The standard error from the collapsed data is 0.34%, misleadingly smaller than that of the beta-binomial estimation of 0.59%.
In this study, we followed advice from the literature: in the case of negative binomial models, estimation of the posterior mean of the dispersion parameter can be affected when a vague prior specification is used to characterize the gamma hyper-parameter.
Table 2 shows the crude and adjusted binomial regression estimations of the association between HLB and the outcome.
The meta-analysis carried out under the Bayesian framework, on the 18 countries and applying binomial likelihood, allowed the estimation of the uncertainty around the between country heterogeneity (τ = 1.73 (95% credible intervals: 0.98, 3.47).
The Nadaraya-Watson regression applying the binomial kernel for the estimation of the functions is (5) g ^ j (x j l ) = ∑ i = 1 n ∑ k = 1 2 λ q − d (x j l, x i j k ) (1 − λ ) d (x j l, x i j k ) y i ∑ i = 1 n ∑ k = 1 2 λ q − d (x j l, x i j k ) (1 − λ ) d (x j l, x i j k ).
The slight outperformance in the negative binomial model relative to the CMP/geometric model (based on log-likelihood comparisons) stems from the negative binomial estimation procedure's allowance for real θ, thus obtaining a more precise estimation of the data over-dispersion.
In this first step – making abstraction of the eventual sample selection bias – we compare the adjusted R-squared from a binomial probit estimation of Eqn. (1), with and without including the set of dummies for the type of mismatch between the education level and the occupational attainment in the last job held in the home country twelve months prior to immigration.
There are other detailed mathematical equations involved in Bayesian estimation of the beta-binomial model for specific cases.
Therefore, the application of a compound distribution —beta-binomial distribution allows estimation of occurrence probability for the summary event rate of chronic diseases/NCDs for the older population across households (39).
After formulating the partial log-likelihood [Equation (1)] of the observed data b, the ML estimation of the parameters is a binomial regression with a CLL link function.
