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Genetics. Jul 2006; 173(3): 1747–1760.
PMCID: PMC1526674

Two-Stage Designs in Case–Control Association Analysis

Abstract

DNA pooling is a cost-effective approach for collecting information on marker allele frequency in genetic studies. It is often suggested as a screening tool to identify a subset of candidate markers from a very large number of markers to be followed up by more accurate and informative individual genotyping. In this article, we investigate several statistical properties and design issues related to this two-stage design, including the selection of the candidate markers for second-stage analysis, statistical power of this design, and the probability that truly disease-associated markers are ranked among the top after second-stage analysis. We have derived analytical results on the proportion of markers to be selected for second-stage analysis. For example, to detect disease-associated markers with an allele frequency difference of 0.05 between the cases and controls through an initial sample of 1000 cases and 1000 controls, our results suggest that when the measurement errors are small (0.005), ~3% of the markers should be selected. For the statistical power to identify disease-associated markers, we find that the measurement errors associated with DNA pooling have little effect on its power. This is in contrast to the one-stage pooling scheme where measurement errors may have large effect on statistical power. As for the probability that the disease-associated markers are ranked among the top in the second stage, we show that there is a high probability that at least one disease-associated marker is ranked among the top when the allele frequency differences between the cases and controls are not <0.05 for reasonably large sample sizes, even though the errors associated with DNA pooling in the first stage are not small. Therefore, the two-stage design with DNA pooling as a screening tool offers an efficient strategy in genomewide association studies, even when the measurement errors associated with DNA pooling are nonnegligible. For any disease model, we find that all the statistical results essentially depend on the population allele frequency and the allele frequency differences between the cases and controls at the disease-associated markers. The general conclusions hold whether the second stage uses an entirely independent sample or includes both the samples used in the first stage and an independent set of samples.

GENOMEWIDE case–control association study is a promising approach to identifying disease genes (Risch 2000). For a specific marker, allele frequency difference between cases and controls may indicate potential association between this marker and disease, although other factors (e.g., population stratification) may account for the observed difference. Allele frequencies among the cases and controls can be obtained either through individual genotyping or through DNA pooling. Although individual genotyping provides more accurate estimates of allele frequencies and allows for the inference of haplotypes and the study of genetic interactions, DNA pooling can be more cost effective in genomewide association studies as individual genotyping needs to collect data from hundreds of thousands of markers for each person.

In the absence of measurement errors associated with DNA pooling, there would be no difference between using DNA pooling or individual genotyping for the estimation of allele frequency. However, one major limitation of the current DNA pooling technologies is indeed the errors associated with measuring allele frequencies in the pooled samples. Recent research suggests that for a given pooled DNA sample, the standard deviation of the estimated allele frequency is between 1 and 4% (cf. Buetow et al. 2001; Grupe et al. 2001; Le Hellard et al. 2002; Sham et al. 2002). Le Hellard et al. (2002) reported that using the SNaPshot method, which is based on allele-specific extension or minisequencing from a primer adjacent to the site of the SNP, the standard deviation ranged from 1 to 4%, depending on the specific markers being tested. Our recent studies have found that the errors of this magnitude may have a large effect on the power of case–control association studies using DNA pooling as the sole source for genotyping (see Zou and Zhao 2004 for unrelated population samples and Zou and Zhao 2005 for family samples). Therefore, a two-stage design where DNA pooling is used as a screening tool followed by individual genotyping for validation in an expanded or independent sample may offer an attractive strategy to balance power and cost (Barcellos et al. 1997; Bansal et al. 2002; Barratt et al. 2002; Sham et al. 2002). In such a design, the first stage evaluates a very large number (e.g., 1 million) of markers using DNA pooling, and only the most promising ones are selected and studied in the second stage through individual genotyping. Similar two-stage designs have been considered by Elston (1994) and Elston et al. (1996) in the context of linkage analysis and by Satagopan and Elston (2003) and Satagopan et al. (2002, 2004) in the context of association studies. However, these studies primarily assumed that individual genotyping is used in both stages, which may not be as cost effective as using DNA pooling in the first stage. Moreover, errors associated with genotyping have never been considered in the literature.

When DNA pooling is used as a screening tool in the first stage, the following issues need to be addressed:

  1. How many markers should be chosen after the first stage so that there is a high probability that all or some of the disease-associated markers are included in the individual genotyping (second) stage?
  2. What is the statistical power that a disease-associated marker is identified when the overall false positive rate is appropriately controlled for?
  3. When the primary goal is to ensure that some of the disease-associated markers are ranked among the top L markers after the two-stage analysis, what is the probability that at least one of the disease-associated markers is ranked among the top?

