Exponential distribution of Vg for all susceptibility variants in the genome

We assume that the Vg of all susceptibility variants in the genome follow an exponential distribution. The probability density function (pdf) of an exponential distribution can be expressed aswhere the rate λ is the only parameter that characterize the distribution. The mean is simply given by 1/λ. Therefore the number of susceptibility variants can be derived given λ:where is the total heritability.

Probability density function for Vg in GWAS and fitting by maximum likelihood

In practice one cannot observe all the susceptibility variants. Suppose a GWAS was performed and some variants were found to be associated with p-values below a certain threshold α. The probability density of the Vg of these observed significant variants is directly proportional to the standard exponential density multiplied by the power:where pwr is a function that returns the corresponding power for a given Vg, assuming allele frequency, prevalence, type I error (p-value cutoff) and sample size are all fixed. Examples of the power-adjusted pdf for different sample sizes are shown in Figure 1. The power was evaluated based on a simple approach described in detail in So and Sham [7]. Briefly, we computed z = ln(OR)/SE(ln OR) under H₁ and estimated the corresponding power. Here we take advantage of the fact that the power corresponding to a given Vg is grossly similar regardless of the risk allele frequency (see Table S1 for some examples). The allele frequency was therefore not taken into account for power calculation but was set at a fixed value (0.5). The problem will however be significantly complicated if we do consider the modest difference in power for different allele frequencies, as the above simple formulation of Vg distribution can no longer be applied. For a continuous outcome, the exact power can be derived by the variance explained alone (see later sections).

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Figure 1. Probability density of the Vg of detected variants in a GWAS with adjustment for power.

We assume an exponential distribution of Vg for susceptibility variants under unlimited sample size. In practice, sample size and power is limited and small-effect variants will be under-represented. Therefore the probability density should be adjusted for power. “Orig density” denotes the original exponential distribution, and the numbers like 2000/2000 denotes the number of cases and controls. The significance threshold was set at 5×10⁻⁷ and prevalence set at 0.001. Lambda equals 2000. The risk allele frequency was assumed at 0.5.

https://doi.org/10.1371/journal.pone.0013898.g001

To scale the above density so that it integrates to 1, we divide the above density by the normalizing factor . Hence the probability density function (pdf) of Vg, corrected for inadequate coverage of small-effect variants due to limited sample size, is given byNote that the denominator is only a function of λ and is not a function of Vg since Vg is integrated over its possible range.

The above pdf can be used for maximum likelihood estimation of λ from a list of “significant” Vg obtained from genome-wide association studies. can be obtained by maximizing the following log-likelihood function:where i refers to the ith observation. The details of deriving the maximum likelihood estimate (MLE) can be found in the appendix.

Distribution fitting by methods of moments

The theoretical population mean of Vg under the power*density curve is given byWe can then find the λ that gives the closest theoretical mean to the observed sample mean of Vg from GWAS. In other words, we solve the following equation,It turns out that the method of moments and maximum likelihood give identical estimates for λ (the proof is given in Text S1). Therefore we can use either method for distribution fitting. The above formulation given by methods of moments is computationally simple.

Relaxing the p-value threshold

Typically in GWAS we only consider the variants that withstand a Bonferroni correction (typical threshold = 0.05/total no. of variants) to be significant. However, for fitting the density curve and estimation of λ, a small number of false positives may be allowed to reduce variance. Here we allow a more liberal threshold so that more variants can be used in distribution fitting and estimation of λ. At the same time, we make use of the local false discovery rate (lfdr) procedure [8] to remove the effect of false positive variants, so that the estimated mean reflect only the effects of truly associated variants.

