## Figures

## Abstract

Because current molecular haplotyping methods are expensive and not amenable to automation, many researchers rely on statistical methods to infer haplotype pairs from multilocus genotypes, and subsequently treat these inferred haplotype pairs as observations. These procedures are prone to haplotype misclassification. We examine the effect of these misclassification errors on the false-positive rate and power for two association tests. These tests include the standard likelihood ratio test (LRT_{std}) and a likelihood ratio test that employs a double-sampling approach to allow for the misclassification inherent in the haplotype inference procedure (LRT_{ae}). We aim to determine the cost–benefit relationship of increasing the proportion of individuals with molecular haplotype measurements in addition to genotypes to raise the power gain of the LRT_{ae} over the LRT_{std}. This analysis should provide a guideline for determining the minimum number of molecular haplotypes required for desired power. Our simulations under the null hypothesis of equal haplotype frequencies in cases and controls indicate that (1) for each statistic, permutation methods maintain the correct type I error; (2) specific multilocus genotypes that are misclassified as the incorrect haplotype pair are consistently misclassified throughout each entire dataset; and (3) our simulations under the alternative hypothesis showed a significant power gain for the LRT_{ae} over the LRT_{std} for a subset of the parameter settings. Permutation methods should be used exclusively to determine significance for each statistic. For fixed cost, the power gain of the LRT_{ae} over the LRT_{std} varied depending on the relative costs of genotyping, molecular haplotyping, and phenotyping. The LRT_{ae} showed the greatest benefit over the LRT_{std} when the cost of phenotyping was very high relative to the cost of genotyping. This situation is likely to occur in a replication study as opposed to a whole-genome association study.

## Synopsis

Localizing genes for complex genetic diseases presents a major challenge. Recent technological advances such as genotyping arrays containing hundreds of thousands of genomic “landmarks,” and databases cataloging these “landmarks” and the levels of correlation between them, have aided in these endeavors. To utilize these resources most effectively, many researchers employ a gene-mapping technique called haplotype-based association in order to examine the variation present at multiple genomic sites jointly for a role in and/or an association with the disease state. Although methods that determine haplotype pairs directly by biological assays are currently available, they rarely are used due to their expense and incongruity to automation. Statistical methods provide an inexpensive, relatively accurate means to determine haplotype pairs. However, these statistical methods can provide erroneous results. In this article, the authors compare a standard statistical method for performing a haplotype-based association test with a method that accounts for the misclassification of haplotype pairs as part of the test. Under a number of feasible scenarios, the performance of the new test exceeded that of the standard test.

**Citation: **Levenstien MA, Ott J, Gordon D (2006) Are Molecular Haplotypes Worth the Time and Expense? A Cost-Effective Method for Applying Molecular Haplotypes. PLoS Genet 2(8):
e127.
doi:10.1371/journal.pgen.0020127

**Editor: **Gonçalo Abecasis, University of Michigan, United States of America

**Received: **March 8, 2006; **Accepted: **June 27, 2006; **Published: ** August 18, 2006

**Copyright: ** © 2006 Levenstien et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Funding: **This work was supported by the National Institutes of Health grant MH44292.

**Competing interests: ** The authors have declared that no competing interests exist.

**Abbreviations:
**CI,
confidence interval;
*DAF*
,
disease allele frequency; KS,
Kolmogorov-Smirnov; LD,
linkage disequilibrium; LRT_{ae},
likelihood ratio test allowing for error; LRT_{std},
standard likelihood ratio test; SNP,
single nucleotide polymorphism

## Introduction

With the advent of the HAPMAP project [1,2], the popularity of haplotype-based case-control genetic association studies has grown markedly. The alleles present at multiple genetic markers across a given chromosome form a haplotype [3]. It has been suggested that association studies utilizing haplotypes formed from single nucleotide polymorphisms (SNPs) may be more powerful than single locus association [4–11].

Methods for explicit determination of phased haplotypes are available [12–18]. However, in practice, phased haplotypes are rarely determined explicitly. Instead statistical methods for gene mapping estimate haplotype frequencies from multilocus genotype data [19–28]. For case-control association studies, the sampling design involves unrelated individuals, and therefore the procedure used to estimate haplotype frequencies treats each individual as an independent observation. As with other procedures of statistical estimation, the accuracy of haplotype frequency estimates depends on several factors including “sample size, number of loci studied, allele frequencies, and locus-specific allelic departures from Hardy-Weinberg and linkage equilibrium” [29]. Furthermore, these factors also affect the accuracy of phased-haplotype inference or phased-haplotype calls [30]. Several researchers have investigated the accuracy of haplotype inference procedures by applying them to real and simulated datasets [18,26,30–37].

Several statistical methods are available to perform tests of haplotype-based case-control association. One method calculates the likelihood of the data in terms of the estimated haplotype frequencies. An alternative method relies on the use of a contingency table containing the case-control counts for each inferred haplotype. The counts in the contingency table can be determined either by inferring phased haplotypes for each individual or by multiplying each haplotype frequency estimate by the total number of haplotypes in the study. Many researchers find the latter method appealing since it applies the same format as the classic genotypic and allelic case-control studies, and explicitly accounts for each phased haplotype. As a result, many researchers employ this method in practice [18,35,38–40]. In the event that all phased haplotypes have been called correctly, this method can provide additional power [41,42]. This situation is analogous to tests of association using allele estimates from individual genotypes as compared with allele frequency estimates from DNA-pooling data [43].

However, misclassifications can lower a study's power and/or affect the false-positive rate. The act of calling haplotype pairs from multilocus genotypes in the phase-ambiguous situation is similar to the act of dichotomizing continuous measures. Royston et al. document a loss in power when dichotomizing continuous predictor variables in a regression analysis [44]. In the context of our study, a misclassification results when the haplotype pair called for an individual is not the true underlying haplotype pair. Non-differential misclassification occurs when the misclassification rates are the same in cases and controls. When non-differential misclassification exists, the test suffers a loss in power, but the false-positive rate remains unchanged [45,46]. In contrast, differential misclassification inflates the test's false-positive rate and may diminish its power [47]. We conjecture that in the absence of differential genotype misclassification, all haplotype misclassification is non-differential when haplotype frequency distributions are the same in cases and controls, i.e., under the null hypothesis.

