Interpreting MetaAnalyses of GenomeWide Association Studies
Metaanalysis is an increasingly popular tool for combining multiple genomewide association studies in a single analysis to identify associations with small effect sizes. The effect sizes between studies in a metaanalysis may differ and these differences, or heterogeneity, can be caused by many factors. If heterogeneity is observed in the results of a metaanalysis, interpreting the cause of heterogeneity is important because the correct interpretation can lead to a better understanding of the disease and a more effective design of a replication study. However, interpreting heterogeneous results is difficult. The standard approach of examining the association pvalues of the studies does not effectively predict if the effect exists in each study. In this paper, we propose a framework facilitating the interpretation of the results of a metaanalysis. Our framework is based on a new statistic representing the posterior probability that the effect exists in each study, which is estimated utilizing crossstudy information. Simulations and application to the real data show that our framework can effectively segregate the studies predicted to have an effect, the studies predicted to not have an effect, and the ambiguous studies that are underpowered. In addition to helping interpretation, the new framework also allows us to develop a new association testing procedure taking into account the existence of effect.
Published in the journal:
. PLoS Genet 8(3): e32767. doi:10.1371/journal.pgen.1002555
Category:
Research Article
doi: 10.1371/journal.pgen.1002555
Summary
Metaanalysis is an increasingly popular tool for combining multiple genomewide association studies in a single analysis to identify associations with small effect sizes. The effect sizes between studies in a metaanalysis may differ and these differences, or heterogeneity, can be caused by many factors. If heterogeneity is observed in the results of a metaanalysis, interpreting the cause of heterogeneity is important because the correct interpretation can lead to a better understanding of the disease and a more effective design of a replication study. However, interpreting heterogeneous results is difficult. The standard approach of examining the association pvalues of the studies does not effectively predict if the effect exists in each study. In this paper, we propose a framework facilitating the interpretation of the results of a metaanalysis. Our framework is based on a new statistic representing the posterior probability that the effect exists in each study, which is estimated utilizing crossstudy information. Simulations and application to the real data show that our framework can effectively segregate the studies predicted to have an effect, the studies predicted to not have an effect, and the ambiguous studies that are underpowered. In addition to helping interpretation, the new framework also allows us to develop a new association testing procedure taking into account the existence of effect.
Introduction
Metaanalysis is a tool for aggregating information from multiple independent studies [1]–[3]. In genomewide association studies (GWASs) [4], the use of metaanalysis is becoming more and more popular because one can virtually collect tens of thousands of individuals that will provide power to identify associated variants with small effect sizes [5]–[7]. Several large scale metaanalyses have been performed for diseases including type 1 diabetes [8], type 2 diabetes [9]–[11], bipolar disorder [12], Crohns disease [13], and rheumatoid arthritis [14], and have identified associations not revealed in the individual studies.
In metaanalyses, the effect size between studies may differ and this difference, or heterogeneity, can be caused by many factors [15]–[18]. If the populations are different between studies, the genetic factors can cause heterogeneity [19], [20]. If the subjects are from different regions, the environmental factors can cause heterogeneity [21]. Even if the true effect size is invariant, the design factors can also cause a phenomenon that looks like heterogeneity, what is often called the statistical heterogeneity [22]. If the linkage disequilibrium structures are different between studies, the collected marker can show heterogeneity [23]. If the studies use different genotyping platforms, different imputation accuracies and different genotyping errors can cause heterogeneity [24].
In current metaanalyses of genomewide association studies, heterogeneity is often observed in the results [9]–[11], [13], [17]. Interpreting the cause of such heterogeneity is important. If the heterogeneity is caused by either genetic or environmental factors, understanding the cause of heterogeneity can help our understanding of the disease mechanism. If the heterogeneity is statistical heterogeneity caused by the design factors, understanding the cause of heterogeneity is crucial in designing a replication study so that we can eliminate the design factors that can hinder the revelation of the true effect in the replication study.
However, interpreting heterogeneous results is difficult. One standard approach is to examine the association pvalues of the studies. The inherent limitation of this approach is that a nonsignificant pvalue is not evidence of the absence of an effect. Thus, a pvalue does not provide the full information necessary for the interpretation whether or not there is an effect in the study. Another standard approach is to plot observed effect sizes and their confidence intervals of all studies [17], [25], [26]. This plot can be overly complicated when the number of studies is large and does not provide a single estimate that represents the existence of an effect in each study. The limitation of both approaches is that they use classical estimates that are calculated using only the data of each single study. That is, they utilize only withinstudy information.
