A brief outline of colocalisation analysis
Usage
An aside: fine mapping under a single causal variant assumption
(Approximate) Bayes Factor colocalisation analyses
1. Introduction
2. The basic coloc.abf function
Proportional testing
1. Using Bayes Factors to compare specific values of \(\eta\)
The difference between proportional and ABF approaches

A brief outline of colocalisation analysis

The coloc package can be used to perform genetic colocalisation analysis of two potentially related phenotypes, to ask whether they share common genetic causal variant(s) in a given region. There are a few key references which this vignette will not duplicate (see below), but, in brief, two approaches are implemented.

First, the proportional approach uses the fact that for two traits sharing causal variants, regression coefficients for either trait against any set of SNPs in the neighbourhood of those variants must be proportional. This test was first proposed by Plagnol et al. ¹ in the context of evaluating whether expression of the gene RPS26 mediated the association of type 1 diabetes to a region on chromosome 12q13 as had recently been proposed. The test addressed a common issue in genetics, and meant researchers could avoid the need to squint at parallel manhattan plots to decide whether two traits share causal variants. The function coloc.test() in this package evolved from code released by Vincent, but no longer available.

However, choosing which SNPs to use for the test is a problem. The obvious choice is to use those most strongly associated with one or other trait to maximise information, but doing so induces bias in the regression coefficients, which in turn leads to increased likelihood of spuriously rejecting the null of colocalisation, ie a quite substantially increased type 1 error rate ². I proposed two alternatives to address this problem, either using a principal component summary of genetic variation in the region to overcome the need to select a small set of test SNPs, implemented in coloc.pcs() and associated functions, or to use the ideas of Bayesian model averaging to average p values over SNP selections, generating posterior predictive p values, implemented in coloc.bma().

Proportional testing, however, requires individual level genotype data, which are not always available. Claudia Giambartolomei and Vincent Plagnol have proposed an alternative method, which makes use of Jon Wakefield’s work on determining approximate Bayes Factors from p values ³ to generate a Bayesian colocalisation analysis ⁴, implemented in the function coloc.summaries(). Note that it is possible to use Bayesian analysis for proportional testing too, in determining the posterior distribution of the proportionality constant \(\eta\).

Table 1: Suitability of functions in this package for different data structures
Function	Approach	Data required	Dependence
			between datasets
coloc.abf	Bayesian	summary statistics	independent
coloc.abf.datasets	Bayesian	raw genotypes	independent
coloc.bma	Proportional, averages	raw genotypes	independent
	over models
coloc.pcs/coloc.test	Proportional, based on	raw genotypes	independent
	PC summary of genotypes

Usage

Let’s simulate a small dataset, and compare the three methods.

library(coloc)
setClass("simdata",
         representation(df1="data.frame",df2="data.frame"))
setValidity("simdata", function(object) {
  n <- nrow(object@df1)
  if(nrow(object@df2)!=n)
    return("nrow of '@df1' should equal nrow of '@df2'")
})

## Class "simdata" [in ".GlobalEnv"]
## 
## Slots:
##                             
## Name:         df1        df2
## Class: data.frame data.frame

setMethod("show", signature="simdata", function(object) {
  cat("pair of simulated datasets, with",ncol(object@df1)-1,"SNPs and",nrow(object@df1),"samples.\n")
})

## [1] "show"

sim.data <- function(nsnps=50,nsamples=200,causals=1:2,nsim=1) {
  cat("Generate",nsim,"small sets of data\n")
  ntotal <- nsnps * nsamples * nsim
  X1 <- matrix(rbinom(ntotal,1,0.4)+rbinom(ntotal,1,0.4),ncol=nsnps)
  Y1 <- rnorm(nsamples,rowSums(X1[,causals]),2)
  X2 <- matrix(rbinom(ntotal,1,0.4)+rbinom(ntotal,1,0.4),ncol=nsnps)
  Y2 <- rnorm(nsamples,rowSums(X2[,causals]),2)
  colnames(X1) <- colnames(X2) <- paste("s",1:nsnps,sep="")
  df1 <- cbind(Y=Y1,X1)
  df2 <- cbind(Y=Y2,X2)
  if(nsim==1) {
    return(new("simdata",
               df1=as.data.frame(df1),
               df2=as.data.frame(df2)))
  } else {
    index <- split(1:(nsamples * nsim), rep(1:nsim, nsamples))
    objects <- lapply(index, function(i) new("simdata", df1=as.data.frame(df1[i,]),
                                             df2=as.data.frame(df2[i,])))
    return(objects)
  }
}

## simulate some data and load the coloc library
set.seed(46411)
data <- sim.data(nsamples=1000,nsim=1)

## Generate 1 small sets of data

library(coloc)

If you are not familiar with the analysis of genetic data in R, I would suggest beginning with the snpStats package, which has functions for reading data in a variety of formats and contains a vignette on data input. Once you have your genetic data loaded as a SnpMatrix, X, and phenotype information in a vector Y, then conversion to a data.frame as used above follows

df <- as.data.frame(cbind(Y=Y, as(X,"numeric")))

An aside: fine mapping under a single causal variant assumption

The Approximate Bayes Factor colocalisation analysis described in the next section essentially works by fine mapping each trait under a single causal variant assumption and then integrating over those two posterior distributions to calculate probabilities that those variants are shared. Of course, this means we can look at each trait on its own quite simply, and we can do that with the function finemap.abf.

First we analyse a single trait from our simulated data to generate summary statistics:

if(requireNamespace("snpStats")) {
library(snpStats)
Y1 <- data@df1$Y
X1 <- new("SnpMatrix",as.matrix(data@df1[,-1]))
p1 <- snpStats::p.value(single.snp.tests(phenotype=Y1, snp.data=X1),df=1)
maf <- col.summary(X1)[,"MAF"]

get.beta <- function(...) {
   tmp <- snpStats::snp.rhs.estimates(..., family="gaussian")
   beta <- sapply(tmp,"[[","beta")
   varbeta <- sapply(tmp, "[[", "Var.beta")
   return(list(beta=beta,varbeta=varbeta))
}
b1 <- get.beta(Y1 ~ 1, snp.data=X1)
} else {
Y1 <- data@df1$Y
X1 <- as.matrix(data@df1[,-1])
tests1 <- lapply(1:ncol(X1), function(i) summary(lm(Y1 ~ X1[,i]))$coefficients[2,])
p1 <- sapply(tests1,"[",4)
maf <- colMeans(X2)/2

get.beta <- function(x) {
   beta <- sapply(x,"[",1)
   varbeta <- sapply(x, "[", 2)^2
   return(list(beta=beta,varbeta=varbeta))
}
b1 <- get.beta(tests1)
}

## Loading required package: survival

## Loading required package: Matrix

## coercing object of mode  numeric  to SnpMatrix

Then we analyse them:

my.res <- finemap.abf(dataset=list(beta=b1$beta, varbeta=b1$varbeta, N=nrow(X1),sdY=sd(Y1),type="quant"))
head(my.res)

##            V.          z.        r.      lABF.   snp prior       SNP.PP
## s1 0.03924328  5.78083831 0.7567274 11.9374068 SNP.1 1e-04 9.372547e-01
## s2 0.03880072  4.01220890 0.7588092  5.3965036 SNP.2 1e-04 1.352631e-03
## s3 0.04455547 -0.06333058 0.7326023 -0.6580400 SNP.3 1e-04 3.174860e-06
## s4 0.04602953  0.83093858 0.7261782 -0.3969407 SNP.4 1e-04 4.122100e-06
## s5 0.04237982 -1.83425483 0.7422945  0.5707526 SNP.5 1e-04 1.084882e-05
## s6 0.04495749 -0.22493863 0.7308390 -0.6377335 SNP.6 1e-04 3.239989e-06

tail(my.res)

##              V.          z.        r.       lABF.    snp  prior
## s46  0.04560292 -0.00230155 0.7280258 -0.65102205 SNP.46 0.0001
## s47  0.05248414 -1.20366960 0.6993259 -0.09426528 SNP.47 0.0001
## s48  0.04101521 -0.03458088 0.7485056 -0.68971972 SNP.48 0.0001
## s49  0.04530207  0.58219760 0.7293344 -0.52983015 SNP.49 0.0001
## s50  0.04467786 -1.08347773 0.7320646 -0.22881057 SNP.50 0.0001
## s110         NA          NA        NA  0.00000000   null 0.9950
##            SNP.PP
## s46  3.197219e-06
## s47  5.579160e-06
## s48  3.075858e-06
## s49  3.609154e-06
## s50  4.876818e-06
## s110 6.100014e-02

The SNP.PP column shows the posterior probability that exactly that SNP is causal. Note the last line in this data.frame does not correspond to a SNP, but to the null model, that no SNP is causal.

Finally, if you do have full genotype data as here, while this is a fast method for fine mapping, it can be sensible to consider multiple causal variant models too. One package that allows you to do this is GUESSFM, described in⁵

(Approximate) Bayes Factor colocalisation analyses

Introduction

The idea behind the ABF analysis is that the association of each trait with SNPs in a region may be summarised by a vector of 0s and at most a single 1, with the 1 indicating the causal SNP (so, assuming a single causal SNP for each trait). The posterior probability of each possible configuration can be calculated and so, crucially, can the posterior probabilities that the traits share their configurations. This allows us to estimate the support for the following cases:

\(H_0\): neither trait has a genetic association in the region
\(H_1\): only trait 1 has a genetic association in the region
\(H_2\): only trait 2 has a genetic association in the region
\(H_3\): both traits are associated, but with different causal variants
\(H_4\): both traits are associated and share a single causal variant

The basic `coloc.abf` function

The function coloc.abf is ideally suited to the case when only summary data are available, and requires, for each trait, either:

p values for each SNP
each SNP’s minor allele frequency
sample size
ratio of cases:controls (if using a case-control trait)

or:

regression coefficients for each SNP
variance of these regression coefficients.

If regression coefficients and their variance are available, please use these, but we can also approximate Bayes Factors from p values and minor allele frequencies, although, note, such approximation can be less accurate when imputed data are used. NB, you can mix and match, depending on the data available from each study.

A wrapper function is avalailable to run all these steps but we will first generate the p values manually to give all the details. We use the snpStats library from Bioconductor to calculate the p values quickly. We will also calculate the coefficients and their standard errors to compare the two approaches (but you should use only one - the coefficients if available, otherwise the p values).

if(requireNamespace("snpStats")) {
library(snpStats)

Y1 <- data@df1$Y
Y2 <- data@df2$Y

X1 <- new("SnpMatrix",as.matrix(data@df1[,-1]))
X2 <- new("SnpMatrix",as.matrix(data@df2[,-1]))

p1 <- snpStats::p.value(single.snp.tests(phenotype=Y1, snp.data=X1),df=1)
p2 <- snpStats::p.value(single.snp.tests(phenotype=Y2, snp.data=X2),df=1)

maf <- col.summary(X2)[,"MAF"]

get.beta <- function(...) {
   tmp <- snpStats::snp.rhs.estimates(..., family="gaussian")
   beta <- sapply(tmp,"[[","beta")
   varbeta <- sapply(tmp, "[[", "Var.beta")
   return(list(beta=beta,varbeta=varbeta))
}
b1 <- get.beta(Y1 ~ 1, snp.data=X1)
b2 <- get.beta(Y2 ~ 1, snp.data=X2)
} else {
Y1 <- data@df1$Y
Y2 <- data@df2$Y

X1 <- as.matrix(data@df1[,-1])
X2 <- as.matrix(data@df2[,-1])

tests1 <- lapply(1:ncol(X1), function(i) summary(lm(Y1 ~ X1[,i]))$coefficients[2,])
tests2 <- lapply(1:ncol(X2), function(i) summary(lm(Y2 ~ X2[,i]))$coefficients[2,])

p1 <- sapply(tests1,"[",4)
p2 <- sapply(tests2,"[",4)

maf <- colMeans(X2)/2

get.beta <- function(x) {
   beta <- sapply(x,"[",1)
   varbeta <- sapply(x, "[", 2)^2
   return(list(beta=beta,varbeta=varbeta))
}
b1 <- get.beta(tests1)
b2 <- get.beta(tests2)
}

## coercing object of mode  numeric  to SnpMatrix
## coercing object of mode  numeric  to SnpMatrix

Note that we are using the second dataset in that case to compute the minor allele frequencies. This is unlikely to make any significant difference but one could have used dataset 1 instead. It is now possible to compute the probabilities of interest.

With coefficients, and std. deviation of Y known, we have the most accurate inference. Note that if you have a case-control trait, set type=“cc” and supply s, the proportion of samples in the dataset that are cases (from which sdY can be derived), instead of giving sdY directly.

my.res <- coloc.abf(dataset1=list(beta=b1$beta, varbeta=b1$varbeta, N=nrow(X1),sdY=sd(Y1),type="quant"),
                    dataset2=list(beta=b2$beta, varbeta=b2$varbeta, N=nrow(X2),sdY=sd(Y2),type="quant"),
                    MAF=maf)

## PP.H0.abf PP.H1.abf PP.H2.abf PP.H3.abf PP.H4.abf 
##  2.92e-05  4.47e-04  6.73e-05  3.08e-05  9.99e-01 
## [1] "PP abf for shared variant: 99.9%"

print(my.res[[1]])

##        nsnps    PP.H0.abf    PP.H1.abf    PP.H2.abf    PP.H3.abf 
## 5.000000e+01 2.915837e-05 4.466024e-04 6.726224e-05 3.079177e-05 
##    PP.H4.abf 
## 9.994262e-01

When std. deviation of Y is unknown, coloc will try and estimate sdY from variance of beta and MAF, but this can be noisy, particularly for small datasets, and will produce a warning:

my.res <- coloc.abf(dataset1=list(beta=b1$beta, varbeta=b1$varbeta, N=nrow(X1),type="quant"),
                    dataset2=list(beta=b2$beta, varbeta=b2$varbeta, N=nrow(X2),type="quant"),
                    MAF=maf)

## Warning in sdY.est(d$varbeta, d$MAF, d$N): estimating sdY from maf and
## varbeta, please directly supply sdY if known

## Warning in sdY.est(d$varbeta, d$MAF, d$N): estimating sdY from maf and
## varbeta, please directly supply sdY if known

## PP.H0.abf PP.H1.abf PP.H2.abf PP.H3.abf PP.H4.abf 
##  2.54e-06  1.48e-04  1.75e-05  2.00e-05  1.00e+00 
## [1] "PP abf for shared variant: 100%"

print(my.res[[1]])

##        nsnps    PP.H0.abf    PP.H1.abf    PP.H2.abf    PP.H3.abf 
## 5.000000e+01 2.535953e-06 1.476540e-04 1.751478e-05 1.997285e-05 
##    PP.H4.abf 
## 9.998123e-01

When only p values are available, we can use the form:

my.res <- coloc.abf(dataset1=list(pvalues=p1,N=nrow(X1),type="quant"),
                    dataset2=list(pvalues=p2,N=nrow(X2),type="quant"),
                    MAF=maf)

## PP.H0.abf PP.H1.abf PP.H2.abf PP.H3.abf PP.H4.abf 
##  1.01e-05  2.61e-04  4.00e-05  2.71e-05  1.00e+00 
## [1] "PP abf for shared variant: 100%"

print(my.res[[1]])

##        nsnps    PP.H0.abf    PP.H1.abf    PP.H2.abf    PP.H3.abf 
## 5.000000e+01 1.014723e-05 2.605269e-04 3.998959e-05 2.705757e-05 
##    PP.H4.abf 
## 9.996623e-01

Here, as we have simulated full genotype data in snpStats format, we can use the wrapper function coloc.abf.datasets() to combine all the steps shown above.

ct.abf <- coloc.abf.datasets(data@df1, data@df2, response1="Y", response2="Y",
                             type1="quant", type2="quant")

## coercing object of mode  numeric  to SnpMatrix
## coercing object of mode  numeric  to SnpMatrix
## PP.H0.abf PP.H1.abf PP.H2.abf PP.H3.abf PP.H4.abf 
##  1.08e-05  2.61e-04  4.25e-05  2.72e-05  1.00e+00 
## [1] "PP abf for shared variant: 100%"

Proportional testing

The code below first prepares a principal component object by combining the genotypes in the two dataset, then models the most informative components (the minimum set required to capture 80% of the genetic variation) in each dataset, before finally testing whether there is colocalisation between these models.

## run a coloc with pcs
pcs <- pcs.prepare(data@df1[,-1], data@df2[,-1])
pcs.1 <- pcs.model(pcs, group=1, Y=data@df1[,1], threshold=0.8)

## selecting 38 components out of 50 to capture 0.8094953 of total variance.

pcs.2 <- pcs.model(pcs, group=2, Y=data@df2[,1], threshold=0.8)

## selecting 38 components out of 50 to capture 0.8094953 of total variance.

ct.pcs <- coloc.test(pcs.1,pcs.2)

The plot shows the estimated coefficients for each principal component modeled for traits 1 and 2 on the x and y axes, with circles showing the 95% confidence region. The points lie close to the line through the origin, which supports a hypothesis of colocalisation.

A little more information is stored in the ct.pcs object:

ct.pcs

##           eta.hat         chisquare                 n p.value.chisquare 
##         1.0346897        35.3484010        38.0000000         0.5465861

str(summary(ct.pcs))

##           eta.hat         chisquare                 n p.value.chisquare 
##         1.0346897        35.3484010        38.0000000         0.5465861 
##  Named num [1:4] 1.035 35.348 38 0.547
##  - attr(*, "names")= chr [1:4] "eta.hat" "chisquare" "n" "p.value.chisquare"

The best estimate for the coefficient of proportionality, \(\hat{\eta}\), is 1.13, and the null hypothesis of colocalisation is not rejected with a chisquare statistic of 5.27 based on 7 degrees of freedom (\(n-1\) where the \(n\) is the number of components tested, and one degree of freedom was used in estimating \(\eta\)), giving a p value of 0.63. The summary() method returns a named vector of length 4 containing this information.

If more information is needed about \(\eta\), then this is available if the bayes argument is supplied:

ct.pcs.bayes <- coloc.test(pcs.1,pcs.2, bayes=TRUE)

## ............

ci(ct.pcs.bayes)

## $eta.mode
## [1] 1.039993
## 
## $lower
## [1] 0.8445606
## 
## $upper
## [1] 1.274574
## 
## $level.observed
## [1] 0.9390264
## 
## $interior
## [1] TRUE

Using Bayes Factors to compare specific values of \(\eta\)

It may be that specific values of \(\eta\) are of interest. For example, when comparing eQTLs in two tissues, or when comparing risk of two related diseases, the value \(\eta=1\) is of particular interest. In proportional testing, we can use Bayes Factors to compare the support for different values of \(\eta\). Eg

## compare individual values of eta
ct.pcs <- coloc.test(pcs.1,pcs.2, bayes.factor=c(-1,0,1))

## ............

bf(ct.pcs)

##           values.against
## values.for           -1            0            1
##         -1 1.000000e+00 4.495343e-33 9.816601e-73
##         0  2.224524e+32 1.000000e+00 2.183727e-40
##         1  1.018683e+72 4.579328e+39 1.000000e+00

## compare ranges of eta
ct.bma <- coloc.bma(data@df1, data@df2, 
                    family1="gaussian", family2="gaussian",
                    bayes.factor=list(c(-0.1,1), c(0.9,1.1)))

## Dropped 0 of 50 SNPs due to LD: r2 > 0.95.

## 50 SNPs remain.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Restricting model space.

## 48 SNPs have single SNP posterior probabilities < 0.01

## Models containing only these SNPs will not be explored.

## Fitting 97 multi SNP models to dataset 1

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Fitting 97 multi SNP models to dataset 2

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Warning in E1ot$app1 - E0ot$app1: Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

## Averaging coloc testing over 3 models with posterior probabilities >= 1e-229 
## ...

bf(ct.bma)

##              values.against
## values.for    c(-0.1, 1) c(0.9, 1.1)
##   c(-0.1, 1)     1.00000  0.08537724
##   c(0.9, 1.1)   11.71272  1.00000000

The difference between proportional and ABF approaches

So what are the differences between proportional and ABF approaches? Which should you choose?

Well, if you only have p values, then you must use ABF. But be aware that a single causal variant is assumed, and that for accurate inference, the causal variant needs to be included, so either very dense or imputed genotyping data is needed. The ABF approach has another big advantage over proportional testing: being Bayesian, it allows you to evaluate support for each hypothesis, whereas with proportional testing, when the null of colocalisation is not rejected, you cannot be sure whether this reflects true colocalisation or a lack of power. The proportional approach is much less affected by spareness of genotyping data, with power only slightly decreased, but type 1 error rates unaffected.

The behaviour of the two approaches only really differs when there are two or more causal variants. When both are shared, proportional testing has the same type 1 error rate, and whilst ABF tends to still favour \(H_4\) (shared variant), it tends to put some more weight on \(H_3\) (distinct variants). But when at least one is specific to one trait and at least one is shared, their behaviour differs more substantially. Of course, neither approach was designed with this case in mind, and so there is no “right” answer, but it is instructive to understand their expected behaviour. Proportional testing tends to reject colocalisation, whilst ABF tends to favour sharing. Until the methods are extended to incorporate this specific case, it can be useful to compare the two approaches when complete and dense genotyping data are available. When the results differ, we have tended to identify a combination of shared and distinct causal variants. The ABF approach can still be applied in this case, if p values are available conditioning on the strongest signal, as demonstrated in our paper ⁴.

You can see more about the ABF approach on this blogpost.

Footnotes

¹http://www.ncbi.nlm.nih.gov/pubmed/19039033

²http://onlinelibrary.wiley.com/doi/10.1002/gepi.21765/abstract

³http://www.ncbi.nlm.nih.gov/pubmed/18642345

⁴http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1004383

⁵http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1005272

Coloc: a package for colocalisation analyses

Chris Wallace

2019-07-07

Table of Contents

A brief outline of colocalisation analysis

Usage

An aside: fine mapping under a single causal variant assumption

(Approximate) Bayes Factor colocalisation analyses

Introduction

The basic `coloc.abf` function

Proportional testing

Using Bayes Factors to compare specific values of \(\eta\)

The difference between proportional and ABF approaches

Footnotes

Contents

Coloc: a package for colocalisation analyses

Chris Wallace

2019-07-07

Table of Contents

A brief outline of colocalisation analysis

Usage

An aside: fine mapping under a single causal variant assumption

(Approximate) Bayes Factor colocalisation analyses

Introduction

The basic coloc.abf function

Proportional testing

Using Bayes Factors to compare specific values of \(\eta\)

The difference between proportional and ABF approaches

Footnotes

Contents

The basic `coloc.abf` function