The objective of this article is to provide answers to these practical questions to facilitate the most efficient use of the two-stage design strategy where DNA pooling is used. In genetic studies, the sample in the first stage can be expanded with a set of new samples in the second-stage analysis, or the second stage may involve only a new set of samples for individual genotyping, so both these strategies are considered in our article. We hope that the principles thus learned will provide an effective and practical guide to genetic-association studies.

This article is organized as follows. We first present our analytical results to treat the above three problems and then conduct numerical calculations under various scenarios to gain an overview and insights on these design issues. Finally, some future research directions are discussed.

METHODS

Genetic models:

We consider two alleles, A and a, at a candidate marker, whose frequencies are p and equation M1, respectively. For simplicity, we consider a case–control study with n cases and n controls. Let equation M2 denote the number of allele A carried by the ith individual in the case group, and equation M3 is similarly defined for the ith individual in the control group. Assuming Hardy–Weinberg equilibrium, each equation M4 or equation M5 has a value of 2, 1, 0 with respective probabilities equation M6, 2pq, and equation M7 under the null hypothesis of no association between the candidate marker and disease. When the candidate marker is associated with disease, we assume that the penetrance is equation M8 for genotype AA, equation M9 for genotype Aa, and equation M10 for genotype aa. Note that these two alleles may be true functional alleles or may be in linkage disequilibrium with true functional alleles. Under this genetic model, the probabilities of having k copies of A among the cases, equation M11, and those among the controls, equation M12, are

equation M13

One-stage designs:

For useful reference, we first formulate the test statistics and derive statistical power on the basis of a one-stage design using either individual genotyping or DNA pooling. These can be considered as special cases or direct extensions of the results in Zou and Zhao (2004).

Individual genotyping:

For individual genotyping, let equation M14 and equation M15 denote the observed numbers of allele A in the case group and the control group, respectively, equation M16 and equation M17 denote the population allele frequencies of allele A in these two groups, and equation M18 and equation M19 denote their maximum-likelihood estimates, where equation M20 and equation M21.

Under the null hypothesis of no association between the candidate marker and disease status, equation M22, and equation M23. On the other hand, under the genetic model introduced above,

equation M24

and

equation M25

The statistic to test genetic association between the candidate marker and disease is

equation M26

where equation M27.

Considering a one-sided test and using a significance level of equation M28, the power of the test statistic equation M29 is

equation M30

where equation M31 is the expected frequency of allele A under the genetic model, Φ is the cumulative standard normal distribution function, and equation M32 is the upper 100αth percentile of the standard normal distribution.

DNA pooling:

For DNA pooling, we consider m pools of cases and m pools of controls each having size s such that n = ms. We assume the following model relating the observed allele frequencies estimated from the pooled samples to the true frequencies of allele A in the samples,

equation M33

where equation M34 denotes the number of allele A carried by the jth individual in the ith case group, and equation M35 is defined similarly (i = 1,  , m; j = 1,  , s), and equation M36 and equation M37 are disturbances with mean 0 and variance equation M38 and are assumed to be independent and normally distributed. Define

equation M39

and

equation M40

Under the null hypothesis of no association, equation M41, and equation M42. On the other hand, under the genetic model introduced above,

equation M43

and

equation M44

We can use the following test statistic to test genetic association based on DNA pooling data,

equation M45

where equation M46.

If we use a one-sided test and a significance level of equation M47, the power of the test statistic equation M48 is

equation M49

Two-stage designs:

How many markers should be selected after the pooling stage?

In the first stage, i.e., the DNA pooling stage, we consider m pools of cases and m pools of controls each having size s such that n = ms. The main objective for the first stage is to select the most promising markers on the basis of pooled DNA data to follow up in the second stage to reduce the overall cost. Therefore, the following problem should be addressed: How many of the M markers initially screened should be selected for second-stage analysis so that the probability that the disease-associated markers are selected is high, e.g., 90%? For simplicity, we assume that the associated markers are independent. Let the desired number of markers be equation M50. As in Satagopan et al. (2002, 2004), we choose those markers that have the largest test statistic.