The local fdr is the posterior probability of H₀ given the test statistic.The posterior probability of H₁ given the test statistic, or Pr(H₁|Z = z), is given by (1-lfdr). Local fdr is calculated for the z-value of every SNP (e.g. from Wald test in logistic regression) in a GWAS. The Vg for each SNP is weighted by 1-lfdr corresponding to that particular SNP. The weighted mean of Vg can be expressed by

Although theoretically we may include all SNPs in distribution fitting, it should be noted that the local fdr around z = 0 is often close to or equal to 1, due to limited sample size. The actual number of truly associated variants is usually larger than that predicted by local fdr when z is around 0. Inclusion of all variants will lead to an underestimation of mean Vg, as the weights for SNPs with z-values around 0 will be inappropriately low. Therefore we set an fdr threshold and only considered z-values corresponding to a fdr lower than a particular value. Here we assume that when we restrict our consideration to z-values below the fdr threshold, we can accurately assess the non-null density of z (denoted f₁(z) by Efron), and hence “recover” the Vg correponding to these non-null z-values. The statistical power (required for the construction of the power x density curve) is calculated according to the z-value cutoff. Local fdr was calculated by the locfdr program by Efron [9].

Fitting the most significant z-values to the alternative density

Another approach is to consider , the density of the non-null z-statistics. Since we assume the Vg of associated variants follow an exponential distribution with parameter λ, we can derive which is also characterized by the parameter λ. We may then fit the most significant z-values to , and estimate the underlying λ. This is similar to our earlier proposal of fitting observed Vg of significant markers to the power x density curve, but this time the z-values instead of Vg are used for distribution fitting.

Correcting for winner's curse

Winner's curse refers to the possible overestimation of effect size of significant genetic variants in association studies. It may be interpreted as a form of selection bias. The selection of SNPs meeting significance threshold alters the probability distribution of odds ratios (or z-values) for the chosen SNPs, resulting in bias in effect size estimates. The selection bias is most prominent when the study has weak power and when the significance threshold is extreme. This is particularly pertinent to genome-wide association studies, in which the association signals are usually weak and very stringent significance thresholds are often imposed to guard against multiple testing. As overestimation of effect size will lead to overestimation of mean Vg, or underestimation of λ, we investigated various statistical methods based on conditional likelihood to correct for the winner's curse.

The details of the winner's curse correction approaches applied here were described by Ghosh et al. [10] and Zhong & Prentice [11]. We consider the test statistic for a single marker in the following form:The above statistic is assumed to be asymptotically normal. For example, when one performs logistic regression to test for SNP associations, the Wald test statistic will take the above form. then denotes the estimated regression coefficient.

In our case, we are only interested in SNPs with z-values exceeding a certain threshold c, where c is chosen such that the family-wise error rate (FWER) or false discovery rate are controlled below a certain level. The selection of SNPs passing the significance cutoff distorts the original normal distribution. The conditional likelihood after selection can be expressed by [10], [12]

Instead of working with the z-values, Zhong and Prentice [11] worked with the regression coefficients β and used an equivalent form of the above conditional likelihood in their paper. Here for clarity, we focused on the formulation using the z-statistics. A number of corrected estimates of the effect size have been proposed based on the above result. Below we briefly describe five estimators considered in our simulations.

The first one is the MLE estimatorwhich is in fact the same as matching the expectation of the sampling distribution of μ to the observed mean.

The second one is the mean of the normalized conditional likelihood,which aims to reduce the mean squared error from a Bayesian point of view. may be treated as a posterior mean with a flat prior on μ.

The third estimator is the mean of the first two estimatorswhich aims to combine the strength of the first and second estimators.

The fourth estimator produces a sampling distribution of z with median at the observed z-value. The estimator can be expressed aswhere z_obs is the observed z-statistic for the SNP under study. Note that Zhong and Prentice [11] dealt with the distribution of regression coefficient β, but since z and β are directly related through the fixed SE (z = β/SE), it does not make any difference if one works with the sampling distribution of z.

The fifth estimator is a weighted average of the uncorrected regression coefficient estimate () and the median corrected coefficient estimate () (note ). The weight depends on the estimated variance of the uncorrected coefficient () and the difference between the corrected and uncorrected coefficients.where .

