RVS 1.17.0

- 1 Introduction
- 2 Setting up Pedigree Data
- 3 Calculating Sharing Probabilities
- 4 Joint analysis of multiple variants
- 5 Partial sharing test
- 6 Appendix
- References

Rare Variant Sharing (RVS) implements tests of association and linkage between rare genetic variant genotypes and a dichotomous phenotype, e.g. a disease status, in family samples (Sherman et al. (2018)). The tests are based on probabilities of rare variant sharing by relatives under the null hypothesis of absence of linkage and association between the rare variants and the phenotype and apply to single variants or multiple variants in a region (e.g. gene-based test).

For this example experiment we will consider four family types. A pair of first
cousins, a pair of second cousins, a triple of first cousins, and a triple of
second cousins. *RVS* comes with several example pedigrees and these four types
can be found in the *samplePedigrees* list.

```
data(samplePedigrees)
# store the pedigrees
fam_type_A <- samplePedigrees$firstCousinPair
fam_type_B <- samplePedigrees$secondCousinPair
fam_type_C <- samplePedigrees$firstCousinTriple
fam_type_D <- samplePedigrees$secondCousinTriple
# re-label the family ids for this example
fam_type_A$famid <- rep('SF_A', length(fam_type_A$id))
fam_type_B$famid <- rep('SF_B', length(fam_type_B$id))
fam_type_C$famid <- rep('SF_C', length(fam_type_C$id))
fam_type_D$famid <- rep('SF_D', length(fam_type_D$id))
```

In order to see the pedigree structure we can use the plot function provided by
the *kinship2* package. In this family we have three second cousins that have
been sequenced.

`plot(fam_type_D)`

In the case of a single variant seen across multiple families, we can compute
the individual sharing probabilities with *RVsharing*, but the sharing
probabilities can no longer be interpreted as a p-value for the sharing pattern
of the variant across the families. The function *multipleFamilyPValue* can be
used to compute the p-value which is defined as the sum of all sharing
probabilities across families at most as large as the sharing probability
observed.

```
# compute the sharing probabilities for all families
fams <- list(fam_type_A, fam_type_B, fam_type_C, fam_type_D)
sharing_probs <- suppressMessages(RVsharing(fams))
signif(sharing_probs, 3)
```

```
## SF_A SF_B SF_C SF_D
## 0.06670 0.01590 0.01180 0.00134
```

```
# compute p-value for this sharing pattern
sharing_pattern <- c(TRUE, TRUE, FALSE, FALSE)
names(sharing_pattern) <- names(sharing_probs)
multipleFamilyPValue(sharing_probs, sharing_pattern)
```

`## [1] 0.002125543`

The *sharing_pattern* vector indicates whether or not the variant is observed
in all sequenced subjects.

The function *multipleVariantPValue* generalizes *multipleFamilyPValue*
across multiple variants. This function takes a *SnpMatrix* instead of a
specific sharing pattern. The behavior of this function could be achieved
by converting every column of a *SnpMatrix* into a sharing pattern across
families and applying *multipleFamilyPValue* across the columns.

The first step is reading in the genotype data. See the *Data Input*
vignette
in the *snpStats* package for examples using different file types. Here we use a
pedigree file in the LINKAGE format. See here for an example of reading genotypes data from a Variant Call Format (VCF) file.

```
pedfile <- system.file("extdata/sample.ped.gz", package="RVS")
sample <- snpStats::read.pedfile(pedfile, snps=paste('variant', LETTERS[1:20], sep='_'))
```

In this data set we have 3 copies of each family type. The sharing probabilities for this set of families are:

```
A_fams <- lapply(1:3, function(i) samplePedigrees$firstCousinPair)
B_fams <- lapply(1:3, function(i) samplePedigrees$secondCousinPair)
C_fams <- lapply(1:3, function(i) samplePedigrees$firstCousinTriple)
D_fams <- lapply(1:3, function(i) samplePedigrees$secondCousinTriple)
fams <- c(A_fams, B_fams, C_fams, D_fams)
famids <- unique(sample$fam$pedigree)
for (i in 1:12)
{
fams[[i]]$famid <- rep(famids[i], length(fams[[i]]$id))
}
sharingProbs <- suppressMessages(RVsharing(fams))
signif(sharingProbs, 3)
```

```
## SF_A1 SF_A2 SF_A3 SF_B1 SF_B2 SF_B3 SF_C1 SF_C2 SF_C3 SF_D1
## 0.06670 0.06670 0.06670 0.01590 0.01590 0.01590 0.01180 0.01180 0.01180 0.00134
## SF_D2 SF_D3
## 0.00134 0.00134
```

