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\begin{center}
{\Large Supplementary material to\\
{\bf Genetic model selection in two-phase analysis for case-control association studies}}

\vskip 0.2in
GANG ZHENG\\
{\it Office of Biostatistics Research, National Heart, Lung and Blood Institute,\\ 
6701 Rockledge Drive, Bethesda, MD 20892-7931, U.S.A.}\\
zhengg@nhlbi.nih.gov

\vskip 0.1in
HON KEUNG TONY NG\\
{\it Department of Statistical Science, Southern Methodist University,\\ 
Dallas, TX 75275-0332, U.S.A}\\
ngh@mail.smu.edu
\end{center}

\baselineskip 20pt
\vskip 0.5in
\centerline{SECTION A: DETERMINING THE GENETIC MODELS}

For a single-nucleotide polymorphism (SNP) with alleles $A$ and $B$, the Hardy-Weinberg disequilibrium coefficient is $\Delta=Pr(AA)-\{Pr(AA)+Pr(AB)/2\}^2$ (Weir, 1996). We denote $\Delta$ in cases and in controls as $\Delta_p$ and $\Delta_q$, respectively. We further denote the penetrance as ($f_0,f_1,f_2$) and genotype relative risks (GRRs) as ($\lambda_1 = f_{1}/f_{0},\lambda_2 = f_{2}/f_{0}$). We have $\Delta_p=f^2_0p^2q^2(\lambda_2-\lambda_1^2)/k^2$ and $\Delta_q=f_0p^2q^2(2\lambda_1-1-\lambda_2-f_0 \lambda_1^2+f_0 \lambda_2)/(1-k)^2$. Four mutually exclusive regions $R_1$, $R_{2}$, $R_{3}$ and $R_4$ are defined in Zheng and Ng (see Figure 1). In region $R_1$, $\Delta_p > 0$ follows from $\lambda_2>\lambda_1^2$. Further, $-\lambda_2(1-f_0)+2\lambda_1-1-f_0\lambda_1^2< -\lambda_1^2(1-f_0)+2\lambda_1-1-f_0\lambda_1^2=-(\lambda_1-1)^2<0$ yields $\Delta_q<0$. In regions $R_2$, $R_3$ and $R_4$, $\Delta_p<0$ from $\lambda_2<\lambda_1^2$. To determine the signs of $\Delta_q$ in these three regions, note that $\Delta_q$ has the same sign as $-\{\lambda_2-(2\lambda_1-1-f_0\lambda_1^2)/(1-f_0)\}$, which is negative for regions above the curve CD (see Figure 1 of Zheng and Ng), and positive for regions below the curve CD. Thus, we obtain the signs of $(\Delta_p,\Delta_q)$ for the four regions as $R_1=(+,-)$, $R_2=(-,-)$, $R_3=(-,-)$ and $R_4=(-,+)$. 


\newpage
\centerline{SECTION B: PERFORMANCE OF MODEL SELECTION}

To examine the performance of the model selection, we conducted a simulation study under the four genetic models (REC = recessive, ADD = additive, MUL = multiplicative, and DOM = dominant) based on 10,000 replicates. Equal numbers  of cases and controls ($r=s=250$) were generated using the multinomial distributions $mul(r; p_0,p_1,p_2)$ and $mul(s; q_0,q_1,q_2)$, respectively. The minor allele frequency (MAF) in the population is 0.1, 0.3, and 0.5, the disease prevalence $K=0.1$, and the GRR $\lambda_2=2.0$ are specified. Distributions of the model selection when Hardy-Weinberg equilibrium (HWE) holds ($F=0$) in the population given the true genetic model are reported in Table 1S. $Z_{\text{HWDTT}}$ is Hardy-Weinberg disequilibrium trend test (HWDTT) of Song and Elston (2006).