All the above correction methods work for binary as well as continuous outcomes. The first four methods just require z-values as input. For the last method, an estimate of the standard error of coefficient is required. As a crude approach, we again assumed a fixed risk allele frequency of 0.5 so that log(OR) and the standard error (SE) can be estimated from z-values. It should be noted that the SE is in fact a function of the risk allele frequency. For the same level of relative risk and the same z-statistic, a larger SE will result if risk allele frequency is lower. Ignoring the risk allele frequency will result in less accurate estimates of SE. Hence it is recommended that one considers the actual risk allele frequencies if they are available. In view of this limitation, the results that are based on the last correction method (i.e the “MSEmedian” estimators) should be regarded as rough estimates only.

Simulation strategy

We simulated z-statistics and investigated the performance of the various estimators in recovering the underlying parameter that generates the statistics. The z-statistics were composed of two groups: the first group corresponded to H₀ with distribution N(0,1) ; the second group corresponded to H₁ (i.e. truly associated variants) with distribution N(δ,1), where δ was determined by the parameter of the distribution of Vg as described below.

We simulated the Vg of associated variants according to the exponential distribution,Given the allele frequency, prevalence and sample size, one may find the relative risk that produces the given Vg and hence a corresponding z-value (denoted δ) can be obtained. Since the same Vg gives rise to similar power and z-values, we simply fix the allele frequency at 0.5.

In each replication we simulated 100,000 z-values, 99.5% of which are null. The non-null z-values were derived from their corresponding exponentially distributed Vg. We assumed a prevalence of 0.001 and a local fdr cut-off of 0.3 throughout. We investigated the performance of a number estimators under three different sample sizes (number of cases = 3500, 5000 and 7000 with equal number of controls) and four different λ (1000, 2000, 3000, 4000). The root mean squared error from 300 simulations was computed. The root mean squared error is given bywhere B is the number of simulations, is the true parameter value and is the estimated parameter value in the i th simulation.

In total fourteen estimators of λ were evaluated. Table 1 provides a summary of the 14 estimators under study. The estimators fell into two main groups. The first group of estimators was obtained from fitting only the variants passing a Bonferroni correction. The family-wise error rate was set at 0.05. The second group of estimators were obtained from fitting all variants below an fdr threshold (set at 0.3), which allowed inclusion of an increased number of markers. In each group, five estimators were obtained by first applying different winner's curse correction methods to the observed effect sizes and then fitting a density x power curve to the corrected levels of Vg. For comparison, we also included in each group an estimator derived from fitting the density x power curve but without any winner's curse corrections. Lastly, we fit the truncated convolution density to the selected z-values in each group.

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Table 1. An overview of the proposed estimators of λ.

https://doi.org/10.1371/journal.pone.0013898.t001

Dealing with continuous traits

Although our previous discussions focus on variance explained of binary outcomes, the method can easily be extended to continuous traits. The only difference lies in the conversion of Vg to z and calculation of power given a level of Vg. In fact fewer assumptions are required for continuous traits. For example, no extra information on the prevalence of disease is required.

A given level of Vg corresponds to a specific z-statistic and the same Vg would always give rise to the same power. In summary, we have the following relations between z and Vg :where n is the sample size.

Recall that we require the derivative to construct the convolution density of z, where g is the function to convert Vg to z. In the case of a continuous outcome, one can easily obtain a closed form expressionThe mathematical details are given in the supplementary methods (Text S1).

Confidence interval (CI) for λ and number of susceptibility variants

Simulation on coverage probabilities of CIs

It is less clear whether these analytic approaches would work if we worked with weighted maximum likelihood. To assess the coverage probability of the proposed analytic CIs, we performed a brief simulation study on a sample size of 3500 cases and 3500 controls and lambdas of 1000, 2000 and 3000. One thousand simulations were carried out. In each simulation run, random z-values are generated as previously described. Ninety-five percent confidence intervals of the parameter λ were computed using two analytic methods (inversion of Fisher information matrix and MLRT), for markers passing Bonferroni correction and for those passing the local fdr threshold of 0.3. We evaluated coverage probabilities and the width of the CIs.

