This study investigated the use of different binomial logistic models as alternatives to the normal model when analysing non-normal aggregate outcomes that are sums of correlated binary responses. The outcome variables provided in the two illustrative examples were preschoolers’ uppercase and lowercase letter naming knowledge with different shapes of non-normal distributions. The binomial, beta-binomial, and mixed binomial models with logit links were examined and compared to each other and to the normal linear model. Results were consistent in both examples. Among the models compared, the beta-binomial and mixed binomial models with overdispersion parameters captured interdependence among correlated binary responses. In addition, the mixed binomial model further explained remaining overdispersion and best fitted the data. Implications including advocating for the use of the binomial models with overdispersion parameters for clustered data were further discussed.