Abstract
Background
The generalized odds ratio (GOR) was recently suggested as a genetic modelfree measure for association studies. However, its properties were not extensively investigated. We used Monte Carlo simulations to investigate typeI error rates, power and bias in both effect size and betweenstudy variance estimates of metaanalyses using the GOR as a summary effect, and compared these results to those obtained by usual approaches of model specification. We further applied the GOR in a real metaanalysis of three genomewide association studies in Alzheimer's disease.
Findings
For biallelic polymorphisms, the GOR performs virtually identical to a standard multiplicative model of analysis (e.g. perallele odds ratio) for variants acting multiplicatively, but augments slightly the power to detect variants with a dominant mode of action, while reducing the probability to detect recessive variants. Although there were differences among the GOR and usual approaches in terms of bias and typeI error rates, both simulation and real databased results provided little indication that these differences will be substantial in practice for metaanalyses involving biallelic polymorphisms. However, the use of the GOR may be slightly more powerful for the synthesis of data from triallelic variants, particularly when susceptibility alleles are less common in the populations (≤10%). This gain in power may depend on knowledge of the direction of the effects.
Conclusions
For the synthesis of data from biallelic variants, the GOR may be regarded as a multiplicativelike model of analysis. The use of the GOR may be slightly more powerful in the triallelic case, particularly when susceptibility alleles are less common in the populations.
Findings
The generalized odds ratio (GOR) was recently suggested as a modelfree measure of effect that might overcome the problem of a genetic model misspecification in metaanalyses of association studies [1]. In the context of casecontrol genetic association studies for a binary trait and under assumption of random sampling, the GOR measures the probability that a case has a higher mutation load (i.e. a larger number of highrisk alleles) than a control divided by the probability that a control has a higher mutation load than a case.
In this note, we highlight advantages and limitations of the use of the GOR as a measure of effect in metaanalyses of bi and triallelic polymorphisms through simulation. Our results are further complemented by a reanalysis of a real metaanalysis of three genomewide association studies covering >311,000 biallelic markers in Alzheimer's disease.
Results
Performance of the GOR in the biallelic model
TypeI error rates
TypeI error rates obtained from metaanalyses employing the GOR as a summary effect size are comparable to the multiplicative and dominant models of analysis (Table 1).
Table 1. TypeI error rates (%) for the biallelic case according to different genetic models of analysis and heterogeneity (τ^{2 }) for α = 5%
Power
Compared to the use of multiplicative approaches, the power to detect variants with a dominant model of action was typically only slightly higher for metaanalyses using the GOR as summary estimate. For variants following a multiplicative pattern of action, all nonrecessive models of analysis were highly comparable. Interestingly, the largest differences observed among the perallele, logadditive trend (LAT) and the GOR were found in true recessive and overdominant models, where the performance of the GOR is slightly inferior for the former, but reasonable better for the latter (Figure 1).
Figure 1. Power (at α = 5%) for the biallelic case for a representative scenario of a variant with modest effect (OR = 1.3) following distinct modes of action (AB, dominant; CD, multiplicative, EF, recessive, and GH, overdominant) under moderate heterogeneity (τ^{2 }= 0.025). The sample size for each study was randomly sampled from a uniform distribution on the interval [5001000] and split equally into cases and controls (i.e. case to control ratio = 1). Color lines depict power estimates under different models of analysis: green (dominant), blue (perallele odds ratio), red (recessive) and black (generalized odds ratio). Results for the logadditive trend were omitted because the striking similarity with the results of the perallele odds ratio. Results are based on 5,000 replications under a randomeffects model (DerSimonianLaird method). f, allelic frequency. Scenarios with alternative magnitudes of heterogeneity or use of a fixedeffects model yielded qualitatively identical results.
Bias in the estimated statistical heterogeneity (τ^{2})
Compared to both perallele and LAT approaches, the median bias in τ^{2 }obtained by the GOR is typically lower in scenarios where the genetic variant is less common in the populations (e.g. minor allele frequency [MAF] = 10%) and acts either dominantly or multiplicatively. For the latter model of action, bias is slightly positive. In addition, for common markers (MAF = 40%) following a dominant model of action, the GOR provides less biased τ^{2 }estimates compared to the specification of a multiplicative model. Importantly, for a common variant (MAF = 40%) acting multiplicatively, metaanalyses using the GOR as an effect size provide upwardly biased estimates of τ^{2 }compared to true underlying average increment in the betweenstudy variance per additional copy of the risk allele (Figure 2). This upward bias in the estimated statistical heterogeneity is also found in both dominant and recessive models of analysis.