Application to real data

The above distribution fitting methods were applied to a few real disease examples. As estimation of the local fdr requires all summary statistics to be available, we focused on studies that had full summary statistics released before. We included three large-scale studies, one on lipid levels [13], one on type 2 diabetes [14] and the other on Crohn's disease [15]. As the SNPs in a GWAS are correlated due to linkage disequilibrium (LD), several markers may be in strong LD with one casual variant and still show statistical significance. The mean effect size or Vg may be inflated as the signals from these proxy markers were counted as well. We experimented a rough pruning procedure to reduce the effect of correlation. The pruning was applied to the diabetes and Crohn's datasets and SNPs were pruned such that they were at least 30 kb apart.

For the Crohn's disease dataset [15], we also tried to estimate the effect sizes from the replication study alone to replace winner's curse corrections. The loci listed in Table 2 of [15] were taken to be the true associated variants. The power was calculated based on the probability of passing both the 1st-stage p-value threshold (<5×10⁻⁵) and the 2nd-stage threshold (<0.0008 or 0.05/63 by Bonferroni correction) given the effect size. λ was estimated by the same distribution fitting procedure as detailed previously.

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Table 2. Root mean squared error of different estimators from simulations when number of cases and controls each equals 3500.

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Fitting other types of distributions to real data

Although we have focused on the exponential distribution in this study, we noted that this is only theoretical and may not exactly be the true model in real scenarios (please see the discussion section for further details). Particularly, the effect size distribution of complex traits may be more skewed towards zero than an exponential distribution, as suggested in Park et al. [6]. Therefore we have also tried to fit other types of distributions to the real data and observe how the parameter estimates will change. We have chosen to fit gamma distributions as an alternative to exponential distribution. This is because the gamma distribution is highly flexible with both shape and scale parameters. When the shape parameter is one, it is equivalent to an exponential distribution. Distributions that are more skewed towards zero than an exponential density have shape parameters less than one. We set different shape parameters and the scale parameter estimates were estimated by fitting the alternative density of z-values (the methodology is described under the section “Fitting the most significant z-values to the alternative density”). The mean of a gamma distribution is simply the product of the shape and scale parameters.

Simulation results for estimators of λ

Tables 2, 3 and 4 shows the overall performance of 14 different estimators as measured by the root mean squared error from 300 simulations. In total 12 scenarios were studied. Firstly, we focus on the comparison of using only the most significant variants passing Bonferroni correction versus using all variants below a certain local fdr threshold. It is clear that by relaxing the p-value threshold and then removing the false positive effects by local fdr weighting, the root mean squared error can be reduced. The mean, bias and SD of different estimators were shown in tables S2 and S3. The variance (or SD) of the estimators decreased when we allowed more markers for fitting with adjustments for false positives using fdr. The improvement was more marked when the effect size was small (i.e. true λ is large) and the sample size was not large. For example when N_case = N_ctrl = 3500 and λ is 4000, the SD of the fdr-adjusted estimators was only about one-third of the SD of the Bonferroni- adjusted counterparts. The bias of the fdr-adjusted estimators were also in general smaller than that of the Bonferroni- adjusted ones, but for the last two (“MSEmedian” and “fitfZ.conv”[i.e. fitting ]) this situation was often reversed.

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Table 3. Root mean squared error of different estimators from simulations when number of cases and controls each equals 5000.

https://doi.org/10.1371/journal.pone.0013898.t003

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Table 4. Root mean squared error of different estimators from simulations when number of cases and controls each equals 7000.

