Phantom heritability and additivity

Just out in prepublication at PNAS is a paper from Eric Lander’s lab, entitled, somewhat provocatively The mystery of missing heritability: Genetic interactions create phantom heritability. This suggests that certain types of gene-gene interactions (often called “epistasis”) could be causing us to overestimate the additive heritability of complex traits, and thus underestimate the proportion of heritability our genetic studies have explained, without being detectable by standard methods.

At its heart, this paper is a challenge to a common assumption used in statistic genetics: the assumption of additive genetic risk. This states that genetic risk factors act independently of each other, with each variant increasing genetic risk by the same amount regardless of what other risk factors are present*. Of course, this is clearly a spherical cow situation, we know that the cell is full of complex interactions of various sorts, and a mutation cannot help but be effected in some way by the rest of the genome. But mathematically the assumption simplifies much of the complexity of statistical genetics, and allows you to do a number of things that would be very hard otherwise. We generally think additivity is a good approximation; it doesn’t matter if it is slightly wrong, and we’d have picked up if it were very wrong.

Zuk et al’s claim is that it is possible that additivity is wrong, that did not pick it up, and that it really does matter. This blog post will discuss the specific arguments that Zuk et al make against additivity. Some of the broader implications of the research is discussed over at Genomes Unzipped, and in particular looking at what this does and doesn’t say about total (not additive) heritability.

Defending Additivity

So why do the methods we use so often assume additivity, even if it seems on the surface to be unrealistic? From my point of view, there are five arguments supporting the additive model. In approximately increasing order of strength:

1) The search for epistatic relationships in GWAS haven’t been successful. Despite most GWAS consortia running tests for pairwise gene-gene interactions at some point during their analyses, there are very few well-replicated interactions in complex disease genetics outside of the well known interactions in the immune-regulating HLA region.

2) Various theoretical genetic considerations make additivity likely. There are a few evolutionary reasons why selection will tend to decrease the effect of interactions if the trait they are contributing to is under selection (see Hill et al for the more detailed arguments). Genetic variants of large effect are likely to be rare, making interactions less likely (as not many mutations occur together), and high frequency effects will tend to very small biological perturbations, which are less likely to show strong interaction. The HLA, with its selection to keep everyone looking immunologically different, is the example of high frequency, large effect variation that proves the interactive rule.

3) Distributions of monozygotic (MZ, identical) and dizygotic (DZ, non-identical) twin correlations in quantitative traits follow patterns predicted under additivity Under additivity, with no shared environment, the correlation between MZ twins is exactly twice that of DZ twins. Data from 86 quantitative traits collected by Hill et al show that the mean MZ correlation is almost exactly twice the mean DZ correlation. It is possible for epistasis and common environmental to perfectly balance out, but this becomes less likely the larger the two effects are, suggesting that they cannot be too large.

4) Very different methods of estimating heritability give consistent answers The additive heritability of height has been calculated using twin and family studies, as well as using Visscher et al’s two genetic methods of variation in sibling relatedness and population-wide identity-by-state (IBS). A range of studies, including adoption, twins-raised-apart, and children-of-twins studies has been done for addiction phenotypes. The WTCCC datasets have been used to measure lower bounders on additive heritability across many diseases using unrelated individuals, giving data to be compared to a range of twin studies. All of these methods have given broadly consistent results, despite all having different potential biases in response to non-additivity. There is, however, quite a bit of wiggle in that “broadly consistent”, due to the large error bars on all the estimates.

5) Excluding the HLA, genetic risk factors for type I diabetes show aggregate behavior highly consistent with additivity. David Clayton performed an analysis looking at interaction in the T1DGC meta-analysis data. He showed that including pairwise interactions gave almost no increase in aggregate predictive power, and that the interaction between non-HLA variants fitted the predictions of the additive model surprisingly well.