In this paper, we propose a framework facilitating the interpretation of the results of a metaanalysis. Our framework is based on a new statistic termed the mvalue which is the posterior probability that the effect exists in each study. Plotting the new statistic together with pvalues in a twodimensional space helps us distinguish between the studies predicted to have an effect, the studies predicted to not have an effect, and the ambiguous studies that are underpowered. We name this plot a PM plot. In this framework, the outlier studies showing distinct characteristics from the other studies are easily identified, as we demonstrate using data from type 2 diabetes and Crohns disease metaanalyses [10], [13]. Our new statistic is fundamentally different from traditional estimates based on the data of single studies. We use all studies simultaneously to calculate the new statistic based on the assumption that the effect sizes are similar if the effect exists. Thus, we utilize crossstudy information as well as withinstudy information.
In addition to helping interpretation, the new framework allows us to develop a new association testing procedure which takes into account the presence or absence of the effect. The new method called the binary effects model is a weighted sum of zscores method [5] assigning a greater weight to the studies predicted to have an effect and a smaller weight to the studies predicted to not have an effect. Application to the Crohns disease data [13] shows that the new method gives more significant pvalues than previous methods at certain loci already identified as associated.
The new method is available at http://genetics.cs.ucla.edu/meta.
Methods
Binary Effects Assumption
In our framework, we use a simplified model to describe heterogeneity among the studies which makes two assumptions. The first assumption is that effect is either present or absent in the studies. This assumption is different from the traditional assumption assuming normally distributed effect sizes [27]–[29]. Our assumption is inspired by the phenomenon that the effect sizes are sometimes observed to be much smaller in some studies than in the others. It is reported that different populations can cause such phenomenon [19], [20], [30], [31]. For example, the homozygosity for APOE 4 variant is known to confer fivefold smaller risk of Alzheimer disease in African Americans than in Asians [19], [30]. The HapK haplotype spanning the LTA4H gene is shown to confer threefold smaller risk of myocardial infraction in the populations of Europeans decent than in African Americans [31]. The HNF4A P2 promoter variants are shown to be associated with type 2 diabetes in Ashkenazi and the results have been replicated [20]. However, in the same study, the same variants did not show associations in four different cohorts of UK population suggesting a heterogeneous effect. Geneenvironmental interactions can also cause such phenomenon. If a study lacks an environmental factor necessary for the interaction, the observed effect size can be much smaller in that study. It is generally agreed that the geneenvironmental interactions exist in many diseases such as cardio vascular diseases [32], respiratory diseases [33], and mental disorders [34].
The second assumption is that if the effect exists, the effect sizes are similar between studies. We call these two assumptions together the binary effects assumption. While other types of heterogeneity structures are possible such as arbitrary effect sizes, for identifying which studies have an effect and which studies do not have an effect, we expect that this model will be appropriate.
MValue
We propose a statistic called the mvalue which is the posterior probability that the effect exists in each study of a metaanalysis. Suppose that we analyze studies together in a metaanalysis. Let () be the observed effect size of study and let be the estimated variance of . It is a common practice to consider the true variance. In the current GWASs, the distribution of is well approximated by a normal distribution due to the large sample sizes. Let denote the observed data.
If there is no effect in study ,where is the probability density function of a normal distribution whose mean is and the variance is . If there is effect in study ,where is the unknown true effect size.
Since we want a posterior probability, the Bayesian framework is a good fit. We assume that the prior for the effect size isA possible choice for in GWASs is 0.2 for small effect and 0.4 for large effect [35], [36].
Let be a random variable which has a value 1 if study has an effect and a value 0 if study does not have an effect. Let be the prior probability that each study will have an effect such thatThen we assume a beta prior on Through this paper, we use the uniform distribution prior ( and ), but other priors can also be chosen.
Let be the vector indicating the existence of effect in all studies. can have different values. Let be the set of those values.
