Allelic frequencies and Hardy weinberg testing
Jerome Goudet
September 2, 2014
1 Allelic frequencies
Let’s start with a few ’words’ about binomial distributionFirst of all, we can reproduce the figure on slide 51
> #reproduces slide 51
> L<-function(p,x,n) choose(n,x)*p^x*(1-p)^(n-x)
> p<-0:1000/1000
> plot(p,L(p,0,4),type="l")
> lines(p,L(p,1,4),col="blue")
> lines(p,L(p,2,4),col="red")
> lines(p,L(p,3,4),col="orange")
> lines(p,L(p,4,4),col="green")
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
p
L(p,
0, 4
)
Easy to extend it to a larger n
> n<-10
> plot(p,L(p,0,n),type="l")
> for (i in 1:n) lines(p,L(p,i,n),col=i)
1
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
p
L(p,
0, n
)
> n<-10
> p<-0.1
> my.sd<-(p*(1-p)/n)^0.5
> (ci<-c(p-1.96*my.sd,p+1.96*my.sd)) #slide 54
[1] -0.08594193 0.28594193
Probability density
> n<-10
> p<-0.5
> dbinom(0:n,n,0.5)
[1] 0.0009765625 0.0097656250 0.0439453125 0.1171875000 0.2050781250
[6] 0.2460937500 0.2050781250 0.1171875000 0.0439453125 0.0097656250
[11] 0.0009765625
How to plot probability densities?
> n<-100
> p<-0.5
> #to illustrate plotting
> par(mfrow=c(2,2))
> plot(0:n,dbinom(0:n,n,p),type="h")
> plot(0:n,dbinom(0:n,n,p),type="p")
> plot(0:n,dbinom(0:n,n,p),type="l")
> plot(0:n,dbinom(0:n,n,p),type="h")
> lines(0:n,dnorm(0:n,mean=n*p,sd=(p*(1-p)*n)^.5))
2
0 20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0:n
dbin
om(0
:n, n
, p)
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0 20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0:n
dbin
om(0
:n, n
, p)
0 20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0:n
dbin
om(0
:n, n
, p)
0 20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0:n
dbin
om(0
:n, n
, p)
With fewer draws (fewer alleles):
> n<-10
> p<-0.5
> plot(0:n,dbinom(0:n,n,p),type="h")
> lines(0:n,dnorm(0:n,mean=n*p,sd=(p*(1-p)*n)^.5))
0 2 4 6 8 10
0.00
0.05
0.10
0.15
0.20
0.25
0:n
dbin
om(0
:n, n
, p)
and with a frequency different from 0.5
> n<-10
> p<-0.1
> plot(0:n,dbinom(0:n,n,p),type="h")
> lines(0:n,dnorm(0:n,mean=n*p,sd=(p*(1-p)*n)^.5))
3
0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
0:n
dbin
om(0
:n, n
, p)
Drawing samples from the binomial distribution:
> #?rbinom
> rbinom(10,100,0.5)
[1] 51 52 47 51 52 43 48 55 43 52
> sample(rep(0:1,each=5),replace=TRUE)
[1] 0 0 1 0 0 1 1 1 0 1
And below is a function to compute the different types of confidence intervalswe covered:
> comp.ci.binom<-function(p=0.5,n=1000,nboot=1000,digits=4){
+ #illustrates slides 55-59
+ x<-rbinom(1,n,p)
+ p.hat<-x/n