Even if additivity is clearly wrong, the above data suggests that genetic risk acts as if additivity applies. Even if the cell is full of interactions, it is believable that the overall effect on genetic risk is well approximately additively, and thus we use it as a working model that fits the data relatively well.

Zuk et al’s challenge

Zuk et al propose a simple, biologically plausible model to contrast with the additive model, called the Limiting Pathways (LP) model. They model disease risk as a set of different values, each of which measures the disruption of a particular disease pathway. If any of these values goes about a set threshold, then you develop disease. A simple example would be modeling diabetes risk as independent disruption of the insulin production and insulin resistance pathways, with symptoms showing if either becomes too damaged.

This model behaves in many ways very similarly to the additive model, as long as there are not too many pathways. Low-level risk variants combine roughly additively, with the interaction occurring at the much higher level of pathways. However, these high-level effects serious disrupt your estimates of additive heritability based on family members, as they cause more closely related individuals to look more similar than you’d expect given additivity. The authors show that twin studies, including twin studies that try to account for some non-linear effects, will seriously overestimated the additive heritability, despite appearing to be consistent with additivity in most normal tests.

They give a few examples of the effects this can have, notably showing that there is an LP model consistent with observed Crohn’s disease twin data that has an additive heritability of 18%, as opposed to the 50% estimate we currently use. This would mean that our latest Crohn’s meta-analysis actually explained 58% of heritability, not 22% as we believed.

What this means

1) Zuk et al show that under the LP model pairwise interactions are very small. Massive sample sizes (N=~500 000) would be required to find interactions between high-frequency GWAS signals. We would not expect to see significant epistatic interactions even if they existed.

2) The authors show that the LP model reacts very similarly to the additive model in response to selection. Either way, these sort of theoretical considerations are pretty weak, given how little we know about the architecture of genetic risk.

3) They also show that the LP model can be parametrized to fit virtually all of Hill’s MZ/DZ correlation pairs. As for why the MZ values are on average twice the DZ values, they dismiss Hill’s data as too noisy and confounded by shared environment to be convincing, though I’d have preferred to see a more detailed numeric treatment of this.

4) The authors show that various twin and family based methods are likely to similarly overestimate additive heritability under the LP model. Visscher’s sibling IBD method is also likely to overestimate additive heritability, though the bias in this case is a lot lower; the fact that it gives a very similar estimate of heritability with twin studies suggests that non-additivity cannot be a large confounder. The IBS-in-unrelated individuals method is not confounded in the same way at all, though it does underestimate heritability due to rare variants. This is good and bad, as it means the heritability estimates that Visscher’s method gives when to the WTCCC data give reliable lower bounds on the additive heritabilities of a number of traits, but the fact that these bounds are quite a bit lower than family methods means there is room for epistasis to hide. For example, the lower bound of Crohn’s disease additive heritability using this method is 22-24%, which is at least higher than the scary example given by the LP model, but leaves room for plenty of interactions, rare variants and badly tagged common signals.

5) The paper doesn’t cite David Clayton’s paper. However, they do give a hypothesis test for whether the additive or LP model applies given risk variants already assigned to pathways. They don’t apply the test, and it will become more complicated when you don’t have variants assigned a priori to pathways. This is a shame, because directly testing the model on known variants is probably the best way to get at what is going on.


This paper doesn’t given any evidence in favor of epistasis in complex disease genetics per se. What it shows is that there exist entirely plausible models of genetic risk that look a lot like additive models in the ways we can measure, but disrupt some of our methods of measuring heritability. As we’ve seen, this paper is not exactly a deathblow to all measures of heritability, and there are still reasons to think that our methods are largely functioning largely as we’d expect (see the post on Genomes Unzipped). However, this does highlight the issue with stating that X percent of heritability has been explained, despite massive uncertainty about what the heritability of the trait is, or even what that value really means.