Our goal is to estimate the mvalue , the posterior probability that the effect exists in study . By the Bayes' theorem,(1)where is a subset of whose elements' th value is 1. Thus, we only need to know for each the posterior probability of ,consisting of the probability of given and the prior probability of .
The prior probability of iswhere is the number of 1's in and is the beta function.
And the probability of given is(2)where is the indices of 0 in and is the indices of 1 in . We can analytically work on the integration to obtainwherewhere is the inverse variance or precision. The summations are all with respect to .
is a scaling factor such thatThe details of the derivation is in Text S1 in Supporting Information S1. As a result, we can calculate for every and therefore obtain for each study .
MCMC
The drawback of the exact calculation of mvalue is that we need to iterate over all which is exponential to . This is not problematic in most of the current metaanalyses of GWASs, but will be problematic in future studies if increases over several tens. Therefore, here we propose a simple Markov Chain Monte Carlo (MCMC) method to estimate mvalue.
We propose the following MetropolisHastings algorithm [37].

Start from a random .

Choose a next .

If , move to . Otherwise, move to with probability .

Repeat from step 2.
The set of moves we use for choosing is . is a simple flipping move of between 0 and 1. is a move that shuffles the values of . This move is introduced to avoid being stuck on one mode in a special case that there are two modes which can happen when the observed direction of the effect is opposite in some studies. At each step, we randomly choose a move from this set assuming a uniform distribution. We allow burnin and sample times. After sampling, samples gives us an approximation of the distribution over , which subsequently gives the approximations of mvalues by the formula (1).
Interpretations and predictions
The mvalue has a valid probabilistic interpretation that it is the posterior probability that the effect exists in each study under our binary effects model. If we are to choose studies predicted to have an effect and studies predicted to not have an effect, a threshold is needed. In this paper, we use the threshold of mvalue for the former and mvalue for the latter. Although this thresholding is arbitrary, the actual level of threshold is often not of importance because outlier studies showing different characteristics from the other studies usually stand out in the plotting framework described below.
Relationship to PPA
The mvalue is closely related to the posterior probability of association (PPA) based on the Bayes factor (BF) [35] in the sense that the presence and absence of effects are essentially describing the same things as the alternative and null models in the association testing. There are two fundamental differences. First, in the usual PPA, the prior probability of association () is given by a point prior which is usually a very small value in GWAS reflecting the fact that the true associations are few. In our framework, we focus on interpreting metaanalysis results after we find associations using metaanalysis. Thus, reflects our belief on the effect conditioned on that the associations are already significant. For this reason, we need not use a very small value but instead choose to use a distribution prior. Second, the PPA is calculated for each study separately. However, the mvalue is calculated using all studies simultaneously utilizing crossstudy information. Thus, if the binary effects assumption approximates the truth, the mvalue is more effective in predicting effects than the PPA or equivalently the BF, as we show by simulations in Results.
PM Plot
We propose plotting the studies' pvalues and mvalues together in two dimensions. This plot, which we call the PM plot, can help interpreting the results of a metaanalysis. Figure 1 shows that how to interpret such a plot. The rightmost (pink) region is where the studies are predicted to have an effect. Often, a study can be in this region even if the pvalue is not very significant. The leftmost (lightblue) region is where the studies are predicted to not have an effect. This suggests that the sample size is large but the observed effect size is close to zero, suggesting a possibility that there exists no effect in that study. The middle (green) region is where the prediction is ambiguous. A study can be in this region because the study is underpowered due to a small sample size. If the sample size increases, the study will be drawn to either the left or the right side.
If the binary effects assumption does not hold, a study can sit in an unexpected region and a careful interpretation is necessary. For example, if the effects are significant but the effect sizes are in opposite direction in some studies, the studies can sit in the unusual top left region. However, such case will be rare and may be a result of the strand errors.
Binary Effects Model
We propose a new type of random effects model metaanalysis approach called the binary effects model. If the binary effects assumption holds, that is, if the effect is either present or absent in the studies, taking into account this pattern of heterogeneity in the association testing procedure can increase power compared to the general RE approach [23]. The binary effects model we propose is the weighted sum of zscores method [5] where the mvalues are incorporated into the weights. Intuitively, this is equivalent to assigning a greater weight to the studies predicted to have an effect and a smaller weight to the studies predicted to not have an effect.
