Bayesian analysis of rare variants in genetic association studies

BAYESIAN MODEL OF MULTIPLE RARE VARIANTS

Suppose that a population-based association study consists of n unrelated individuals, phenotyped for a binary disease trait y_i (i.e. if diseased, y_i = 1; otherwise, y_i = 0), and genotyped for m rare variants in a candidate gene or functional genomic region. We denote the genotypes of variant j by A_jA_j, A_ja_j, or a_ja_j, where a_j is the minor allele with the observed frequency p_j<1%. The relation between the disease status and the genotypes of m rare variants can be expressed by a generalized linear regression

(1)

where the link function h is the logit or probit function, µ is the intercept, β_j is the main effect for the jth variant, and x_ij is the main-effect predictor for the ith individual at the jth variant. For an additive model, x_ij = 0, 1, or 2 for A_jA_j, A_ja_j, or a_ja_j, and for a dominant model, x_ij = 0 or 1 for A_jA_j or A_ja_j and a_ja_j, respectively. For a rare variant, the additive model is approximately equivalent to the dominant model because the frequency of a_ja_j is extremely low.

The association between the disease and the variants may be examined by testing β_j = 0, j = 1,…,m. For rare variants, however, such an analysis is underpowered because a single variant explains very low genetic variation and typically is undetectable. Under the additive model and Hardy-Weinberg equilibrium, for example, the genetic variance of the jth variant is , equal to 9.5×10⁻³ when the frequency p_j and the odds ratio exp(β_j) equal 1 and 2%, respectively. A solution would be to create a “genetic score” that combines information across multiple rare variants for each individual. The genetic score is then treated as a single predictor, allowing us to detect the overall association of the variants with the disease. We construct the genetic score as a linear function of the separate main-effect predictors, i.e. , and set up a generalized linear model:

(2)

In this model, the common coefficient β represents the overall effect for the m rare variants, and the α_j's can be interpreted as the relative effects or weights of the individual variants. To investigate the overall association, we test the hypothesis β = 0.

Instead of presetting α_j's as in existing methods, it would be better to estimate them from the data. But we cannot simply use classical framework (i.e. setting uniform distributions on the α_j's), since this would result in a nonidentifiable model and thus be equivalent to estimating a separate coefficient for each of the variants [Gelman, 2004; Gelman and Hill, 2007]. However, we can set up an informative prior for the α_j's, so that the model is identifiable. We use the Student-t distribution for the α_j's:

(3)

with the scale s_α set to a low value such as 0.5 [Gelman et al., 2008]. This prior distribution constrains α_j's to be fairly close to the prior mean µ_j's, but allows for different values. An alternative prior is to use the normal distribution with a fixed variance, i.e. [Gelman and Hill, 2007]. However, we prefer the Student-t distribution because it estimates the variances from the data and thus may better deal with disparate effects. The prior means µ_j can be specified as the relative importance of the individual variants based on our prior knowledge or initial analysis (see Discussion). In this study, we incorporate no prior information into the model by setting µ_j = 1 for all j = 1,…,m. We found that the method is fairly robust to any small changes for the scale s_α (for example, from 0.2 to 0.8).

The common coefficient β usually can be estimated classically. However, low allelic frequencies can yield very small variance Var(T_i), for which the classical procedure often results in numerically instable estimate. To overcome this problem, we use a weakly informative prior to constrain β in a reasonable range. Following Gelman et al. [2008], we place a Student-t distribution with center 0, degree of freedom 1, and scale 2.5 on β:

(4)

COMPUTATION

Our Bayesian generalized linear model can be fitted using Markov chain Monte Carlo algorithms that fully explore the joint posterior distribution of the parameters by alternatively sampling each parameter from its conditional posterior distribution. However, it is desirable to have a faster computation that provides a point estimate (i.e. the posterior mode) of β and α_j's and their standard errors (and thus the P-values) by maximizing the marginal posterior . Such an approximate calculation has been routinely applied in statistical practice [Gelman et al., 2008]. We develop our algorithm by modifying the standard iterative weighted least squares (IWLS) for fitting classical generalized linear models. We have implemented these computations by altering the glm function in R (the general statistical package) that fits classical generalized linear models.

