Bayesian analysis updates inference as more data becomes available. Typically, Bayesian inference uses simulation approaches such as Markov Chain Monte Carlo (MCMC) but an approximation approach, the Integrated Nested Laplace Approximation (INLA), is also available. Although the simulation-based methods are theoretically accurate, they can be computationally expensive. The goal of the study is to compare INLA and two MCMC algorithms (in the software JAGS and STAN) using ATTACC/ACTIV-4a trial data of patients who were hospitalized for Covid-19 but not critically ill. By fitting Bayesian hierarchical generalized mixed models with categorical, binary and time-to-event outcomes, the posterior distributions of the treatment effect are compared. INLA requires noticeably less computational time compared to STAN and JAGS (seconds compared to hours). All the 95% CIs for the treatment effect estimated using INLA overlapped with the simulation-based methods.