization methods. The bias obtained by numerical integration method is smaller and coverage probability of 95%confidence interval is higher than that obtained by other methods. So the robustness of numerical integration approximation methods is better in the analysis of longitudinal binary data with large sample sizes.When the data contains the missing values and when missing is completely at random or missing at random, numerical integration approximation methods are more accurate and stable than other methods when missing is a small proportion in whole data. In the case of large proportion of missing, the bias obtained by RSPL and MSPL methods is smaller than that obtained by other methods, and the coverage probability of 95% confidence interval is higher by RSPL and MSPL as well. These indicate that linearization methods are more accurate and stable in the case of small proportion of missing.In the case study, because the sample size is large and the proportion of missing is low, numerical integration approximation methods should be the best parameter estimation method. And there seems to be no significant differences by various numerical integration methods in estimating log (Odds Ratio) between two groups as well as the corresponding 95% confidence intervals.Conclusion:When using generalized linear mixed methods to analyze longitudinal data with binary outcomes, we should choose appropriate parameter estimation method based on sample size, covariance structure and missing data. When sample sizes are large and there are no missing data or the proportion of missing is low, numerical integration approximation methods perform better than linearization methods; On the other hand, linearization methods are better when sample sizes are relatively small. When proportion of missing is high, RSPL and MSPL are more accurate and stable in the analysis of longitudinal data with binary outcomes than other methods.