Although there have been several studies aimed at evaluating the accuracy of haplotype inference and haplotype frequency estimation procedures [26,29,30,32,35,37], to our knowledge, no systematic study of the effects of haplotype misclassification has been documented. Thus, the purpose of this work is to address the effects of haplotype misclassification on the false-positive rate and power of commonly used tests of haplotype-based association. Specifically, this research aims to (1) classify the nature of the misclassification present in calling phased haplotypes; (2) determine the appropriateness of using the asymptotic χ^{2} distribution and permutation methods to evaluate the significance of the test statistics we employ; and (3) compare the power of our test statistic which accounts for haplotype misclassification with the power of the standard likelihood ratio test statistic when the costs are fixed.

## Methods

### Test Statistics

In order to detect an association between a haplotype pair and disease status, we employed two statistical tests on 2 × *n* contingency tables where *n* is the number of haplotype pair categories found by inference. These tests include the standard likelihood ratio test (LRT_{std}) and a likelihood ratio test that employs a double-sampling approach to allow for the misclassification inherent in the haplotype inference procedure (LRT_{ae}). The LRT_{std} is a likelihood ratio statistic that treats the called haplotype pairs as observations, and as a result, the likelihood is the multinomial distribution where the called haplotype pairs are the categories [48]. The LRT_{ae} statistic is a likelihood ratio statistic that employs a double-sampling procedure to account for the misclassification present in haplotype inference. On all the individuals in the study, there is a fallible measure [49,50], the haplotype pairs inferred from the multilocus genotypes, and on a subset of these individuals, there is a second measure that is considered to be infallible [49,50], molecular haplotypes. By comparing the fallible data with infallible data, the LRT_{ae} procedure estimates the misclassification rates present in the fallible data and incorporates this information into the likelihood calculation [51]. The details regarding the LRT_{std} and LRT_{ae} statistics including notation and computation are provided in Protocol S1.

### Permuted and Asymptotic *p*-Values

We applied two methods for evaluating the *p*-value or statistical significance of each statistic. The first method relies on using the central χ^{2} distribution to find the *p*-value since, according to statistical theory under the null hypothesis of no association, twice the natural logarithm of the likelihood ratio follows the central χ^{2} distribution asymptotically for large sample sizes [3,48]. In addition, it has been shown that when Cochran's rule is followed (more than five observations in each cell of the contingency table), the presence of non-differential misclassification does not affect the distribution of the likelihood ratio test statistics under the null hypothesis of no association [46,51]. The second method employs permutation testing to generate the distribution of the test statistic under the null hypothesis and to determine its statistical significance. In this article, *p*-values found with the former and latter approaches are referred to as asymptotic *p*-values and permutation *p*-values, respectively.

### Description of Data Generation and Analysis

To investigate the behavior of these test statistics for a variety of situations, we applied these statistical tests to many simulated datasets. Figure 1 illustrates the procedure we used to simulate the data and to evaluate the false-positive rate or type I error and power at fixed significance levels for each statistic. For the analysis of each replicate dataset simulated, the multilocus genotype data from cases and controls were pooled to infer haplotype pairs for each individual. These inferred haplotypes are sufficient for the computation of LRT_{std}; however, LRT_{ae} requires additional information in the form of molecular haplotypes for a subset of the individuals in the study. Two alternative procedures for selecting individuals for the double sample (individuals with molecular haplotypes in addition to genotypes) were employed. In one selection scheme, individuals were selected randomly. In the other selection scheme, individuals possessing the most ambiguity in their statistically inferred haplotype pairs were prioritized in selecting the double sample. Specifically, we double-sampled those individuals with the smallest posterior probabilities associated with their inferred haplotype pair up to a posterior probability threshold, *δ,* of 0.85 or until the number of individuals specified by the maximum double-sample proportion was reached. Therefore, under this second scheme, the number of individuals double-sampled varied between replicate datasets. In this article, the former and latter procedures for determined the double sample are referred to as random and threshold double-sample selection, respectively.

(A) shows type I error, and (B) shows power by way of data simulation.

### Two-SNP Scenario

#### Evaluation of false-positive rate for permutation and asymptotic *p*-values.

For the simplest non-trivial case, the scenario in which the haplotype under evaluation includes two SNPs, we applied a fractional factorial design [52] to perform a comprehensive study of type I error. For the type I error, haplotype pairs were inferred using both SNPHAP v 1.3.1 (see Electronic Database Information) and PHASE v 2.1.1 [27] (see also Electronic Database Information). Table 1 contains the fractional factorial design settings for the study of type I error for the scenario involving two SNP markers. We consider a 1/2(2* ^{k}*) fractional factorial design, where

*k*= 6. Because of redundancy, we were able to reduce the number of experimental runs from 32 to 18. For instance, under the null hypothesis of no association, a run with 1,000 cases and 250 controls is equivalent to a run with 250 cases and 1,000 controls (with all other factors having equal settings to those for the first run). During each run, 10,000 replicate datasets were simulated. We performed the 18 runs with both of the two alternative procedures for selecting the double sample: random and threshold double-sample selection. For the threshold double-sample selection method,

*δ*was 0.85, and the maximum double-sample proportion was set to the value of

*α*in the fractional factorial design.

Fractional Factorial Design Parameter Settings for the Study of Type I Error Assuming the Haplotype under Investigation Contains Two SNP Markers

To evaluate each test statistic's ability to maintain the correct type I error, we examined the distribution of the *p*-values computed for data simulated under the null hypothesis of no association. We performed two goodness-of-fit tests, the Kolmogorov-Smirnov (KS) and the Anderson-Darling (AD) tests [53] to determine whether the *p*-values deviate significantly from the standard uniform distribution, and examined the false-positive rate for significant thresholds of 0.05, 0.01, and 0.001.

#### Evaluation of power for fixed cost.

We also evaluated the behavior of these statistics under the hypothesis that an unobserved disease locus exists in linkage disequilibrium (LD) with the haplotype under study. Table 2 contains the factorial design settings for the power study in the scenario involving two SNP markers. The factorial design includes three factors: disease model, genotype relative risk [54] for the homozygote genotype *(R _{2})*, and the disease allele frequency

*(DAF)*. Each factor contains two levels. For the disease model factor, the two levels are a dominant disease model and a multiplicative disease model. The dominant disease model requires that

*R*whereas the multiplicative disease model requires that

_{2}= R_{1}*R*, where

_{2}= R_{1}^{2}*R*and

_{1}*R*are the genotype relative risks for the heterozygote and homozygote genotypes, respectively. Specifically, the genotype relative risks are defined as the following. If the penetrances,

_{2}*f*are defined by

_{i},*f*=Pr(affected|

_{i}*i*copies of disease allele), where

*i*= 0, 1, or 2, the genotype relative risks,

*R*and

_{1}*R*, are defined by

_{2}*R*

_{1}=

*f*

_{1}/

*f*

_{0}and

*R*

_{2}=

*f*

_{2}/

*f*

_{0}, respectively [54].

