#### Definition of Test Statistics

We focus first on situations where there are no covariates to be adjusted for. Assume that there are *r* cases and *s* controls in a CCGAS, and that there are two alleles, G and g, at a given SNP locus with the possible genotypes gg, Gg, and GG. The notations for genotype counts in the case and control groups are given in Table 1. Based upon Table 1, the general form of the CATT can be written as

- (1)

where is a genotype score vector for the coding of genotypes gg, Gg, and GG, and , *i*= 0, 1, 2. The genotype score vector *x* is chosen by the investigator. It should be pointed out that the CATT given by (1) is equivalent to the score statistic testing for the null hypothesis derived from the following standard logistic regression that models the effect of genotypes represented by *x*,

- (2)

Based upon (2), the three commonly assumed genetic models–recessive, additive, and dominant–correspond to the following assignments of the genotype score vector *x*: , , and , respectively. Among the three CATT tests, *Z*_{x=A}, called CATTA, which is derived according to an additive model, is usually preferred over *Z*_{x=D} and *Z*_{x=R}, as it does not rely on the assumption of a high-risk allele (assuming a two-sided test is performed), and thus this is the version of CATT that is generally used in CCGAS. The p-value for *Z*_{x} can be obtained according to the standard normal distribution.

Table 1. Notation for genotype frequencies. | gg | Gg | GG | Total |
---|

Case | *r*_{0} | *r*_{1} | *r*_{2} | *r* |

Control | *s*_{0} | *s*_{1} | *s*_{2} | *s* |

Total | *n*_{0} | *n*_{1} | *n*_{2} | *n* |

If the true underlying disease model is known, the CATT test *Z*_{x} is the most efficient. But in reality, the true disease model is unknown. For a more robust test that enjoys a good performance over a wide range of disease models, the following test statistic, called MAX, has been proposed (Freidlin et al., 2002; Sladek et al., 2007; Li et al., 2008a, 2008b):

There are several ways to evaluate the significance level of MAX. For example, the multiple-integration procedure, which is available in R, can be used (Conneely & Boehnke, 2007), and it is computationally feasible in the context of GWAS. A more computationally challenging approach is through a permutation procedure (Sladek et al., 2007). Li et al. (2008a) derived an analytic upper bound that is reasonably accurate for small p-values.

Another robust test is the 2-df χ^{2} test. Using the notations listed in Table 1, we can define the Chi-2df test as

- (3)

The significance level of the Chi-2df test can be evaluated through the 2-df χ^{2} distribution.

#### The Formula for Power Calculation

Under a given disease model, we denote the expected genotype frequencies of (gg, Gg, GG) for cases and controls as (*p*_{0}, *p*_{1}, *p*_{2}) and (*q*_{0}, *q*_{1}, *q*_{2}), respectively. The analytic power calculation for CATTA *Z*_{A} can be found in Freidlin et al. (2002) and Pfeiffer & Gail (2003).

Under the disease model with the expected genotype frequencies (*p*_{0}, *p*_{1}, *p*_{2}) and (*q*_{0}, *q*_{1}, *q*_{2}) in cases and controls, respectively, asymptotically follows a multinormal distribution with mean vector and covariance matrix . The mean vector is given by

- (4)

with the score vector*x* chosen as (0, 0, 1), (0, 0.5, 1), and (0, 1, 1) for , , and , respectively, and with . The definition for the covariance matrix is more complicated and is presented in the Appendix, along with its detailed derivations.

The power of the MAX test for the alternative hypothesis **H**_{1} can be written as

- (5)

where and is the covariance matrix.

