This study is motivated by population-based family data collected from National Genetic Household Surveys (NGHS), in which complex samples (i.e. sample collected with stratified multistage cluster sampling), partially from the same family, are selected. There are two levels of correlations of the collected genetic data in NGHS. The first level of correlation is induced by the hierarchical geographic clustered sampling of households and the second level of correlation is induced by biological inheritances from individuals sampled in the same household. Moreover, national household surveys often apply differential selection probabilities of selecting households and persons within the sampled households.

NGHS from various countries, such as the Canadian Health Measures Survey [3], Health 2000 Survey from Finland [4] and the US National Health and Nutrition Examinations Survey [5], have collected blood samples, from which DNA can be extracted for genetic analyses. For example, the NHANES III, a nationally representative survey of the U.S. population conducted by National Center for Health Statistics (NCHS), has genotyped candidate genes for participants 12 years and older. NHANES III employed a complex sample design, involving stratified multistage cluster sampling, to select participants [6, 7]. Multiple blood-related individuals are often sampled from the same household. On average 1.6 persons are sampled per household [8].

There are at least three complications to analyze data collected in NGHS: 1) differential population weights, 2) hierarchical geographical correlation among families, and 3) genetic correlation within families. As known, genetic variability can differ by race [9] and social-economic factors [10]. The same factors, in complex sample designs, are often used to define sampling strata. Different selection probabilities are applied to each stratum and oversampling is often conducted to increase the efficiency of the estimates for certain subgroups (e.g. African Americans were oversampled in NHANES III). As a result, *differential population weights* related to genetic variability are associated with participants in the study. The second complication of *geographic correlation* in genetic variations is induced because race and social-economic status of individuals vary by location of residence, and multi-stage clustered sampling is often geography-based. Lastly, g*enetic correlation* is induced by genetic inheritance from biological parents within the household. Therefore, inflated variances of the estimates for the genetic quantities due to differential population weights, geographical intracluster correlation among households, and the genetic correlation within households are resulted. Consequently, the sample distribution with respect to genetic factors can be considerably different from the underlying population distribution. Researchers who implement complex sample designs should perform analyses with adjustments for the sample design complications. If analysis for simple random samples (SRS), instead, is performed on data collected with complex sample designs, the inference may be invalid.

Testing Hardy-Weinberg Equilibrium (HWE) of marker genotype frequencies has been widely recommended as a crucial step in genetic association studies [11–13]. The Hardy- Weinberg principal states that without disturbing genetic-related factors (e.g., non-random mating, selection, migration, or mutation) the genotype frequencies at an autosomal locus will attain equilibrium (i.e., HWE) in a single generation and maintain this equilibrium in future generations. Testing for HWE is useful because for non-codominant loci that are in HWE, genotype frequencies can be estimated from allele frequencies or vice versa, and more powerful genetic association studies are possible [14]. A variety of test methods have been developed to test HWE in SRS [15–20]. There are, however, limited literatures in developing HWE tests for genetic data collected in NGHS, considering both levels of correlations and differential weights.

Methods developed by She et al. [1] and Li et al. [2] appears to be the only two HWE tests available that consider all of the three complications. Interestingly, both methods are derived along different directions. In brief, tests developed by She et al. [1] are essentially corrected Pearson Chi-Square (CCS) tests. The exact joint distribution of the three genotypes from father, mother, and offspring were derived under HWE. Assuming a diallelic locus (i.e. alleles *A* and *a*) of autosomal genes, there are 10 possible genotype combinations among parents-child triads when ignoring the mating orders of father and mother. If HWE holds, the observed number of families in each of the 10 categories follows a multinomial distribution. Departure from HWE was then tested with the corrected Pearson chi-square statistics, including first-order correction, second-order correction, or a Satterthwaite F-version second-order correction [21, 22]. By contrast, Li et al. [2] proposed a quasi-score (QS) test based on quasi-generalized estimating equations (GEE). Different from the regular GEE where the covariance matrix are often unknown, covariance matrix of the observed genotypes in the quasi-GEE are derived from the condensed coefficients of identity (CCI) [23], which appropriately measures the second level of genetic correlation among family members within families. The first level of correlation due to hierarchical geographic sampling of families is considered in the variance estimation of the estimated quasi- scores with respect to the fixation index (correlations between any pair of alleles within individuals, characterizing the departure from the HWE) using Taylor linearization methods.

In Li *et al*. [2], the QS method is claimed to advantage over the CCS method because the CCS method is limited to one type of family structure, i.e., 2 parents and 1 offspring (2P1O), while the QS method allows for a wide range of family structures. Trio (2P1O), however, is one of the common study designs in genetic association studies, e.g. transmission disequilibrium test. For the analysis of genetic data from 2P1O, Li et al. [2] didn’t make thorough comparisons between the two methods analytically or numerically. In addition, the QS and the CCS methods appear to be the only two existing methods that take account of the two levels of clustering effects induced by the national genetic household surveys (NGHS) (i.e. geographic correlation among families within PSUs and genetic correlation within families). Which method should be recommended in the analysis of NGHS genetic data collected from 2P1O families? For survey practitioners, it is important to select a proper test that maintains the nominal levels, but with higher power.

In this paper, we examine and compare the performance of two methods, in terms of the sizes and powers, via Monte-Carlo simulation studies under a variety of complex sample designs with differential weighting schemes and varying magnitudes of the correlation effects. It is observed that the QS method maintains the nominal size relatively well and consistently achieves higher power than the CCS method in testing HWE using data collected from 2 parents and 1 offspring in NGHS. In Section “Methods”, we outline the detailed methodology of the CCS and the QS tests. Both methods are compared in Section “Results” via simulation studies on the finite sample performance and applied to a real data example from the Hispanic Health and Nutrition Examination Survey with generated genotype data. Finally, the paper is wrapped up in Section “Conclusions”.