reports/lmm_example.md
In sahirbhatnagar/ggmix: Variable Selection in Linear Mixed Models for SNP Data
Fitting Linear Mixed Model
Load Packages
# install.packages("gaston")
library(gaston)
Set Simulation Parameters
# number of subjects
n <- 1000
# number of genotypes
p <- 1e4
# number of causal genotypes
p_causal <- 50
# Signal to noise ratio
signal_to_noise_ratio <- 2
# vector of allele frequencies from which to sample
probs <- c(0.05, 0.1, 0.3, 0.4)
Generate Sample Data with Missing Genotypes
set.seed(345321)
geno <- replicate(p, rbinom(n, 2, sample(probs, 1)))
dim(geno)
## [1] 1000 10000
geno[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 1 1 0
## [2,] 1 0 0 1 0
## [3,] 1 1 0 0 0
## [4,] 2 1 1 0 0
## [5,] 1 2 1 0 0
geno[sample(1:p, 100)] <- NA
geno[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 1 1 0
## [2,] 1 0 0 1 0
## [3,] 1 1 0 0 0
## [4,] 2 1 1 0 0
## [5,] 1 2 1 0 0
Convert to BED Matrix
This is so that we can use the functions in the gaston package for fitting mixed models
DT <- gaston::as.bed.matrix(geno)
# can access the data by
as.matrix(DT)[1:5, 1:5]
## M_1 M_2 M_3 M_4 M_5
## 1 0 0 1 1 0
## 2 1 0 0 1 0
## 3 1 1 0 0 0
## 4 2 1 1 0 0
## 5 1 2 1 0 0
# see the contents of DT
slotNames(DT)
## [1] "ped" "snps" "bed"
## [4] "p" "mu" "sigma"
## [7] "standardize_p" "standardize_mu_sigma"
# to access the different contents of DT use @
DT@snps[1:5,]
## chr id dist pos A1 A2 N0 N1 N2 NAs N0.f N1.f N2.f NAs.f callrate
## 1 NA M_1 NA NA NA NA 480 411 101 8 NA NA NA NA 0.992
## 2 NA M_2 NA NA NA NA 349 477 167 7 NA NA NA NA 0.993
## 3 NA M_3 NA NA NA NA 461 442 85 12 NA NA NA NA 0.988
## 4 NA M_4 NA NA NA NA 781 200 9 10 NA NA NA NA 0.990
## 5 NA M_5 NA NA NA NA 886 95 3 16 NA NA NA NA 0.984
## maf hz
## 1 0.30897177 0.41431452
## 2 0.40835851 0.48036254
## 3 0.30971660 0.44736842
## 4 0.11010101 0.20202020
## 5 0.05132114 0.09654472
DT@ped[1:5,]
## famid id father mother sex pheno N0 N1 N2 NAs N0.x N1.x N2.x NAs.x
## 1 1 1 0 0 0 0 6431 2916 652 1 0 0 0 0
## 2 2 2 0 0 0 0 6450 2873 677 0 0 0 0 0
## 3 3 3 0 0 0 0 6441 2899 660 0 0 0 0 0
## 4 4 4 0 0 0 0 6374 2986 640 0 0 0 0 0
## 5 5 5 0 0 0 0 6339 2978 683 0 0 0 0 0
## N0.y N1.y N2.y NAs.y N0.mt N1.mt N2.mt NAs.mt callrate hz callrate.x
## 1 0 0 0 0 0 0 0 0 -Inf -2916 NaN
## 2 0 0 0 0 0 0 0 0 NaN Inf NaN
## 3 0 0 0 0 0 0 0 0 NaN Inf NaN
## 4 0 0 0 0 0 0 0 0 NaN Inf NaN
## 5 0 0 0 0 0 0 0 0 NaN Inf NaN
## hz.x callrate.y hz.y callrate.mt hz.mt
## 1 NaN NaN NaN NaN NaN
## 2 NaN NaN NaN NaN NaN
## 3 NaN NaN NaN NaN NaN
## 4 NaN NaN NaN NaN NaN
## 5 NaN NaN NaN NaN NaN
# p contains the alternate allele frequency
# mu is equal to 2*p and is the expected value of the genotype (coded in 0, 1, 2)
# sigma is the genotype standard error
DT@p[1:10]