When we call the function on the genotypes from a *snpMatrix* as follows, it
converts them into a sharing pattern assuming the rare variant is the allele
with the lowest frequency in the family sample:

`result <- multipleVariantPValue(sample$genotypes, sample$fam, sharingProbs)`

`## Ignoring 0 variants not present in any subject`

`signif(result$pvalues, 3)`

```
## variant_A variant_B variant_C variant_D variant_E variant_F variant_G variant_H
## 4.44e-03 6.67e-02 1.00e+00 1.00e+00 2.52e-04 1.59e-02 1.00e+00 1.00e+00
## variant_I variant_J variant_K variant_L variant_M variant_N variant_O variant_P
## 1.38e-04 1.18e-02 1.00e+00 1.00e+00 1.80e-06 1.34e-03 1.00e+00 1.00e+00
## variant_Q variant_R variant_S variant_T
## 4.93e-05 1.00e+00 4.44e-05 1.00e+00
```

The argument *minorAllele* can be used to specify which allele of each variant is the rare variant.

Correcting for multiple testing reduces the cutoff below which a p-value is
considered significant. With a large set of variants, it will be impossible
to reject the null hypothesis for many of the variants with limited information
(e.g. a variant seen in a single small family) because the smallest p-value
achievable for that variant is larger than the cutoff.
*multipleVariantPValue* provides a filtering option based on the potentia
p-values. Potential p-values are the lower bound on the p-value for each
variant (Bureau et al. (2014)). In this example we omit any variant whose potential
p-values exceeds the cutoff obtained by applying the Bonferroni correction for
the number of variants with a sufficiently low potential p-value. In this way,
we both increase the p-value cutoff (and hence the power of the test) while
maintaining the family-wise error rate at 0.05 and reduce computation time.

`result <- multipleVariantPValue(sample$genotypes, sample$fam, sharingProbs, filter='bonferroni', alpha=0.05)`

`## Ignoring 0 variants not present in any subject`

The effects of this filter can be seen here. The blue curves show the potential p-value, sorted from most to least significant. It is important to note that these potential p-values are independent of the actual sharing pattern among affected subjects, and therefore of the subsequent testing of variant sharing. The red curve shows the Bonferroni cut-off depending on how many variants are tested. Only 11 variants are included, as sharing here could produce a sharing p-value below the Bonferroni cutoff. The black points show the observed sharing p-values, with six variants being significant after multiple comparisons correction.

```
pvals <- result$pvalues
ppvals <- result$potential_pvalues
ppvals_sub <- ppvals[names(pvals)] # subset potential p-values
plot(-log10(ppvals[order(ppvals)]), ylab="-log10 p-value", col="blue", type="l", xaxt="n", xlab="variants", ylim=c(0,8))
xlabel <- sapply(names(ppvals)[order(ppvals)], function(str) substr(str, nchar(str), nchar(str)))
axis(1, at=1:length(ppvals), labels=xlabel)
points(-log10(pvals[order(ppvals_sub)]), pch=20, cex=1.3)
bcut <- 0.05/(1:20)
lines(1:20,-log10(bcut),col="red",type="b",pch=20)
```