Let be the zscore of study . The common form of the weighted sum of zscores statistic for the fixed effects model isIn many cases, the weight approximates to the form where is the sample size and is the minor allele frequency [23]. When the minor allele frequency is similar between studies, the weight approximates to the popular form of [5].
The binary effects model statistic we propose isOur method is an empirical approach that uses estimated from the data as the prior weight for each study. Since the mvalue is estimated using all studies, our approach can be thought of as gathering information from all studies and distributing back to each study in the form of weight. We choose this approach because of its simple formulation.
Since is not independent of , the statistic does not follow a normal distribution. Thus, the pvalue is obtained using sampling which can be inefficient. We use two ideas to expedite the sampling. First, we propose an importance sampling procedure which is more efficient than the standard sampling. Second, we use an efficient approximation of mvalue. See Text S2 and S3 in Supporting Information S1 for details.
Simulation Framework
In order to evaluate our methods, we use the following simulation approach. Assuming a minor allele frequency, a relative risk, and the number of individuals of cases and controls, a straightforward simulation approach is to sample alleles for cases and alleles for controls according to the expected minor allele frequencies in the cases and controls respectively [38]. However, since we perform extensive simulations assuming thousands of individuals, we use an approximation approach that samples the minor allele count from a normal distribution and rounds it to the nearest nonnegative integer.
Web Resources
The URL for methods presented herein is as follows:
http://genetics.cs.ucla.edu/meta
Results
Simulation of MValues
We demonstrate a simple simulation example showing how mvalue behaves depending on the presence and absence of the effect and the sample size. First, we make the following assumptions throughout all of the experiments in this paper. We assume that the minor allele frequency of the collected marker is 0.3. We assume that the equal number of cases and controls are collected and refers to the total number of individuals as sample size . We also assume a very small disease prevalence when we calculate the expected minor allele frequencies for cases and controls given a relative risk . For the details how the expected values are calculated, see Han and Eskin [23]. Note that these assumptions are not critical factors affecting our simulation results. In all experiments, the random effects model (RE) denotes the RE method of Han and Eskin [23]. We omit the results of the conventional RE method [15] because they are highly conservative [23]. Throughout this paper, we use the following priors for calculating mvalues. We use for the prior of the effect size (). We use the uniform distribution prior, , for the prior of the existence of effect ().
In this simulation example, we assume four different types of studies. The first type is a large study having an effect ( and ). The second type is a small study having an effect ( and ). The third type is a large study not having an effect ( and ). The fourth type is a small study not having an effect ( and ). We generate two studies per each type, constructing a simulated metaanalysis set of total eight studies. We accept this simulation set only if none of eight studies' pvalues exceeds the genomewide threshold () but the metaanalysis pvalue calculated by the RE approach exceeds the genomewide threshold. Otherwise, we repeat. We construct 1,000 metaanalysis sets.
Given this simulated data, we plot the histogram of mvalues for each type of studies separately in Figure 2. Figure 2A shows that almost all (99.9%) of large studies with an effect are concentrated on large mvalues (), showing that the mvalues effectively predict that the effect exists in the studies. Figure 2C shows that a large amount (78.6%) of large studies without an effect are concentrated on small mvalues (). Figure 2B and 2D show that when the sample size is small, mvalue tends to the midrange regardless of the effect, suggesting that the studies are underpowered to determine the presence of an effect.
Comparison of PValue, MValue, and BF
In this experiment, we compare the pvalue, mvalue, and BF by measuring how well they predict which studies have an effect and which studies do not have an effect. We assume a metaanalysis of 10 studies where the effect is either present () or not. We randomly pick the number of studies having an effect () from a uniform distribution ranging from 1 to 9, and randomly decide which studies have an effect. We randomly pick the sample size of each study from a uniform distribution between 500 and 2,000. Given the sample sizes and the effect sizes, we generate a metaanalysis study set. We accept the metaanalysis set only if none of the studies' pvalues exceeds the genomewide threshold () and the metaanalysis pvalue exceeds the genomewide threshold. We repeat until we construct 1,000 metaanalysis sets.