+ sd.p<-(p.hat*(1-p.hat)/n)^0.5
+ norm.ci<-round(c(p.hat-1.96*sd.p,p.hat+1.96*sd.p),digits=digits) #wald ci
+ exact.ci<-round(binom.test(x,n)$conf.int[1:2],digits=digits) #exact from binomial
+ bx<-numeric(nboot)
+ ox<-numeric(n);ox[1:x]<-1
+ for (i in 1:nboot) bx[i]<-sum(sample(ox,replace=TRUE))
+ boot.ci<-round(quantile(bx,c(0.025,0.975))/n,digits=digits) #empirical bootstrap ci
+ return(list(p.hat=p.hat,norm.ci=norm.ci,boot.ci=boot.ci,exact.ci=exact.ci))
+ }
> comp.ci.binom()
$p.hat
[1] 0.506
$norm.ci
[1] 0.475 0.537
$boot.ci
2.5% 97.5%
4
0.477 0.538
$exact.ci
[1] 0.4745 0.5374
> set.seed(13) #so that we all get the same values
> comp.ci.binom()
$p.hat
[1] 0.47
$norm.ci
[1] 0.4391 0.5009
$boot.ci
2.5% 97.5%
0.437 0.500
$exact.ci
[1] 0.4387 0.5015
> comp.ci.binom(n=10,p=0.1)
$p.hat
[1] 0.2
$norm.ci
[1] -0.0479 0.4479
$boot.ci
2.5% 97.5%
0.0 0.5
$exact.ci
[1] 0.0252 0.5561
1.1 Variance of allele frequencies from genotypic data
Here, we will verify that the variance of allele frequencies is not binomial whenthe genotypes are not in Hardy weinberg equilibrium:
> bootstrap<-function(x,nboot=1000){
+ n<-length(x)
+ b<-numeric(nboot)
+ for (i in 1:nboot){
+ dum<-sample(n,replace=TRUE)
+ b[i]<-sum(x[dum])/2/n
+ }
+ return(var(b))
+ }
> obs<-rep(2:0,c(25,50,25))
> n<-length(obs)
> pA<-sum(obs)/2/n
> pA*(1-pA)/2/n
5
[1] 0.00125
> bootstrap(obs)
[1] 0.001405668
> obs1<-rep(2:0,c(50,0,50))
> n<-length(obs1)
> pA<-sum(obs1)/2/n
> pA*(1-pA)/2/n
[1] 0.00125
> bootstrap(obs1)
[1] 0.002673211
2 Hardy Weinberg testing
2.1 χ2 tests
To run a chi-sqare test of goodness of fit in R, enter the argument p correspond-ing to the expected frequencies of the different genotypes
> #slide 96
> ng<-c(6,3,1)
> phat<-(ng[1]+ng[2]/2)/sum(ng)
> chisq.test(ng,p=c(phat^2,2*phat*(1-phat),(1-phat)^2))
Chi-squared test for given probabilities
data: ng
X-squared = 0.4, df = 2, p-value = 0.8187
How large should the chi-square statistics be to reject HW at the 5% level?
> qchisq(0.95,df=1)
[1] 3.841459
Remember, non centrality parameter for the chi-square test is nf2. Howlikely am I to detect departure from HW if f = 0.125 with a sample of 100individuals?
> n<-100
> f<-0.125
> pchisq(qchisq(0.95,df=1),df=1,ncp=n*f^2,lower=FALSE)
[1] 0.239527
and indeed:
> x<-seq(0.2,20,0.1)
> plot(x,dchisq(x,df=1,ncp=n*f^2),type="h")
> abline(v=qchisq(0.95,df=1),col="red")
6
0 5 10 15 20
0.0
0.1
0.2
0.3
0.4
x
dchi
sq(x
, df =
1, n
cp =
n *
f^2)
You would need 500 genotypes to detect with good power a fairly high in-breeding coefficient!
