## Introduction

With the large amount of genetic variants that are available for use in association analyses nowadays, association studies, designed using either population data or familial data, have been proved useful in mapping of genes underlying complex diseases (Kruglyak & Lander, 1995; Risch & Merikangas, 1996; Schork et al., 2001; Cordell & Clayton, 2005). Traditional population-based studies are easier to collect data in practice, but often challenged for the population stratification problem that causes spurious results. To adjust for population stratification, methods using information of unlinked markers, for example, were proposed by several researchers (Devlin & Roeder, 1999; Pritchard & Rosenberg, 1999; Reich & Goldstein, 2001). Instead of use of statistical methods to adjust for population stratification in population-based association studies, on the other hand, family-based association studies avoid the stratification problem by use of the matched data structure generated from the familial data. For example, for case-parent triad families a matched case-pseudocontrol data structure can be created (Falk & Rubinstein, 1987). Based on this data structure a family-based association test, the transmission/disequilibrium test (TDT), was proposed to avoid the stratification problem (Spielman et al., 1993). Methods extending the TDT to allow for various situations, such as multi-allele locus (Sham & Curtis, 1995; Bickeboller & Clerget-Darpoux, 1995; Kaplan et al., 1997), missing parental data (Spielman & Ewens, 1998; Horvath & Laird, 1998; Weinberg, 1999), or genotyping errors (Gordon et al., 2001; Douglas et al., 2002), have been widely proposed. Pros and cons of the population-based and family-based association approaches have been extensively discussed in the literature (Gauderman et al., 1999; Teng & Risch, 1999; McGinnis et al., 2002; Tabor et al., 2002; Cardon & Palmer, 2003).

By specifying the genetic model (mode of inheritance) of a disease, Schaid & Sommer (1993) showed that the TDT is identical to the score test derived from a conditional likelihood under the additive model. Because score tests are powerful in local alternatives (Cox & Hinkley, 1974), the TDT can be regarded as an optimal test for the additive model. Analogously, for dominant and recessive models, the optimal score tests can also be derived based on the corresponding conditional likelihoods. In practice, however, since the genetic model for a complex disease under investigation is usually unknown, developing a robust test that has relatively stable power over all plausible genetic models is thus required.

For binary disease traits, Zheng et al. (2002) developed TDT-type robust association tests using case-parent triad data. When dealing with complex diseases, the phenotype of an individual is likely to be measured as a quantitative trait, such as bone mineral density used in diagnosis of bone disorders (Deng et al., 2002) and bronchial responsiveness or the numbers of eosinophils in airway tissues used in allergic asthma studies (Zhang et al., 1999). For these quantitative traits, several association tests have been proposed by researchers (Allison, 1997; Rabinowitz, 1997; Xiong et al., 1998; Abecasis et al., 2000; Monks & Kaplan, 2000; Sun et al., 2000; Liu et al., 2002; Alcais & Abel, 2003; Diao & Lin, 2006). In particular, the likelihood-based QTDT/orthogonal model proposed by Abecasis et al. (2000) and the TDT-based method proposed by Monks & Kaplan (2000) are feasible in wider family structures than other methods and are widely used (Li et al., 2008). Along with the idea of Li et al. (2008), the two tests are allele-based association tests which means that they set genotype scores according to the count of alleles and so provide evidence of association for an allele at a marker locus. Thus they are expected to have powerful performances for the additive model rather than for other models. In practical studies, when a genetic model cannot be certainly specified, we may consider the additive model as an intermediate solution. However, such a measure still cannot avoid suffering the potential loss of power in some situations. A method that is robust against the influence on testing power due to misspecification of genetic models is therefore required. In this paper, we will apply two robust procedures to establish robust candidate-gene association tests for quantitative traits using parent-offspring triad data.

We will first demonstrate the feasibility of using conditional likelihood of parent-offspring triad data to extract association information between a candidate-gene and the quantitative trait, and then derive the optimal score tests under three typical genetic models. Based on the robust procedures suggested by Gastwirth (1966, 1985); Davies (1977) and Freidlin et al. (1999), we will construct two robust statistics for assessing putative association between a candidate gene and a quantitative trait. The quantitative trait under investigation is assumed to have a distribution which belongs to the exponential family rather than a normal distribution. Statistical powers of the proposed robust tests are compared with the optimal score test under the correct model and with the score tests under incorrect models. According to the simulation results, the proposed robust tests do exhibit robustness with acceptable powers.