We examine each of 10,000 studies included in the simulated 1,000 metaanalysis sets. For each study, we calculate the pvalue, mvalue, and BF. We use the asymptotic BF of Wakefield [39] assuming the same prior distribution about the effect size as the mvalue. Then we evaluate the performance of each statistic as follows. To evaluate the performance of mvalue, we fix an arbitrary threshold so that we predict the studies having mvalue to have an effect. Since we know the underlying truth if the effect exists or not in each study, we can measure what proportion of the studies actually having an effect is correctly predicted to have an effect (true prediction rate) and what proportion of the studies actually not having an effect is incorrectly predicted to have an effect (false prediction rate). Then we change the threshold to draw a curve between the true prediction rate and the false prediction rate, which is often called the receiveroperatingcharacteristic (ROC) curve. We do the same analysis for pvalue and BF.
Figure 3A shows that mvalue is superior to pvalue and BF in predicting the studies having an effect. This is because mvalue can utilize the crossstudy information when the binary effects assumption holds. The performances of pvalue and BF are almost identical.
Next, we evaluate the performance of the statistics in predicting studies not having an effect. The experiment is exactly the same as the previous experiment except that, given a threshold , we predict the studies having mvalue to not have an effect. We similarly draw the ROC curves for the three statistics. True and false prediction rates are defined similarly for the objective of predicting the studies not having an effect.
Figure 3B shows that the mvalue is even more superior to the other statistics in this experiment than in the previous experiment. The pvalue shows the most inferior performance. This is expected because pvalue is designed for detecting the presence of an effect but not for detecting the absence of an effect. That is, a nonsignificant pvalue is not evidence of the absence of an effect but can be the result of a small sample size. On the other hand, the BF testing for the absence of an effect is just the reciprocal of the BF testing for the presence of an effect. Thus, the same BF can be used for both purposes. Although the BF performs better than the pvalue, the mvalue is even more superior. The relative performance gain of the mvalue compared to the BF is due to the crossstudy information utilized.
PM Plot: Type 2 Diabetes Data
We apply our PM plot framework to the real data of the metaanalysis of type 2 Diabetes (T2D) of Scott et al. [10]. The metaanalysis consists of three different GWAS investigations, the FinlandUnited States Investigation on NIDDM Genetics (FUSION) [10], the Diabetes Genetics Initiative (DGI) [11], and the WTCCC [9], [40].
In their analysis, two SNPs are shown to have a heterogeneous effect, rs8050136 and rs9300039. Ioannidis et al. [17] provide an insightful explanation about the heterogeneity at rs8050136. The WTCCC/UKT2D groups identified evidence for T2D and body mass index (BMI) associations with a set of SNPs including rs8050136 in the FTO region [40]. On the other hand, in the DGI study, the SNP rs8050136 was not significant. The explanation that Ioannidis et al. suggest is that the observed association at rs8050136 (FTO) may be mediated by its association with obesity. In fact, DGI is the only study where the BMI is matched between cases and controls, and the T2D association appears to be mediated through a primary effect on adiposity [11]. Thus, although the truth is unknown, the explanation of Ioannidis et al. is reasonable. Compared to rs8050136, the cause of heterogeneity at rs9300039 is less understood. It is suggested that the heterogeneity might reflect the different tag polymorphisms used in the studies [17].
To gain insights on these studies, we apply our PM plot. Figure 4A shows the forest plot, the plot showing only the pvalues, and the PM plot for rs8050136. In the PM plot, DGI appears to be well separated from the other two studies, even though its mvalue () is not below the threshold (). Thus, the PM plot visualizes that DGI can have a different characteristic from the others. Such a separation is not clear in the plot showing only the pvalues. In the plot showing only the pvalues, DGI is close to FUSION since FUSION is also not very significant (). However, the mvalue of FUSION is much greater () than that of DGI. This suggests that the effect is much more likely to exist in the FUSION study than in the DGI study.
Figure 4B shows the plots for rs9300039. The PM plot shows a different pattern from the PM plot of rs8050136. In this PM plot, every study has an mvalue greater than 0.5. Thus, no study shows evidence of no effect. Comparing the plots of rs8050136 and rs9300039 gives an interesting observation. In the plot showing only the pvalues, both SNPs show a specific pattern of pvalues that a single study is considerably more significant than the other two. However, despite of this similarity in the pattern of pvalues, the two SNPs' PM plots look different enough that can lead us to different interpretations. This shows that our PM plot can provide information that is not apparent in the analysis of only the pvalues.