The good news is that people are thinking about the models that underlie our statistical genetics techniques. We know that the assumptions you make can drastically change (for instance, whether you think genetic risk prediction will be useful in complex disease will depend strongly on what model of genetic risk you use). With the new wealth of genetic data and methods in our hands, now is the time to go back and start assessing and comparing the models that underlie everything we do.

The authors of the paper promise that this is merely the first in a series of papers about missing heritability so expect the conversation to continue soon.

* There is actually a whole world of complexity here, which starts with “additive on what scale?” and ends with your crying in a corner with a sign hanging from your neck reading “NOTHING IS TRUE”. So we’ll talk about that another time.

The photo is of the Brown Lady of Raynham Hall, a well known phantom, and is taken from wikipedia

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6 Responses to Phantom heritability and additivity

  1. Thanks for the great post. I agree that we need a model and that as long as the model works well it fulfills the function and hence we keep it. I also believe that papers like this one, which challenge the model, are needed to keep the field healthy. A model is not the truth. It’s what we need to describe what we see as best as we can.

    BTW, did you see this recent paper in Science:
    These are worms, not humans, of course, but still, quite interesting .

  2. Clearly, the emotional attachment to the spherical cow estimates runs very strong. Half of the time the argument is reduced to referring to a paper with a title “Data and Theory Point to Mainly Additive Genetic Variance for Complex Traits” without ever mentioning papers with titles like “Genetic architecture of complex traits: Large phenotypic effects and pervasive epistasis” or “Genetic architecture of growth traits revealed by global epistatic interactions”.

    Data: Epistasis is ubiquitous and dominant in yeasts (On the classification of epistatic interactions. Genetics (2010), 184: 827–37).

    Theory: Humans are unlikely to be much less complex than yeasts.

    Data: No one ever saw pure additivity on a micro level (metabolic pathways and protein-protein interactions).

    Theory: If it is not true on a micro level, it is unlikely to be true on a macro level. If anything, more non-linearity with rising level of complexity is likely.

  3. Dr Robert Peers

    What is missing, most of all, in twin and family studies, is not imaginary heritability, but a simple food frequency questionnaire, without which one can only speculate that environmental exposures are similar in contrasted families or twins. The widespread failure to examine dietary exposures in basic (quantitative) genetic studies is simply bad science, and all such papers should be retracted, on the grounds of misleading incompleteness and premature conclusions. When the work is properly done, they could be re-submitted. Molecular genetics of common disease rests on a very shaky foundation, compared with a sound nutritional hypothesis–first proposed in 1956, by Oxford biochemist Dr Hugh Sinclair, who suggested that the diseases afflicting Westerners were due to fatty diet. Dairy, meat and bakery fats–when eaten out of proportion to “oily” polyunsaturated fatty acids in fish, seed oils, nuts etc.–created “faulty cell membranes” deficient in polyunsaturates. Recent research proves he was right: fatty diet displaces polyunsaturates from both plasma membrane and mitochondrial membrane, so causing insulin resistance (pre-diabetes) and chronic oxidation/inflammation. The result is a highly co-morbid cluster of chronic diseases (diabetes, vascular rot, tumour formation, arthritis, autoimmunity, hearing and visual loss, vascular dementia, Parkinson’s disease)–often rolled into the one patient! In addition, fatty MATERNAL diet is now known to cause offspring anxiety (1 in 4 Westerners), leading to addictions, obesity, worse diabetes, high cardiac risk, osteoporosis, immune failure and depression. The same anxiety is common in (Western) schizophrenia and bipolar, and actually drives these (only partly genetic) psychoses. To explain the extensive cross-morbidity of all these disorders, genetics would need an unlikely combination of multiple weakly-acting genes for EACH disorder, each team of genes inherited independently of each other. Game over?

  4. @Dr Robert Peers

    I think you have missed the point of twin study methodology somewhat – we are comparing MZ to DZ twins precisely to avoid such confounders of environment etc, and get at the variance due to genetics. The aim is to estimate how much impact genetics has after you have factored out environment.

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