### Notation and model

Consider a SNP with alleles *D* and *d* and frequencies *p* and *q* = 1 - *p*, respectively. In a case-control design, *r* cases and *s* controls are independently sampled from a population. The genotype counts of three genotypes *G*_{0} = *dd*, *G*_{1} = *Dd*, and *G*_{2} = *DD* are denoted as (*r*_{0}, *r*_{1}, *r*_{2}) in cases and (*s*_{0}, *s*_{1}, *s*_{2}) in controls, which follow multinomial distributions *mul*(*r*: *p*_{0}, *p*_{1}, *p*_{2}) and *mul*(*s*: *q*_{0}, *q*_{1}, *q*_{2}), respectively. Denote the disease prevalence as *k* and penetrances as *f*_{
i
}= P(case|*G*_{
i
}) for *i* = 0, 1, 2. By the Bayes Theorem, *p*_{
i
}= *g*_{
i
}*f*_{
i
}/*k* and *q*_{
i
}= *g*_{
i
}(1 - *f*_{
i
})/(1 - *k*), where *g*_{
i
}= P(*G*_{
i
}). Without loss of generality, assume that *D* has high risk. Then the null hypothesis of no association can be stated as *H*_{0}: *f*_{0} = *f*_{1} = *f*_{2} = *k*. The alternative hypothesis is *H*_{1}: *f*_{0} ≤ *f*_{1} ≤ *f*_{2} with at least one inequality. The genotype relative risks (GRRs) are defined as *λ*_{1} = *f*_{1}/*f*_{0} and *λ*_{2} = *f*_{2}/*f*_{0}. The recessive, additive, and dominant models are referred to as *λ*_{1} = 1, *λ*_{1} = (1 + *λ*_{2})/2, and *λ*_{1} = *λ*_{2}, respectively [2–4].

### Trend tests and robust tests

To test association using case-control data, the Cochran-Armitage trend test (CATT) has been proposed [2–4], which can be written as

{Z}_{x}=\frac{{n}^{1/2}{\displaystyle \sum _{i=0}^{2}{x}_{i}\left(s{r}_{i}-r{s}_{i}\right)}}{{\left[rsn\left\{n{\displaystyle \sum _{i=0}^{2}{x}_{i}^{2}{n}_{i}-{\left({\displaystyle \sum _{i=0}^{2}{x}_{i}{n}_{i}}\right)}^{2}}\right\}\right]}^{1/2}},

(1)

where (*x*_{0}, *x*_{1}, *x*_{2}) = (0, *x*, 1) and 0 ≤ *x* ≤ 1. Given *x*, *Z*_{
x
}follows asymptotically *N*(0,1). The choice of *x* is 0, 1/2, and 1 for the recessive, additive/multiplicative, and dominant models, respectively [5]. In practice, however, the true genetic model is unknown. Hence the robust tests, maximin efficiency robust test (MERT) and maximum test (MAX), can be applied, which are given by MERT = (*Z*_{0} + *Z*_{1})/{2(1 + *ρ*)}^{1/2} and MAX = max(|*Z*_{0}|, |*Z*_{1/2}|, |*Z*_{1}|), where *ρ* = [*n*_{0}*n*_{2}/{(*n*_{0} + *n*_{1})(*n*_{1} + *n*_{2})}]^{1/2} [4]. Note that Pearson's association test can also be used. However, Zheng et al. [6] showed that the MAX is often more powerful than the Pearson chi-squared test for a case-control design. Comparison of MERT and MAX can be found in Freidlin et al. [7]. The MAX and MERT have also been applied to other designs for GAW14 [8, 9].

### Ranking markers with multiple statistics

When the genetic model is unknown, the three CATTs (*Z*_{0}, *Z*_{1/2}, *Z*_{2}) are calculated for each of *M* SNPs. Then the *p*-values of MERT and MAX can be obtained for ranking. However, computing the *p*-value of MAX needs extensive simulation. Thus, alternatively, the minimum of the *p*-values (min *p*) of the three CATTs can be used for ranking. Rather than ranking *M* SNPs based on any single CATT, we propose ranking the SNPs by the MERT and the minimum of the *p*-values. We expect that ranking SNPs based on this approach would be more robust compared to ranking by a single CATT when the ranks by the three CATTs are quite different.