PM Plot: Crohns Disease Data
We apply our plotting framework to the data of the recent metaanalysis of Crohns disease of Franke et al. [13]. This metaanalysis consists of six different GWAS comprising 6,333 cases and 15,056 controls, and even more samples in the replication stage. In this study, 39 associated loci are newly identified increasing the number of associated loci to 71. We apply our framework to six loci where a high level of heterogeneity is observed. Han and Eskin [23] showed that at these six loci, RE gave more significant pvalues than the fixed effects model (FE).
Figure 5 shows the PM plots of two loci. See Figure S1 for the plots of all six loci. The names of the studies follow the names used in Franke et al. [13]. At these two loci, rs3024505 and rs17293632, the mvalue of WTCCC is close to the threshold for predicting no effect. A possible explanation is that the different marker sets could have caused the statistical heterogeneity at these loci. WTCCC [40] used the Affymetrix platform while others used the Illumina platform. Although we do not further investigate this hypothesis, it is true that the PM plots visualize an interesting outlier behavior of WTCCC at these loci. Such an observation is not clear in both the forest plot and the plot showing only pvalues. In the plot showing only pvalues, studies having nonsignificant pvalues are all clustered and WTCCC is only one of them. In the forest plot, WTCCC is not the only study showing a small effect size at both loci. For example, at rs3024505, NIDDKNJ shows a smaller effect size than WTCCC. However, the mvalue of WTCCC is much smaller than NIDDKNJ's because of the large sample size. Such an interaction between the sample size and the prediction can also be inferred from the forest plot since the forest plot includes the confidence interval. However, it is difficult to numerically quantify the effect of sample size on the prediction by visually examining the forest plot.
Binary Effects Model: False Positive Rate
We estimate the false positive rate of the new binary effects model (BE). Assuming the null hypothesis of no association, we construct 5 studies of sample size 1,000 to build a metaanalysis set. We calculate the metaanalysis pvalue of BE using our importance sampling procedure with 10,000 samples. We also calculate the metaanalysis pvalues of FE and RE. We build 100 million sets of metaanalysis and estimate the false positive rate as the proportion of the simulated sets whose pvalue exceeds a threshold. We vary the threshold levels from 0.05 to . Table 1 shows that all methods including BE control the false positive rates accurately, at all threshold levels examined. When we increase the number of studies from 5 to 10, the results are essentially the same and the false positive rates are controlled (Data not shown).
Binary Effects Model: Power
We compare the power of BE to the powers of FE and RE. Assuming a metaanalysis of five studies of an equal sample size 1,000, we construct 10,000 metaanalysis sets. The power of each method is estimated as the proportion of the metaanalysis sets whose metaanalysis pvalue calculated by each method exceeds the genomewide threshold ().
We measure power in two different situations. First, we assume a situation that the effect is either present or absent. We decrease the number of studies having an effect () from 5 to 2. We increase the relative risk as decreases, using for respectively, in order to show the relative performance between methods.
Figure 6 shows that except for the case that there is no heterogeneity (), BE is the most powerful among all methods. BE is more powerful than RE, even though both are a random effects model, possibly because it learns the fact that some studies do not have an effect from the data. When there is no heterogeneity (), FE achieves the highest power and BE achieves the lowest power.
Second, we assume a classical setting where the effect sizes follow a normal distribution. Assuming that the mean effect size of , we sample the log of effect size of each study from a normal distribution having the mean and the standard deviation where is the parameter we vary. As increases, the heterogeneity increases. We measure the power of each method varying from zero to one. Figure 7 shows that in this situation, BE is generally less powerful than RE. The power difference between BE and RE is the greatest when the heterogeneity is small. As the heterogeneity increases, BE shows a similar power to RE.
Binary Effects Model: Real Data
We apply BE to the real data of Crohns disease of Franke et al. [13]. Han and Eskin [23] showed that out of 69 associated loci analyzed, RE gave more significant pvalues than FE at six loci where high level of heterogeneity is observed. We calculate the pvalues at these loci using BE and compare to the pvalues of FE and RE.