Figure 2. Bias (%) in the betweenstudy variance (τ^{2 }) estimate (statistical heterogeneity) for the biallelic case for a representative scenario of a variant with modest effect (OR = 1.3) following distinct modes of action (AB, dominant; CD, multiplicative, EF, overdominant) under moderate heterogeneity (τ^{2 }= 0.025). The sample size for each study was randomly sampled from a uniform distribution on the interval [5001000] and split equally into cases and controls (i.e. case to control ratio = 1). Color lines depict bias estimates under different models of analysis: green (dominant), blue (perallele odds ratio), red (recessive) and black (generalized odds ratio). Results for the logadditive trend model of analysis were omitted because the striking similarity with the results of the perallele odds ratio. Scenarios with a genuine variant acting recessively are not displayed for simplicity, since all models of analysis (except the recessive) are unable to capture the underlying τ^{2 }even when the frequency of risk alleles is high. Scenarios with alternative magnitudes of heterogeneity yielded qualitatively identical results. Results are based on the median value from 5,000 replications. f, allelic frequency.
Bias in the estimated genetic effect size
The GOR provides nearly unbiased summary effects for less common variants (MAF = 10%) acting dominantly, regardless of the metaanalytical model and τ^{2}. Conversely, when the variant follows a multiplicative model of action and is common (MAF = 40%), GORbased metaanalyses overestimate the true underlying increase in the effect size per additional copy of the risk allele (on average 20%) [Additional file 1: Supplementary tables S1S2].
Additional file 1. Supplementary tables S1 through S10.
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Performance of the GOR in the triallelic model
TypeI error rates
The performance of each model of analysis depends on the underlying betweenstudy variability, allele frequencies and metaanalytical model, but typeI error rates for LAT and GORbased metaanalyses are comparable, whereas false discoveries tend to be higher for the perallele approach when statistical heterogeneity is present (i.e. τ^{2 }>0). However, the extent of these differences is smaller in randomeffects calculations [Additional file 1: Supplementary tables S3S4].
Power: two alleles acting on the same direction
When at least one of the riskalleles is less common in the populations (f = 10%), and both exhibit either a dominant or multiplicative mode of action, power obtained by using the GOR as a summary effect is higher than that provided by either the perallele or LAT approaches (Figure 3).
Figure 3. Power (at α = 5%) for the triallelic case for a representative scenario of two alleles (A_{2 }and A_{3}) acting on the same direction with modest effect (OR = 1.3) following distinct modes of action (AB, dominant; CD, multiplicative, EF, recessive) under moderate heterogeneity (τ^{2 }= 0.025). The sample size for each study was randomly sampled from a uniform distribution on the interval [5001000] and split equally into cases and controls (i.e. case to control ratio = 1). Color lines depict power estimates under different models of analysis: orange (logadditive trend), blue (DunnŠidákcorrected perallele odds ratio) and black (generalized odds ratio). Results are based on 5,000 replications under a randomeffects model (DerSimonianLaird method). f, allelic frequency. Scenarios with alternative magnitudes of heterogeneity or use of a fixedeffects model yielded qualitatively identical results.
Power: two alleles acting on opposite directions
When prior evidence on the direction of the effects of the susceptibility alleles is available, similar power is achieved with the use of the perallele, LAT and GOR, regardless of the metaanalytical model, f and statistical heterogeneity [Additional files 1: Supplementary tables S5S7].
On the other hand, when no prior evidence on the direction of effects is available (e.g. initial screenings), the perallele model of analysis displays a superior performance compared to the use of either the LAT or GORbased approaches. In particular, compared to both GOR and LAT approaches, the gain in power for metaanalyses using the perallele OR may range from 1.5 to 10fold depending on the number of combined studies (Figure 4).