\vskip 0.2in
\begin{center}
Table 1Sa. Distributions (percentages) of model selections using $Z_{\text{HWDTT}}$ under HWE in the population ($F=0$) based on 10,000 simulations with $c = \Phi^{-1}(0.95)$.\\
\begin{tabular}{ccccc}
\hline
MAF &  Model & REC & ADD/MUL & DOM\\
\hline
0.1 & REC      & 20.71 & 78.95 & 0.34\\
0.3 & REC      & 67.40 & 32.60 & 0.00\\
0.5 & REC      & 66.76 & 33.24 & 0.00\\
\hline
0.1 & ADD      & 1.91 & 90.29 & 7.80\\
0.3 & ADD      & 2.11 & 88.36 & 9.53\\
0.5 & ADD      & 2.45 & 89.17 & 8.38\\
\hline
0.1 & MUL      & 3.21 & 92.24 & 4.55\\
0.3 & MUL      & 5.39 & 89.93 & 4.68\\
0.5 & MUL      & 5.29 & 90.32 & 4.39\\
\hline
0.1 & DOM      & 0.03  & 62.71 & 37.26\\
0.3 & DOM      & 0.00  & 32.28 & 67.72\\
0.5 & DOM      & 0.01  & 35.23 & 64.76\\
\hline
\end{tabular}
\end{center}
\vskip 0.2in

Table 1Sa shows that, when $F=0$, under REC, the model selection procedure has 66\% correct selection for common allele frequencies (MAF = 0.3 and 0.5) and only about 20\% for the rare allele (MAF = 0.1). Under either ADD or MUL, the probabilities to select the correct genetic model are about 90\%. The percentage for selecting a true DOM is about 60\%-67\% for common alleles and 37\% for rare allele. 

The performance of model selection when a more extreme cutoff point \linebreak $c=\Phi^{-1}(0.975)$ is used is present in Table 1Sb. The results show that using more extreme cutoff point $c$ increases the percentages of correct selection of ADD and MUL but decreases the percentages of correct selection of REC and DOM. Therefore, it would have a trade-off effect in terms of power performance for the two-phase procedure. In this article, we use $c=\Phi^{-1}(0.95)$ and an optimal cutoff point or data-driven cut-off point will be studied in the future. 
 
\vskip 0.2in
\begin{center}
Table 1Sb. Distributions (percentages) of model selections using $Z_{\text{HWDTT}}$ under HWE in the population ($F=0$) based on 10,000 simulations with $c=\Phi^{-1}(0.975)$.\\ 
\begin{tabular}{ccccc}
\hline
MAF &  Model & REC & ADD/MUL & DOM\\
\hline
0.1 & REC      & 11.43 & 88.47 & 0.00\\
0.3 & REC      & 54.90 & 45.10 & 0.00\\
0.5 & REC      & 53.71 & 46.29 & 0.00\\
\hline
0.1 & ADD      & 0.72 & 95.48 & 3.80\\
0.3 & ADD      & 1.05 & 93.60 & 5.35\\
0.5 & ADD      & 0.96 & 94.31 & 4.73\\
\hline
0.1 & MUL      & 1.26 & 96.50 & 2.24\\
0.3 & MUL      & 2.55 & 95.21 & 2.24\\
0.5 & MUL      & 2.70 & 95.33 & 1.97\\
\hline
0.1 & DOM      & 0.00  & 73.22 & 26.78\\
0.3 & DOM      & 0.00  & 44.20 & 55.80\\
0.5 & DOM      & 0.02  & 47.91 & 52.07\\
\hline
\end{tabular}
\end{center}

%\vskip 0.2 in

\newpage

\begin{center}
Figure 1S. Percentages of correct model selections with HWE ($F=0$) or without HWE ($F=0.05$) in the population.\\
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When departure from HWE occurs, we considered $F=0.05$. Figure 1S plots the percentage of the correct selections over the MAF (0.1 to 0.5 with increments of 0.05) for the four genetic models. For ADD and MUL, departure from HWE has little effect on model selection. For REC and DOM, the selection is slightly affected, especially for the rare allele frequencies. Interestingly, departure from HWE could also improve model selection for the recessive model. 