Table 2 shows that at all six loci where RE gave more significant pvalues than FE, BE gives even more significant pvalues. The reason why BE gives more significant pvalues can be explained by examining the PM plots of these loci in Figure 5 and Figure S1. The PM plots show that at these loci, some studies show high mvalues and some studies show low mvalues, suggesting a bimodal distribution of effect size. Thus, the situation is very similar to the case that the effect is either present or not, in which case BE achieves higher power than RE as shown in Figure 6.
Binary Effects Model: Accuracy of Importance Sampling
We measure how accurately the importance sampling procedure of BE estimates the pvalue depending on the number of samples used. We calculate the BE pvalue for the same dataset in 100 different runs to estimate the variance of the pvalue estimate. Our criterion of interest is the ratio between the standard deviation of our estimate and the target pvalue. For this, we use the 69 associated loci in the Crohns disease data of Franke et al. [13] that were previously analyzed in Han and Eskin [23]. We measure the ratio for each locus and average over all loci. We do this varying the number of samples from 1,000 to 1,000,000.
Table 3 shows that as the number of samples used for importance sampling increases, the accuracy increases. The pattern of accuracy increase is what we would usually expect in a sampling procedure; standard deviation is decreased approximately by the square root of the sample size increase. When the number of samples is 1,000, the ratio is roughly 0.5. A ratio of 0.5 is large, but can be enough for initial screening if we would apply an adaptive sampling that samples larger number of samples only for loci that are at least moderately significant (e.g. ).
Binary Effects Model: Computational Efficiency
We measure the computational efficiency of the importance sampling procedure of BE. In our software, we implemented an adaptive sampling procedure that samples smaller number first () and then larger number () for the loci that are at least moderately significant. In the machine equipped with Intel Xeon 1.68 GHz CPU, when we use 1,000 samples in the importance sampling, calculating BE pvalues of 1,000 loci for the metaanalysis of 10 studies takes 100 seconds. Thus, to calculate BE pvalues of one million loci assuming that 1,000 loci among them are moderately significant, it will take approximately 30 hours which is a feasible amount of time. If the number of samples is increased to achieve better accuracy, such as and , the procedure will still be efficient if one uses multiple computers or a cluster since the procedure is parallelizable.
Discussion
We introduce a framework facilitating the interpretation of metaanalysis results based on a new statistic representing the posterior probability that the effect exists in each study. Our framework utilizes crossstudy information and is shown to help interpretations in the simulations and the real data. The new statistic also allows us to develop a new association testing procedure called the binary effects model.
In the current metaanalyses of genomewide association studies, heterogeneity is often observed and our framework will be a useful tool for interpreting such results. We expect that our framework will be even more useful in the future metaanalyses. As the number of studies in a metaanalysis grows, the chance of heterogeneity will increase [6]. Also, a metaanalytic approach can often be applied to a broader area such as to multiple diseases with similar etiology, in which case the heterogeneity is more likely to occur. Moreover, the majority of the current metaanalyses only use the fixed effects model (FE). The use of a random effects model (RE) approach [23] such as the binary effects model presented herein will increase the number of identified associations showing heterogeneity, since an RE approach is more powerful than FE for detecting associations with heterogeneity.
One limitation of our approach is that although the new statistic can predict the studies having an effect and the studies not having an effect, it does not distinguish the true heterogeneity and the statistical heterogeneity [22]. Discriminating between the two can be very difficult based on the observed data and might often be possible only by external data such as the replication studies. In that sense, our method can help discriminating them because one can come up with a hypothesis based on mvalues that the heterogeneity is caused by specific design factors and then control the factors in the replication stage. The heterogeneity will disappear in the replication stage if it was due to the design factors.
Similarly to other Bayesian approaches [35], [36], the prior choice in our method can have a nonnegligible effect on the predictions. For the prior of the effect size , it is important to set a reasonable value based on the prior information about the effect size. See Stephens and Balding [35] for the general guideline for this choice. For the prior of the probability that the effect exists , we used the uniform distribution () in this paper. However, different priors can also be used for different situations. If one expects that most of the studies have an effect, an asymmetric prior such as can be used. If one is certain that the studies having an effect and the studies not having an effect are mixed, a bellshape prior such as can be used. See Figure S2 for the plots of the possible choices of priors.
Supporting Information
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Štítky
Genetika Reprodukční medicínaČlánek vyšel v časopise
PLOS Genetics
2012 Číslo 3
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