Figure 4. Power (at α = 5%) for the triallelic case for a representative scenario of two alleles (A_{2 }and A_{3}) acting in opposite directions (A_{2 }is protective, whereas A_{3 }is the susceptibility allele) with modest effect (OR = 0.77 for allele A_{2 }and OR = 1.3 for allele A_{3}) following distinct modes of action (AB, dominant; CD, multiplicative, EF, recessive) under moderate heterogeneity (τ^{2 }= 0.025). The sample size for each study was randomly sampled from a uniform distribution on the interval [5001000] and split equally into cases and controls (i.e. case to control ratio = 1). Color lines depict power estimates under different models of analysis: orange (logadditive trend), blue (DunnŠidákcorrected perallele odds ratio) and black (generalized odds ratio). Results are based on 5,000 replications under a randomeffects model (DerSimonianLaird method). f, allelic frequency. Scenarios with alternative magnitudes of heterogeneity or use of a fixedeffects model yielded qualitatively identical results.
Power: when only one allele displays a significant effect
Power is comparable among the GOR, LAT and perallele odds ratio when only one allele displays a significant effect. This is specially true when the highrisk allele is less common in the populations (f = 10%), particularly when f (A_{2}) = f (A3) = 10%. Overall, for common variants acting multiplicatively, the best performance is achieved with both GOR and LAT. When the risk allele is either recessive or dominant and is common, the best approach may depend on the frequency of the remaining alleles, but power is comparable among the three tested approaches whenever f (A_{2}) ≅ f (A3) [Additional file 1: Supplementary tables S8S10].
Real application
Results for the seven "top hits" variants associated with lateonset Alzheimer's disease are presented in Table 2. As expected, the largest association signal arose from the variant rs41377151, located at the 3' end of the apolipoprotein CI (APOC1) gene within the Apolipoprotein E (APOE)/APOC1 gene cluster on chromosome 19q13.3. This polymorphism is only 10.9 kb away from rs7412 variant (Arg176Cys) [2], which is one of the alleles that dictate the APOE ε status [3]. In addition, the remaining signals are also commensurate with results from previous [4] and more recent, large investigations [2,5,6].
Table 2. Summary results according to different models of analysis for the seven strongest association signals obtained by a metaanalysis of three independent genomewide association studies in Alzheimer's disease (TGen data sets, Reiman dise et al. 2007)
In agreement with our simulationbased results, plots of summary ORs and Pvalues (Figure 5) based on real data suggest a good concordance between GOR and both LAT and perallele approaches, followed by the dominant and recessive models, respectively.
Figure 5. BlandAltman plots depicting agreement between randomeffects modelbased summary odds ratios (Panels A, C, E and G) and Pvalues (Panels B, D, F and H) obtained by traditional genetic models of analysis and the generalized odds ratio (GOR). Pvalues are given on a log10 scale. Dashed lines represent 95% confidence limits of agreement, computed as Δ ± 1.96(standard deviation of Δ). Within each panel, the spearman coefficient (ρ) is shown, and the summarizes the correlation between estimates obtained from GOR and other models of analysis.
Discussion
The GOR was suggested as a modelfree approach for the synthesis of genetic association studies. The rational is that the GOR provides more flexibility for the true underlying genetic effect to describe the difference between two cumulative distribution functions of the latent variables, particularly when the assumption of proportional odds is violated. Furthermore, an additional advantage is that this ordinal measure of association is easily interpretable in practice [1].
Recent metaanalyses have applied the GOR claiming that this might be considered a different genetic model or an independent approach compared to the specification of traditional genetic model of analysis [7,8]. However, here we show that, since the GOR inherently assumes an ordinal mutation load (e.g. 1, 2 and 3 for genotypes A_{1}A_{1}, A_{1}A_{2}, and A_{2}A_{2}, respectively), this measure of assocation performs like a multiplicative model of analysis for biallelic polymorphisms. For diallelic variants, our simulations show that GORbased results are highly correlated to those obtained by both LAT and perallele ORs, resulting in similar typeI error rates and power compared to these traditional multiplicative models of analysis. In addition, a real metaanalysis of three GWAs in Alzheimer's disease indicates that limited. For example, under a fixedeffects framework and assumption of a threshold of P<10^{5 }(probably realistic due to the small samples sizes available), the total number of markers considered promising for further replication [9] would be 10, 13, 13, 14 and two for the perallele, LAT, GOR, dominant and recessive approaches, respectively. Under a randomeffects model, the correspondent numbers would be two for the recessive model and 8 for the remaining approaches.