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Figures 2 and 3 show the boxplots of the different estimators. For the estimators in figure 2, variants were included in fitting if they passed the Bonferroni threshold. The last two estimators (“MSEmedian” and “fitfZ.conv”) were nearly unbiased and had comparable variance. Figure 3 shows estimators that were fdr-adjusted. Overall speaking, the median and fitfZ.conv estimators had the lowest bias among all. The estimator without winner's curse correction was clearly downward biased, no matter which kind of inclusion threshold was used.

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Figure 2. Boxplots of different estimators of λ, with inclusion threshold based on Bonferroni correction.

https://doi.org/10.1371/journal.pone.0013898.g002

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Figure 3. Boxplots of different estimators of λ, with inclusion threshold based on local fdr.

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Considering the overall performance of all estimators, three estimators stood out as the best three in 10 out of 12 scenarios. They included “truncfdr.corr.median”, “truncfdr.corr.MSEmedian” and “truncfdr.fitfZ.conv” (please refer to table 1 for a description of the estimators). As expected, all three were estimators using fdr adjustment. Among winner's curse correction approaches, the median and MSE median estimators outperformed others in the current study. Fitting directly by also had favorable performance.

Simulation results for confidence intervals of λ

Table 5 shows the properties of CIs obtained from 1000 simulations, including coverage probabilities, mean and SD of the width of CI, and the value of the upper and lower 95% CI averaged from the simulations. Under the three simulated scenarios, all four CIs had coverage probabilities close to the theoretical value (0.95), although for “MLRT weighted” (maximum likelihood ratio test using weighted likelihood) the coverage probability seemed to be slightly lower than 0.95. By employing the weighted likelihood approach, we were able to include more variants and not surprisingly, the width of CI was much shorter than that from a standard likelihood approach. The SD of the CI width was also smaller.

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Table 5. Simulation results for different estimates of 95% confidence intervals (CI).

https://doi.org/10.1371/journal.pone.0013898.t005

Results for real disease data

For Crohn's disease, we have tried an approach that did not require any winner's curse corrections. The effect sizes of the true associated variants were extracted from replication studies as described in [15]. λ was estimated at 1196 and hence the estimate of mean Vg was 0.0836%. The other results were summarized in table 6. Only the results based on fdr adjustment were shown since they are superior to Bonferroni-corrected estimators. Readers may refer to table S4 for the Bonferroni-corrected results for reference. The estimated values of λ range from about a few hundreds to around two thousand. Pruning of SNPs did not change the estimates substantially, though we noted an increase in λ for the Crohn's dataset after pruning. The estimated number of susceptibility variants is simply given by the heritability multiplied by λ. The heritabilities for type 2 DM and Crohn's disease were taken to be 0.424 [16] and 0.55 [17]. The heritabilities of HDL, LDL and TG were taken to be 0.63, 0.36 and 0.37 respectively according to Abney et al. [18]. As shown in table 6, the estimated number of susceptibility variants underlying the studied traits ranged from around four hundred to a thousand. Table 7 showed the estimated number of variants when we assumed a gamma distribution that allows a greater number of SNPs with smaller effects. A smaller shape parameter implies that the distribution is more skewed towards zero. As one would expect, the estimated number of variants went up as the distribution is increasingly skewed towards zero. The resulting estimates were about two to three times of the original estimate (based on exponential assumption) when the shape parameter decreased to 0.3.

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Table 6. Estimates and confidence intervals of lambda for a few complex traits, variants included according to fdr threshold.

https://doi.org/10.1371/journal.pone.0013898.t006

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Table 7. Estimated number of susceptibility variants assuming a gamma distribution of effect sizes.

https://doi.org/10.1371/journal.pone.0013898.t007

Limitations

To accurately infer the number of total susceptibility variants in the genome from GWAS data is a very difficult problem, and the methodology presented here should only be interpreted as a rough estimate based on the exponential distribution assumption. We stress that the work presented here is not a perfect solution to the problem. Instead, we made our best attempt to estimate the number of variants, and more importantly, we hoped to provide a useful and rigorous statistical framework to attack the problem.

