Maximum Approximated Likelihood Estimation in Generalized Linear Multilevel Model for Nominal Response with Covariates Subject to Measurement Error

Ignoring measurement errors in generalized linear mixed models (GLMMs) as well as other regression models will inevitably lead to significant biases, deviations, and incorrect inferences in the estimation of the parameters. In the presence of measurement error, numerous approaches have been proposed to rectify this issue. Furthermore, the application of the frequentist approach of GLMMs is intricate due to the emergence of an intractable numerical integration process. In this study, the complete likelihood inference of a multinomial logit random effects model with covariate measurement error in conjunction with replicate measures is suggested via the multivariate Gauss-Hermite quadrature approximation. To achieve this objective, the likelihood function will be evaluated and approximated for two distinct scenarios; the proposed method which incorporates the classical additive structural measurement error model for the error-prone covariate and the naive method. We will describe and compare different upshots of parameter estimation in these two different situations. The results of performing the proposed method, assessed through simulation, show that the proposed method performs well when correcting for measurement error in terms of bias, empirical standard error, root of mean squared error and coverage ratio. The application of the proposed method is further highlighted with real-world data based on a multilevel study concerning the prevalence of contraceptive methods used by women in Bangladesh.