## [1] 0.30897177 0.40835851 0.30971660 0.11010101 0.05132114 0.05393145
## [7] 0.28556911 0.04984894 0.30502513 0.39635996
DT@mu[1:10]
## [1] 0.61794355 0.81671702 0.61943320 0.22020202 0.10264228 0.10786290
## [7] 0.57113821 0.09969789 0.61005025 0.79271992
DT@sigma[1:10]
## [1] 0.6607853 0.6950724 0.6350671 0.4338020 0.3110142 0.3155285 0.6256291
## [8] 0.3053246 0.6458457 0.6949752
plot(2*DT@p, DT@mu)
abline(a=0,b=1, col = "red")
If the Hardy-Weinberg equilibrium holds, sigma should be close to $\sqrt{2*p(1-p)}$. This is illustrated on the figure below
plot(DT@p, DT@sigma, xlim=c(0,1))
t <- seq(0,1,length=101);
lines(t, sqrt(2*t*(1-t)), col="red")
Standardized SNP matrix
# this will center the columns of DT to mean 0, and standard deviation sqrt(2p(1-p))
gaston::standardize(DT) <- "p"
X <- as.matrix(DT)
X[1:5, 1:5]
## M_1 M_2 M_3 M_4 M_5
## 1 -0.9456415 -1.1749151 0.5819959 1.7615748 -0.32893
## 2 0.5846625 -1.1749151 -0.9472912 1.7615748 -0.32893
## 3 0.5846625 0.2636678 -0.9472912 -0.4974395 -0.32893
## 4 2.1149665 0.2636678 0.5819959 -0.4974395 -0.32893
## 5 0.5846625 1.7022506 0.5819959 -0.4974395 -0.32893
Dealing With Missing Values
In standardized matrices, the NA values are replaced by zeroes, which amount to impute the missing genotypes by the mean genotype.
X[is.na(X)] <- 0
X[1:5,1:5]
## M_1 M_2 M_3 M_4 M_5
## 1 -0.9456415 -1.1749151 0.5819959 1.7615748 -0.32893
## 2 0.5846625 -1.1749151 -0.9472912 1.7615748 -0.32893
## 3 0.5846625 0.2636678 -0.9472912 -0.4974395 -0.32893
## 4 2.1149665 0.2636678 0.5819959 -0.4974395 -0.32893
## 5 0.5846625 1.7022506 0.5819959 -0.4974395 -0.32893
The object X is what will be used as the data matrix in the LMM analysis. We also need to create the kinship matrix, which we do next.
Calculate Kinship Matrix
If $X_s$ is a standardized $n \times p$ matrix of genotypes, a Genetic Relationship Matrix of individuals can be computed as [GRM = \frac{1}{p-1} X_s X_s^\top] where $p$ is the number of SNPs and $n$ is the number of individuals. This computation is done by the gaston::GRM function. Note that we could also use
(1 / (p - 1)) * tcrossprod(X)
to calculate the kinship (covariance) matrix, but the gaston::GRM function is faster.
Note that the gaston::GRM function internally standardizes the genotype data, which is why we provide it the object DT. The object X will be used for fitting the model. We specify autosome.only = FALSE because we don't have that information.
