A four-parameter beta-binomial model is developed as a competitor to the three traditional binomial-error models under the strong true-score theory.The developed model (Model 3) may be considered a reduced form of Lords two-term approximation to the compound binomial-beta model (Model 4), as well as a more general form relative to Carlin and Rubins three-parameter beta-binomial model (Model 2) and the standard beta-binomial model (Model 1).Solutions are derived for the parameter estimates for Models 3 and 4.Formulas under Models 2, 3, and 4 are obtained for three Bayes quantities, the Bayes point estimator for true score (the nonlinear regression function of true score ζ on observed test score x), the variance of the posterior distribution of ζ | x, and the Bayes risk of the point estimator.In addition, the chain effect of the compound-binomial function on the jagged regression estimates and negative values of posterior variance for lower test scores in the setting of Model 4 are explored.Model 3 fits well for 66 out of 75 data sets having a variety of test score distributions.Results indicate that Model 3 is capable of performing as well as Model 4, and both models yield substantial improvement in fit over Models 1 and 2.The empirical Bayes point estimates for the true scores in Model 3,(δ)N((ψ)3, X , n), are monotonic in test scores, which is an advantage of Model 3 over Model 4.This study shows that the convergence of ENW((δ)N ((ψ)3, x, n)) → W(δ G (Ψ3, x, n)) is a function of test length (n) and sample size (N).Guidelines for constructing PEB bootstrap confidence intervals for the mean function and for a particular ζ | x are also given.