The proposed method of analysis applies a standard two-sample t-test to genetic scores obtained from allele counts of variants weighted according to their observed frequency in both cases and controls. This avoids the need for permutation testing and allows for rapid analysis. Simulation studies confirm that the method is valid and demonstrate that combining information from both common and rare variants can, in at least some situations, provide more power than considering each separately. Furthermore, it confirms that weighting the scores from different variants can further increase power. The use of a smooth weighting function means that all types of variants are subjected to the same method of analysis. A weighting factor allows the user to choose a weighting scheme appropriate for the type of trait being studied.
Of course, as pointed out previously,6
a variety of different functions could be used to generate weights. We have chosen a parabolic function that can easily be adjusted and that produces weights relative to a value of one for variants with MAF = 0.5. It is possible, however, that more or less sharply curved functions or sigmoid functions might offer some advantages. Most importantly, we are of the opinion that cases and controls should be treated equally to avoid the need for simulation, and that the function should not fall off too sharply with very small values of MAF.
It is easy to speculate that different values for the weighting factor might be appropriate for different situations. A high value, which gave more weight to very rare variants, might be helpful for a disease that appeared often to result from mutations with large effect size, typically a rare disease with Mendelian inheritance. Conversely, one might speculate that a common syndrome that might be expected to arise from the cumulative effects of common variants could be more appropriately analyzed with a relatively low value, although of course rare variants might still exert important effects. As the analyses are quick to perform, it might be reasonable to analyze datasets using a number of different values for the weighting factor, provided that appropriate corrections are then made for multiple testing. The different results obtained using different weighting factors might then allow one to make some inferences about the nature of the effects influencing susceptibility to the trait in terms of the relative contribution of common and rare variants in the gene under consideration.
The method described clearly assumes some kind of additive contribution from different variants; however, it is unclear how well it would perform with variants with recessive effects. It might be possible, in principle, to devise some kind of alternative weighting scheme aimed specifically to detect associations using a recessive model.
In contrast to some approaches, no special treatment is required to deal with linkage disequilibrium (LD) between variants. If this is present it is not expected to affect the validity of the test. In essence this is because all information is combined at the level of the individual subject before being entered into the analysis. Hence, if there is nonindependence of genotypes within a subject, the fact that observations for different subjects are independent of each other is not affected. And so, the total scores are still expected to follow a random distribution under the null hypothesis. To illustrate this, we could consider the situation in which two variants are in complete LD with each other. This would be equivalent to having information from just one variant, but counting it twice for each subject, which would have exactly the same effect as assigning twice the weight to that variant. Thus, LD relationships can be seen as having equivalent effects to varying the weights assigned to variants. As such, they would not influence the validity of the analysis in the sense that they would not impact on the number of statistically significant results expected to occur by chance. They might, however, have an effect on power. If a large number of common variants were in LD with each other, then their contributions to the score might tend to overshadow contributions from individual variants. In such situations, it might be beneficial to identify this and in some way scale down the weights of variants belonging to such LD groups.
As was also noted for the previously described method,6
this implementation implicitly assumes that it is the rare allele of each variant that may be associated with the disease. This allows the effects of different variants to be combined within an individual and also implies that significance testing can be one-sided. This assumption may be reasonable for rare variants when the phenotype being studied reduces fitness. However, the method as it stands could not be applied to a quantitative trait in which there was no a priori assumption as to the direction of effect of each allele.