When the minor allele frequency (MAF) is known in the population, then an
exact sharing probability can be calculated using the *alleleFreq* parameter.
Here we analyze the sensitivity of our p-values to the population MAF, using
the 3 most significant variants. Note that variants which don’t reach their
potential p-values, indicating some families only have partial sharing, are
much more sensitive to the MAF.

```
# calculate p-values for each MAF
freq <- seq(0,0.05,0.005)
variants <- names(sort(result$pvalues))[1:3]
pvals <- matrix(nrow=length(freq), ncol=length(variants))
pvals[1,] = sort(result$pvalues)[1:3]
for (i in 2:length(freq))
{
sharingProbs <- suppressMessages(RVsharing(fams, alleleFreq=freq[i]))
pvals[i,] <- multipleVariantPValue(sample$genotypes[,variants], sample$fam, sharingProbs)$pvalues
}
```

```
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
```

```
colnames(pvals) <- variants
# plot p-values as a function of MAF
plot(NULL, xlim=c(min(freq),max(freq)), ylim=c(0,max(pvals)), type='l',
xlab="minor allele frequency", ylab="p-value",
main="sensitivity of p-value to allele frequency in three variants")
lines(freq, pvals[,1], col="black")
lines(freq, pvals[,2], col="red")
lines(freq, pvals[,3], col="blue")
legend(min(freq), max(pvals), legend=colnames(pvals), col=c("black", "red", "blue"), lty=1)
```

The power of single variant analyses is limited due to the small number of families where a rare variant is seen (often a single family). Even if no individual variant has a significant p-value, it is possible for multiple variants considered across multiple families to exhibit an unusual amount of sharing. The procedure to test multiple rare variants depends whether the variants are far apart or close together in a short genomic region, spanning a single gene for instance.

The *enrichmentPValue* function can compute a single p-value for all
variants seen in all families assuming the variants are independent. This assumption is reasonable when variants are sufficiently far apart to be unlinked, such as rare deletions scattered over the whole genome as analyzed by Fu et al. (2017). The computation is implemented using a binary tree algorithm described by Fu et al. (2017). When calculating this p-value, note that a very
small p-value may result in a very long computation time. Because of this, we
can pass a minimum p-value threshold, where the greater of this threshold and
the actual p-value will be returned.

`enrichmentPValue(sample$genotypes, sample$fam, sharingProbs, 0.001)`

`## Ignoring 0 variants not present in any subject`

`## [1] 0.001`

Joint analysis of rare variants within a gene (typically single nucleotide variants and short indels, possibly filtered based on functional annotations) is another approach to increase statistical power. Here the assumption of independence of rare variants does not hold when variants are seen in the same family, and the solution described by Bureau et al. (2018) and implemented in the *RVgene* function is to keep the variant with the sharing pattern having the lowest probability (usually the variant shared by the largest number of affected relatives in the family). The gene-based analysis with the *RVgene* function is illustrated in section 4.2 along with another new feature: the partial sharing test.

Phenocopies, diagnosis error and intra-familial genetic heterogeneity in complex disorders result in disease susceptibility variants being shared by a subset of affected subjects. In order to detect such causal variants, a partial sharing test was defined by Bureau et al. (2018) where the p-value is the probability of sharing events as or more extreme as the observed event. A more extreme sharing event is defined as having lower probability and involving more carriers of the variant.

In order to perfore the partial sharing test, the *RVgene* function requires the lists *pattern.prob.list* of vectors of sharing probabilities and *N.list* of number of carriers for all possible affected carrier subsets in each family in the sample being analyzed. The arguments of the *RVsharing* function allowing the computation of sharing probabilities by a subset of affected subjects are described here. The elements of both of these lists must have the same names as the pedigree objects in the *ped.listfams* argument. When all affected subjecs in a family are final descendants, the sharing probabilities and number of carriers for all subsets can be generated automatically. Here is an exanple with three second cousins:

```
carriers = c(15,16,17)
carrier.sets = list()
for (i in length(carriers):1)
carrier.sets = c(carrier.sets, combn(carriers,i,simplify=FALSE))
fam15157.pattern.prob = sapply(carrier.sets,function (vec) RVsharing(samplePedigrees$secondCousinTriple,carriers=vec))
```

`## Probability subjects 15 16 17 among 15 16 17 share a rare variant: 0.001342`

`## Probability subjects 15 16 among 15 16 17 share a rare variant: 0.009396`

`## Probability subjects 15 17 among 15 17 16 share a rare variant: 0.009396`

`## Probability subjects 16 17 among 16 17 15 share a rare variant: 0.009396`

`## Probability subjects 15 among 15 16 17 share a rare variant: 0.3235`

`## Probability subjects 16 among 16 15 17 share a rare variant: 0.3235`

`## Probability subjects 17 among 17 15 16 share a rare variant: 0.3235`

`fam15157.N = sapply(carrier.sets,length)`

When the *splitPed* option is *TRUE*, the generation of all carrier subsets is performed within the *RVsharing* function, which then returns the vector of sharing probabilities for all subsets. So the following code is equivalent to the *sapply* of *RVsharing* above:

`fam15157.pattern.prob = RVsharing(samplePedigrees$secondCousinTriple,splitPed=TRUE)`

```
## Founder 1 subped size 17
## Founder 4 subped size 5
## Founder 6 subped size 5
## Founder 8 subped size 5
## Founder 10 subped size 3
## Founder 12 subped size 3
## Founder 14 subped size 3
```

`## Probability every subset of subjects among 15 16 17 share a rare variant:`

`## [1] 0.001342 0.009396 0.009396 0.009396 0.323500 0.323500 0.323500`

While this code applies to any configuration of affected final descendants, symmetries in the relationships of these third cousins results in equal sharing probabilities for multiple subsets. Subsets with the same probabilities are equivalent, and the optional argument *nequiv.list* can be used to indicate the number of equivalent subset for each sharing probability. While shorter vectors in *pattern.prob.list* and *N.list* result in more efficient computation, identification of the equivalent subsets is not easily automated, and will usually require custom code for each pedigree in a sample. With three second cousins we can use:

```
fam15157.pattern.prob = c(RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15,16,17)),
RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15,16)),
RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15)))
```

`## Probability subjects 15 16 17 among 15 16 17 share a rare variant: 0.001342`

`## Probability subjects 15 16 among 15 16 17 share a rare variant: 0.009396`

`## Probability subjects 15 among 15 16 17 share a rare variant: 0.3235`

```
fam15157.N = 3:1
fam15157.nequiv = c(1,3,3)
```

It is then easy to check that the distribution sums to 1:

`sum(fam15157.pattern.prob*fam15157.nequiv)`

`## [1] 1`

When some affected subjects are not final descendants, some subsets are incompatible with a variant being IBD among carriers. Assume individual 3, the grand-father of subject 15 in family 15157, is also affected and his genotype is available.

```
fam15157 <- samplePedigrees$secondCousinTriple
fam15157$affected[3] = 1
plot(fam15157)
```

Then the carrier subsets (15,16,17), (15,16) and (15,17) involving subject 15 but not 3 are incompatible with sharing IBD and must be removed from the list of subsets. The code then becomes:

```
carriers = c(3,15,16,17)
carrier.sets = list()
for (i in length(carriers):1)
carrier.sets = c(carrier.sets, combn(carriers,i,simplify=FALSE))
carrier.sets
```

```
## [[1]]
## [1] 3 15 16 17
##
## [[2]]
## [1] 3 15 16
##
## [[3]]
## [1] 3 15 17
##
## [[4]]
## [1] 3 16 17
##
## [[5]]
## [1] 15 16 17
##
## [[6]]
## [1] 3 15
##
## [[7]]
## [1] 3 16
##
## [[8]]
## [1] 3 17
##
## [[9]]
## [1] 15 16
##
## [[10]]
## [1] 15 17
##
## [[11]]
## [1] 16 17
##
## [[12]]
## [1] 3
##
## [[13]]
## [1] 15
##
## [[14]]
## [1] 16
##
## [[15]]
## [1] 17
```

```
carrier.sets = carrier.sets[-c(5,9,10)]
fam15157.pattern.prob = sapply(carrier.sets,function (vec) RVsharing(fam15157,carriers=vec,useAffected=TRUE))
```

`## Probability subjects 3 15 16 17 among 3 15 16 17 share a rare variant: 0.001121`

`## Probability subjects 3 15 16 among 3 15 16 17 share a rare variant: 0.007848`

`## Probability subjects 3 15 17 among 3 15 16 17 share a rare variant: 0.007848`

`## Probability subjects 3 16 17 among 3 15 16 17 share a rare variant: 0.003363`

`## Probability subjects 3 15 among 3 15 16 17 share a rare variant: 0.05493`

`## Probability subjects 3 16 among 3 15 16 17 share a rare variant: 0.02354`

`## Probability subjects 3 17 among 3 15 16 17 share a rare variant: 0.02354`

`## Probability subjects 16 17 among 3 15 16 17 share a rare variant: 0.004484`

`## Probability subjects 3 among 3 15 16 17 share a rare variant: 0.1648`

`## Probability subjects 15 among 3 15 16 17 share a rare variant: 0.2152`

`## Probability subjects 16 among 3 15 16 17 share a rare variant: 0.2466`

`## Probability subjects 17 among 3 15 16 17 share a rare variant: 0.2466`

`fam15157.N = sapply(carrier.sets,length)`

Notice the use of the option *useAffected=TRUE* with affected subjects who are not final descendants. Again, one can check that the distribution sums to 1:

`sum(fam15157.pattern.prob)`

`## [1] 1`

Precomputed sharing probabilities and numbers of carriers can be used directly to obtain p-values of observed sharing events, by summing the probability of all events as or more extreme as the one observed (both in terms of sharing probability and number of carriers), i.e. this is a one-sided exact test. For instance, if subjects 3, 16 and 17 share a rare variant, the probability of that event is

`pobs = RVsharing(fam15157,carriers=c(3,16,17),useAffected=TRUE)`

`## Probability subjects 3 16 17 among 3 15 16 17 share a rare variant: 0.003363`

The p-value of that sharing event is then:

`sum(fam15157.pattern.prob[fam15157.pattern.prob<=pobs & fam15157.N >= 3])`

`## [1] 0.004484305`

The *RVgene* function enables these computations with more than one family harboring the same or different variants. Once the vectors of sharing probabilities and number of carriers have been computed for all families in the sample, they need to be stored in lists. We return to the original second cousin triple family and add a first and second cousin triple family. Then we create the lists of pattern probabilities, number of equivalent subsets and number of carriers in the subsets.

```
fam15157.pattern.prob = c(RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15,16,17)),
RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15,16)),
RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15)))
```

`## Probability subjects 15 16 17 among 15 16 17 share a rare variant: 0.001342`

`## Probability subjects 15 16 among 15 16 17 share a rare variant: 0.009396`

`## Probability subjects 15 among 15 16 17 share a rare variant: 0.3235`

```
fam15157.N = 3:1
fam15157.nequiv = c(1,3,3)
fam28003.pattern.prob = c(RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(36,104,110)),
RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(36,104)),
RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(104,110)),
RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(36)),
RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(104)))
```

`## Probability subjects 36 104 110 among 36 104 110 share a rare variant: 0.00277`

`## Probability subjects 36 104 among 36 104 110 share a rare variant: 0.00831`

`## Probability subjects 104 110 among 104 110 36 share a rare variant: 0.04155`

`## Probability subjects 36 among 36 104 110 share a rare variant: 0.3352`

`## Probability subjects 104 among 104 36 110 share a rare variant: 0.3019`

```
fam28003.N = c(3,2,2,1,1)
fam28003.nequiv = c(1,2,1,1,2)
ex.pattern.prob.list = list("15157"=fam15157.pattern.prob,"28003"=fam28003.pattern.prob)
ex.nequiv.list = list("15157"=fam15157.nequiv,"28003"=fam28003.nequiv)
ex.N.list = list("15157"=fam15157.N,"28003"=fam28003.N)
```

We now turn to the analysis of variants observed in the simulated genomic sequence of the gene *PEAR1* in a sample of related affected subjects. The processing of the sequence data results in Variant Call Format (VCF) files, which can be read into R with the function *readVcf* from the *variantAnnotation* package. Two *VCF* objects obtained with *readVcf* from VCF files of sequence data for the second cousin triple and first and second cousin triple families are contained in the *famVCF* data. These VCF files are converted to *snpMatrix* objects using the *genotypeToSnpMatrix* function.

```
data(famVCF)
fam15157.snp = VariantAnnotation::genotypeToSnpMatrix(fam15157.vcf)
```

`## non-single nucleotide variations are set to NA`

`## Warning in .local(x, ...): non-diploid variants are set to NA`

`fam28003.snp = VariantAnnotation::genotypeToSnpMatrix(fam28003.vcf)`

`## non-single nucleotide variations are set to NA`

`## Warning in .local(x, ...): non-diploid variants are set to NA`

*RVgene* requires lists of the *snpMatrix* and *pedigree* objects for these two families. The names given to the elements of these lists are not used by *RVgene* and are thus arbitrary. Family IDs are extracted from the *famid* element of the *pedigree* objects. Please note that currently *RVgene* does not accept a *pedigreeList*, but only a plain list of *pedigree* objects.

```
ex.SnpMatrix.list = list(fam15157=fam15157.snp$genotypes,fam28003=fam28003.snp$genotypes)
ex.ped.obj = list(fam15157=samplePedigrees$secondCousinTriple,fam28003=samplePedigrees$firstAndSecondCousinsTriple)
```

In the sequence segment, one can specify which variants are rare and possibly satisfy other filtering criteria (e.g. coding variants) using the *sites* argument. Here, we will focus on two sites: 92 where the three second cousins of family 15157 share the rare allele and 119 where the two first cousins of family 28003 share the rare allele but not their second cousin.

```
sites = c(92,119)
ex.SnpMatrix.list[["fam15157"]][,sites[1]]@.Data
```

```
## rs187756653
## 1 00
## 2 00
## 3 00
## 4 00
## 5 00
## 6 00
## 7 00
## 8 00
## 9 00
## 10 00
## 11 00
## 12 00
## 13 00
## 14 00
## 15 02
## 16 02
## 17 02
```

`ex.SnpMatrix.list[["fam28003"]][,sites[2]]@.Data`

```
## rs199628333
## 3 00
## 4 00
## 6 00
## 7 00
## 11 00
## 12 00
## 13 00
## 15 00
## 27 00
## 28 00
## 36 01
## 103 00
## 104 02
## 109 00
## 110 02
```

Finally, the call to *RVgene* returns the P-value of the exact rare variant sharing test allowing for sharing by a subset of affected subjects (p), the P-value of the exact rare variant sharing test requiring sharing by all affected subjects (pall) and the minimum achievable p-value if all affected subjects were carriers of a rare variant (potentialp).

`RVgene(ex.SnpMatrix.list,ex.ped.obj,sites,pattern.prob.list=ex.pattern.prob.list,nequiv.list=ex.nequiv.list,N.list=ex.N.list,type="count")`

```
## $p
## [1] 0.000159884
##
## $pall
## [1] 0.001342282
##
## $potentialp
## [1] 3.718232e-06
```

In this case, we assume the variant is rare enough so that the probability of
more than one founder introducing it to the pedigree is negligible. This is
the default scenario for *RVsharing*.

We define the following random variables:

*\(C_i\): Number of copies of the RV received by subject \(i\),

*\(F_j\): Indicator variable that founder \(j\) introduced one copy of the RV into the pedigree,

For a set of \(n\) subjects descendants of \(n_f\) founders we want to compute the probability \[\begin{eqnarray*} P[\mbox{RV shared}] &=& P[C_1 = \dots = C_n = 1 | C_1 + \dots + C_n \geq 1] \nonumber \\[0.5em] &=& \frac{P[C_1 = \dots = C_n = 1 ]}{P[C_1 + \dots + C_n \geq 1]} \nonumber \\[0.5em] &=& \frac{\sum_{j=1}^{n_f} P[C_1 = \dots = C_n = 1 | F_j] P[F_j]} {\sum_{j=1}^{n_f} P[C_1 + \dots + C_n \geq 1 | F_j]P[F_j]}, \label{sharingp} \end{eqnarray*}\] where the expression on the third line results from our assumption of a single copy of that RV among all alleles present in the \(n_f\) founders. The probabilities \(P[F_j] = {1 \over n_f}\) cancel from the numerator and denominator.

By default, *RVsharing* will compute the probability that all of the final
descendants share the variant given that it is seen in at least one of them.
Final descendants are defined as subjects of the pedigree with no children.
This event can be customized with the *carriers* and *useAffected* arguments.