> n<-500
> f<-0.125
> pchisq(qchisq(0.95,df=1),df=1,ncp=n*f^2,lower=FALSE)
[1] 0.7981762
Let’s go back to our function calculating the different ci:
> comp.ci.binom<-function(p=0.5,n=1000,nboot=1000,digits=4){
+ x<-rbinom(1,n,p)
+ p.hat<-x/n
+ sd.p<-(p.hat*(1-p.hat)/n)^0.5
+ norm.ci<-round(c(p.hat-1.96*sd.p,p.hat+1.96*sd.p),digits=digits)
+ exact.ci<-round(binom.test(x,n)$conf.int[1:2],digits=digits)
+ bx<-numeric(nboot)
+ ox<-numeric(n);ox[1:x]<-1
+ for (i in 1:nboot) bx[i]<-sum(sample(ox,replace=TRUE))
+ boot.ci<-round(quantile(bx,c(0.025,0.975))/n,digits=digits)
+ return(list(p.hat=p.hat,norm.ci=norm.ci,boot.ci=boot.ci,exact.ci=exact.ci))
+ }
and a new function to separate alleles for data of the type presented in theFBI dataset:
> getallele<-function (loc)
+ {#one locus only
+ x <- length(loc)
+ if (max(loc, na.rm = TRUE) < 1e+06)
+ modulo = 1000
+ if (max(loc, na.rm = TRUE) < 10000)
+ modulo <- 100
+ if (max(loc, na.rm = TRUE) < 100)
+ modulo <- 10
7
+ firstal <- loc%/%modulo
+ secal <- loc%%modulo
+ y <- array(dim = c(x, 2))
+ y[ , 1] <- as.matrix(firstal)
+ y[ , 2] <- as.matrix(secal)
+ return(y)
+ }
This function gives a confidence interval on observed heterozygosity
> hwt<-function(genotypes,nperm=2000){
+ #estimates observed heterozygosity,
+ #and gives a confidence interval of Ho under HWE
+ #as well as the probability to observed a value
+ #as low(pl) or as high (pu) as the observed
+ ho<-function(alleles){
+ #return mean observed heterozygosity of argument alleles
+ Ho <- alleles[ , 1] == alleles[ , 2]
+ mHo<-1-mean(Ho)
+ return(mHo)
+ }
+ #remove missing values
+ geno<-genotypes[!is.na(genotypes)]
+ #separate alleles
+ alleles<-getallele(geno)
+ n<-length(geno)
+ #calculate ho
+ oHo<-ho(alleles)
+ #shuffle alleles
+ bHo<-numeric(nperm)
+ for (i in 1:nperm)
+ {
+ dum<-sample(alleles);
+ bHo[i]<-ho(cbind(dum[1:n],dum[(n+1):(2*n)]))
+ }
+ #ci of Ho given data under HW
+ ci<-quantile(bHo,c(0.025,0.975))
+ pl<-sum(bHo<=oHo)/(nperm+1)
+ pu<-sum(bHo>=oHo)/(nperm+1)
+ p2<-pl+pu
+ hist(bHo,breaks=seq(0,1,0.02));abline(v=oHo,col="red")
+ return(list(Ho=oHo,ci=ci,pl=pl,pu=pu))
+ }
How is this working?
> dat<-rep(c(11,12,22),c(25,50,25))
> hwt(dat)
$Ho
[1] 0.5
8
$ci
2.5% 97.5%
0.4 0.6
$pl
[1] 0.5492254
$pu
[1] 0.6031984
Histogram of bHo
bHo
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
250
300
Not anymore in HWE:
> dat<-rep(c(11,12,22),c(40,20,40))
> hwt(dat)
$Ho
[1] 0.2
$ci
2.5% 97.5%
0.4 0.6
$pl
[1] 0
$pu
[1] 0.9995002
9
Histogram of bHo
bHo
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
250
300
350
generating our own data with function sim.genot from package hierfstat
> library(hierfstat)
> (dathw<-sim.genot(nbal=20,nbpop=1))
Pop loc.1 loc.2
1 1 205 1103
2 1 512 1111
3 1 503 411
4 1 302 1111
5 1 203 1904
6 1 505 404
7 1 1405 808
8 1 1405 1111
9 1 512 1111
10 1 205 1111
11 1 502 1119
12 1 1202 408
13 1 1214 1111
14 1 505 1104
15 1 203 1119
16 1 202 404
17 1 1402 302
18 1 202 911
19 1 1019 1103
20 1 1214 2011
> hwt(dathw[,2])
$Ho
[1] 0.8
$ci
2.5% 97.5%
0.60 0.95
$pl
10
[1] 0.6186907
$pu
[1] 0.6166917
Histogram of bHo
bHo
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
010
020
030
040
0
Now illustrate slides 106 and nexts
> library(hwde) #if needed, run install.packages("hwde") if not yet installed
> x<-rep(c(11,12,22),c(89,9,2))
> hwexact(89,9,2)
[1] 0.04825197
> hwt(x)
$Ho
[1] 0.09
$ci
2.5% 97.5%
0.09 0.13
$pl
[1] 0.04047976
$pu
[1] 0.996002
> hwt(x,nperm=5000)
$Ho
[1] 0.09
$ci
2.5% 97.5%
0.09 0.13
$pl
11
[1] 0.04919016
$pu
[1] 0.9970006
Histogram of bHo
bHo
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
020
040
060
080
010
0012
00
> x<-rep(c(11,12,22),c(88,11,1))
> hwexact(88,11,1)
[1] 0.3405211
> hwt(x,nperm=5000)
$Ho
[1] 0.11
$ci
2.5% 97.5%
0.09 0.13
$pl
[1] 0.3321336
$pu
[1] 0.955009
> hwt(x,nperm=10000)
$Ho
[1] 0.11
$ci
2.5% 97.5%
0.09 0.13
$pl
[1] 0.3455654
$pu
[1] 0.950205
12
Histogram of bHo
bHo
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
050
010
0015
0020
0025
0030
00
> x<-rep(c(11,12,22),c(89,9,2))
> hwt(x,nperm=10000)
$Ho
[1] 0.09
$ci
2.5% 97.5%
0.09 0.13
$pl
[1] 0.04719528
$pu
[1] 0.9973003
Histogram of bHo
bHo
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
010
0020
0030
0040
0050
0060
00
3 Multiple tests
Distribution of p-values for a t-test when the null hypothesis is true
13
> dat<-matrix(rnorm(20000),ncol=1000)
> res<-apply(dat,2,fun<-function(x)t.test(x[1:10],x[11:20])$p.value)
> par(mfrow=c(2,2))
> hist(res)
> plot((1:1000)/1000,sort(res),xlab="uniform quantiles",ylab="empirical quantiles")
> abline(c(0,1))
> plot(-log10((1:1000)/1000),-log10(sort(res)),
+ xlab="uniform quantiles",ylab="empirical quantiles");abline(c(0,1))
> par(mfrow=c(1,1))
Histogram of res
res
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
020
6010
0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
uniform quantiles
empi
rical
qua
ntile
s
●●
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0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
1.0
2.0
uniform quantiles
empi
rical
qua
ntile
s
Let’s apply this to FBI dataset and departure form HWE
> fbi<-read.table("http://www2.unil.ch/popgen/teaching/SISG14/fbiwh02.txt",header=T)
> #hwt(fbi[fbi[,1]==1,2]) #pop 1, loc 1
> #hwt(fbi[fbi[,1]==2,2]) #pop 2, loc 1
> #tapply(fbi[,2],fbi[,1],hwt) # all loc at once
> #matrix(unlist(tapply(fbi[,2],fbi[,1],hwt)),nrow=5)
> (res<-matrix(unlist(apply(fbi[,-1],2,fun<-function(x) tapply(x,fbi[,1],hwt))),nrow=5))
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 0.7857143 0.8078818 0.7368421 0.883333333 0.7755102 0.7536946 0.8722222
