Estimated associations between an outcome variable and misclassified covariates tend to be biased when the methods of estimation that ignore the classification error are applied. Available methods to account for misclassification often require the use of a validation sample (i.e, a gold standard). But in practice, such gold standard may be unavailable or impractical. We propose a Bayesian approach to adjust for misclassification in a binary covariate in fixed and random effect logistic models when a gold standard is not available. This Markov Chain Monte Carto (MCMC) approach uses two imperfect measures of a dichotomous exposure under the assumption of conditional independence and non-differential misclassification. This approach is validated with several simulation studies. We illustrate the proposed approach to adjust for misclassification with respect to oxygenation status in a multi-center trial of patients with pneumonia, where 16 per cent of patients are classified discordantly by two assessments. The estimated log odds of inpatient care and the corresponding standard deviation are much larger in our proposed method compared to the models ignoring misclassification. We also applied the proposed Bayesian method to the EDCAP trial to assess the intervention effect allowing for misclassification with respect to risk status. Ignoring misclassification produces downwardly biased estimates and underestimates uncertainty. The public health significance of this study is that the proposed approach can correct for the bias of an estimated association when a covariate is misclassified and no gold standard is available, which is common problem in epidemiology studies.