Bivariate correlation coefficients (BCCs) are often calculated to gauge the relationship between two variables in medical research. In a family-type clustered design where multiple participants from same units/families are enrolled, BCCs can be defined and estimated at various hierarchical levels (subject level, family level and marginal BCC). Heterogeneity usually exists between subject groups and, as a result, subject level BCCs may differ between subject groups. In the framework of bivariate linear mixed effects modeling, we define and estimate BCCs at various hierarchical levels in a family-type clustered design, accommodating subject group heterogeneity. Simplified and modified asymptotic confidence intervals are constructed to the BCC differences and Wald type tests are conducted. A real-world family-type clustered study of Alzheimer disease (AD) is analyzed to estimate and compare BCCs among well-established AD biomarkers between mutation carriers and non-carriers in autosomal dominant AD asymptomatic individuals. Extensive simulation studies are conducted across a wide range of scenarios to evaluate the performance of the proposed estimators and the type-I error rate and power of the proposed statistical tests. Abbreviations: BCC: bivariate correlation coefficient; BLM: bivariate linear mixed effects model; CI: confidence interval; AD: Alzheimer's disease; DIAN: The Dominantly Inherited Alzheimer Network; SA: simple asymptotic; MA: modified asymptotic.