[2,] 0.7095238 0.7438424 0.6650718 0.755555556 0.7602041 0.7142857 0.8166667
[3,] 0.8144048 0.8522167 0.7751196 0.866666667 0.8673469 0.8226601 0.9111111
[4,] 0.7871064 0.6971514 0.7516242 0.994502749 0.1034483 0.3028486 0.6501749
[5,] 0.2733633 0.3618191 0.3098451 0.007996002 0.9255372 0.7476262 0.4537731
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] 0.8775510 0.8768473 0.7611111 0.8316327 0.8128079 0.8938547 0.8673469
[2,] 0.8112245 0.8325123 0.7166667 0.7448980 0.7438424 0.8156425 0.8010204
[3,] 0.9081633 0.9211823 0.8333333 0.8469388 0.8423645 0.9106145 0.9030612
[4,] 0.7736132 0.4777611 0.2963518 0.9080460 0.7821089 0.9165417 0.7086457
[5,] 0.3008496 0.6046977 0.7556222 0.1264368 0.2718641 0.1299350 0.3678161
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] 0.8029557 0.8666667 0.8877551 0.8817734 0.7777778 0.6974359 0.7142857
[2,] 0.7635468 0.8277778 0.8316327 0.8323892 0.6777778 0.6256410 0.6650246
[3,] 0.8620690 0.9222222 0.9234694 0.9162562 0.8001389 0.7435897 0.7735222
14
[4,] 0.3663168 0.4132934 0.6981509 0.6161919 0.9020490 0.7061469 0.4532734
[5,] 0.7051474 0.6776612 0.3853073 0.4672664 0.1319340 0.3568216 0.6076962
[,22] [,23] [,24] [,25] [,26] [,27] [,28]
[1,] 0.6871508 0.72959184 0.8226601 0.8047619 0.7733990 0.7607656 0.8000000
[2,] 0.6312849 0.71938776 0.7733990 0.7333333 0.7536946 0.7224880 0.7285714
[3,] 0.7430168 0.82653061 0.8768473 0.8380952 0.8572660 0.8230861 0.8333333
[4,] 0.4822589 0.07146427 0.4207896 0.7876062 0.1174413 0.3438281 0.7626187
[5,] 0.5962019 0.94802599 0.6491754 0.2618691 0.9130435 0.7246377 0.2903548
[,29] [,30] [,31] [,32] [,33] [,34] [,35]
[1,] 0.7635468 0.6937799 0.7846890 0.66995074 0.5789474 0.7047619 0.72906404
[2,] 0.6748768 0.6507177 0.7081340 0.56157635 0.5502392 0.6761905 0.72906404
[3,] 0.7931034 0.7655502 0.8181818 0.67980296 0.6602871 0.7809524 0.83743842
[4,] 0.8450775 0.3343328 0.7761119 0.95952024 0.1594203 0.2168916 0.02798601
[5,] 0.1964018 0.7241379 0.2873563 0.06246877 0.8755622 0.8285857 0.97801099
[,36] [,37] [,38] [,39]
[1,] 0.7751196 0.8133971 0.7970297 0.7740385
[2,] 0.6985646 0.7416268 0.7128713 0.7163462
[3,] 0.8133971 0.8516746 0.8217822 0.8269231
[4,] 0.7411294 0.7036482 0.8615692 0.5362319
[5,] 0.3153423 0.3578211 0.1799100 0.5242379
Histogram of bHo
bHo
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
010
020
030
040
050
0
Now plot the p-values for pl:
> n<-dim(res)[2]
> plot(-log10((1:n)/n),-log10(sort(res[4,])),
+ xlab="uniform quantiles",ylab="empirical quantiles")
> abline(c(0,1))
15
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0.0 0.5 1.0 1.5
0.0
0.5
1.0
1.5
uniform quantiles
empi
rical
qua
ntile
s
No evidence for heterozygote deficiency whatsoever, perhaps some evidenceof a little excess. Normal you think?
