Medical Journal of the Islamic Republic Of Iran

en Hierarchical Bootstrapping for Quadratic Exponential Models in Genetic Epidemiology: Familial Aggregation Analysis with Small Samples and Unequal Family Sizes Biostatistics and Epidemiology Biostatistics and Epidemiology Original Research Original Research     Background: Familial aggregation analysis evaluates whether disease clustering within families exceeds chance expectations. A key challenge is irreproducibility, which arises when a model cannot be consistently applied to families with different family sizes. Although the quadratic exponential model is widely used for correlated binary outcomes, its performance is affected by irreproducibility and small-sample bias.    Methods: This study compares the marginal approach, designed to address irreproducibility, with a hierarchical bootstrap procedure. Simulations were conducted under exchangeable and non exchangeable dependence structures using the quadratic exponential framework, focusing on samples with unequal family sizes. Performance was evaluated using RMSE and empirical coverage. The two methods were also applied to data from the Tehran Eye Study.     Results: In exchangeable settings, both methods produced reasonable estimates for baseline disease odds; however, the hierarchical bootstrap provided more stable inference. It also performed better in estimating the incremental effect associated with an additional affected relative (γ) in samples of 200 families (coverage: 94.8% vs 92.5%). In non-exchangeable scenarios, the bootstrap approach showed better performance for key parameters, including conditional child–parent odds (95.1% vs 93.5%) and γ (93.9% vs 93%). Results from the Tehran Eye Study were consistent with these findings (93.9% vs 93.6% for γ and 94.7% vs 92.6% for<m:omath><m:r> </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>γ</m:r></m:e><m:sub><m:r>M</m:r></m:sub></m:ssub></m:omath> ).   Conclusion: These findings suggest that hierarchical bootstrapping may provide a useful alternative for inference in familial aggregation studies with heterogeneous family sizes.   Familial aggregation, Hierarchical bootstrap, Quadratic exponential model, Irreproducibility, Tehran Eye Study 970 977 http://mjiri.iums.ac.ir/browse.php?a_code=A-10-10190-1&slc_lang=en&sid=1 Mohammad Hassan Rakhshani Rakhshanimh@medsab.ac.ie 2003194753284600100964 2003194753284600100964 No Department of Biostatistics and Epidemiology, School of Health, Sabzevar University of Medical Sciences, Sabzevar, Iran Hojjat Zeraati zeraatih@tums.ac.ir 2003194753284600100965 2003194753284600100965 No Department of Epidemiology and Biostatistics, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran Hassan Hashemi hhashemi@noorvision.com 2003194753284600100966 2003194753284600100966 No Noor Research Center for Ophthalmic Epidemiology, Noor Eye Hospital, Tehran, Iran Akbar Fotouhi afotouhi@tums.ac.ir 2003194753284600100967 2003194753284600100967 Yes Department of Epidemiology and Biostatistics, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran