Journal club of one: ”Eliciting priors and relaxing the single causal variant assumption in colocalisation analyses”

This paper (Wallace 2020) is about improvements to the colocalisation method for genome-wide association studies called coloc. If you have an association to trait 1 in a region, and another association with trait 2, coloc investigates whether they are caused by the same variant or not. I’ve never used coloc, but I’m interested because setting reasonable priors is related to getting reasonable parameters for genetic architecture.

The paper also looks at how coloc is used in the literature (with default settings, unsurprisingly), and extends coloc to relax the assumption of only one causal variant per region. In that way, it’s a solid example of thoughtfully updating a popular method.

(A note about style: This isn’t the clearest paper, for a few reasons. The structure of the introduction is indirect, talking a lot about Mendelian randomisation before concluding that coloc isn’t Mendelian randomisation. The paper also uses numbered hypotheses H1-H4 instead of spelling out what they mean … If you feel a little stupid reading it, it’s not just you.)

coloc is what we old QTL mappers call a pleiotropy versus linkage test. It tries to distinguish five scenarios: no association, trait 1 only, trait 2 only, both traits with linked variants, both traits with the same variant.

This paper deals with the priors: What is the prior probability of a causal association to trait 1 only p_1, trait 2 only p_2, or both traits p_{12} , and are the defaults good?

They reparametrise the priors so that it becomes possible to get some estimates from the literature. They work with the probability that a SNP is causally associated with each trait (which means adding the probabilities of association q_1 = p_1 + p_{12} ) … This means that you can look at single trait association data, and get an idea of the number of marginal associations, possibly dependent on allele frequency. The estimates from a gene expression dataset and a genome-wide association catalog work out to a prior around 10 ^ {-4} , which is the coloc default. So far so good.

How about p_{12} ?

If traits were independent, you could just multiply q_1 and q_2. But not all of the genome is functional. If you could straightforwardly define a functional proportion, you could just divide by it.

You could also look at the genetic correlation between traits. It makes sense that the overall genetic relationship between two traits should inform the prior that you see overlap at this particular locus. This gives a lower limit for p_{12} . Unfortunately, this still leaves us dependent on what kinds of traits we’re analysing. Perhaps, it’s not so surprising that there isn’t one prior that universally works for all kinds of pairs of trait:

Attempts to colocalise disease and eQTL signals have ranged from underwhelming to positive. One key difference between outcomes is the disease-specific relevance of the cell types considered, which is consistent with variable chromatin state enrichment in different GWAS according to cell type. For example, studies considering the overlap of open chromatin and GWAS signals have convincingly shown that tissue relevance varies by up to 10 fold, with pancreatic islets of greatest relevance for traits like insulin sensitivity and immune cells for immune-mediated diseases. This suggests that p_{12} should depend explicitly on the specific pair of traits under consideration, including cell type in the case of eQTL or chromatin mark studies. One avenue for future exploration is whether fold change in enrichment of open chromatin/GWAS signal overlap between cell types could be used to modulate p_{12} and select larger values for more a priori relevant tissues.

Literature

Wallace, Chris. ”Eliciting priors and relaxing the single causal variant assumption in colocalisation analyses.” PLoS Genetics 16.4 (2020): e1008720.

Adrian Bird on genome ecology

I recently read this essay by Adrian Bird on ”The Selfishness of Law-Abiding Genes”. That is a colourful title in itself, but it doesn’t stop there; this is an extremely metaphor-rich piece. In terms of the theoretical content, there is not much new under the sun. Properties of the organism like complexity, redundancy, and all those exquisite networks of developmental gene regulation may be the result of non-adaptive processes, like constructive neutral evolution and intragenomic conflict. As the title suggests, Bird argues that this kind of thinking is generally accepted about things like transposable elements (”selfish DNA”), but that the same logic applies to regular ”law-abiding” genes. They may also be driven by other evolutionary forces than a net fitness gain at the organismal level.