Our algorithm simultaneously estimates the parameters α_j's and β using an iterative procedure. We initialize the algorithm by setting β to the value estimated from the Simple-Sum method or setting . Then, at each step of the algorithm, we first update α_j's conditional on the current estimate by using the modified IWLS algorithm of Yi and Banerjee [2009] and Yi et al. [2010] to fit the hierarchical generalized linear model:

(5)

We then update β conditional on the current estimates 's by fitting the hierarchical model using the modified IWLS algorithm [Yi and Banerjee, 2009; Yi et al., 2010]:

(6)

Instead of doing a full IWLS for each of these two models, we can perform one step of weighted least squares at each iteration, thus taking less computer time to ultimately achieve convergence by not wasting time getting hyper-precise estimates at each step of the algorithm. We apply the criterion in the glm function to assess convergence, i.e.

where and are deviances at the tth iteration for the models [4] and [5], respectively, and is a small value (say 10⁻⁸). In practice, our algorithm converges rapidly. At convergence of the algorithm, we obtain all the outputs produced by the glm function, including the latest estimate , their standard deviations, and the P-values for testing β = 0.

RELATIONSHIP WITH EXISTING METHODS

The basic procedure of rare variant analysis is to construct a weighted combination (genetic score) of m rare variants, , that summarizes the information across the variants for each individual i, and then estimate the association between the phenotype y_i and the genetic score T_i using a generalized linear model, , or other testing statistics. Our method differs from existing methods in estimating the weights α_j's (along with the overall effect β) from the data using a hierarchical modeling framework rather than simply presetting them to fixed values. This would produce higher weights for more ‘important’ variants.

Presetting the weights α_j's to different values results in different existing methods: (1) If , we have and thus the method is the Simple-Sum [Han and Pan, 2010; Morris and Zeggini, 2010]; (2) If we take , where I(x) is an indicator variable taking 1 if x>0, and 0 otherwise, the method becomes the Collapsing approach [Li and Leal, 2008]; (3) If α_j = 1/sd(x_ij|y_i = 0), j = 1,…,m, where sd(x_ij|y_i = 0) is the estimated standard deviation of x_ij in unaffected individuals, the model is similar to the Weighted-Sum approach [Madsen and Browning, 2009]; (4) If setting α_j to the posterior probability of being functional for each variant j, T_i corresponds to that of Price et al. [2010]. These posterior probabilities can be calculated using bioinformatics tools such as PolyPhen [Adzhubei et al., 2010; Price et al., 2010].

From the above procedure, we can see that the term α_jβ actually corresponds to the genetic effect β_j of the jth variant. With the fixed weights, therefore, the individual genetic effect β_j is proportional to the corresponding weight α_j. This important result reveals the underlying assumptions of the existing methods. The Simple-Sum method implicates that the effects of all variants are identical; obviously, this is an unrealistic assumption. The Weighted-Sum method first standardizes the main-effect predictors and then assumes identical coefficients for all variants in the standardized model. This corresponds to the implicit assumption . The approach of Price et al. [2010] implicitly assumes that the effect of a variant is proportional to the posterior functional probability. Therefore, all the existing methods implement pooling of multiple rare variants according to certain assumptions about the genetic effects of variants. Although powerful in certain situations, these methods can be inefficient if the underlying assumption is not true. In contrast, the proposed method does not require any assumptions about the relative importance of individual variants and thus could be more robust than the existing methods.

BAYESIAN VERSIONS OF EXISTING METHODS

Although various testing statistics have been proposed for the existing Collapsing, Simple-Sum, and Weighted-Sum methods [Han and Pan, 2010; Li and Leal, 2008; Madsen and Browning, 2009; Price et al., 2010], we implement these methods using logistic regressions:

with for the Collapsing, for the Simple-Sum, and for the Weighted-Sum. The All-Variants method jointly estimates the individual effects of all variants:

and the Single-Variant separately estimates the effects of individual variants:

These logistic regressions can be nonidentifiable when the variance Var(T_i) or Var(x_ij) is small [Li and Leal, 2008]. We overcome this problem by placing the weakly informative prior [Gelman et al., 2008], , , and . We fit these models using the modified IWLS algorithm of Yi and Banerjee [2009] and Yi et al. [2010]. This improves the performance of these previous approaches and has the advantage of always producing stable estimates.

