Ordered categorical data summarized in a 2 x K table usually consist of two-sample multinomial or K-sample binomial observations. In analysing these data, we usually assign scores to the K columns and perform a testing for the equality of two multinomial distributions in the former case and no trend among K binomial proportions in the latter case. Among the most popular score tests are the Wilcoxon rank sum test and the Armitage's linear trend test. In this paper we extend the score tests to be used for clustered data under diverse study designs. Our methods do not require correct specification of the dependence structure within clusters. The proposed tests are based on the asymptotic normality for large number of clusters and are a generalization of the standard tests used for independent data. Simulation studies are conducted to investigate the finite-sample performance of the new methods. The proposed methods are applied to real-life data.