\newpage
\centerline{SECTION C: CORRELATION OF THE TWO ANALYSES}


Denote $U_x=\left(\hat{p}_2+x\hat{p}_1\right)-\left(\hat{q}_2+x\hat{q}_1\right)$ for $x=0,1/2,1$. Then, the Cochran-Armitage trend test (CATT) $Z_x$ can be written as $Z_x= U_x/\left\{\widehat{var}_{H_0}(U_x)\right\}^{1/2}$, where 
\begin{equation}
var(U_x)=\left\{(p_2+x^2p_1)-(p_2+xp_1)^2\right\}/r + \left\{(q_2+x^2q_1)-(q_2+xq_1)^2\right\}/s\label{varus}
\end{equation}
and its consistent estimate under the null is 
$$
\widehat{var}_{H_0}(U_x)=\left\{(n_2+x^2n_1) -(n_2+xn_1)^2/n\right\}/(rs)
$$ 
(Slager and Schaid, 2001; Freidlin {\it et al.}, 2002).  From Weir (1996), $E(\hat{\Delta}_p)=\Delta_p+O(r^{-1})$ and $var(\hat{\Delta}_p)=\{p^2(1-p)^2+(1-2p)^2\Delta_p-\Delta_p^2\}/r$, where $p$ is the frequency of allele A in cases. Similarly results can be obtained for controls. When HWE holds in population, under the null hypothesis of no association, $\Delta_p=\Delta_q=0$, it follows that $var_{H_0}(\hat{\Delta}_p)=p^2(1-p)^2/r$ and $var_{H_0}(\hat{\Delta}_q)=q^2(1-q)^2/s$ and $p=q$.  Assume that $Z_{\text{HWDTT}}$ and $Z_x$ follow a bivariate normal distribution with the null correlation $\rho_{x}=corr_{H_0}(Z_{\text{HWDTT}},Z_x)=cov_{H_0}(Z_{\text{HWDTT}},Z_x)$. Asymptotically, we have
\begin{equation}
cov_{H_0}(Z_{\text{HWDTT}},Z_x)
=\{var_{H_0}(\hat{\Delta}_p-\hat{\Delta}_q)var_{H_0}(U_x)\}^{-1/2} cov_{H_0}(\hat{\Delta}_p-\hat{\Delta}_q, U_x),\label{corr}
\end{equation}
where $var_{H_0}(\hat{\Delta}_p-\hat{\Delta}_q)=var_{H_0}(\hat{\Delta}_p)+var_{H_0}(\hat{\Delta}_q)=p^2(1-p)^2(1/r+1/s)$, $var_{H_0}(U_x)$ is given by (\ref{varus}) with $p_i=q_i$, and 
\begin{eqnarray*}
&&cov_{H_0}(\hat{\Delta}_p, U_x) = E\left(
{\hat p}_{2}^2 + x {\hat p}_{1} {\hat p}_{2} - {\hat p}_{2}^{3} - x {\hat p}_{1} {\hat p}_{2}^{2} -  {\hat p}_{1} {\hat p}_{2}^{2} -  x {\hat p}_{1}^{2}{\hat p}_{2} - {\hat p}_{1}^{2}{\hat p}_{2}/4 - x {\hat p}_{1}^{3}/4 \right)\\
&& - E\left({\hat p}_{2} + x {\hat p}_{1}\right) E \left({\hat p}_{2} - {\hat p}_{2}^{2} - {\hat p}_{1} {\hat p}_{2} - {\hat p}_{1}^{2}/4 \right)\\
&=& p_{2}^{2} + \frac{1}{r} p_{2}(1-p_{2}) + x \left(1 - \frac{1}{r} \right) p_{1} p_{2}\\
&-& \left\{ p_{2}^{3} + \frac{3}{r} p_{2}^{2} (1- p_{2}) + \frac{1}{r^{2}} p_{2}(1-p_{2})(1-2p_{2}) \right\} \\
&-& (1 + x) \left\{ p_{1} p_{2}^{2} + \frac{1}{r} p_{1}p_{2} (1- 3p_{2}) - \frac{1}{r^{2}} p_{1} p_{2} (1-2p_{2}) \right\}\\
&-& \left( x + \frac{1}{4} \right) \left\{ p_{1}^{2} p_{2} + \frac{1}{r} p_{1}p_{2} (1- 3p_{1}) - \frac{1}{r^{2}} p_{1} p_{2} (1-2p_{1}) \right\}\\
&-& \frac{x}{4} \left\{ p_{1}^{3} + \frac{3}{r} p_{1}^{2} (1- p_{1}) + \frac{1}{r^{2}} p_{1}(1-p_{1})(1-2p_{1}) \right\} \\
&-& (p_{2} + x p_{1}) \left\{ 
p_{2} - p_{2}^{2} - \frac{1}{r} p_{2} (1-p_{2}) - \left( 1- \frac{1}{r} \right) p_{1} p_{2} - \frac{1}{4} \left(p_{1}^{2} + \frac{1}{r} p_{1} (1-p_{1})  \right)\right\}.
\end{eqnarray*}
A similar expression can be obtained for $cov_{H_0}(\hat{\Delta}_q, U_x)$, where $p_i$ are replaced by $q_i$ and $r$ by $s$. Substituting $p_1=2p(1-p)$ and $p_2=p^2$, $cov_{H_0}(\hat{\Delta}_p, U_0)=p^2(-2+2r+5p-3p^2+2rp^2-4rp)/(2r^2)$, $cov_{H_0}(\hat{\Delta}_p, U_{1/2})=-p(1-3p+2p^2)/(4r^2)$, and $cov_{H_0}(\hat{\Delta}_p, U_{1})=-p(-5p+7p^2+1-4rp^2+2rp-3p^3+2rp^3)/(2r^2)$, where $p$ is the allele frequency of $A$ in cases. After simplification, we have
\begin{eqnarray}
&&cov_{H_0}(\hat{\Delta}_p, U_0)=p^2(1-p)^2/r+O(r^{-2}),\nonumber\\
&&cov_{H_0}(\hat{\Delta}_p, U_{1/2})=O(r^{-2}),\nonumber\\
&&cov_{H_0}(\hat{\Delta}_p, U_1)=-p^2(1-p)^2/r+O(r^{-2}).\nonumber
\end{eqnarray} 
The expressions for $cov_{H_0}(\hat{\Delta}_q, U_{x})$ can be obtained from $cov_{H_0}(\hat{\Delta}_p, U_{x})$ by changing the signs of $cov_{H_0}(\hat{\Delta}_p, U_{x})$ and $r$ to $s$ ($p=q$ under $H_0$). As $n\rightarrow \infty$, assume that $r/n\rightarrow \omega\in (0,1)$. Then, from (\ref{corr}), the values of $\rho_{x}=corr_{H_0}(Z_{\text{HWDTT}}, Z_x)$ are given by 
\begin{eqnarray}
&&corr_{H_0}(Z_{\text{HWDTT}}, Z_0)=\left(\frac{1-p}{1+p}\right)^{1/2}+O(n^{-1}),\nonumber\\
&&corr_{H_0}(Z_{\text{HWDTT}}, Z_{1/2})=O(n^{-1}),\nonumber\\
&&corr_{H_0}(Z_{\text{HWDTT}}, Z_1)=-\left(\frac{p}{2-p}\right)^{1/2}+O(n^{-1}).\nonumber
\end{eqnarray} 
Note that $Z_{\text{HWDTT}}$ and $Z_{1/2}$ are asymptotically independent. When $p \rightarrow 1$, $Z_{\text{HWDTT}}$ and $Z_{0}$ are asymptotically independent while $Z_{\text{HWDTT}}$ and $Z_{1}$ are perfectly negatively correlated. When $p \rightarrow 0$, $Z_{\text{HWDTT}}$ and $Z_{1}$ are asymptotically independent while $Z_{\text{HWDTT}}$ and $Z_{0}$ are perfectly positively correlated.     