For markers not associated with disease, the test statistic can be approximated by

equation M51

where equation M52, equation M53, equation M54, equation M55, and equation M56 and w are mutually independent. Whereas for markers associated with disease through the genetic model introduced above, the test statistic can be approximated by

equation M57

where equation M58, and equation M59 and w are mutually independent.

Let equation M60 be the test statistics corresponding to the equation M61 disease-associated markers, equation M62 be those corresponding to the equation M63 null markers, and equation M64 are the corresponding ordered test statistics. Let equation M65 denote the probability that the specified equation M66 of the equation M67 truly associated markers are among the top equation M68 markers. Furthermore, denote

equation M69

and

equation M70

Note that equation M71, where

equation M72
equation M73

and equation M74, equation M75, and equation M76 are defined as equation M77, equation M78, and equation M79 with allele frequency equation M80 and penetrances equation M81, equation M82, and equation M83 at the truly associated marker j in place of p, equation M84, equation M85, and equation M86, respectively, equation M87. In addition, equation M88, equation M89. For convenience, we denote the distribution and density functions of equation M90 by equation M91 and equation M92 and the distribution and density functions of equation M93 by equation M94 and equation M95, respectively. Then it can be shown that the joint density function of equation M96 is

equation M97

where

equation M98

and

equation M99

Moreover, the joint density of equation M100 is

equation M101

Hence,

equation M102
(1)

where

equation M103

Therefore, the probability that equation M104 of the equation M105 disease-associated markers are among the top equation M106 markers is given by

equation M107
(2)

From this expression, we can determine the value of equation M108 such that equation M109 is higher than or equal to a given level, e.g., 90%.

For a given equation M110, let equation M111 denote the number of disease-associated markers included in the top equation M112 markers; then its expectation is equation M113. Therefore, we can determine the value of equation M114 through this formula such that the average number of disease-associated markers included in the top equation M115 markers isequation M116; i.e., equation M117 disease-associated markers are selected on average.

The above Equations 1 and 2 are exact but somewhat complicated. In the following, we derive their asymptotic expressions so that we can obtain simpler analytical results. It is easy to see that we need only to consider Equation 1.

For a fixed proportion equation M118, let equation M119 denote the normal distribution quantile corresponding to equation M120, that is, equation M121. Then from the asymptotic property of order statistics, we have

equation M122
(3)

and

equation M123
(4)

when equation M124 tends to infinity, where equation M125 denotes the integer part of equation M126, and equation M127 denotes convergence almost sure.

If we write equation M128, then we have

equation M129
(5)

where

equation M130
(6)

and

equation M131

Note that the total number of markers equation M132 is usually extremely large, the number of disease-associated markers equation M133 is extremely small compared to M, and

equation M134

Therefore, taking equation M135 top markers is equivalent to taking the top markers in the proportion of equation M136.

In particular, when the number of disease-associated markers is equation M137, we can obtain an analytical expression for the selected proportion equation M138 necessary to attain the desired probability that the disease-associated marker is selected. In fact, when equation M139, from Equations 5 and 6, we have

equation M140

Therefore, if we require the probability that the truly associated marker is included in the selected subset from the first stage is at least equation M141, i.e., equation M142, then

equation M143

where equation M144 is the normal distribution quantile corresponding to equation M145. Clearly, the above formula is equivalent to

equation M146

So the proportion equation M147 should satisfy equation M148. Therefore, a conservative selection of the proportion equation M149 is the maximum of equation M150 over various genetic models and allele frequencies.

It should be noted that the above selection approach for markers is through comparing the values of the test statistics at all the markers and no statistical inference is conducted. If statistical tests are performed to select the promising markers, then one would keep those markers showing stronger statistical significance in the first stage. However, the two methods are actually asymptotically equivalent. This is because, if we take equation M151 (where equation M152 is the upper 100equation M153th percentile of the standard normal distribution corresponding to the significance level equation M154 for each marker tested in the first stage), that is, equation M155, which means that the selected proportion of markers is the same as the significance level for testing each marker in the first stage, then the asymptotic probability of the specified equation M156 of equation M157 truly associated markers being selected given in Equation 5 is in fact the statistical power of detecting the specified equation M158 of equation M159 truly associated markers. So for the case of independent markers, selecting the markers through comparing the values of their test statistics is asymptotically equivalent to selecting the markers through statistical tests, a conclusion similar to that of Satagopan et al. (2004) who considered individual genotyping in the first stage. In other words, the selection approach based on statistical tests is the limiting case of that based on comparing the values of test statistics at the markers when the number of total markers is very large.