He gives a couple of possible examples: toxin–antitoxin gene pairs, RNA editing and MeCP2 (that’s probably Bird’s favourite protein that he has done a lot of work on). He gives this possible description of MeCP2 evolution:

Loss of MeCP2 via mutation in humans leads to serious defects in the brain, which might suggest that MeCP2 is a fundamental regulator of nervous system development. Evolutionary considerations question this view, however, as most animals have nervous systems, but only vertebrates, which account for a small proportion of the animal kingdom, have MeCP2. This protein therefore appears to be a late arrival in evolutionary terms, rather than being a core ancestral component of brain assembly. A conventional view of MeCP2 function is that by exerting global transcriptional restraint it tunes gene expression in neurons to optimize their identity, but it is also possible to devise a scenario based on self-interest. Initially, the argument goes, MeCP2 was present at low levels, as it is in non-neuronal tissues, and therefore played little or no role in creating an optimal nervous system. Because DNA methylation is sparse in the great majority of the genome, sporadic mutations that led to mildly increased MeCP2 expression would have had a minimal dampening effect on transcription that may initially have been selectively neutral. If not eliminated by drift, further chance increases might have followed, with neuronal development incrementally adjusting to each minor hike in MeCP2-mediated repression through compensatory mutations in other genes. Mechanisms that lead to ‘constructive neutral evolution’ of this kind have been proposed. Gradually, brain development would accommodate the encroachment of MeCP2 until it became an essential feature. So, in response to the question ‘why do brains need MeCP2?’, the answer under this speculative scenario would be: ‘they do not; MeCP2 has made itself indispensable by stealth’.

I think this is a great passage, and it can be read both as a metaphorical reinterpretation, and as substantive hypothesis. The empirical question ”Did MeCP2 offer an important innovation to vertebrate brains as it arose?”, is a bit hard to answer with data, though. On the other hand, if we just consider the metaphor, can’t you say the same about every functional protein? Sure, it’s nice to think of p53 as the Guardian of the Genome, but can’t it also be viewed as a gangster extracting protection money from the organism? ”Replicate me, or you might get cancer later …”

The piece argues for a gene-centric view, that thinks of molecules and the evolutionary pressures they face. This doesn’t seem so be the fashionable view (sorry, extended synthesists!) but Bird argues that it would be healthy for molecular cell biologists to think more about the alternative, non-adaptive, bottom-up perspective. I don’t think the point is to advocate that way of thinking to the exclusion of the all other. To me, the piece reads more like an invitation to use a broader set of metaphors and verbal models to aid hypothesis generation.

There are too may good quotes in this essay, so I’ll just quote one more from the end, where we’ve jumped from the idea of selfish law-abiding genes, over ”genome ecology” — not in the sense of using genomics in ecology, but in the sense of thinking of the genome as some kind of population of agents with different niches and interactions, I guess — to ”Genetics Meets Sociology?”

Biologists often invoke parallels between molecular processes of life and computer logic, but a gene-centered approach suggests that economics or social science may be a more appropriate model …

I feel like there is a circle of reinforcing metaphors here. Sometimes when we have to explain how something came to be, for example a document, a piece of computer code or a the we do things in an organisation, we say ”it grew organically” or ”it evolved”. Sometimes we talk about the genome as a computer program, and sometimes we talk about our messy computer program code as an organism. Like viruses are just like computer viruses, only biological.

Literature

Bird, Adrian. ”The Selfishness of Law-Abiding Genes.” Trends in Genetics 36.1 (2020): 8-13.

Journal club of one: ”Genomic predictions for crossbred dairy cattle”

A lot of dairy cattle is crossbred, but genomic evaluation is often done within breed. What about the crossbred individuals? This paper (VanRaden et al. 2020) describes the US Council on Dairy Cattle Breeding’s crossbred genomic prediction that started 2019.

In short, the method goes like this: They describe each crossbred individual in terms of their ”genomic breed composition”, get predictions for each them based on models from all the breeds separately, and then combine the results in proportion to the genomic breed composition. The paper describes how they estimate the genomic breed composition, and evaluated accuracy by predicting held-out new data from older data.

The genomic breed composition is a delightfully elegant hack: They treat ”how much breed X is this animal” as a series of traits and run a genomic evaluation on them. The training set: individuals from sets of reference breeds with their trait value set to 100% for the breed they belong to and 0% for other breeds. ”Marker effects for GBC [genomic breed composition] were then estimated using the same software as for all other traits.” Neat. After some adjustment, they can be interpreted as breed percentages, called ”base breed representation”.

As they already run genomic evaluations from each breed, they can take these marker effects and then animal’s genotypes, and get one estimate for each breed. Then they combine them, weighting by the base breed representation.

Does it work? Yes, in the sense that it provides genomic estimates for animals that otherwise wouldn’t have any, and that it beats parent average estimates.

Accuracy of GPTA was higher than that of [parent average] for crossbred cows using truncated data from 2012 to predict later phenotypes in 2016 for all traits except productive life. Separate regressions for the 3 BBR categories of crossbreds suggest that the methods perform equally well at 50% BBR, 75% BBR, and 90% BBR.

They mention in passing comparing these estimates to estimates from a common set of marker effects for all breeds, but there is no detail about that model or how it compared in accuracy.

The discussion starts with this sentence:

More breeders now genotype their whole herds and may expect evaluations for all genotyped animals in the future.

That sounds like a reasonable expectation, doesn’t it? Before what they could do with crossbred genotypes was to throw it away. There are lots of other things that might be possible with crossbred evaluation in the future (pulling in crossbred data into the evaluation itself, accounting for ancestry in different parts of the genome, estimating breed-of-origin of alleles, looking at dominance etc etc).

My favourite result in the paper is Table 8, which shows:

Example BBR for animals from different breeding systems are shown in Table 8. The HO cow from a 1964 control line had 1960s genetics from a University of Minnesota experimental selection project and a relatively low relationship to the current HO population because of changes in breed allele frequencies over the past half-century. The Danish JE cow has alleles that differ somewhat from the North American JE population. Other examples in the table show various breed crosses, and the example for an animal from a breed with no reference population shows that genetic contributions from some other breed may be evenly distributed among the included breeds so that BBR percentages sum to 100. These examples illustrate that GBC can be very effective at detecting significant percentages of DNA contributed by another breed.

Literature

VanRaden, P. M., et al. ”Genomic predictions for crossbred dairy cattle.” Journal of Dairy Science 103.2 (2020): 1620-1631.

Robertson on genetic correlation and loss of variation

It’s not too uncommon to see animal breeding papers citing a paper by Alan Robertson (1959) to support a genetic correlation of 0.8 as a cut-off point for what is a meaningful difference. What is that based on?

The paper is called ”The sampling variance of the genetic correlation coefficient” and, as the name suggests, it is about methods for estimating genetic correlations. It contains a section about the genetic correlation between environments as a way to measure gene-by-environment interaction. There, Robertson discusses experimental designs for detecting gene-by-environment interaction–that is, estimating whether a genetic correlation between different environments is less than one. He finds that you need much larger samples than for estimating heritabilities. It is in this context that the 0.8 number comes up. Here is the whole paragraph:

No interaction means a genetic correlation of unity. How much must the correlation fall before it has biological or agricultural importance? I would suggest that this figure is around 0.8 and that no experiment on genotype-environment interaction would have been worth doing unless it could have detected, as a significant deviation from unity, a genetic correlation of 0.6. In the first instance, I propose to argue from the standpoint of a standard error of 0.2 as an absolute minimum.

That is, in the context of trying to make study design recommendations for detecting genotype-by-environment interactions, Robertson suggests that a genetic correlation of 0.8 might be a meaningful difference from 1. The paper does not deal with designing breeding programs for multiple environments or the definition of traits, and it has no data on any of that. It seems to be a little bit like Fisher’s p < 0.05: Suggest a rule of thumb, and risk it having a life of its own in the future.

In the process of looking up this quote, I also found this little gem, from ”The effect of selection on the estimation of genetic parameters” (Robertson 1977). It talks about the problems that arise with estimating genetic parameters in populations under selection, when many quantitative genetic results, in one way or another, depend on random mating. Here is how it ends:

This perhaps points the moral of this paper. The individuals of one generation are the parents of the next — if they are accurately evaluated and selected in the first generation, the variation between families will be reduced in the next. You cannot have your cake and eat it.

Literature

Robertson, A. ”The sampling variance of the genetic correlation coefficient.” Biometrics 15.3 (1959): 469-485.

Robertson, A. ”The effect of selection on the estimation of genetic parameters.” Zeitschrift für Tierzüchtung und Züchtungsbiologie 94.1‐4 (1977): 131-135.

Virtual animal breeding journal club: ”An eQTL in the cystathionine beta synthase gene is linked to osteoporosis in laying hens”

The other day the International Virtual Animal Breeding Journal Club, organised by John Cole, had its second meeting. I presented a recent paper about using genetic mapping and gene expression to find a putative causative gene for a region associated with bone strength in layer chickens. This from colleauges I know and work with, but I wasn’t involved in this work myself.

Here is the paper:

De Koning, Dirk-Jan, et al. ”An eQTL in the cystathionine beta synthase gene is linked to osteoporosis in laying hens.” Genetics Selection Evolution 52.1 (2020): 1-17.

Here are my slides:

Ian Dunn and DJ de Koning were both on the call to answer some questions and give the authors’ perspective, which, again, I thought was very useful. I hope this becomes a recurring theme of the journal club.

I chose the paper because I think it’s a good example of the QTL–eQTL paradigm of causative gene identification. We got some discussion about that. Conclusions: You never really know whether an association with gene expression is causal or reactive, unless there’s some kind of experimental manipulation. We all want more annotation, more functional genomics and more genome sequences. I can’t argue with that.

Here is the a review of layer chicken bone biology referred to in the slides, if you want to look into that:

Whitehead, C. C. ”Overview of bone biology in the egg-laying hen.” Poultry science 83.2 (2004): 193-199.

If you want to follow the journal club, see the Google group and Twitter account for announcements.

Virtual animal breeding journal club: ”Structural equation models to disentangle the biological relationship between microbiota and complex traits …”

The other day was the first Virtual breeding and genetics journal club organised by John Cole. This was the first online journal club I’ve attended (shocking, given how many video calls I’ve been on for other sciencey reasons), so I thought I’d write a little about it: both the format and the paper. You can look the slide deck from the journal club here (pptx file).

The medium

We used Zoom, and that seemed to work, as I’m sure anything else would, if everyone just mute their microphone when they aren’t speaking. As John said, the key feature of Zoom seems to be the ability for the host to mute everyone else. During the call, I think we were at most 29 or so people, but only a handful spoke. It will probably get more intense with the turn taking if more people want to speak.

The format

John started the journal club with a code of conduct, which I expect helped to set what I felt was a good atmosphere. In most journal clubs I’ve been in, I feel like the atmosphere has been pretty good, but I think we’ve all heard stories about hyper-critical and hostile journal clubs, and that doesn’t sound particularly fun or useful. On that note, one of the authors, Oscar González-Recio, was on the call and answered some questions.

The paper

Saborío‐Montero, Alejandro, et al. ”Structural equation models to disentangle the biological relationship between microbiota and complex traits: Methane production in dairy cattle as a case of study.” Journal of Animal Breeding and Genetics 137.1 (2020): 36-48.

The authors measured methane emissions (by analysing breath with with an infrared gas monitor) and abundance of different microbes in the rumen (with Nanopore sequencing) from dairy cows. They genotyped the animals for relatedness.

They analysed the genetic relationship between breath methane and abundance of each taxon of microbe, individually, with either:

  • a bivariate animal model;
  • a structural equations model that allows for a causal effect of abundance on methane, capturing the assumption that the abundance of a taxon can affect the methane emission, but not the other way around.

They used them to estimate heritabilities of abundances and genetic correlations between methane and abundances, and in the case of the structural model: conditional on the assumed causal model, the effect of that taxon’s abundance on methane.

My thoughts

It’s cool how there’s a literature building up on genetic influences on the microbiome, with some consistency across studies. These intense high-tech studies on relatively few cattle might build up to finding new traits and proxies that can go into larger scale phenotyping for breeding.

As the title suggests, the paper advocates for using the structural equations model: ”Genetic correlation estimates revealed differences according to the usage of non‐recursive and recursive models, with a more biologically supported result for the recursive model estimation.” (Conclusions)

While I agree that a priori, it makes sense to assume a structural equations model with a causal structure, I don’t think the results provide much evidence that it’s better. The estimates of heritabilities and genetic correlations from the two models are near indistinguishable. Here is the key figure 4, comparing genetic correlation estimates:

saborio-montero-fig4

As you can see, there are a couple of examples of genetic correlations where the point estimate switches sign, and one of them (Succinivibrio sp.) where the credible intervals don’t overlap. ”Recursive” is the structural equations model. The error bars are 95% credible intervals. This is not strong evidence of anything; the authors are responsible about it and don’t go into interpreting this difference. But let us speculate! They write:

All genera in this case, excepting Succinivibrio sp. from the Proteobacteria phylum, resulted in overlapped genetic cor- relations between the non‐recursive bivariate model and the recursive model. However, high differences were observed. Succinivibrio sp. showed the largest disagreement changing from positively correlated (0.08) in the non‐recursive bivariate model to negatively correlated (−0.20) in the recursive model.

Succinivibrio are also the taxon with the estimated largest inhibitory effect on methane (from the structural equations model).

While some taxa, such as ciliate protozoa or Methanobrevibacter sp., increased the CH4 emissions …, others such as Succinivibrio sp. from Proteobacteria phylum decreased it

Looking at the paper that first described these bacteria (Bryan & Small 1955),  Succinivibrio were originally isolated from the cattle rumen, and their name is because ”they ferment glucose with the production of a large amount of succinic acid”. Bryant & Small made a fermentation experiment to see what came out, and it seems that the bacteria don’t produce methane:

succ_table2

This is also in line with a rRNA sequencing study of high and low methane emitting cows (Wallace & al 2015) that found lower Succinivibrio abundance in high methane emitters.

We may speculate that Succinivibrio species could be involved in diverting energy from methanogens, and thus reducing methane emissions. If that is true, then the structural equations model estimate (larger genetic negative correlation between Succinivibrio abundance and methane) might be better than one from the animal model.

Finally, while I’m on board with the a priori argument for using a structural equations model, as with other applications of causal modelling (gene networks, Mendelian randomisation etc), it might be dangerous to consider only parts of the system independently, where the microbes are likely to have causal effects on each other.

Literature

Saborío‐Montero, Alejandro, et al. ”Structural equation models to disentangle the biological relationship between microbiota and complex traits: Methane production in dairy cattle as a case of study.” Journal of Animal Breeding and Genetics 137.1 (2020): 36-48.

Wallace, R. John, et al. ”The rumen microbial metagenome associated with high methane production in cattle.” BMC genomics 16.1 (2015): 839.

Bryant, Marvin P., and Nola Small. ”Characteristics of two new genera of anaerobic curved rods isolated from the rumen of cattle.” Journal of bacteriology 72.1 (1956): 22.

Preprint: ”Genetics of recombination rate variation in the pig”

We have a new preprint posted, showing that recombination rate in the pig is lowly heritable and associated with alleles at RNF212.

We developed a new method to estimate recombinations in 150,000 pigs, and used that to estimate heritability and perform genome-wide association studies in 23,000.

Here is the preprint:

Johnsson M*, Whalen A*, Ros-Freixedes R, Gorjanc G, Chen C-Y, Herring WO, de Koning D-J, Hickey JM. (2020) Genetics of recombination rate variation in the pig. BioRxiv preprint. https://doi.org/10.1101/2020.03.17.995969 (* equal contribution)

Here is the abstract:

Background In this paper, we estimated recombination rate variation within the genome and between individuals in the pig for 150,000 pigs across nine genotyped pedigrees. We used this to estimate the heritability of recombination and perform a genome-wide association study of recombination in the pig.

Results Our results confirmed known features of the pig recombination landscape, including differences in chromosome length, and marked sex differences. The recombination landscape was repeatable between lines, but at the same time, the lines also showed differences in average genome-wide recombination rate. The heritability of genome-wide recombination was low but non-zero (on average 0.07 for females and 0.05 for males). We found three genomic regions associated with recombination rate, one of them harbouring the RNF212 gene, previously associated with recombination rate in several other species.

Conclusion Our results from the pig agree with the picture of recombination rate variation in vertebrates, with low but nonzero heritability, and a major locus that is homologous to one detected in several other species. This work also highlights the utility of using large-scale livestock data to understand biological processes.

Things that really don’t matter: megabase or megabasepair

Should we talk about physical distance in genetics as number of base pairs (kbp, Mbp, and so on) or bases (kb, Mb)?