%Note that the right hand side (RHS) of (\ref{cutoff}) is zero when $\alpha^*=0$ and greater than $\alpha/M$ when $\alpha^* > 10\alpha/9$. Further, the RHS of (\ref{cutoff}) is a continuous increasing function of $\alpha^*$. Thus, the adjusted significance level $\alpha^*$ satisfying (\ref{cutoff}) uniquely exists in $(0, 10\alpha/9]$. 

\newpage
\centerline{SECTION D: POWER COMPARISON}

We conducted simulation studies to compare the power performance of several robust tests: the CATT with the additive score, the two-phase analysis with model selection, and MERT and MAX under the four genetic models. The chi-squared test and the likelihood ratio test, that are known to have similar or less power than the MAX were not considered. The goal of our simulation is to assess the improvement in efficiency robustness that the two-phase method provided.  

We first assumed that HWE holds in the population. In the simulation studies, we used the GRR $\lambda_2=2$, MAF = 0.1, 0.3, 0.5, disease prevalence $K=0.01, 0.1$, and both equal and unequal numbers of cases and controls. These choices of parameter values cover most practical applications. The results of equal and unequal sample sizes in cases and controls are reported in Tables 2S and 3S. In both tables, we used the adjusted significance levels $\alpha^{*}$ for $Z_{\text{model}}$ and $\alpha=0.05$ for the other tests (CATT, MERT, and MAX). The adjusted level $\alpha^*$ was calculated for each simulated dataset.