One difficulty in dealing with GWAS data is that the SNPs are correlated, while most distribution fitting methods in statistics apply to independent observations. Often clusters of SNPs may show high significance but may just contain one or few true independent signals. As a result of the “redundant” significant signals, the mean Vg may be overestimated and λ underestimated. In addition the strong correlation may distort the assumed exponential distribution of effect sizes. One way to alleviate this problem is to prune the SNPs before analyses so that they are roughly independent. On the other hand, pruning reduces the number of markers available for distribution fitting and will increase the variability of the estimates, especially when the effect sizes are small and there are few markers considered significant even in the entire set of genotyped SNPs. There is no perfect solution to this sort of bias-variance tradeoff. However, as the sample size becomes larger, more SNPs will pass the significance threshold and there will be adequate SNPs for stable parameter estimates even if the SNPs are pruned prior to analysis.

Also, the significant associations in a GWAS may not be causal variants themselves, but maybe merely tagging them. We have to assume that the variance explained by significant results in a GWAS is close to that of the causal variants if they were typed.

Current GWAS technologies mainly aim at capturing common variants. As a result, the best thing we can do is to infer the total number of underlying based on the effect sizes of common variations. The contribution of rare variants and structural variations to complex diseases is largely unexplored in the literature. In addition, gene-gene or gene-environmental interactions may also play a role in the etiology of complex diseases, but our current understanding regarding interactions is still very limited. To make predictions on the total number of susceptibility variants in the genome, inevitably we have made the assumption that the other variants that are not captured share a similar distribution of Vg as common variants. Of course this assumption may not work very well in practice. Nonetheless, we think this limitation may become less of a concern in the future when studies accumulate and technologies improve. While now we can only fit the Vg of common variants, with more rare and structural variants discovered in the future, their effect sizes may also be fit using the proposed framework. Interactions may be regarded as multi-locus genotypes and their Vg can be readily calculated and distribution fitting can be done as well. The framework we presented does not pose any restrictions on the actual nature of the susceptibility variants. With accumulation of more studies we will hopefully be able to obtain more reliable estimates considering a greater variety of genetic variations.

We assumed an exponential distribution of Vg in this study, which is justified by theories in adaptation [3]. The exponential distribution is simple with only one parameter, hence the estimation can be quite reliable despite modest sample size. Employing more flexible models with more parameters would allow better fitting when the true distribution is not exponential. However, we will need much larger sample sizes such that there are enough variants passing the significance threshold (Bonferroni or fdr) that are available for fitting. Otherwise, the estimate will be unstable.

Despite the advantages, we must point out that the exponential assumption is a theoretical prediction after all, and may not be exactly true in practice. For example, Park et al.[6] suggested recently that the effect size distributions for complex traits may be more skewed to the left than an exponential distribution. As a result, we may have underestimated the total number of susceptibility loci. In view of this limitation, we have also tried to fit gamma distributions with shape parameters less than one, which allow more variants with small effect sizes than an exponential distribution. When the shape parameter dropped to 0.3, the estimated number of loci increased by about 2 to 3 times compared to the case when the exponential distribution is fitted. The new estimates are mostly in the range of one to two thousands. This gives a rough idea of the range of estimates if the distribution of effect size is more left-skewed.

As first mentioned in the introduction, Park et al. [6] recently have worked on a similar problem. It is notable that Park et al. have proposed a way to deal with the problem in a non-parametric manner without relying on estimate of total heritability. However, it should be noted that we are considering the entire range of effect sizes rather than the observed range, which is a bit trickier. The unobserved effect sizes are likely to be smaller than the observed ones and cannot be modeled easily. Note that non-parametric methods are usually highly flexible. They can fit the observed data-points well (and may perform well within the range of effect sizes observed). However, flexible models also increase the risk of overfitting. They may be less reliable at predicting future data-points as they may magnify the “noise” or random fluctuations in the original data. The danger of overfitting is exaggerated when we have to extrapolate beyond the observed spectrum of effect sizes. The risk of overfitting is also heightened when the number of loci (i.e. data-points) available for fitting is small in the first place, as explained earlier.