Factorial Design Parameter Settings for the Study of Power Assuming the Haplotype under Investigation Contains Two SNP Markers

As with the study of type I error, we inferred the haplotypes for the power simulations with both SNPHAP v 1.3.1 and PHASE v 2.1.1. The proportion of individuals double-sampled, α, for the LRT_{ae} method (random double-sample selection) was set at 0.75. For the threshold double-sample selection, *δ* was set to 0.85, and the maximum double-sample proportion was 0.75. In the power simulations, the conditional haplotype frequencies were found from the specified disease model parameters by the method described previously [7,55] (also see the PAWE Web site at http://linkage.rockefeller.edu/derek/pawe1.html). However, we selected a specific haplotype to be in LD with the disease locus. During each run, 1,000 replicate datasets consisting of 500 cases and 500 controls were simulated. For these simulations, the disease prevalence was 0.025; the LD between the disease locus and the linked haplotype was 0.9 (measured by D′ [56]); and the population haplotype frequencies were 0.05, 0.15, 0.25, and 0.55. The selection of the specific haplotype in LD with the disease locus depended on the *DAF*. The haplotype occurring with a frequency most similar to that of the disease allele was selected. Thus, the haplotypes with frequencies of 0.05 and 0.25 were selected as the variant in LD with the disease when the *DAF* was set at 0.07 and 0.27, respectively. As with the evaluation of the false-positive rate, we performed all eight runs from the factorial design using both random and threshold double-sample selection.

To compare the power of the two test statistics, we evaluated the power of the statistics under fixed cost conditions. Since the LRT_{ae} requires the additional cost associated with obtaining molecular haplotypes on a subset of the samples, we reduced the number of samples when the LRT_{ae} statistic was applied so that the same total cost would be incurred as for the runs with the LRT_{std}. The reduced sample size for the LRT_{ae} sample was computed using Equation 1,
where *N ^{DS}* is the sample size for the LRT

_{ae};

*N*is the sample size for the LRT

_{std};

*C*is the cost of phenotyping,

_{p}*C*is the cost of genotyping;

_{g}*r*is the cost ratio of molecular haplotyping to genotyping

*(C*; and

_{mh}/C_{g})*α*is the proportion of individuals in the LRT

_{ae}sample that have molecular haplotypes determined (double-sampling proportion). We consider the phenotyping costs,

*C*, to include costs associated with ascertainment and diagnosis. We illustrate fixed-costs sample sizes for the following example. With settings of

_{p}*C*= 25,

_{p}/C_{g}*r*= 5,

*α*= 0.75, and

*N*= 1,000 for the LRT

_{std}method, the corresponding total sample size for the LRT

_{ae}method,

*N*is 874. The reader should note that the reduced sample size results from the additional cost incurred by double-sampling 75% of the total sample for the LRT

^{DS},_{ae}method. If

*C*= 1,000, note that this term will dominate the expression in Equation 1, and the fixed-cost sample size,

_{p}/C_{g}*N*will not differ greatly from the sample size for the LRT

^{DS},_{std},

*N*. All power simulations were performed under fixed-cost conditions. Since the double-sample proportion,

*α,*varies from replicate to replicate when the threshold double-sample selection method is employed, we first performed several test runs to determine the mean double-sample proportion, . Using , we computed

*N**, the total sample size for the LRT

^{DS}_{ae}determined from the expectation of

*α*. Here, the asterisk is added to indicate that total sample size is computed using the expected value of

*α,*as compared with the random double-sample selection, in which

*α*is a fixed quantity. For a specific disease model, we performed a comprehensive study of the power difference between the LRT

_{ae}and LRT

_{std}for the situation of a haplotype comprising two SNPs.

### Multi-SNP Scenario

#### Evaluation of false-positive rate and power for fixed costs.

Through additional simulations, we investigated the behavior of these statistics when applied to haplotypes comprising larger numbers of SNPs. Because these simulations required additional computational time, we only used SNPHAP v 1.3.1 (see Electronic Database Information) for inferring haplotypes. Our simulations were based on haplotype frequencies from two datasets: (1) a dataset of molecular haplotypes with very high levels of pair-wise LD between markers [14] and (2) a dataset of multilocus genotypes from the *TAP2* gene within the major histocompatibility complex, a region with low pair-wise LD between markers [1,2] (see also Electronic Database Information), hereafter referred to as the Horan and the HapMap *TAP2* datasets, respectively. Figure 2 displays the inter-marker LD for each of these two datasets using GOLD plots [57]. For the Horan dataset, we determined the generating population haplotype frequencies for our simulations directly using the counting method [3]. For the HapMap *TAP2* dataset, we found the generating population haplotype frequencies for our simulations indirectly using SNPHAP v 1.3.1 (see Electronic Database Information). In the latter case, haplotype frequencies were estimated from the parents of each trio in the Yoruba population group from the International HapMap Project. For the type I error simulation studies, 1,000 replicate datasets containing 250 cases and 250 controls were simulated. For the type I error runs based on the Horan data and the HapMap *TAP2* data, we simulated haplotypes comprising 15 SNPs and ten SNPs, respectively, whereas for the power runs, we simulated haplotypes comprising five SNPs [1,2,14]. Figure 2 specifies the SNPs we used from each of the datasets in the type I error and power runs. For the Horan dataset, we provide the SNP markers' positions (relative to the transcription start site of the *GH1* gene) whereas for the HapMap *TAP2* dataset, we provide the name of the SNP marker. As a result, we simulated haplotypes using 17 haplotype variants with frequencies greater than 0.01 for both the Horan and HapMap *TAP2* type I error simulations. In addition, we simulated haplotypes using five and ten haplotype variants with frequencies greater than 1/(2*s*), where *s* is the total number of individuals, for the Horan and HapMap *TAP2* power simulations, respectively. For each scenario, we normalized the frequencies so that they summed to unity. As with the power studies for the two-SNP scenario, the selection of the specific haplotype in LD with the disease locus depended on the *DAF*. The rationale for the selection procedure is provided in the Results section addressing multi-SNP power. For multi-marker type I error and power studies, we employed both the random and threshold double-sample selection methods in computing the LRT_{ae} statistic. When the random double-sample selection method was used, the double-sample proportion, *α,* was 0.75. When the threshold double-sample method was used, the setting of *δ* = 0.85 was used, and the maximum proportion of individuals included in the double-sample was 0.75.