\begin{table}
Table 2S. Comparison of powers of the robust tests under HWE: cases and controls have equal sample sizes and GRR $\lambda_2=2$. LSRE(1,2,3) are empirical large sample efficiencies of $Z_{\text{model}}$ relative to $Z_{1/2}$,  MAX,  and $Z_{\text{MERT}}$, respectively.

\begin{center}
\vskip 0.2in
{\footnotesize
\begin{tabular}{cccccccccccc} \hline
 & &  &  &  &  &  &  &&\multicolumn{3}{c}{LSRE}\\
Model & $r$ & $s$ & MAF & $K$ & $Z_{1/2}$  & MAX & $Z_{\text{MERT}}$ & $Z_{\text{model}}$ & 1 & 2 & 3\\
\hline
REC	& 500&500& 0.1 & 0.01 & 9.52& 19.2&15.7& 21.4&2.25&1.11&1.37\\
REC	& 500&500& 0.1 & 0.1  & 10.7& 23.6&19.4& 26.1&2.44&1.10&1.34\\
REC	& 250&250& 0.3 & 0.01 & 47.7& 65.0&55.9& 68.4&1.44&1.05&1.22\\
REC	& 250&250& 0.3 & 0.1  & 54.0& 74.4&64.0& 77.1&1.43&1.04&1.20\\
REC	& 250&250& 0.5 & 0.01 & 86.3& 87.2&92.6& 85.5&1.07&1.01&1.09\\
REC	& 250&250& 0.5 & 0.1  & 92.5& 96.5&91.4& 97.9&1.04&1.00&1.06\\
\hline
MUL	& 500&500& 0.1 & 0.01 & 70.0& 64.9&61.6& 66.6&0.95&1.03&1.08\\
MUL	& 500&500& 0.1 & 0.1  & 79.5& 75.3&72.0& 76.7&0.96&1.02&1.07\\
MUL	& 250&250& 0.3 & 0.01 & 74.0& 69.3&72.7& 72.3&0.98&1.04&0.99\\
MUL	& 250&250& 0.3 & 0.1  & 82.1& 77.9&81.1& 80.5&0.98&1.03&0.99\\
MUL	& 250&250& 0.5 & 0.01 & 78.7& 74.6&78.7& 77.6&0.99&1.04&0.99\\
MUL	& 250&250& 0.5 & 0.1  & 85.5& 81.3&85.6& 84.7&0.99&1.04&0.99\\
\hline
ADD	& 500&500& 0.1 & 0.01 & 80.5& 76.7&70.5& 78.1&0.97&1.02&1.11\\
ADD	& 500&500& 0.1 & 0.1  & 88.1& 85.0&79.4& 86.2&0.98&1.01&1.09\\
ADD	& 250&250& 0.3 & 0.01 & 77.9& 74.0&75.4& 76.5&0.98&1.03&1.01\\
ADD	& 250&250& 0.3 & 0.1  & 85.4& 81.8&83.3& 84.0&0.98&1.03&1.01\\
ADD	& 250&250& 0.5 & 0.01 & 76.3& 72.4&76.6& 75.8&0.99&1.05&0.99\\
ADD	& 250&250& 0.5 & 0.1  & 83.5& 79.9&83.9& 82.7&0.99&1.03&0.99\\
\hline
DOM	& 500&500& 0.1 & 0.01 & 99.7& 99.6&95.9& 99.6&1.00&1.00&1.04\\
DOM	& 500&500& 0.1 & 0.1  & 99.9& 99.9&99.9& 99.9&1.00&1.00&1.00\\
DOM	& 250&250& 0.3 & 0.01 & 92.7& 95.0&86.5& 95.4&1.03&1.00&1.10\\
DOM	& 250&250& 0.3 & 0.1  & 96.3& 98.0&92.0& 98.0&1.02&1.00&1.06\\
DOM	& 250&250& 0.5 & 0.01 & 67.0& 81.2&69.6& 82.2&1.23&1.01&1.18\\
DOM	& 250&250& 0.5 & 0.1  & 74.1& 87.6&76.3& 88.2&1.19&1.01&1.16\\
\hline
\end{tabular}}
\end{center}
\end{table}



\begin{table}

Table 3S. Comparison of powers of the robust tests under HWE: cases and controls have different sample sizes, and GRR $\lambda_2=2$. LSRE(1,2,3) are empirical large sample efficiencies of $Z_{\text{model}}$ relative to $Z_{1/2}$,  MAX,  and $Z_{\text{MERT}}$, respectively.