SIMULATION DESIGN

We consider different combinations of the factors that may affect the performance of the methods:

(a) Sample size: We simulate n = 500, 1,000, and 2,000 individuals with an equal number of affected and unaffected.

(b) Number of rare variants: We simulate m = 20, 40, and 80 rare variants.

(c) Minor allelic frequencies and genotypes: We sample m variants independently because correlation between rare variants is low [Pritchard, 2001; Pritchard and Cox, 2002]. For the jth variant, we sample the MAF p_j uniformly from the region [0.001, 0.01], as variants with MAF<0.001 would be indistinguishable in our presumed sample sizes. Assuming the Hardy-Weinberg equilibrium for each variant, we thus generate the genotypes from the multinomial distribution:

(d) Genetic model: We evaluate our method using the additive genetic model for each variant. For rare variants, the additive model is approximately equivalent to the dominant model, and detection of recessive effects requires extremely large sample [Li and Leal, 2008].

(e) Number of functional variants and genetic effects: For m = 20, 40 and 80, we set all the variants to be functional or randomly sample 40% of the simulated variants as nonfunctional. For each functional variant, we simulate the odds ratio exp(β_j) to be 1 (for type-I error rate) or uniformly from the region [1.05, OR_u] (for power analysis). To ensure that the overall effect of all variants is reasonably low, we determine the upper bound OR_u by controlling the total liability heritability, which approximates (as derived in the next paragraph), where p_ave is the average MAF, m_f is the number of functional variants, and exp(β_ave) is the average odds ratio, equal to (1.05+OR_u)/2. For example, OR_u = 2.0 when m_f = 40 and . We consider the total liability heritability from 0.7 to 8%. Finally, we consider the most complicated case in which the effects of the functional rare variants are in opposite directions; for each functional variant, we first simulate the odds ratio exp(β_j) uniformly from the region [1.05, OR_u], and then change the sign of β_j with the probability of 0.3 or 0.5.

Given the coefficients β_j and the genotypic codes x_ij, we can simulate the disease phenotype y_i using two methods. The first method is to directly sample y_i from the binomial distribution: . This procedure is repeated until we obtain n/2 affected and n/2 unaffected individuals. The second is to use the latent-data formulation of the logistic regression; the logistic model is equivalent to the model, , [see Gelman and Hill, 2007]. Thus, we first sample n latent normal phenotypes w_i and then set n/2 individuals with the 50% largest w_i as affected (i.e. y_{i =}1) and the other n/2 individuals as unaffected. The latent-data formulation allows us to calculate the proportion of the latent-data variance explained by the variants, i.e. the liability heritability [Wray et al., 2010], . As described above, this formulation can be used to control the total heritability when we simulate the coefficients β_j.

For each set of parameters, 1,000 replicated data sets are simulated, and each is analyzed using our hierarchical model approach and the Collapsing, Simple-Sum, Weighted-Sum, All-Variants, and Single-Variant methods. For each analysis, we use the additive genetic model. We calculate power to detect overall association β=0 at significance levels of α = 0.001 and α = 2.5×10⁻⁶. These thresholds correspond to candidate gene studies [Price et al., 2010] or a genome-wide study of about 20,000 fairly independence human genes [Madsen and Browning, 2009], respectively. We also examine type-I error rate at a significance level of α = 0.05. For the All-Variants and Single-Variant methods, the overall association is examined by testing whether at least one β_j = 0 for all j = 1,…,m, and for simplicity we do not adjust the significance level for multiple testing. Therefore, we overestimate the power for the All-Variants and Single-Variant methods.

TYPE-I ERROR RATE

As shown in Figure 1, the type-I error rates are well controlled for the proposed method and the Collapsing, Simple-Sum, and Weighted-Sum methods. The proposed method slightly inflates the type-I error rate when n = 500 and m = 20. However, there is no constant trend that the proposed method generates higher type-I error rate than those of the previous methods. The type-I error rates for the All-Variants and Single-Variant methods are unacceptably high, and significantly increases with increasing number of variants, indicating the need for multiple-testing correction.