(A) shows the LD for 15 SNP markers within the proximal promoter region of human pituitary expressed growth hormone (GH1), and (B) shows the LD for ten SNP markers within the *TAP2* gene. In (A), the SNP markers are listed as their position relative to the transcription start site of the *GH1* gene whereas in (B), the SNP markers are listed by their National Center for Biotechnology Information (NCBI) reference SNP (rs) numbers. Physical distances are provided. All SNP markers displayed were included in the type I error study whereas only the SNP markers accompanied by an asterisk (*) were included in the power study.

#### Identifying the nature of haplotype pair misclassification.

For all the simulations performed, we recorded the details of the misclassifications that occurred. Specifically, for every replicate, we computed the misclassification rates,
where *j*′≠*j* are integers ranging from 1 to the maximum number of haplotype pairs [51]. Previous research studying genotype misclassification rates in tests of genotypic association provides the motivation for ascertaining these values [58,59]. This notation is also used in Protocol S1.

## Results

### Two-SNP Scenario

Our results for type I error and power were almost identical from the simulations utilizing SNPHAP v 1.3.1 and PHASE v 2.1.1 for the haplotype inference. Although we present graphs and tables that display the results provided by SNPHAP v 1.3.1 for the haplotype inference, the reader should note that similar results were found using PHASE v 2.1.1.

#### Evaluation of false-positive rates for permutation and asymptotic *p*-values.

The type I error simulations demonstrated that the approach for determining statistical significance is critical for maintaining the correct false-positive rate. Although the KS and Anderson-Darling (AD) test results indicated that the distribution of permutation *p*-values was consistent with the standard uniform distribution, they also indicated that the distributions of asymptotic *p*-values did not resemble the standard uniform distribution. These results were reinforced by the false-positive rates we found. For all the simulation runs displayed in Table 1, Figure 3 shows the false-positive rate for a significance threshold of 0.05 for LRT_{std} and LRT_{ae} (using the random and threshold double-sample selection methods) association tests in which statistical significance was indicated by permutation and asymptotic *p*-values. The graph shows that asymptotic *p*-values for LRT_{ae} are anti-conservative whereas those for LRT_{std} fluctuate between conservative and anti-conservative values. In contrast, the permutation *p*-values for both statistics consistently maintain the nominal significance level of 0.05. We found that the asymptotic and permuted *p*-values demonstrated similar behavior for significance thresholds of 0.01 and 0.001 (unpublished data). SNPHAP v 1.3.1 was used for the haplotype inference for the simulation results displayed in the graph. These results are not surprising since several simulation parameter settings have expected cell counts of less than five counts, violating Cochran's rule [60].

The *p*-values were determined by both permutation and the asymptotic central χ^{2} distribution. The 18 runs correspond to the combinations of parameter settings described in Table 1. For all 18 runs, LRT_{ae} was computed with the random and threshold double-sample selection methods. When the threshold double-sample method was used to compute LRT_{ae}, the setting of *δ* = 0.85 was used, and the maximum proportion of individuals included in the double sample was the value for *α* specified by the fractional factorial design. SNPHAP v 1.3.1 was used for the haplotype inference for the simulation results displayed in the graph.

#### Evaluation of power for fixed cost.

Based on the results for the false-positive rates, we conclude that power can only be evaluated using the permutation *p*-values. We compare the power of LRT_{ae} (using the random and threshold double-sample selection methods) to LRT_{std}. Table 3 presents summary statistics for the power difference (LRT_{ae} power − LRT_{std} power) at various significance levels for the two cost ratios *C _{p}/C_{g}* = 25 and

*C*= 1,000 using the eight parameter settings from the factorial design (Table 2). Note that in all runs, we set the cost ratio of molecular haplotyping to genotyping,

_{p}/C_{g}*r,*to be 5, and the proportion of individuals to be double-sampled, α, to be 0.75 (for the random double-sample selection method). The values reported correspond to the simulations utilizing SNPHAP v 1.3.1.

Summary Statistics for Power Difference (LRT_{ae} − LRT_{std}) at Various Significance Levels

For the random double-sample selection method, the minimum power difference observed occurred when *C _{p}/C_{g}* = 25 for a dominant disease model with

*R*= 2 and

_{2}*DAF*= 0.27 at a significance level of 0.01. For these settings, the LRT

_{ae}power was 0.544 and LRT

_{std}power was 0.606. The maximum power difference observed occurred when

*C*= 1,000 for a dominant disease model with

_{p}/C_{g}*R*= 3.5 and

_{2}*DAF*= 0.07 at a significance level of 0.001. For these settings, the LRT

_{ae}power was 0.910 and LRT

_{std}power was 0.775.

For the threshold double-sample selection method, the minimum power difference observed occurred when *C _{p}/C_{g}* = 25 for a dominant disease model with

*R*= 2 and

_{2}*DAF*= 0.27 at a significance level of 0.05. For these settings, the LRT

_{ae}power was 0.821 and LRT

_{std}power was 0.831. The maximum power difference observed occurred when

*C*= 1,000 for a dominant disease model with

_{p}/C_{g}*R*= 2 and

_{2}*DAF*= 0.07 at a significance level of 0.05. For these settings, the LRT

_{ae}power was 0.573 and LRT

_{std}power was 0.411.

#### Power difference as a function of double-sample proportion and cost ratio.

In the spirit of response surface analysis for factorial design [52], we performed a more thorough analysis of the parameter settings that provided the maximum power difference with LRT_{ae} computed with the random double-sample selection method. These parameter settings are a dominant disease model with *R _{2}* = 3.5 and

*DAF*= 0.07. These setting provided the additional benefit of power results greater than 75% for both the LRT

_{ae}and LRT

_{std}methods at the 0.05, 0.01, and 0.001 significance levels for both cost ratios of

*C*= 25 and

_{p}/C_{g}*C*= 1,000. The analysis involved computation of the LRT

_{p}/C_{g}_{ae}with the random double-sample selection method. Figure 4 displays the two-dimensional contour plot of the power difference between the LRT

_{ae}and the LRT

_{std}as a function of

*r,*the cost ratio of molecular haplotyping to genotyping, and

*α,*the proportion of individuals double-sampled. These power differences are computed for the fixed parameter settings of

*C*= 25 (Figure 4A) and

_{p}/C_{g}*C*= 1,000 (Figure 4B) at significance level = 0.001 for the disease model described immediately above. The values of