There are both biological and statistical arguments in favor of considering the alternative hypothesis to be that in general it is the rarer allele of each variant that is associated with disease. The biological argument is that if one begins with the reference sequence and then generates a variant at random, then one is more likely to produce a disease than to prevent one. Additionally, if a randomly generated variant should happen to be beneficial and to confer a survival advantage, then, over time, selection pressures will increase its frequency until it ultimately becomes common. Thus, one may expect that, on average, rare variants will be more likely to be associated with deleterious phenotypes. There is also a statistical argument for basing the test on the assumption that rare variants will be more likely to show association with a rare phenotype, even if it is nondeleterious or even advantageous. To begin with an example, suppose that a particular phenotype has prevalence 0.01 and that a variant with allele frequency 0.001 in the population produces a ten-fold increase in risk of manifesting this rare phenotype. It is simple to calculate that in samples of cases with this phenotype and of controls, we would expect allele frequencies of 0.0099 and 0.00091, respectively. With a sample size of 500 of each we might expect to observe the variant in ten cases and one control. Now, suppose that we have a different variant also with frequency 0.001 but which is “protective” so that it produces a relative risk (RR) of 0.1 rather than ten. In this situation, we calculate the expected allele frequencies in cases and controls to be 0.0001 and 0.001. With the same sample size we might observe the variant once amongst the controls and not at all in the cases. Thus, the excess of the rare variant associated with the rare phenotype amongst subjects with the rare phenotype, is larger than the excess of the rare variant associated with the common phenotype observed amongst subjects with the common phenotype. If we were to count up both variants together we would still expect to find an overall excess of rare alleles amongst subjects with the rare phenotype in spite of the fact that both variants produce an equal and opposite effect on risk. This particular example represents just one instance of a general phenomenon, which is that if one assumes an equal and opposite effect on risk of a pair of variants with equal frequency, then there will be more enrichment of the “risk” variant amongst “cases” than there is enrichment of the “protective” variant amongst “controls.” This statistical effect continues to be active as the MAF of the variants increases. At higher values for the MAF, an additional complication occurs: the rarer allele becomes so enriched amongst cases that when the frequency is jointly estimated from cases and controls, this allele actually becomes designated as the “common” allele, in spite of the fact that in the population as a whole it is rarer. That is, the allele that is rarer in the population becomes the allele that is more common in the case control sample. Even taking this phenomenon into account, for pairs of variants with RR equal to 10 or 0.1, one still expects to observe an excess of more rare alleles amongst cases with true values of MAF up to 0.24. For values of RR of 2 and 0.5, one expects an overall excess of rare alleles for all values of MAF up to 0.42; and for values of RR of 1.5 and 0.7, one expects this up to values of MAF of 0.45. Thus, when variants within a gene affect risk there is a consistent phenomenon that means that, over a wide a range of genetic models, one expects to observe an overall excess of rare alleles amongst subjects having a rare phenotype. This statistical effect applies even before one considers the biological argument that one expects rare variants, a priori, to be deleterious.
It was also noted previously6
that weighting could be based not on allele frequency, but on the presumed effect of the variant on gene function. This could be equally incorporated into the score test as we describe it, the only additional feature being that we would suggest that the t
-test be used for significance testing rather than permutation, provided that cases and controls were treated symmetrically. A further possibility would be to produce a combined weight based on both allele frequency and presumed effect. One simple approach would be to simply multiply the weight based on effect by the weight derived from frequency. Such techniques could mean that for example, a rare variant producing a nonsynonymous coding change would be assigned a higher weight than either a common nonsynonymous variant or a rare synonymous variant.
Tests such as these can be applied at the level of a single gene, a region within a gene, or a set of genes comprising a pathway. It is up to the user to define the region of interest and to make decisions about such matters as to whether or not to include intergenic variants and intronic variants, whether to focus on a particular transcript or particular exon, and what assumptions to make about how to define regulatory regions. Sometimes the same variant will be defined to be included in the analysis of two or more different genes but this does not pose any particular problem for the method.
Which functions and/or weighting schemes in fact produce the best performance when applied to real data can only be properly assessed when more such data becomes available for analysis. As such data emerges over the next few years it will be helpful to undertake a formal comparison of different approaches. For now, it seems reasonable to suggest that a weighting factor of around 10 might be appropriate for analyses of diseases in which it is suspected that both common and rare variants might contribute to risk.