Let’s move to SNPs data. Distribution of p-values for HWE for a given fvalue
> #library(hwde)
> ntests<-10000
> ssize<-100
> pval<-numeric(ntests)
> pvale<-numeric(ntests)
> for (i in 1:ntests){
+ p<-runif(1,.05,.95)
+ #start with an f of 0
+ f<-0.0
+ #expected genotype freq given p & f
+ freqs<-c(p^2+p*(1-p)*f,2*p*(1-p)*(1-f),(1-p)^2+p*(1-p)*f)
+ dat<-table(factor(sample(2:0,size=ssize,replace=TRUE,prob=freqs),levels=c("2","1","0")))
+ #observed allele freq
+ phat<-(2*dat[1]+dat[2])/2/ssize
+ #observed f
+ fhat<-1-dat[2]/ssize/2/phat/(1-phat)
+ pval[i]<-pchisq(ssize*fhat^2,1,lower=FALSE) #slide 92
+ pvale[i]<-hwexact(dat[1],dat[2],dat[3])
+ }
> plot(pvale,pval,xlab="Exact p-value",ylab="Chi-square p-value")
> abline(c(0,1))
16
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Exact p−value
Chi
−sq
uare
p−
valu
e
QQ-plot for the two sets of estimated p-values:
> par(mfrow=c(1,2))
> plot(-log10((1:ntests)/ntests),-log10(sort(pval)),
+ xlab="quantiles from uniform (-log10)",ylab="empirical quantiles chisq (-log10)")
> abline(c(0,1))
> plot(-log10((1:ntests)/ntests),-log10(sort(pvale)),
+ xlab="quantiles from uniform (-log10)",ylab="empirical quantiles exact (-log10)")
> abline(c(0,1))
> par(mfrow=c(1,1))
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0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
quantiles from uniform (−log10)
empi
rical
qua
ntile
s ex
act (
−lo
g10)
Now onto the AMD dataset
> amd <- read.table("AMD.txt", header=T) #dataset need to be uncompressed first
> head(amd[,1:10]) #what's in there?
SNP rsID chromosome pos X11008349 X13805906 X18038619
1 SNP_A-1738457 <NA> 1_random 328296 1 2 2
2 SNP_A-1658232 rs6424140 1_random 1435232 3 3 3
3 SNP_A-1718890 rs1496555 1 2103664 0 3 3
4 SNP_A-1678466 rs1338382 1 2708522 3 3 3
17
5 SNP_A-1676440 rs10492936 1 2719853 1 1 1
6 SNP_A-1662392 rs10489589 1 2734227 1 2 1
X18577706 X18621262 X19614947
1 3 3 3
2 3 3 3
3 3 3 2
4 3 3 3
5 1 1 1
6 1 1 2
> control1<-amd[which(amd$chromosome==1),101:150] #extract data from chrom1 only and for the controls
> ncontrols<-dim(control1)[2]
> co1<-apply(control1,1,fun<-function(x) sum(x==1)) #homAA
> co2<-apply(control1,1,fun<-function(x) sum(x==2)) #het
> co3<-apply(control1,1,fun<-function(x) sum(x==3)) #homBB
> mco<-apply(control1,1,fun<-function(x) sum(x==0)) #missing
> control.counts<-cbind(co1,co2,co3)
> pvalc<-apply(control.counts,1,fun<-function(a) hwexact(a[1],a[2],a[3]))
> pvalcn1<-pvalc[which(pvalc<1.0)]
> n<-length(pvalcn1)
> plot(-log10(1:n/n),-log10(sort(pvalcn1)));abline(c(0,1)) #slide 132
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Removing SNPs where not all individuals have been typed:
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> pvalcnmn1<-pvalc[completen1]
> (n<-length(pvalcnmn1))
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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
02
46
810
−log10(1:n/n)
−lo
g10(
sort
(pva
lcnm
n1))
Which are the outliers?
> outliers<-which(-log10(pvalcnmn1)>4.0)
> control.counts[rownames(control.counts) %in% names(outliers),]
co1 co2 co3
904 0 46 4
916 0 43 7
4794 10 40 0
9140 13 37 0
> amd[rownames(amd) %in% names(outliers),1:3]
SNP rsID chromosome
904 SNP_A-1692270 rs620913 1
916 SNP_A-1699173 rs1002378 1
4794 SNP_A-1691743 rs1971378 1
9140 SNP_A-1671923 rs1796846 1
19