_{p}/C_{g}*r*considered in the contour plots are 1, 5, 10, 25, and 50 whereas the values of

*α*considered are 0.25, 0.50, 0.75, and 1.0. One should note that

*α*= 1.0 indicates that all individuals in the study are double-sampled regardless of phase ambiguity. Simulations were performed with 1,000 replicates and 10,000 permutations for each combination of parameters, and SNPHAP v 1.3.1 was used for the haplotype inference. The sample size for the LRT

_{std},

*N,*was 1,000 (equal numbers of cases and controls). Figure 4A shows that the LRT

_{ae}provides a power advantage over the LRT

_{std}when

*r*is less than 10 and

*α*is greater than 0.5. The maximum power gain is 0.16 and occurs when

*r*and

*α*are 1.0. Conversely, when the

*r*is greater than 10, LRT

_{ae}is less powerful than LRT

_{std}for these parameter settings. The maximum power loss is 0.58 and occurs when

*r*is 50 and

*α*is 1.0. Note that for these values, the total sample available for the LRT

_{ae}method,

*N*(Equation 1), is 342 whereas the total sample available for the LRT

^{DS}_{std}method,

*N,*is 1,000.

Various settings for the cost ratio of molecular haplotyping to genotyping *(r)* and the proportion of individuals double-sampled *(α)* are shown. Power is compared at the 0.001 significance level. The cost ratio of phenotyping to genotyping for (A) is 25 while the cost ratio of phenotyping to genotyping for (B) is 1,000. The generation of haplotype frequencies for the cases and controls was based on a dominant disease model with *R _{2}* = 3.5 and DAF = 0.07, as well as population haplotype frequencies of 0.05, 0.25, 0.15, and 0.55. The haplotype with frequency of 0.05 was placed in LD (D′ = 0.9) with the disease locus. LRT

_{ae}was computed with the random double-sample selection method only. Haplotype pairs were inferred using SNPHAP v 1.3.1.

Figure 4B illustrates that LRT_{ae} is always at least as powerful as the LRT_{std} when *C _{p}/C_{g}* = 1,000. We observe the minimum power gain of 0.02 when

*r*is 50 and

*α*is 0.25 and the maximum power gain of 0.17 when

*r*and

*α*are 1.0. Furthermore, Figure 4B indicates that for any cost ratio,

*r,*increasing the double-sampling proportion,

*α,*always increases the power gain with the maximum power gain occurring when

*α*= 1.0.

### Multi-SNP Scenario

#### Evaluation of false-positive rates for permutation and asymptotic *p*-values.

Table 4 displays our estimates of the false-positive rates using a significance threshold of 0.05 and the results of the KS test for the Horan and HapMap *TAP2* dataset-based simulations. Again, only the permuted *p*-values resemble the standard uniform distribution. In addition, the permuted *p*-values maintained the nominal significance level whereas the asymptotic *p*-values are anti-conservative. The false-positive rate estimates for significance thresholds of 0.01 and 0.001 displayed similar characteristics (unpublished data).

False-Positive Rate Estimates for Simulations with Generating Population Haplotype Frequencies Based on the Horan and HAPMAP *TAP2* Datasets

#### Evaluation of power for fixed cost.

In our power study for haplotypes comprising five SNPs, we again used the disease model parameter settings that provided the maximum power difference (LRT_{ae} power − LRT_{std} power) for the two-SNP factorial design (Table 2) with LRT_{ae} computed using random double-sample selection. These parameter settings are a dominant disease model with *R _{2}* = 3.5 and

*DAF*= 0.07. We based the population haplotype frequencies on the Horan and HapMap

*TAP2*datasets as described in the Methods section. For each dataset, we selected the haplotype with frequency closest to 0.05 as the haplotype in LD with the disease locus. By this choice of haplotype, we approximated the frequency of the linked haplotype for the two-SNP scenario (see Material and Methods section) when

*DAF*= 0.07. As with the two-SNP power study, the LD between the disease locus and the linked haplotype was 0.9 (measured by D′) [56]. The cost ratio of molecular haplotyping to genotyping

*(r)*was 5. When the random double-sample selection method was used to compute LRT

_{ae}, the double-sample proportion

*(α)*was 0.75. When the threshold double-sample method was used to compute LRT

_{ae}, the setting of

*δ*= 0.85 was used, and the maximum proportion of individuals included in the double-sample was 0.75.

For the Horan dataset, the power estimates for the LRT_{std} and the LRT_{ae} were almost identical at the 0.05, 0.01, and 0.001 significance levels for cost ratios *(C _{p}/C_{g})* of both 1,000 and 25 (unpublished data). The high pair-wise intermarker LD present in the Horan dataset causes the haplotype inference to occur with almost complete fidelity. In the absence of misclassification, the LRT

_{ae}statistic reduces to the LRT

_{std}. Therefore, the high degree of similarity in power for these statistics is not surprising.

For the HAPMAP *TAP2* dataset, Table 5 displays the power estimates and the corresponding 95% confidence intervals (CIs) for the LRT_{std} and LRT_{ae} methods at the 0.05, 0.01, and 0.001 significance levels assuming fixed costs. When *C _{p}/C_{g}* = 1,000, the LRT

_{ae}provides a substantial power benefit over the LRT

_{std}with the power difference ranging from 6% and 7% at a significance level of 0.05, to 14% and 21% at a significance level of 0.001 for random double-sample selection and threshold double-sample selection, respectively. When

*C*= 25, the advantage of the LRT

_{p}/C_{g}_{ae}over the LRT

_{std}is still substantial for threshold double-sample selection, but more modest for random double-sample selection. For the three significance levels under investigation, the power difference ranged from 7% to 22%, and 1% to 3.5% for threshold and random double-sample selection, respectively.

Power Estimates for Simulations with Generating Population Haplotype Frequencies Based on the HAPMAP *TAP2* Datasets

We found that the median power gain of the LRT_{ae} over the LRT_{std} for the threshold double-sample selection method was consistently greater than that for the random double-sample selection method for the runs associated with the factorial design settings displayed in Table 2 and the HAPMAP *TAP2* power simulations (see Tables 3 and 5). Furthermore the power gain for the threshold double-sample selection method occurred for either setting of *C _{p}/C_{g}*. For the threshold double-sample selection method,
was small (less than 21%) in our simulations so that our computed

*N** values had a minimum of 963 individuals.

