Assessing Goodness-of-Fit Tests Based on Pairwise Concordant Marginal Information for Generalized Linear Mixed Models Under Second-Order Serial Error Dependence

Traditional goodness-of-fit tests for binary longitudinal data can perform poorly when response-pattern tables are sparse. Concordant information-based tests mitigate this issue by using lower-dimensional marginal information, but their performance under second-order serial dependence has not been systematically examined. Building on this framework, we first discuss a third-order concordant marginal formulation and identify an important limitation in the binary setting: for binary responses, we show that each third-order concordance residual is exactly one half of the sum of the three corresponding pairwise concordance residuals. Thus, the third-order concordance formulation contains no additional information beyond the pairwise concordance residuals, and its rank deficiency in larger binary designs follows from this redundancy. We therefore focus on the second-order concordance statistic and related limited-information diagnostics under AR(2) and MA(2) error structures. Specifically, the AR(2) simulations cover all parameter pairs on the specified grid that satisfy the stationarity conditions, whereas the MA(2) simulations include all 162 grid points satisfying |ϕ1|+|ϕ2|≤0.9 and ϕ2≠0, which form a symmetric subset of the invertible parameter region. We assess Type I error rates and empirical power across different sample sizes and dependence configurations. The results show that second-order serial dependence, especially negative dependence patterns, can substantially affect test performance. These findings clarify the relative performance of concordance-based diagnostics under the AR(2) and MA(2) parameter settings examined in this study.