Nonetheless, other important considerations in metaanalysis of genetic association studies involving biallelic polymorphism are biases in the estimated effect size [10] and heterogeneity [11]. In this respect, the most negative aspect of using the GOR as a measure of association in practice is that this measure provides inflated effects for biallelic variants following a multiplicative model of action. Although this inflation may be only mild for less common markers (i.e. median bias of ~5% for variants with MAF = 10%), the average upward bias in the observed genetic effect augments with increasing MAFs, reaching up to 20% for MAFs around 40%.
On the other hand, our data showed that the use of the GOR may be advantageous in metaanalyses involving triallelic polymorphisms as long as genotypes can be correctly ordered in terms of mutation load. In fact, a reasonable gain in power in the order of 2 to 15% may be achieved for the detection of association signals from variants with small frequencies (e.g. f ~10%) compared to the use of perallele or LAT odds ratios. The observation that higher power might be obtained with GOR in scenarios with a larger number of alleles of low frequency may serve as hypothesisgenerating information to extent the use of the GOR to metaanalysis of different types of genetic variants. For example, a special case might the use of the GOR in metaanalysis of structural variants such as copynumber variations (CNVs), which tend to exhibit a substantial number of alleles, yielding a correspondent large number of possible genotype categories [12]. Since the GOR handles categories with zero counts [13], and a different number of genotypes may be considered per study (for instance, in the case of specific allele sizes in some populations), the properties of the GOR in metaanalysis of CNVs is a topic worth of further investigation.
In summary, although there are differences in the statistical properties among the investigated approaches for biallelic variants, the absolute magnitude of these differences may be actually small and likely to be of very limited practical significance. An exception might be the use of the GOR in metaanalyses involving triallelic polymorphisms with less common alleles, since GOR uses of the complete genotypic distribution (e.g. the GOR less affected by zero cells). For these scenarios, the use of the GOR as a measure of effect may be slightly more powerful than traditional measures. However, the performance of GORbased metaanalyses will depend on some knowledge about the direction of the effects when there are two alleles modulating the risk of disease in opposite directions.
Material and methods
Simulation procedures and scenarios
We simulated metaanalyses of association studies using approaches that rely on multinomial distributions described in detail elsewhere (autosomal markers) [9,10]. HardyWeinberg equilibrium is assumed to hold for the whole population, whereas the susceptibility alleles are considered the causal variants or surrogate markers in tight linkage disequilibrium (r^{2 }= 1.0). For the biallelic case, we simulated data assuming the susceptibility variant A_{2 }(minor allele) and nonrisk allele A_{1}.
Under a threeallele model, we denote A_{1}, A_{2 }and A_{3 }as the possible alleles with frequencies f(A_{1}), f(A_{2}) and f(A_{3}^{)}, respectively, yielding six possible distinct genotypes (A_{1}A_{1}, A_{1}A_{2}, A_{1}A_{3}, A_{2}A_{2}, A_{2}A_{3 }and A_{3}A_{3}).
For each possible combination of the parameters presented in Table 3 we considered metaanalyses that included two up to 30 studies (casetocontrol ratio of 1:1).
Table 3. Parameters and simulated scenarios (trait prevalence = 10%).
For the triallelic case, three possible scenarios were considered: (i), among the alleles, two were susceptibility variants (e.g. both increase the susceptibility for the trait with the same magnitude), (ii) two alleles were associated with the trait, but in opposite directions (i.e. one increases, while the other decreases the risk for the trait in a similar magnitude) and (iii) only one out of the three alleles displays significant effects on the trait. We further assumed that the mechanism of action is similar for both alleles when there are two alleles with genuine effects on the trait (e.g. both act multiplicatively, or both act dominantly, and so forth). For scenarios with two alleles modulating the risk of disease, two additional situations of practical interest were investigated: (iia) the two alleles are associated with the susceptibility of disease in opposite directions and investigators have no prior evidence on the direction of these effects (e.g. initial agnostic screenings) and (iib) two alleles are associated with the susceptibility of disease in opposite directions, but investigators posses prior evidence on the direction of the effects (e.g. metaanalyses from the literature). For consistency, allele A_{2 }is coined to be the protective variant, whereas allele A_{3 }is the susceptibility one in these scenarios.