^{DS}## Discussion

In practice, few researchers employ molecular haplotyping techniques in genetic case-control studies. The absence of a high-throughput procedure relative to current SNP genotyping technologies is arguably the main reason that this methodology is not more widely used. Another related reason is the cost in terms of both the time and money associated with employing this methodology. Our research suggests that the additional costs involved in molecular haplotyping may be worth the effort, especially if the cost of phenotyping is high relative to the cost of genotyping for a study. Ji et al. found analogous results for the effects of genotype misclassification on genotypic test of association [61]. In practice, this situation arises for replication studies. A genome-wide scan involving thousands of SNP markers along with subsequent fine mapping in an initial set of case and control individuals may identify a number of promising regions for follow-up studies. These follow-up or replication studies involve recruiting an independent sample of cases and controls for which only SNPs in the promising regions will be genotyped [62]. In replication studies for complex traits, the cost ratio of phenotyping to genotyping may be on the order of thousands. For these situations, the LRT_{ae} for testing haplotype association should provide the most utility. It is interesting to note, however, that applying the threshold double-sample selection method provided comparable powers for both high and low phenotyping to genotyping cost ratios. This finding suggests that this selection strategy may provide additional power for an initial genome-wide association study, as well as for a replication study.

One potential limitation of these test statistics that we selected is the increase in degrees of freedom associated with using haplotype pairs rather than individual haplotypes. In general, larger degrees of freedom may result in a loss of power. That is, methods that fully account for uncertainty in the phase-assignment process [11,63,64] may be more powerful than LRT_{ae} because the LRT_{ae} method examines haplotype pairs rather than single haplotypes and therefore has more degrees of freedom. We chose these statistics for the following reasons: (1) The most general misclassification model involves modeling errors in haplotype pairs rather than in individual haplotypes [51,65,66]. (2) When haplotype pair frequencies deviate from Hardy-Weinberg Equilibrium in either case or control sample populations, test statistics that use single haplotype frequencies may increase false-positive rates and/or lose power [67,68]. (3) In contrast with methods that use single haplotype frequencies, the Cochran-Armitage Linear Test of Trend maintains the nominal false-positive rate and does not lose power [68–70]. To our knowledge, a version of this test that incorporates double-sampling procedures to correct for haplotype miscalls does not currently exist.

A point for further research involves identifying the scenarios that produce differential and non-differential haplotype pair misclassification, respectively, as well as identifying the effects of each kind of misclassification on type I error and power. Under the null hypothesis that haplotype frequency distributions are equal in case and control populations, theoretical and simulation studies (including ours in this work) suggest that misclassification is non-differential. Under the alternative hypothesis, it is conceivable that haplotype pair misclassification rates may be different in case and control populations. Although recent research [47,71] indicates that differential misclassification increases the type I error, the effects of differential misclassification on the power of these statistics are unclear.

Although the current perception may be that molecular haplotyping costs are not cost-effective, recent publications suggest that for relatively small regions of the genome, accurate molecular haplotyping is no more expensive than performing fluorescent polymerase chain reactions [18]. In addition, current techniques are able to provide molecular haplotypes for an entire chromosome at a cost ratio *(C _{mh}/C_{g})* of approximately 5 (C. Ding, personal communication). Finally, as technology improves, the costs associated with molecular haplotyping will likely decrease, and the throughput will likely increase.

### Conclusion

In this work, our simulations showed that the misclassification present in calling phased haplotypes from multilocus genotypes using statistical methods is complete. That is, each misclassified haplotype pair is consistently misclassified as the same incorrect haplotype pair throughout the entire dataset. In addition, our simulations under the null hypothesis of no association demonstrate that applying the theoretical χ^{2} distribution to evaluate the significance of test statistics produces conservative and anticonservative *p*-values whereas applying permutation methods consistently produces *p*-values that maintain the nominal false-positive rate. Consequently, permutation methods should be exclusively used to determine statistical significance for the tests we perform. As expected, the LRT_{ae} provides the greatest advantage in terms of power over the LRT_{std} in situations in which more haplotype misclassification errors are present. These situations arise when the haplotype under investigation comprises many SNP markers with low pair-wise intermarker LD.

For fixed costs, the power gain of the LRT_{ae} over the LRT_{std} varied depending on the relative costs of genotyping, molecular haplotyping, and phenotyping. In general, the LRT_{ae} showed the greatest benefit over the LRT_{std} when the cost of phenotyping was very high relative to the cost of genotyping. This situation is likely to occur in a candidate gene replication study as opposed to a genome-wide association study. For intermediate phenotyping to genotyping cost ratios (e.g., *C _{p}/C_{g}* = 25), the LRT

_{ae}may still provide a power advantage if the cost ratio of molecular haplotyping to genotyping is low (

*C*< 10 for

_{mh}/C_{g}*α*≥ 0.5). Currently, inexpensive long-range PCR methods for molecular haplotyping are under development. As technology improves leading to less-expensive molecular haplotyping methods, the LRT

_{ae}will become applicable to a wider set of circumstances.

### Electronic Database Information

The documentation for SNPHAP and PHASE can be found at http://www-gene.cimr.cam.ac.uk/clayton/software and http://www.stat.washington.edu/stephens/software.html respectively.

The documentation for PAWE can be found at http://linkage.rockefeller.edu/derek/pawe1.html.

Data for the estimation of haplotype frequencies from SNP markers within the *TAP2* gene were downloaded from http://www.hapmap.org/downloads/index.html.en (HapMap public release #16c.1).

LRT_{ae} software is available for free download from ftp://linkage.rockefeller.edu/software/lrtae.

## Supporting Information

### Protocol S1. Mathematical Formulation of LRT_{ae} and LRT_{std} Statistics

doi:10.1371/journal.pgen.0020127.sd001

(415 KB DOC)

## Author Contributions

MAL, JO, and DG conceived and designed the experiments. MAL performed the experiments. MAL analyzed the data. MAL contributed reagents/materials/analysis tools. MAL and DG wrote the paper.