I got into a discussion about this recently, and I said I’d continue the struggle on my blog. Here it is. Let me first say that I don’t think this matters at all, and if you make a big deal out of this (or whether ”data” can be singular, or any of those inconsequential matters of taste we argue about for amusement), you shouldn’t. See this blog post as an exorcism, helping me not to trouble my colleagues with my issues.

What I’m objecting to is mostly the inconsistency of talking about long stretches of nucleotides as ”kilobase” and ”megabase” but talking about short stretches as ”base pairs”. I don’t think it’s very common to call a 100 nucleotide stretch ”a 100 b sequence”; I would expect ”100 bp”. For example, if we look at Ensembl, they might describe a large region as 1 Mb, but if you zoom in a lot, they give length in bp. My impression is that this is a common practice. However, if you consistently use ”bases” and ”megabase”, more power to you.

Unless you’re writing a very specific kind of bioinformatics paper, the risk of confusion with the computer storage unit isn’t a problem. But there are some biological arguments.

A biological argument for ”base”, might be that we care about the identity of the base, not the base pairing. We note only one nucleotide down when we write a nucleic acid sequence. The base pair is a different thing: that base bound to the one on the other strand that it’s paired with, or, if the DNA or RNA is single-stranded, it’s not even paired at all.

Conversely, a biochemical argument for ”base pair” might be that in a double-stranded molecule, the base pair is the relevant informational unit. We may only write one base in our nucleotide sequence for convenience, but because of the rules of base pairing, we know the complementing pair. In this case, maybe we should use ”base” for single-stranded molecules.

If we consult two more or less trustworthy sources, The Encylopedia of Life Sciences and Wiktionary, they both seem to take this view.

eLS says:

A megabase pair, abbreviated Mbp, is a unit of length of nucleic acids, equal to one million base pairs. The term ‘megabase’ (or Mb) is commonly used inter-changeably, although strictly this would refer to a single-stranded nucleic acid.

Wiktionary says:

A length of nucleic acid containing one million nucleotides (bases if single-stranded, base pairs if double-stranded)

Please return next week for the correct pronunciation of ”loci”.

Literature

Dear, P.H. (2006). Megabase Pair (Mbp). eLS.

Using R: Animal model with hglm and Stan (with Cholesky trick)

A few weeks ago I posted about fitting the quantitative genetic animal model with MCMCglmm and R-INLA. Since then, I listened to a talk by Lars Rönnegård, one of the creators of the hglm package, and this paper was published in GSE about animal models in Stan.

hglm

The hglm package fits hierarchical generalised linear models. That includes the animal model with pedigree or genomic relatedness. Hierarchical generalised linear models also allow you to model the dispersion of random effects, which lets you do tricks like variance QTL mapping (Rönnegård & Valdar 2011), breeding values for variances (Rönnegård et al. 2010) or genomic prediction models with predictors of marker variance (Mouresan, Selle & Rönnegård 2019). But let’s not get ahead of ourselves. How do we fit an animal model?

Here is the matrix formulation of the animal model that we skim through in every paper. It’s in this post because we will use the design matrix interface to hglm, which needs us to give it these matrices (this is not a paper, so we’re not legally obliged to include it):

\mathbf{y} = \mu + \mathbf{X} \mathbf{b} + \mathbf{Z} \mathbf{a} + \mathbf{e}

The terms are the the trait value, intercept, fixed coefficients and their design matrix, genetic coefficients and their design matrix, and the residual. The design matrix Z will contain one row and column for each individual, with a 1 to indicate its position in the phenotype table and pedigree and the rest zeros. If we sort our files, it’s an identity matrix.

The trick with the genetic coefficients is that they’re correlated, with a specific known correlation structure that we know from the pedigree (or in genomic models, from markers). It turns out (Lee, Nelder & Pawitan 2017, chapter 8) that you can change the Z matrix around so that it lets you fit the model with an identity covariance matrix, while still accounting for the correlations between relatives. You replace the random effects for relatedness with some transformed random effects that capture the same structure. One way to do this is with Cholesky decomposition.

\mathbf{Z_{fudged}} = \mathbf{Z_0} \mathbf{L}

As an example of what the Cholesky decomposition does, here is slice of the additive relationship matrix of 100 simulated individuals (the last generation of one replicate of these simulations) and the resulting matrix from Cholesky decomposition.