The statistical power of the two-stage design:

After a set of promising markers is identified through DNA pooling, these markers will be individually genotyped in the second stage. In this subsection, we first derive the statistical power of the two-stage design to detect the disease-associated markers. In the next subsection, we investigate the possibility of at least one disease-associated marker being ranked among the top after the second stage. In addition to the 2equation M160 individuals used in the pooling stage, we also consider an additional sample of size 2equation M161. Under the null hypothesis equation M162, i.e., the marker is not associated with disease, the test statistic for markers tested in the second stage can be written approximately as

equation M163

where equation M164 and equation M165 is independent of equation M166 and w, which were defined above in the discussion of pooled DNA analysis.

Similarly, for markers associated with disease under the genetic model introduced above, the test statistic for markers tested in the second stage can be written approximately as

equation M167

where equation M168, and equation M169 is independent of equation M170 and w, which were defined above in the discussion of pooled DNA analysis.

Under the null hypothesis of no association, equation M171 has a joint bivariate normal distribution equation M172, where

equation M173

Under the alternative hypothesis equation M174, equation M175 has a joint bivariate normal distribution equation M176, where

equation M177

and

equation M178

For a given sample size equation M179 and significance level α1 or power 1 − β1 in the first stage (or a given proportion of markers to be selected for second-stage analysis), we can determine a critical value equation M180 by solving equation M181 or equation M182. Then for the overall significance level equation M183 for testing equation M184 markers and an additional sample of size equation M185, we can determine the critical value equation M186 in the second stage by solving

equation M187

where equation M188 is the density function of equation M189 under equation M190, which is given by

equation M191

where equation M192 is the determinant of the matrix equation M193, and equation M194is the inverse of equation M195.

The probability that a disease-associated marker is identified by the two-stage design is then given by

equation M196

where equation M197 is the density function of equation M198 under equation M199, which is given by

equation M200

In the above two-stage design, the sample in the first stage is reused in the second stage, and this introduces correlation between the two test statistics, equation M201 and equation M202. Therefore, we call this two-stage scheme the two-stage dependent design in the following discussion. On the other hand, we may use two separate samples in the two stages with one sample used for screening and another independent sample used for individual genotyping. In this scenario, the two test statistics, equation M203 and equation M204, are independent. Hereafter we call such a two-stage scheme the two-stage independent design. For the two-stage independent design, the type I error rate and power are simply the products of those in both stages. That is,

equation M205

and

equation M206

The chance of at least one marker associated with disease being ranked among the top L markers after individual genotyping:

We suppose that, among the equation M207 markers selected from the first stage, there are equation M208 markers associated with disease and equation M209 null markers. Without loss of generality, we assume that they are equation M210 and equation M211, respectively. In this case, let equation M212 and equation M213 denote equation M214 and equation M215, respectively. Let equation M216 be the test statistic for the jth truly associated marker, equation M217 be the test statistic for the jth null marker in the second stage, and equation M218 and equation M219 be their order statistics. Then in the second stage, the probability that none of the truly associated markers are ranked among the top equation M220 markers is

equation M221
(7)

where

equation M222

and

equation M223

Like Equation 1, an exact expression for calculating the probability equation M224 can be derived (appendix). Therefore, the probability that at least one truly associated marker is ranked among the top equation M225 markers is obtained by equation M226. Because the exact formula is quite complicated, we provide an approximate one below to simplify the calculation of this probability. First note that equation M227, where

equation M228

and

equation M229

equation M230. We denote the distribution function of equation M231 by equation M232. Also, let equation M233 denote the joint distribution function of equation M234, equation M235.

Now for a fixed proportion equation M236, we have

equation M237

when equation M238 is large, where equation M239 is a normal distribution quantile corresponding to equation M240; that is, equation M241, and equation M242 denotes the integer part of equation M243 as before. Denote equation M244 and then equation M245. Therefore, we substitute equation M246 for equation M247 in Equation 7. This means that as long as equation M248, we think that no truly associated markers are ranked among the top equation M249 markers, regardless of the null markers chosen from the first stage. On the other hand, we have demonstrated that in the first stage, selecting a proportion equation M250 of the markers through comparing the values of the test statistics is asymptotically equivalent to selecting the significant markers through statistical tests with significance level equation M251; that is, the critical value can be taken as equation M252. Therefore, we obtain

equation M253
(8)

where equation M254 is given in Equation 6, and

equation M255

For the two-stage independent design, the probability of at least one truly associated marker being ranked among the top equation M256 markers after the second stage can be easily obtained as

equation M257

where

equation M258

and

equation M259

An approximation to equation M260 is

equation M261
(9)