As a result, we have focused on the more restrictive parametric models in the current study, and in particular the exponential model. This is not because the exponential model is perfect, but it is the only parametric model with theoretical support and represents the best “educated guess” of effect size distribution in our view. One may change the exponential assumption to other models such as gamma distributions, as was done in this study. With greater sample sizes for association studies and more loci available for fitting, one may be able to reliably fit statistical distributions with more free parameters, for example the gamma distribution (which is very flexible) with free shape and rate parameters. The reliance on the exponential distribution can then be relaxed. The framework presented in this paper can potentially be extended to deal with other types of distributions.

It should be noted that our methodology requires an external estimate of the total heritability. As there are many different statistical approaches and designs that are available for heritability studies, one may need to be aware of the limitations of different heritability estimates. For example, twin and adoption studies can separate shared environmental from shared genetic factors in families, while family-based studies (e.g. those based on siblings and parent-offsprings only) cannot make a distinction between the two types of shared effects.

We have proposed an fdr-based approach to relax the significance threshold so that more markers are available for fitting. Simulations showed that this approach had superior performance when compared to the conventional way of including only the markers that passed the Bonferroni threshold. In practice, confounding factors such as population stratification may be present to increase the false positive rate. Many methods such as principal component analysis are available for correcting population stratification [19]. If major residual confounding is suspected, for example the genomic inflation factor (λ) [20] remains high after correction by principal component analysis, then one should be cautious in relaxing the significance threshold. In such cases, one should also be alert to the most significant markers being false positives, and extensive replications are necessary to confirm the results. One may also consider correcting the results by genomic control [20].

Goldstein [21] presented an alternative approach to estimate the number of variants underlying height in a recent commentary. His approach is different from ours and Park et al. [6]. In [6] and our study, the authors considered the probability density function of effect sizes and obtained parameters estimates by distribution fitting. Goldstein considered the entire range of effect sizes as we do. However, he ranked the SNP from largest to smallest effect sizes and plotted the effect size (in variance explained) (on y-axis) against the rank of each SNP (on x-axis). An exponential function curve was fit to the observed data-points with least-squares regression. Essentially the rank of SNP is the predictor variable and the effect size is the outcome variable. The number of variants is solved using integration. This is not the same as the probably more intuitive and conventional distribution fitting methods in which one fits a distribution to the histogram of effect sizes (i.e. y-axis is the density, x-axis is the effect size). Goldstein's method was only described briefly and the statistical methodology itself was not the major focus of the commentary. No detailed rationale for the approach or simulations was presented. Hence we are unable to make rigorous and comprehensive comparisons of our approach to that described in Goldstein. However, it is clear that in Goldstein's approach, the power of study is not considered and as admitted by the author, it is assumed that all SNPs yet to be found have smaller effects than the weakest one discovered to date. This will lead to an overestimation of the number of variants.

Estimation of the number of variants underlying a trait is a very challenging task. We believe that for most complex diseases, the Vg of susceptibility variants are likely to be rather small. It may be possible to investigate the aggregate effect of these variants, for example Purcell et al. [22] showed a large number of SNPs (with p values up to 0.5) from a schizophrenia GWAS demonstrate predictive power collectively. It will be much harder to estimate the number of variants making up this combined effect as the Vg of each variant is probably tiny. For instance, it may be difficult to distinguish between 100 variants having an average Vg of 0.01% versus 10 variants having an average Vg of 0.1%.

In conclusion, we have developed a novel statistical approach to estimate the number of susceptibility variants in the genome. The performance of different proposed estimators were tested by simulations and applied to some real data examples. Despite the limitations, we believe this study is a useful step towards the understanding of the genetic architecture of complex diseases.