\vskip 0.1in
\begin{center}
{\footnotesize
\begin{tabular}{cccccccccccc}
\hline
 & &  &  &  &  &  &  & & \multicolumn{3}{c}{LSRE}\\
Model & $r$ & $s$ & MAF & $K$ & $Z_{1/2}$ & MAX & $Z_{\text{MERT}}$& $Z_{\text{model}}$& 1 & 2 & 3 \\
\hline
REC	& 400&600& 0.1 & 0.01 & 10.2& 20.6&17.8& 23.6&2.30&1.15&1.32\\
REC	& 400&600& 0.1 & 0.1  & 11.3& 25.9&21.6& 29.8&2.64&1.15&1.38 \\
REC	& 200&300& 0.3 & 0.01 & 47.1& 65.2&56.1& 68.5&1.45&1.05&1.22 \\
REC	& 200&300& 0.3 & 0.1  & 55.1& 75.0&65.3& 76.9&1.39&1.03&1.18 \\
REC	& 200&300& 0.5 & 0.01 & 85.8& 91.5&84.4& 91.9&1.07&1.00&1.09 \\
REC	& 200&300& 0.5 & 0.1  & 91.4& 96.3&90.5& 96.3&1.05&1.00&1.06\\
\hline
MUL	& 400&600& 0.1 & 0.01 & 71.1& 66.7&62.9& 68.5&0.96&1.03&1.09 \\
MUL	& 400&600& 0.1 & 0.1  & 78.9& 75.1&71.6& 76.5&0.97&1.02&1.07\\
MUL	& 200&300& 0.3 & 0.01 & 73.1& 68.2&71.9& 71.3&0.98&1.05&0.99\\
MUL	& 200&300& 0.3 & 0.1  & 81.1& 77.1&80.5& 79.8&0.98&1.03&0.99\\
MUL	& 200&300& 0.5 & 0.01 & 76.4& 71.6&76.5& 74.8&0.98&1.05&0.98\\
MUL	& 200&300& 0.5 & 0.1  & 84.6& 80.9&84.4& 83.3&0.98&1.03&0.99\\
\hline
ADD	& 400&600& 0.1 & 0.01 & 81.4& 78.0&71.4& 79.0&0.97&1.01&1.11\\
ADD	& 400&600& 0.1 & 0.1  & 88.3& 85.5&79.5& 86.2&0.98&1.01&1.08\\
ADD	& 200&300& 0.3 & 0.01 & 76.8& 72.6&74.2& 75.6&0.98&1.04&1.02\\
ADD	& 200&300& 0.3 & 0.1  & 84.5& 80.9&82.4& 83.4&0.99&1.03&1.01\\
ADD	& 200&300& 0.5 & 0.01 & 74.8& 69.3&75.0& 73.6&0.98&1.06&0.98\\
ADD	& 200&300& 0.5 & 0.1  & 82.4& 78.6&82.6& 81.3&0.99&1.03&0.98\\
\hline
DOM	& 400&600& 0.1 & 0.01 & 99.5&99.4&95.7& 99.4&1.00&1.00&1.04 \\
DOM	& 400&600& 0.1 & 0.1  & 99.9&99.9&98.3& 99.9&1.00&1.00&1.02 \\
DOM	& 200&300& 0.3 & 0.01 & 91.5& 93.9&85.2& 94.6&1.03&1.01&1.11\\
DOM	& 200&300& 0.3 & 0.1  & 95.7& 97.2&90.5& 97.6&1.02&1.00&1.08\\
DOM	& 200&300& 0.5 & 0.01 & 62.7& 77.6&64.7& 79.1&1.26&1.02&1.22\\
DOM	& 200&300& 0.5 & 0.1  & 71.9& 85.3&73.5& 86.1&1.20&1.01&1.17\\
\hline
\end{tabular}}
\end{center}
\end{table}


\begin{table}
Table 4S. Comparison of powers of the other robust tests without HWE ($F=0.05$): cases and controls have different sample sizes, disease prevalence $K=0.1$, and GRR $\lambda_2=2$. LSRE(1,2,3) are empirical large sample efficiencies of $Z_{\text{model}}$ relative to $Z_{1/2}$,  MAX,  and $Z_{\text{MERT}}$, respectively.