RESULTS

To see how many markers should be chosen from the pooling stage, we conduct some calculations using Equation 5 first under various genetic models and allele frequencies. The following four genetic models are considered: a dominant model with equation M262, equation M263; a recessive model with equation M264, equation M265; a multiplicative model with equation M266, equation M267, equation M268; and an additive model with equation M269, equation M270, and equation M271 (Risch and Teng 1998; Zou and Zhao 2004). The population frequency of allele A is varied from 0.05, 0.2, to 0.7. We take the sample size to be equation M272 and assume that the number of the disease-associated markers is equation M273.

Table 1 provides the probabilities of equation M274 truly associated markers being among the top 1/1000 markers when we assume the same genetic model and allele frequency at each disease-associated marker and no measurement errors. It is clear from Table 1 that for most cases, the probability that all truly associated markers are among the top 1/1000 markers is high. The probability that these top markers include only some of the truly associated markers is often very low. An explanation is that when there is a signal that the marker is associated with disease, the corresponding test statistic should often be large when the sample size is reasonably large. So the chance for such a marker to be ranked low is rather small. The exceptional cases are the recessive models with small allele frequencies or dominant models with large allele frequencies. This is because the allele frequency difference between the cases and controls is often small in these scenarios and the sample sizes are not large enough to distinguish the signals from noises. However, we can observe from the table that the probability of at least one truly associated marker being among the top 1/1000 markers is uniformly very large except for the recessive models with small allele frequencies. The conclusion still holds for the case in which genetic models and allele frequencies are different at each truly associated marker or the case of different sample sizes (data not shown). So in the following analysis, we consider the chance that at least one truly associated marker is among the top equation M275 of the markers.

TABLE 1
The probability of equation M352equation M353 disease-associated markers ranked among the top 1/1000 markers for the case of the same genetic model and allele frequency at each truly associated marker

Figure 1 presents the probability of at least one truly associated marker being included among the top equation M276 of the markers for a fixed population allele frequency, p and allele frequency difference between the case and control groups, equation M277 [where equation M278 is taken as 0.01; when equation M279 is taken to be other values, the results are similar (data not shown)]. It can be observed from Figure 1 that for given p and equation M280, the probabilities are almost the same under different genetic models. This shows that the probability that at least one truly associated marker is included among the top markers depends on the genetic model and allele frequency mostly through the population allele frequency and allele frequency difference between the case and control groups. Because the exact genetic model is often unavailable to researchers, this fact makes it possible to select the proportion equation M281 on the basis of the assumed population allele frequency and allele frequency difference between the cases and controls at the candidate marker. Note that the effect of the number of truly disease-associated markers on the probability that at least one such marker is included is not very small (data not shown). So we require that the value of equation M282 is chosen so that the probability is >80% for the case of having only one truly associated marker and not <99% for the case of five truly associated markers. For the case of five truly associated markers, the allele frequency differences at four markers are assumed to be at least 0.03. Note that when the number of truly associated markers equation M283 is greater than five, the probability that at least one truly associated marker is included is larger.

Figure 1.
The probability of the truly associated marker being included among the top equation M334 of the markers under different genetic models for the same population allele frequency (0.20) and allele frequency difference between the case and control groups (0.05). From ...

Figure 2 gives the probability that the disease-associated marker is included among the top equation M284 = 6.7% of the markers for various population allele frequencies and allele frequency differences between the cases and controls when there is only one truly associated marker. Figure 2 shows that when the error rate is 0.01, choosing equation M285 can detect the truly associated marker with an allele frequency difference of 0.05 with >80% chance. Furthermore, when there are five disease-associated markers, to detect at least one such marker with >99% chance, the selection proportion should be 7% (data not shown). Therefore, to detect the disease markers with an allele frequency difference of 0.05 at one marker, the selection proportion of 7% is recommended when the error rate is 0.01 and the sample consists of 1000 cases and 1000 controls. To select the truly associated markers with an allele frequency difference of 0.03 at one marker, the proportion equation M286 should be ~29% (data not shown). If the error rate is reduced to 0.005, the proportion equation M287 can be reduced to 3 or 19% to select the truly associated markers with an allele frequency difference of 0.05 or 0.03 at one marker, respectively. The required proportions for including at least one truly associated marker with an allele frequency difference of equation M288, 0.05, 0.07, or 0.10 are summarized in Table 2 when the sample size is equation M289. Generally, the effect of sample size on selecting the disease-associated markers is not very large, especially for the extreme allele frequencies (data not shown). However, it can be seen from Table 2 that reducing the measurement errors can greatly reduce the required proportion equation M290. Therefore, it is important to reduce the measurement errors in the first stage.