So instead of

\mathbf{a} \sim N(0, \mathbf{A} \sigma)

We can fit

\mathbf{a_{fudged}} \sim N(0, \mathbf{I} \sigma)

This lets us fit the animal model with hglm, by putting in a modified Z matrix.

Assuming we have data frames with a pedigree and a phenotype (like, again, from these simulations):

library(AGHmatrix)
library(hglm)

A  <- Amatrix(ped)

Z0  <- diag(1000)
L <- t(chol(A))
Z  <- Z0 %*% L
X <- model.matrix(~1, pheno)

model <- hglm(y = pheno$pheno,
              X = X,
              Z = Z,
              conv = 1e-8)

est_h2  <- model$varRanef / (model$varRanef + model$varFix)

(I found the recommendation to decrease the convergence criterion from the default for animal models in a YouTube video by Xia Chen.)

Stan

When we turn to Stan, we will meet the Cholesky trick again. Stan is a software for Markov Chain Monte Carlo, built to fit hierarchical linear models, and related high-dimensional models, more effectively than other sampling strategies (like Gibbs). rstan is a helpful package for running Stan from within R.

Nishio & Arakawa (2019) recently published a Stan script to fit an animal model, comparing Stan to a Gibbs sampler (and a related MCMC sampler that they also didn’t publish the code for). If we look into their Stan model code, they also do a Cholesky decomposition to be able to use an identity matrix for the variance.

First, they decompose the additive relationship matrix that the program takes in:

transformed data{
  matrix[K,K] LA;
  LA = cholesky_decompose(A);
}

And then, they express the model like this:

vector[N] mu;
vector[K] a;
a_decompose ~ normal(0, 1);
a = sigma_G * (LA * a_decompose);
mu = X * b + Z * a;
Y ~ normal(mu, sigma_R);

We can add this line to the generated quantities block of the Stan program to get heritability estimates directly:

real h2;
h2 = sigma_U / (sigma_U + sigma_E)

Here, we’ve saved their model to a stan file, and now we can run it from R:

pheno$scaled_pheno <- as.vector(scale(pheno$pheno))

model_stan <- stan(file = "nishio_arakawa.stan",
                   data = list(Y = pheno$scaled_pheno,
                               X = X,
                               A = A,
                               Z = Z0,
                               J = 1,
                               K = 1000,
                               N = 1000))

est_h2_stan <- summary(model_stan, pars = "h2")$summary

Important note that I always forget: It's important to scale your traits before you run this model. If not, the priors might be all wrong.

The last line pulls out the summary for the heritability parameter (that we added above). This gives us an estimate and an interval.

The paper also contains this entertaining passage about performance, which reads as if it was a response to a comment, actual or anticipated:

R language is highly extensible and provides a myriad of statistical and graphical techniques. However, R language has poor computation time compared to Fortran, which is especially well suited to numeric computation and scientific computing. In the present study, we developed the programs for GS and HMC in R but did not examine computation time; instead, we focused on examining the performance of estimating genetic parameters and breeding values.

Yes, two of their samplers (Gibbs and HMC) were written in R, but the one they end up advocating (and the one used above), is in Stan. Stan code gets translated into C++ and then compiled to machine code.

Stan with brms

If rstan lets us run Stan code from R and examine the output, brms lets us write down models in relatively straightforward R syntax. It’s like the MCMCglmm of the Stan world. We can fit an animal model with brms too, by directly plugging in the relationship matrix:

model_brms <- brm(scaled_pheno ~ 1 + (1|animal),
                  data = pheno,
                  family = gaussian(),
                  cov_ranef = list(animal = A),
                  chains = 4,
                  cores = 1,
                  iter = 2000)

Then, we can pull out the posterior samples for the variability, here expressed as standard errors, compute the heritability and then get the estimates (and interval, if we want):

posterior_brms <- posterior_samples(model_brms,
                                    pars = c("sd_animal", "sigma"))

h2_brms  <- posterior_brms[,1]^2 /
    (posterior_brms[,1]^2 + posterior_brms[,2]^2)

est_h2_brms <- mean(h2_brms)

(Code is on GitHub: both for the graphs above, and the models.)

Interpreting genome scans, with wisdom

Eric Fauman is a scientist at Pfizer who also tweets out interpretations of genome-wide association scans.