If the argument *carriers* is provided, then the probability of all carriers
having the variant given it is seen in at least one subject in the union of the final descendants and the carriers will be computed.

If the argument *useAffected* is TRUE and the pedigree has a slot for
*affected*, then the probability of all carriers having the variant given
it is seen in at least one affected will be computed.

These two arguments can be used individually or in combination, the only restriction is that carriers must be a subset of affected.

```
ped <- samplePedigrees$firstCousinTriple
ped$affected[9] <- 0
plot(ped)
```

`p <- RVsharing(ped)`

`## Probability subjects 9 10 11 among 9 10 11 share a rare variant: 0.01176`

`p <- RVsharing(ped, useAffected=TRUE)`

`## Probability subjects 10 11 among 10 11 share a rare variant: 0.06667`

`p <- RVsharing(ped, carriers=c(7,9,10))`

`## Probability subjects 7 9 10 among 7 9 10 11 share a rare variant: 0.01064`

`p <- RVsharing(ped, carriers=c(10,11), useAffected=TRUE)`

`## Probability subjects 10 11 among 10 11 share a rare variant: 0.06667`

*RVsharing* also allows for estimating sharing probabilities through Monte
Carlo simulation. The primary use of this feature is for calculating sharing
probabilities under non standard assumptions about the founders. However,
this feature is available for the standard assumptions as well. To run a
monte carlo simulation, specify all parameters as normal and additionally
provide the *nSim* parameter specifying how many simulations should be run.

`p <- RVsharing(samplePedigrees$firstCousinPair, alleleFreq=0.01)`

`## Probability subjects 7 8 among 7 8 share a rare variant: 0.07631`

`p <- RVsharing(samplePedigrees$firstCousinPair, alleleFreq=0.01, nSim=1e5)`

`## Probability subjects 7 8 among 7 8 share a rare variant: 0.07528`

This method allows for more complex relationships among the founders to be
given. *RVsharing* allows for a complete distribution among the founders to
be passed in as the parameter *founderDist*. This function should accept a
single argument, N, and should return a vector of length N with values in
{0,1,2} representing the number of copies of the variant each founder has.

```
# assumption that 1 founder introduces variant
fDist <- function(N) sample(c(rep(0,N-1), 1))
p <- RVsharing(samplePedigrees$firstCousinPair, nSim=1e5, founderDist=fDist)
```

`## Probability subjects 7 8 among 7 8 share a rare variant: 0.0672`

`p <- RVsharing(samplePedigrees$firstCousinPair)`

`## Probability subjects 7 8 among 7 8 share a rare variant: 0.06667`

Bureau, Alexandre, Ferdouse Begum, Margaret A Taub, Jacqueline Hetmanski, Margaret M Parker, Hasan Albacha-Hejazi, Alan F Scott, et al. 2018. “Inferring Disease Risk Genes from Sequencing Data in Multiplex Pedigrees Through Sharing of Rare Variants.” *Http://Biorxiv.org/Cgi/Content/Short/285874v1*.

Bureau, Alexandre, Samuel G Younkin, Margaret M Parker, Joan E Bailey-Wilson, Mary L Marazita, Jeffrey C Murray, Elisabeth Mangold, Hasan Albacha-Hejazi, Terri H Beaty, and Ingo Ruczinski. 2014. “Inferring Rare Disease Risk Variants Based on Exact Probabilities of Sharing by Multiple Affected Relatives.” *Bioinformatics (Oxford, England)* 30 (15): 2189–96. https://doi.org/10.1093/bioinformatics/btu198.

Fu, Jack, Terri H Beaty, Alan F Scott, Jacqueline Hetmanski, Margaret M Parker, Joan E Bailey Wilson, Mary L Marazita, et al. 2017. “Whole Exome Association of Rare Deletions in Multiplex Oral Cleft Families.” *Genetic Epidemiology* 41 (1): 61–69. https://doi.org/10.1002/gepi.22010.

Sherman, Thomas, Jack Fu, Robert B Scharpf, Alexandre Bureau, and Ingo Ruczinski. 2018. “Detection of rare disease variants in extended pedigrees using RVS,” November. https://doi.org/10.1093/bioinformatics/bty976.