\vskip 0.1in
\begin{center}
{\footnotesize
\begin{tabular}{ccccccccccc}
\hline
 & &  &  &  &  &  &  & \multicolumn{3}{c}{LSRE}\\
Model & $r$ & $s$ & MAF & $Z_{1/2}$ & MAX & $Z_{\text{MERT}}$& $Z_{\text{model}}$& 1 & 2 & 3 \\
\hline

REC	& 500&500& 0.1  & 15.78&31.96&26.73&37.73&2.39&1.18&1.41\\
REC	& 250&250& 0.3  & 61.39&79.18&69.76&81.80&1.33&1.03&1.17\\
REC	& 250&250& 0.5  & 93.61&97.01&92.59&97.16&1.04&1.00&1.05\\
\hline
DOM	& 500&500& 0.1  & 99.92&99.92&98.60&99.95&1.00&1.00&1.01\\
DOM	& 250&250& 0.3  & 96.41&97.96&92.80&98.05&1.02&1.00&1.06\\
DOM	& 250&250& 0.5  & 76.55&87.92&78.34&88.81&1.16&1.01&1.13\\
\hline
\end{tabular}}
\end{center}
\end{table}

Define the large sample relative efficiencies, LSRE1, LSRE2, and LSRE3, as the ratios of the large sample power of $Z_{\text{model}}$ over those of CATT ($Z_{1/2}$), MAX, and MERT ($Z_{\text{MERT}}$), respectively. Then LSRE$>$1 indicates that $Z_{\text{model}}$ is more powerful than the other test. The results demonstrate that $Z_{\text{model}}$ is always more powerful than MAX with LSRE from 100\% to 115\%. Compared to MERT, $Z_{\text{model}}$ is also often more powerful (LSRE from 98\%- 138\%). Under the ADD and MUL models with common allele frequencies (MAF=0.3, 0.5), $Z_{\text{model}}$ is about 2\% less powerful than MERT. Compared to CATT, the two-phase analysis is always more powerful than $Z_{1/2}$ under the REC and DOM models with LSRE from 100\% to 264\% but $Z_{1/2}$ is always more powerful than $Z_{\text{model}}$ under the ADD and DOM models. The latter is not surprising as some of the ADD/MUL models are classified to the REC or DOM in the simulations. However, under the ADD/MUL, the LSRE is still at least 95\%. Based on simulation studies, in general, $Z_{\text{model}}$ is preferable to $Z_{1/2}$ and $Z_{\text{MERT}}$. Previous research showed that MAX has greater efficiency robustness for testing genetic association using case-control studies when the underlying genetic model is uncertain (Freidlin {\it et al.}, 2002). Our simulation studies indicate that the two-phase analysis with model selection has even greater efficiency robustness than MAX although the difference in power is only about 2 to 3\%. 

When HWE does not hold, we used $F=0.05$ and only focused on the recessive and dominant models because, from Figure 1S, model selections for these two models may be affected by departure from HWE. The results are reported in Table 4S, which are consistent with those in Tables 2S and 3S.

\vspace{3cm}
\centerline{SECTION E: SAS MACRO FOR CALCULATION OF $\alpha^*$}

The SAS macro for calculation of $\alpha^{*}$ is listed below. This is also avaliable from the authors upon request. 
      
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\centerline{REFERENCES}
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\item Freidlin, B., Zheng, G., Li, Z., and Gastwirth, J.L. (2002). Trend tests for case-control studies of genetic markers: power, sample size and robustness. {\it Hum. Hered.} {\bf 53}, 146-152.

\item Slager, S.L. and Schaid, D.J. (2001). Case-control studies of genetic markers: Power and sample size approximations for Armitage's test for trend. {\it Hum. Hered.} {\bf 52}, 149-153.

\item Song, K. and Elston, R.C. (2006). A powerful method of combining measures of association and Hardy-Weinberg disequilibrium for fine-mapping in case-control studies. {\it Stat. Med.} {\bf 25}, 105-126.

\item Wang, K. and Sheffield, V.C. (2005). A constrained-likelihood approach to marker-trait association studies. {\it Am. J. Hum. Genet.} {\bf 77}, 768-780.


\item Weir, B.S. (1996). {\it Genetic Data Analysis II: Methods for Discrete Population Genetic Data}. Sinauer Associations Inc: Massachusetts.

\item Zheng, G., Freidlin, B., and Gastwirth, J.L. (2006). Comparison of robust tests for genetic association using case-control studies. IMS Lecture Notes-Monograph Series (2nd special issue in honor of E. L. Lehmann), 320-336.

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