Biallelic polymorphisms
Assessment of bias
The percentage bias was computed as and for genetic effect sizes and betweenstudy variance, respectively, where is the (average) observed summary effect, μ is the true average genetic effect across populationspecific genetic effects, τ^{2 }is the true betweenstudy variance and is the methodofmomentsbased estimate of τ^{2}. Both and μ are captured as the natural logarithm of the odds ratio (Table 3). Use of alternative bias estimators (e.g. mean squared error) yielded qualitatively analogous results (data not shown).
Triallelic polymorphisms
Metaanalyses involving threeallele polymorphisms may rely on a diversity of approaches to summarize effects across studies. However, because the assumption of multiplicative effects yields, on average, the lowest rates of falsepositive results in biallelic markers [9,10], we compared the GOR to two approaches that assume a multiplicative mode of action: the perallele OR, which yields to three correlated odds ratios (OR[A_{3 }vs A_{1}], OR[A_{3 }vs A_{2}] and OR[A_{2 }vs A_{1}]) and the logadditive trend approach.
The generalized odds ratio
For a binary trait (e.g. casecontrol studies), GOR measures the probability that a randomly sampled case has a genotype with a higher mutation load (i.e. a larger number of highrisk alleles) than a randomly sampled control divided by the probability that a randomly sampled case has a genotype with lower mutation load than a randomly sampled control [1].
The GOR for a binary trait and an mallelic variant can be computed as [13]:
where J is the total number of genotypes (categories) given the number of alleles, i.e., J = m(m+1)/2, m is the number of alleles, (i.e. the proportion of the subjects with genotype j, for j = 1,..,J, in which the higher the value of j, the higher the mutation load) in the group i (i = 0 or 1 for controls and cases, respectively). In the present investigation, the largesample variance for GOR was computed from the asymptotic standard error of the GoodmanKruskal γ [1]. Stata and R codes to compute the GOR and its largesample variance are available from the first author upon request.
Mutational load order
The order of the jth genotypic category (i.e. mutational load) for the GOR and logadditive trend is anticipated to impact statistical power. Hence, for the situation iia (initial agnostic screenings), we set as genotypic order and for situation iib (metaanalyses from the literature with prior information on the direction of effects).
Assessment of power and typeI error
Empirical power and typeI error rates (i.e. falsepositive discoveries) were computed as the proportion of simulations that gave a twosided Pvalue < 5%. Because there are three correlated OR estimates for the triallelic case for the perallele model, we corrected the α level using the DunnŠidák procedure. Specifically, power and typeI error rates for the perallele model (triallelic case) were computed as the proportion of the simulations that gave one or more Pvalues < α_{corrected}, where .
Real application
We compared results based on the GOR as a summary effect to those obtained by usual approaches of model specification in a real metaanalysis of three independent genomewide studies in lateonset Alzheimer's disease. After standard control measures, a total of 311,915 biallelic polymorphisms were scored in 1411 participants (961 cases and 560 controls). Detailed description on the samples, genotyping platforms and diagnostics criteria are available elsewhere [4]. Results from individual studies were corrected for residual inflation of the test statistic using genomic control methods [14].
Metaanalysis methods
Metaanalyses were carried out under both fixed and randomeffects models, represented by the general inversevariance and DerSimonianLaird methods, respectively [15,16]. For the real application, statistical heterogeneity was test using the Cochran's Q test [11], and quantified using the I^{2 }index [17].
All simulations were performed in Stata 11.1 package (Stata Corporation), whereas the metaanalysis of real data sets were carried out in PLINK [18].
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
TVP carried out the computational experiments, tabulated the data and drafted the manuscript. TVP and RCMN conceived the study. RCMN participated in its design and coordination and helped to draft the manuscript. Both authors read and approved the final manuscript.
Acknowledgements
TVP is funded by grants from the Fundação de Amparo à Pesquisa do Estado de São Paulo (State of São Paulo Research Foundation, FAPESP). The authors are deeply indebted to the two anonymous reviewers for their extensive and valuable comments on the manuscript.
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