#### Power Comparison

We assume that the case and control sample sizes are *r*=*s*= 1, 000. We first conduct the comparison under a single-marker disease risk model. We let the minor allele frequency (MAF) *f* for a particular SNP in the study population be in the range of 5–50%. For the MAF = f, we let the genotype frequencies of (gg, Gg, GG) in the control population (*q*_{0}, *q*_{1}, *q*_{2}), have the values (*q*_{0}, *q*_{1}, *q*_{2}) = ((1 −*f*)^{2}, 2*f*(1 −*f*), *f*^{2}). This is reasonable for the study of a rare disease in a source population where Hardy-Weinberg equilibrium holds. Let the odds ratios (ORs) for having 1 copy and 2 copies of the high-risk alleles be *R*_{1} and *R*_{2}, respectively. We have *R*_{2}=*R*^{2}_{1} > 1 for an additive model (in the logit scale), *R*_{2}=*R*_{1} > 1 for a dominant model, and *R*_{2} >*R*_{1}= 1 for a recessive model. Given (*R*_{1}, *R*_{2}), we know that the genotype frequencies of (gg, Gg, GG) in the case population (*p*_{0}, *p*_{1}, *p*_{2}) are (*q*_{0}, *q*_{1}*R*_{1}, *q*_{2}*R*_{2})/(*q*_{0}+*q*_{1}*R*_{1}+*q*_{2}*R*_{2}).

In addition to the single-marker disease risk model, we compare the power of the three single-marker tests under the following 2-marker haplotype risk model. Suppose the disease risk is conferred by haplotypes consisting of two linked markers, with marker #1 having allele types B and b, and with marker #2 having allele types C and c. We designate the haplotype BC as the high-risk variant (corresponding to the high-risk allele in the single-marker risk model). As with the single-marker risk model, we can define the haplotype risk model as dominant, recessive, and additive. For example, if *R*_{1} and *R*_{2} denote the ORs for having one copy and two copies of the high-risk haplotype, respectively, we have *R*_{2}=*R*^{2}_{1} for the additive haplotype risk model. To simplify the power comparison setup, we let *p*_{1}be the BC haplotype frequency in the study population and assume the other three 2-marker haplotypes have the same haplotype frequency. We further assume the independence of the two haplotypes within a subject in the study population. In the Appendix, we provide the formula for calculation of (*p*_{0}, *p*_{1}, *p*_{2}) and (*q*_{0}, *q*_{1}, *q*_{2}), the genotype frequencies of (bb, Bb, BB) in the case and the control populations, respectively.

Figure 1 shows the power curves of the above-considered association tests under each of three commonly assumed single-marker risk models (additive, dominant, and recessive) at a significance level of 0.05. From Figure 1, we can see that MAX is always more powerful than Chi-2df, and in some cases it is associated with up to a 5% power increase. Comparing CATTA with MAX, we see that CATTA is slightly more powerful than MAX under the additive model, but in most cases the advantage is negligible. Under the recessive model, MAX (as well as the Chi-2df) is noticeably superior to CATTA. Under the dominant model, it is interesting to see that neither CATTA nor MAX dominates the other. CATTA is more favorable when the risk allele is relatively rare, while MAX becomes more attractive as the risk allele frequency gets larger.

In addition to the three commonly used disease models, we also compared the power under a single-marker risk model with all possible combinations of two odds ratios *R*_{1} and *R*_{2}, with each ranging from 1 to 1.5. Figure 2 summarizes the power comparison results. Clearly, there is no test that can outperform the others in all of the single-marker risk models considered. When the risk allele is relatively rare (say, MAF less than 0.2), all three tests have comparable power under various single-marker models, although CATTA outperforms the others in most of the (*R*_{1}, *R*_{2}) region. As the risk allele gets more common, CATTA becomes less powerful than both MAX and Chi-2df under the single-marker risk model when*R*_{1} >*R*_{2}, although whether this kind of disease risk model is reasonable is debatable. MAX and Chi-2df have similar performances under all the considered choices of risk models and MAFs, with MAX performing more favorably under the risk model when *R*_{1} < *R*_{2}, and less favorably when *R*_{1} >*R*_{2}.

Power comparison results under the 2-marker haplotype risk models are given in Figure 3. Similar to what we observed in Figure 1, MAX appears to have the most robust performance. Although MAX is slightly less powerful than CATTA under the additive haplotype risk model, it has a noticeable power advantage over CATTA (more than 10% higher) under the dominant haplotype risk model. Also, from Figure 3, we notice that MAX is mostly better than the Chi-2df, although the percentage increase in power is limited.