## References

- 1. Altshuler D, Brooks LD, Chakravarti A, Collins FS, Daly MJ, et al. (2005) A haplotype map of the human genome. Nature 437: 1299–1320.
- 2. The International HapMap Consortium (2003) The International HapMap Project. Nature 426: 789–796.
- 3.
Ott J (1999) Analysis of human genetic linkage. 3rd edition. Baltimore: Johns Hopkins University Press. 382 p.
- 4. Ellis NA, Kirchhoff T, Mitra N, Ye TZ, Chuai S, et al. (2006) Localization of breast cancer susceptibility loci by genome-wide SNP linkage disequilibrium mapping. Genet Epidemiol 30: 48–61.
- 5. Martin ER, Lai EH, Gilbert JR, Rogala AR, Afshari AJ, et al. (2000) SNPing away at complex diseases: Analysis of single-nucleotide polymorphisms around APOE in Alzheimer disease. Am J Hum Genet 67: 383–394.
- 6. Fallin D, Cohen A, Essioux L, Chumakov I, Blumenfeld M, et al. (2001) Genetic analysis of case/control data using estimated haplotype frequencies: Application to APOE locus variation and Alzheimer's disease. Genome Res 11: 143–151.
- 7. De La Vega FM, Gordon D, Su X, Scafe C, Isaac H, et al. (2005) Power and sample size calculations for genetic case/control studies using gene-centric SNP maps: application to human chromosomes 6, 21, and 22 in three populations. Hum Hered 60: 43–60.
- 8. Clark AG (2004) The role of haplotypes in candidate gene studies. Genet Epidemiol 27: 321–333.
- 9. Morris RW, Kaplan NL (2002) On the advantage of haplotype analysis in the presence of multiple disease susceptibility alleles. Genet Epidemiol 23: 221–233.
- 10. Akey J, Jin L, Xiong M (2001) Haplotypes vs single marker linkage disequilibrium tests: What do we gain? Eur J Hum Genet 9: 291–300.
- 11. Zaykin DV, Westfall PH, Young SS, Karnoub MA, Wagner MJ, et al. (2002) Testing association of statistically inferred haplotypes with discrete and continuous traits in samples of unrelated individuals. Hum Hered 53: 79–91.
- 12. Hoppe B, Haupl T, Gruber R, Kiesewetter H, Burmester GR, et al. (2006) Detailed analysis of the variability of peptidylarginine deiminase type 4 in German patients with rheumatoid arthritis: A case-control study. Arthritis Res Ther 8: R34.
- 13. Hoppe B, Heymann GA, Tolou F, Kiesewetter H, Doerner T, et al. (2004) High variability of peptidylarginine deiminase 4 (PADI4) in a healthy white population: Characterization of six new variants of PADI4 exons 2–4 by a novel haplotype-specific sequencing-based approach. J Mol Med 82: 762–767.
- 14. Horan M, Millar DS, Hedderich J, Lewis G, Newsway V, et al. (2003) Human growth hormone 1 (GH1) gene expression: Complex haplotype-dependent influence of polymorphic variation in the proximal promoter and locus control region. Hum Mutat 21: 408–423.
- 15. Burgtorf C, Kepper P, Hoehe M, Schmitt C, Reinhardt R, et al. (2003) Clone-based systematic haplotyping (CSH): A procedure for physical haplotyping of whole genomes. Genome Res 13: 2717–2724.
- 16. Ding C, Cantor CR (2003) Direct molecular haplotyping of long-range genomic DNA with M1-PCR. Proc Natl Acad Sci U S A 100: 7449–7453.
- 17. Douglas JA, Boehnke M, Gillanders E, Trent JM, Gruber SB (2001) Experimentally-derived haplotypes substantially increase the efficiency of linkage disequilibrium studies. Nat Genet 28: 361–364.
- 18. Proudnikov D, LaForge KS, Kreek MJ (2004) High-throughput molecular haplotype analysis (allelic assignment) of single-nucleotide polymorphisms by fluorescent polymerase chain reaction. Anal Biochem 335: 165–167.
- 19. Long JC, Williams RC, Urbanek M (1995) An E-M algorithm and testing strategy for multiple-locus haplotypes. Am J Hum Genet 56: 799–810.
- 20. Hawley ME, Kidd KK (1995) HAPLO: A program using the EM algorithm to estimate the frequencies of multi-site haplotypes. J Hered 86: 409–411.
- 21.
Terwilliger JD, Ott J (1994) Handbook of human genetic linkage. Baltimore: Johns Hopkins University Press. 307 p.
- 22. Xie X, Ott J (1993) Testing linkage disequilibrium between a disease gene and marker loci [abstract]. Am J Hum Genet 53(Suppl): 1107.
- 23. Zhao JH, Curtis D, Sham PC (2000) Model-free analysis and permutation tests for allelic associations. Hum Hered 50: 133–139.
- 24. Zhao JH, Sham P (2002) Faster haplotype frequency estimation using unrelated subjects. Hum Hered 53: 36–41.
- 25. Excoffier L, Slatkin M (1995) Maximum-likelihood estimation of molecular haplotype frequencies in a diploid population. Mol Biol Evol 12: 921–927.
- 26. Stephens M, Donnelly P (2003) A comparison of Bayesian methods for haplotype reconstruction from population genotype data. Am J Hum Genet 73: 1162–1169.
- 27. Stephens M, Smith NJ, Donnelly P (2001) A new statistical method for haplotype reconstruction from population data. Am J Hum Genet 68: 978–989.
- 28. Clark AG (1990) Inference of haplotypes from PCR-amplified samples of diploid populations. Mol Biol Evol 7: 111–122.
- 29. Fallin D, Schork NJ (2000) Accuracy of haplotype frequency estimation for biallelic loci, via the expectation-maximization algorithm for unphased diploid genotype data. Am J Hum Genet 67: 947–959.
- 30. Niu T (2004) Algorithms for inferring haplotypes. Genet Epidemiol 27: 334–347.
- 31. Heid IM, Lamina C, Bongardt F, Fischer G, Klopp N, et al. (2005) [How about the uncertainty in the haplotypes in the population-based KORA studies?]. Gesundheitswesen 67(Suppl 1): S132–136.
- 32. Sabbagh A, Darlu P (2005) Inferring haplotypes at the NAT2 locus: The computational approach. BMC Genet 6: 30.
- 33. Kang H, Qin ZS, Niu T, Liu JS (2004) Incorporating genotyping uncertainty in haplotype inference for single-nucleotide polymorphisms. Am J Hum Genet 74: 495–510.
- 34. Adkins RM (2004) Comparison of the accuracy of methods of computational haplotype inference using a large empirical dataset. BMC Genet 5: 22.
- 35. Xu H, Wu X, Spitz MR, Shete S (2004) Comparison of haplotype inference methods using genotypic data from unrelated individuals. Hum Hered 58: 63–68.