Background: There is a GWASbot twitter account which posts Manhattan plots with links for various traits from the UK Biobank. The bot was made by the Genetic Epidemiology lab at the Finnish Institute for Molecular Medicine and Harvard. The source of the results is these genome scans (probably; it’s little bit opaque); the bot also links to heritability and genetic correlation databases. There is also an EnrichrBot that replies with enrichment of chromatin marks (Chen et al. 2013). Fauman’s comments on some of the genome scans on his Twitter account.

Here are a couple of recent ones:

And here is his list of these threads as a Google Document.

This makes me thing of three things, two good, and one bad.

1. The ephemeral nature of genome scans

Isn’t it great that we’re now at a stage where a genome scan can be something to be tweeted or put en masse in a database, instead of published one paper per scan with lots of boilerplate. The researchers behind the genome scans say as much in their 2017 blog post on the first release:

To further enhance the value of this resource, we have performed a basic association test on ~337,000 unrelated individuals of British ancestry for over 2,000 of the available phenotypes. We’re making these results available for browsing through several portals, including the Global Biobank Engine where they will appear soon. They are also available for download here.

We have decided not to write a scientific article for publication based on these analyses. Rather, we have described the data processing in a detailed blog post linked to the underlying code repositories. The decision to eschew scientific publication for the basic association analysis is rooted in our view that we will continue to work on and analyze these data and, as a result, writing a paper would not reflect the current state of the scientific work we are performing. Our goal here is to make these results available as quickly as possible, for any geneticist, biologist or curious citizen to explore. This is not to suggest that we will not write any papers on these data, but rather only write papers for those activities that involve novel method development or more complex analytic approaches. A univariate genome-wide association analysis is now a relatively well-established activity, and while the scale of this is a bit grander than before, that in and of itself is a relatively perfunctory activity. [emphasis mine] Simply put, let the data be free.

That being said, when starting to write this post, first I missed a paper. It was pretty frustrating to find a detailed description of the methods: after circling back and forth between the different pages that link to each other, I landed on the original methods post, which is informative, and written in a light conversational style. On the internet, one would fear that this links may rot and die eventually, and a paper would probably (but not necessarily …) be longer-lasting.

2. Everything is a genome scan, if you’re brave enough

Another thing that the GWAS bot drives home is that you can map anything that you can measure. The results are not always straightforward. On the other hand, even if the trait in question seems a bit silly, the results are not necessarily nonsense either.

There is a risk, for geneticists and non-geneticists alike, to reify traits based on their genetic parameters. If we can measure the heritability coefficient of something, and localise it in the genome with a genome-wide association study, it better be a real and important thing, right? No. The truth is that geneticists choose traits to measure the same way all researchers choose things to measure. Sometimes for great reasons with serious validation and considerations about usefulness. Sometimes just because. The GWAS bot also helpfully links to the UK Biobank website that describes the traits.

Look at that bread intake genome scan above. Here, ”bread intake” is the self-reported number of slices of bread eaten per week, as entered by participants on a touch screen questionnaire at a UK Biobank assessment centre. I think we can be sure that this number doesn’t reveal any particularly deep truth about bread and its significance to humanity. It’s a limited, noisy, context-bound number measured, I bet, because once you ask a battery of lifestyle questions, you’ll ask about bread too. Still, the strongest association is at a region that contains olfactory receptor genes and also shows up two other scans about food (fruit and ice cream). The bread intake scan hits upon a nugget of genetic knowledge about human food preference. A small, local truth, but still.

Now substitute bread intake for some more socially relevant trait, also imperfectly measured.

3. Lost, like tweets in rain

Genome scan interpretation is just that: interpretation. It means pulling together quantitative data, a knowledge of biology, previous literature, and writing an unstructured text, such as a Discussion section or a Twitter thread. This makes them harder to organise, store and build on than the genome scans themselves. Sure, Fauman’s Twitter threads are linked from the above Google Document, and our Discussion sections are available from the library. But they’re spread out in different places, they mix (as they should) evidence with evaluation and speculation, and it’s not like we have a structured vocabulary for describing genetic mechanisms of quantitative trait loci, and the levels of evidence for them. Maybe we could, with genome-wide association study ontologies and wikis.