Figure 2.
The probability of the truly associated marker being included among the top 6.7% of the markers when the number of disease-associated markers is equation M339. The sample size is equation M340, the error rate is equation M341, and the number of pools formed for either the cases or the controls ...
TABLE 2
The recommended proportion equation M372 of markers selected from the first stage for including at least one truly associated marker with an allele frequency difference of equation M373 at one marker

To investigate the statistical power of the two-stage design, we set the sample size in the first stage to be equation M291 and the supplemental sample size in the second stage to be equation M292. Note that the main purpose in the first stage is to screen for those truly associated markers. Therefore, we hope that the probability of the truly associated markers being included is large. Thus, we set the power to be 95% in the pooling stage. The significance level of the two-stage design for a single-marker test is taken to be equation M293, a level suggested by Risch and Merikangas (1996) for genomewide association studies. The results for the two-stage dependent design under the previous four genetic models are presented in Table 3. Clearly, the power depends on the genetic model and allele frequency. In general, the power is very high for the sample sizes we consider here. The exceptions are the recessive models with a small allele frequency or dominant models with a large allele frequency. From Table 3, we can see that the measurement errors in DNA pooling have little impact on the statistical power of the two-stage design. Our previous studies showed that such an effect can be large for a one-stage design, especially when the error rates are not small (Zou and Zhao 2004). Our finding shows that the impact of measurement errors on the case–control association studies can almost be neglected by using the two-step design, although a larger measurement error will lead to more markers to be selected in the first stage. Compared to the one-stage design, the two-stage strategy has slightly smaller power due to the selection in the first stage (data not shown). When the two-stage independent design is used, the power is higher than that of the two-stage dependent design (Table 4). In our calculation, we assume that the same numbers of the cases and the controls are typed at the second stage for both designs, which implies that more efforts are needed for the two-stage independent design to collect additional cases and controls compared to the two-stage dependent design. Our calculation shows that if we ignore the correlation between the two stages for a two-stage dependent design, then we will slightly overestimate the power. On the other hand, from Table 4, the two-stage independent design is more affected by the measurement errors than the two-stage dependent design but less affected than the one-stage pooling scheme.

TABLE 3
The power of the two-stage dependent design for the sample sizes of equation M377 and equation M378
TABLE 4
The power of the two-stage independent design for the sample sizes of equation M390 in the first stage and equation M391 in the second stage

Table 5 gives the statistical power of the two-stage dependent design for the fixed allele frequency and allele frequency difference between the cases and controls (where equation M294 is still taken as 0.01). It can be observed from Table 5 that for given p and equation M295, the power is almost the same under different genetic models. This shows that the power of the two-stage design depends on the genetic model and allele frequency almost only through the population allele frequency and allele frequency difference between the case and control groups. As before, this observation is useful in practice because, although the genetic models are often unknown to us, we can estimate the sample size to attain the desired significance level and power under different genetic models as long as the allele frequencies in the general population and the allele frequency differences between the cases and controls can be assumed.

TABLE 5
The power of the two-stage dependent design for the fixed allele frequency and allele frequency difference between the case and the control groups

We use the approximate Equation 8 to calculate the probability of at least one truly associated marker being ranked among the top equation M296 markers after the second stage for the two-stage dependent design. Likewise, the probabilities are almost the same under different genetic models for the same population allele frequency and allele frequency difference between the case and control groups (data not shown). As an example, we consider a recessive model with a population allele frequency of 0.2 and allele frequency difference of 0.05. The results are presented in Figure 3. It can be seen that there is a high probability for the top 50 markers to include at least one truly associated marker when 1% of the markers are selected from the first stage, even though the measurement errors are not small. However, this probability may not be high for detecting disease-associated markers with small allele frequency differences, e.g., 0.03 (data not shown). Essentially, the chance that at least one truly associated marker is ranked among the top equation M297 markers after the second stage is higher for markers with larger allele frequency differences. The conclusion is similar for the two-stage independent design (data not shown). In general, the probabilities are not larger for the two-stage independent design than those for the two-stage dependent design. This can be understood by noting the positive correlation between the two stages for the two-stage dependent design that leads to the smaller value of the right-hand side of Equation 8 than equation M298.