- 36. Zhang J, Vingron M, Hoehe MR (2005) Haplotype reconstruction for diploid populations. Hum Hered 59: 144–156.
- 37. Marchini J, Cutler D, Patterson N, Stephens M, Eskin E, et al. (2006) A comparison of phasing algorithms for trios and unrelated individuals. Am J Hum Genet 78: 437–450.
- 38. Hindorff LA, Psaty BM, Carlson CS, Heckbert SR, Lumley T, et al. (2006) Common genetic variation in the prothrombin gene, hormone therapy, and incident nonfatal myocardial infarction in postmenopausal women. Am J Epidemiol 163: 600–607.
- 39. Maksymowych WP, Reeve JP, Reveille JD, Akey JM, Buenviaje H, et al. (2003) High-throughput single-nucleotide polymorphism analysis of the IL1RN locus in patients with ankylosing spondylitis by matrix-assisted laser desorption ionization-time-of-flight mass spectrometry. Arthritis Rheum 48: 2011–2018.
- 40. Hoehe MR, Kopke K, Wendel B, Rohde K, Flachmeier C, et al. (2000) Sequence variability and candidate gene analysis in complex disease: Association of mu opioid receptor gene variation with substance dependence. Hum Mol Genet 9: 2895–2908.
- 41.
Little RJA, Rubin DB (1987) Statistical analysis with missing data. New York: John Wiley. 278 p.
- 42.
Cox DR, Hinkley DV (1974) Theoretical statistics. Boca Raton (Florida): Chapman and Hall/CRC. 511 p.
- 43. Barratt BJ, Payne F, Rance HE, Nutland S, Todd JA, et al. (2002) Identification of the sources of error in allele frequency estimations from pooled DNA indicates an optimal experimental design. Ann Hum Genet 66: 393–405.
- 44. Royston P, Altman DG, Sauerbrei W (2006) Dichotomizing continuous predictors in multiple regression: A bad idea. Stat Med 25: 127–141.
- 45. Gordon D, Finch SJ, Nothnagel M, Ott J (2002) Power and sample size calculations for case-control genetic association tests when errors are present: Application to single nucleotide polymorphisms. Hum Hered 54: 22–33.
- 46. Mote VL, Anderson RL (1965) An investigation of the effect of misclassification on the properties of chi-square-tests in the analysis of categorical data. Biometrika 52: 95–109.
- 47. Clayton DG, Walker NM, Smyth DJ, Pask R, Cooper JD, et al. (2005) Population structure, differential bias and genomic control in a large-scale, case-control association study. Nat Genet 37: 1243–1246.
- 48.
Agresti A (2002) Categorical data analysis. Hoboken (New Jersey): John Wiley and Sons. 710 p.
- 49. Tenenbein A (1970) A double sampling scheme for estimating from binomial data with misclassifications. J Am Stat Assoc 65: 1350–1361.
- 50. Tenenbein A (1972) A double sampling scheme for estimating from misclassified multinomial data with applications to sampling inspection. Technometrics 14: 187–202.
- 51.
Gordon D, Yang Y, Haynes C, Finch SJ, Mendell NR, et al. (2004) Increasing power for tests of genetic association in the presence of phenotype and/or genotype error by use of double-sampling. Stat Appl Genet Mol Biol. 3. Article 26.
- 52.
Box GEP, Hunter WG, Hunter JS (1978) Statistics for experimenters: An introduction to design, data analysis, and model building. New York: John Wiley and Sons. 653 p.
- 53.
DeGroot MH, Schervish MJ (2001) Probability and statistics. 3rd edition. Reading (Massachusetts): Addison-Wesley. 816 p.
- 54. Schaid DJ, Sommer SS (1993) Genotype relative risks: Methods for design and analysis of candidate-gene association studies. Am J Hum Genet 53: 1114–1126.
- 55.
Sham P (1997) Statistics in human genetics. New York: J. Wiley and Sons. 290 p.
- 56. Lewontin RC (1964) The interaction of selection and linkage. I. General considerations; heterotic models. Genetics 49: 49–67.
- 57. Abecasis GR, Cookson WO (2000) GOLD—Graphical overview of linkage disequilibrium. Bioinformatics 16: 182–183.
- 58. Kang SJ, Finch SJ, Haynes C, Gordon D (2004) Quantifying the percent increase in minimum sample size for SNP genotyping errors in genetic model-based association studies. Hum Hered 58: 139–144.
- 59. Kang SJ, Gordon D, Finch SJ (2004) What SNP genotyping errors are most costly for genetic association studies? Genet Epidemiol 26: 132–141.
- 60. Cochran WG (1952) The chi-square test of goodness of fit. Ann Math Stat 23: 315–345.
- 61.
Ji F, Yang Y, Haynes C, Finch SJ, Gordon D (2005) Computing asymptotic power and sample size for case-control genetic association studies in the presence of phenotype and/or genotype misclassification errors. Stat Appl Genet Mol Biol. 4. Article 37.
- 62. Skol AD, Scott LJ, Abecasis GR, Boehnke M (2006) Joint analysis is more efficient than replication-based analysis for two-stage genome-wide association studies. Nat Genet 38: 209–213.
- 63. Stram DO, Leigh Pearce C, Bretsky P, Freedman M, Hirschhorn JN, et al. (2003) Modeling and E-M estimation of haplotype-specific relative risks from genotype data for a case-control study of unrelated individuals. Hum Hered 55: 179–190.
- 64. Schaid DJ, Rowland CM, Tines DE, Jacobson RM, Poland GA (2002) Score tests for association between traits and haplotypes when linkage phase is ambiguous. Am J Hum Genet 70: 425–434.
- 65. Sobel E, Papp JC, Lange K (2002) Detection and integration of genotyping errors in statistical genetics. Am J Hum Genet 70: 496–508.
- 66. Douglas JA, Skol AD, Boehnke M (2002) Probability of detection of genotyping errors and mutations as inheritance inconsistencies in nuclear-family data. Am J Hum Genet 70: 487–495.
- 67. Sasieni PD (1997) From genotypes to genes: Doubling the sample size. Biometrics 53: 1253–1261.
- 68. Czika W, Weir BS (2004) Properties of the multiallelic trend test. Biometrics 60: 69–74.
- 69. Armitage P (1955) Tests for linear trends in proportions and frequencies. Biometrics 11: 375–386.
- 70. Cochran WG (1954) Some methods for strengthening the common chi-squared tests. Biometrics 10: 417–451.
- 71. Moskvina V, Craddock N, Holmans P, Owen MJ, O'Donovan MC (2006) Effects of differential genotyping error rate on the type I error probability of case-control studies. Hum Hered 61: 55–64.