ANALYSIS OF FUNCTIONAL VARIANTS

We first investigated powers of the methods for the relatively simple scenario in which all rare variants are functional and affect disease risk in the same direction. As shown in Figures 2 and 3, the results for different sample sizes, different numbers of rare variants and different liability heritabilities display similar patterns of empirical powers. A notable result is that the proposed method is consistently more powerful than the other methods. Because our method estimates the weights of multiple variants from the data, our model fits the data better and generates a genetic score that better summarizes the information across the variants. Thus, the proposed method improves the power to detect the overall association between rare variants and disease.

As expected, the power drastically increases with the sample size and the total liability heritability explained by the variants. These relationships hold rather generally for the methods that we examined. Given sample size and total liability heritability, the power slightly decreases with the number of variants. This is likely the results that more variants generate a lower upper bound OR_u of odds ratios for individual effects (thus smaller individual effects) and thus their information may be more difficult to be summarized. Our simulations showed that with small sample sizes (n = 500, 1,000) the All-Variants and Single-Variant methods have no power to detect the association between rare variants and disease. These are expected because these methods test for the effects of single variants each of which has little variation. For larger sample size (n = 2,000), the powers of the All-Variants and Single-Variant methods go up at the significance level of α = 0.001 with no multiple-testing correction, but rapidly decrease to near zero with a more stringent significance level (Fig. 3).

Our results showed that some of the previous methods produce similar power as the proposed method in some situations, masking the real difference between these methods. To investigate whether the proposed method provides any advantages in these situations, we calculated the median value of P-values for simulation replicates with P-value <0.001. A notable outcome of this analysis is that the proposed method uniformly yields much lower P-values than the previous methods (Fig. 4). This finding indicates that our method usually provides stronger evidence of association if the variants really influence the disease.

INCLUSION OF NONFUNCTIONAL VARIANTS

Nonfunctional variants do not contribute to disease risk. Therefore, the inclusion of nonfunctional variants in the analysis introduces noisy variation in the model and may influence the performance of the methods. Our simulations showed that the power decreases when nonfunctional variants are included (Figs. 5 and 6). This is true for all the methods that we examined. However, we found that the previous methods lose more power than the proposed method. This probably results from the fact that the previous methods use equal or inappropriate weights for functional and nonfunctional variants, thereby ineffectively summarizing the information across the multiple rare variants. In contrast, our method estimates weights from the data and thus can set lower or even zero weights to nonfunctional variants, providing a better genetic score.

Although the power decreases with inclusion of nonfunctional variants, the general conclusions obtained earlier still hold. The proposed method is uniformly more powerful (Figs. 5 and 6) and generates much lower P-values than the previous methods (Fig. 7).

ANALYSIS OF RARE VARIANTS WITH OPPOSITE EFFECTS

We finally investigated empirical power of the methods in the complicated scenario in which the effects of the functional rare variants influence disease in opposite directions. With 30% (70%) of functional variants increasing (decreasing) disease risk, the previous methods have some power to detect the association when sample size is large (n = 2,000) (Fig. 8). But the power rapidly decreases with a more stringent significance level (Fig. 9). For the worst case where 50% (50%) of functional variants increase (decrease) disease risk, the Collapsing, Simple-Sum, and Weighted-Sum methods have no power to detect the association even when sample size and odds ratios are large. These results are expected because these methods simply pool all variants together, using equal weights for disease-causing and disease-protective variants. Therefore, the information across multiple rare variants is canceled and the true association signal is completely hidden. As expected, the All-Variants and Single-Variant methods perform similarly as the previous cases that we studied.

The most striking finding of this study is that our method is still powerful even when multiple rare variants have opposite effects on disease risk (Figs. 8 and 9). This remarkable feature is certainly the result of the unique property of the proposed method. Our method estimates weights from the data and thus yields different weights for disease-causing and protective variants, avoiding cancellation of individual-variant variation. Compared to the simpler case, however, for this complicated case, the proposed method is less powerful and is more sensitive to the number of variants. This is expected because the increasing complexity certainly reduces the accuracy of statistical inference.