Figure 3.
The probability of at least one truly associated marker being ranked among the top equation M343 markers after the second stage for the two-stage dependent design where the sample sizes are equation M344 and equation M345, the error rate is equation M346, and the number of pools formed for the cases or ...

DISCUSSION

In this article, we have investigated the two-stage design with DNA pooling used in the first-stage screening. Three related problems have been considered: (i) How many markers should be chosen from the first stage?, (ii) What is the overall statistical power when the two-stage design is used?, and (iii) What is the probability that at least one of the disease-associated markers is ranked among the top after the second stage? Our analyses show that the answers to these questions are dependent on the genetic models and allele frequencies essentially through the population allele frequencies and allele frequency differences between the case and the control groups at the candidate markers. For the first problem, we have derived the proportion of markers that needs to be selected to include the truly associated markers. For instance, when the measurement errors are small (0.005), 3% of the markers need to be selected to include a disease-associated marker with an allele frequency difference of 0.05 between the case and control groups for a sample consisting of 1000 cases and 1000 controls. When the measurement errors are not small, multiple pools can be formed to reduce measurement errors. For the second problem, we have derived the formula for calculating the statistical power of a two-stage strategy. We find that the measurement errors in pooled DNA have little effect on the power when the two-stage design, especially the two-stage dependent design, is used, contrary to the single-stage pooling scheme. Recalling our conclusion that reducing measurement errors can greatly reduce the selection proportion of markers in the pooling stage, we see that for a two-stage design, measurement errors have a large impact only on the first stage. Once the markers are selected, the effect of measurement errors can be very small. Three strategies, the two-stage dependent design, the two-stage independent design, and the one-stage design, have been compared. Overall, the two-stage independent design has the highest power, and the one-stage design with individual genotyping has slightly higher power than the two-stage dependent design. However, their difference in power is not large. On the other hand, the one-stage design will be either too expensive (for individual genotyping) in genomewide search or seriously affected by measurement errors (for DNA pooling). Furthermore, for the two-stage independent design, extra sample collection is needed, although the genotyping cost is the same as in the two-stage dependent design. In fact, if in our calculations, we use exactly the same number of individuals as that in the two-stage dependent design with 500 used to screen and the other 500 for follow-up analyses, the statistical power for such a two-stage independent design can be much lower than that of the two-stage dependent design. For example, the power under the multiplicative model with a population allele frequency of 0.05 and a measurement error rate of 0.005 is 0.209 for the above two-stage independent design but 0.599 for the two-stage dependent design. For the third problem, our studies show that the chance that at least one truly associated marker selected from the first stage is ranked among the top markers after the second stage is high when the allele frequency differences are not <0.05 for samples of reasonable sizes, even though the measurement errors are not small.

It is of practical interest how to allocate the sample sizes in the two stages to maximize the power (or minimize the total cost) for a given cost (or given power), as Satagopan et al. (2002), Satagopan and Elston (2003), and Satagopan et al. (2004) have done. For example, let equation M299 be the total cost, equation M300 be the cost of recruiting an individual, equation M301 be the cost of measuring allele frequency at a single marker for a DNA pool, equation M302 be the cost of genotyping a single marker for an individual, and equation M303 be the other cost such as administration. Then we have

equation M304

for the two-stage dependent design, and

equation M305

for the two-stage independent design. In particular, we take the number of total markers to be equation M306, the number of the truly disease-associated markers to be equation M307, and the number of pool pairs to be equation M308. Further, we take equation M309 (unit: United States dollar), equation M310, equation M311, equation M312, equation M313, and equation M314. Then our preliminary calculation results showed that for the given cost, the optimal design that leads to highest power is to allocate exactly (nearly) the same sample size to each stage for the two-stage dependent (independent) design (Y. Zuo, J. Wang, G. Zou, H. Zhao and H. Liang, unpublished results). For the two-stage dependent design, this means that all individuals should be used at both stages and no additional sample is needed at the second stage. This is similar to the two-stage individual genotyping design with sample size constraint (Satagopan et al. 2004) but is different from the design with individual genotyping at both stages in which the optimal design maximizing power is to allocate ~25% of the individuals to the first stage and the remaining individuals to the second stage (Satagopan et al. 2002; Satagopan and Elston 2003). Clearly, an overall investigation is needed in this regard. This warrants our further research.

To simplify our analyses, we have assumed independence among the markers. This would be reasonable when the marker density is low. However, for a genomewide association study, the marker density is high and adjacent markers may be highly correlated. But it is not evident how to model the correlation among markers. One way to avoid this difficulty is to study many subsets of the whole marker set such that they cover the entire genome yet the markers are independent. However, this is clearly less than satisfactory due to the loss of information in the data. On the other hand, this question can be examined empirically to assess the effect of correlations among markers on our results. For example, we have investigated the effect of correlation on the selection of markers in the first stage through the HapMap data. We considered the SNPs on the 500K SNP Array and used the HapMap data to approximate the level of correlations among SNPs. The HapMap data consist of 270 individuals from four populations, and the information for the 500K data can be downloaded from http://www.affymetrix.com/support/downloads/data/500K_HapMap270.zip (For the missing alleles, we imputed them by the corresponding frequencies of the existing alleles). For simplicity, we have considered only the first 300 markers and let the 140th marker be disease associated to illustrate the impact of marker dependence and a more thorough investigation will be reported in future articles. Assuming a dominant model with equation M315, the allele frequency difference between the case and control groups is equation M316. We considered the sample sizes of the two pools to be equation M317. Using the results established before under the independence assumption, we found that if we took the top equation M318, and equation M319 of the markers when equation M320, equation M321, equation M322, and equation M323, respectively, then we would have the chance of equation M324 to select the disease-associated marker (i.e., 140th marker) in the first stage. When we applied these equation M325's obtained under the independence assumption to the HapMap data, we observed that in 10,000 simulations, we had the chances of equation M326, equation M327, equation M328, and equation M329 to include the disease-associated marker when equation M330, equation M331, equation M332, and equation M333, respectively. This shows that the correlation among markers can reduce the chance that the truly disease-associated marker is selected but such reduction is not large. Further, the impact of correlation is larger (smaller) for less (more) stringent requirement on the chance of including the disease-associated marker under the independence assumption (data not shown). Clearly, to eliminate the effect of correlation, the best way is to develop similar methods to those given in this article incorporating the correlations among markers, and this will be addressed in our future work.

Throughout this article, we have assumed that measurement errors exist in the DNA pooling stage but not in the individual genotyping stage. How genotyping errors at both stages can affect the efficiency of the two-stage scheme also warrants future research.

Note that family based data are often used in genetic epidemiological studies in addition to population-based data. Association studies using pooled DNA family data have been considered for the one-stage scheme (e.g., Risch and Teng 1998; Zou and Zhao 2005). The research on the two-stage designs using family data is no doubt an interesting topic for future research.

Acknowledgments

The authors are grateful to the associate editor Yunxin Fu and to the two reviewers for their constructive comments and suggestions that led to great improvement of the original manuscript. This work was supported in part by grants DMS0234078 from the National Science Foundation (to Y. Zuo), 70221001 and 10471043 from the National Natural Science Foundation of China (to G. Zou), and GM59507 from the National Institutes of Health (to H. Zhao).

APPENDIX: THE CALCULATION OF THE PROBABILITY THAT NONE OF THE TRULY ASSOCIATED MARKERS ARE RANKED AMONG THE TOP L MARKERS

Clearly, equation M409 can be written as

equation M410
(A1)

We have known equation M411 ~ equation M412, equation M413, and equation M414 ~ equation M415, equation M416. We denote the distribution and density functions of equation M417 by equation M418 and equation M419, respectively. The distribution and density functions of equation M420 are still denoted as equation M421 and equation M422, respectively. Further, let equation M423 denote the joint distribution of equation M424, equation M425, and equation M426 denote the joint distribution of equation M427, equation M428. Moreover, equation M429 and equation M430 denote the corresponding density functions. Then it can be shown that

equation M431
(A2)
equation M432
(A3)
equation M433
(A4)

and

equation M434
(A5)

where

equation M435

with equation M436, and equation M437, and

equation M438

with equation M439 being some equation M440 numbers of equation M441, and

equation M442

and equation M443 and equation M444.

Combining (A1) and (A2)–(A5), we can obtain equation M445. Thus, the probability that at least one truly associated marker is ranked among the top equation M446 markers can be calculated by equation M447.

References

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