kin <- gaston::GRM(DT, autosome.only = FALSE)
kin[1:5,1:5]
## 1 2 3 4 5
## 1 0.994037390 -0.020637164 -2.610047e-03 1.937922e-02 -0.0050506615
## 2 -0.020637164 0.970255971 1.689422e-03 -1.211568e-02 0.0014872842
## 3 -0.002610047 0.001689422 9.864578e-01 -9.975582e-05 0.0006858376
## 4 0.019379225 -0.012115683 -9.975582e-05 9.979130e-01 0.0061614369
## 5 -0.005050662 0.001487284 6.858376e-04 6.161437e-03 1.0134803057
Principal Components
From the GRM, we can compute the Principal components. The eigenvectors are normalized. The Principal Components (PC) can be computed by multiplying them by the square root of the associated eigenvalues
eiK <- eigen(kin)
# deal with a small negative eigen value
eiK$values[ eiK$values < 0 ] <- 0
PC <- sweep(eiK$vectors, 2, sqrt(eiK$values), "*")
dim(PC)
## [1] 1000 1000
plot(PC[,1], PC[,2])
Simulate Phenotype
p_causal SNPs are randomly assigned to a Uniform(0.9,1.1) distribution.
beta <- rep(0, p)
beta[sample(1:p, p_causal)] <- runif(p_causal, min = 0.9, max = 1.1)
y.star <- X %*% beta
error <- stats::rnorm(n)
k <- as.numeric(sqrt(stats::var(y.star)/(signal_to_noise_ratio*stats::var(error))))
Y <- y.star + k*error
Run Univariate LMM
There are two packages we can use to fit uni/multivariate LMMs.
gaston package
# make design matrix with intercept
x1 <- cbind(1, X[,1,drop=FALSE])
# with 1 random effect this function is faster than lmm.aireml
# in gaston::lmm.diago you provide the eigen decomposition
# in gaston::lmm.aireml you provide the kinship matrix
fit <- gaston::lmm.diago(Y, x1, eigenK = eiK)
## Optimization in interval [0, 1]
## Optimizing with p = 0
## [Iteration 1] Current point = 0 df = 28.579
## [Iteration 2] Current point = 0.54494 df = 2.00363
## [Iteration 3] Current point = 0.585132 df = -0.018836
## [Iteration 4] Current point = 0.584762 df = -1.81023e-06
# equivalently you can also fit using
# fit <- gaston::lmm.aireml(Y, x1, K = kin, verbose = FALSE)
# the second coefficient is x1, the first is the intercept
(z_score <- fit$BLUP_beta[2]/sqrt(fit$varbeta[2,2]))
## [1] 0.4642883
# pvalue
2*pnorm(z_score, lower.tail = F)
## [1] 0.6424412
# random effect variance
fit$tau
## [1] 44.29653
# error variance
fit$sigma2
## [1] 31.45491
# error sd
sqrt(fit$sigma2)
## [1] 5.608468
coxme package
# install.packages("coxme")
library(coxme)
# need an ID variable
dat <- data.frame(Y, x=X[,1], id = 1:n)
# provide the kinship matrix
gfit1 <- lmekin(Y ~ x + (1|id), data=dat, varlist=kin)
gfit1
## Linear mixed-effects kinship model fit by maximum likelihood
## Data: dat
## Log-likelihood = -3572.846
## n= 1000
##
##
## Model: Y ~ x + (1 | id)
## Fixed coefficients
## Value Std Error z p
## (Intercept) 0.3185476 0.1720343 1.85 0.064
## x 0.1320349 0.2748453 0.48 0.630
##
## Random effects
## Group Variable Std Dev Variance
## id Vmat.1 6.790932 46.116756
## Residual error= 5.440202
Session Info
## setting value
## version R version 3.4.1 (2017-06-30)
## system x86_64, linux-gnu
## ui X11
## language en_US
## collate en_US.UTF-8
## tz Canada/Eastern
## date 2017-10-12
##
## package * version date source
## backports 1.1.0 2017-05-22 cran (@1.1.0)
## base * 3.4.1 2017-07-08 local
## bdsmatrix * 1.3-2 2014-08-22 CRAN (R 3.4.1)
## compiler 3.4.1 2017-07-08 local
## coxme * 2.2-5 2015-06-15 CRAN (R 3.4.1)
## datasets * 3.4.1 2017-07-08 local
## devtools 1.13.3 2017-08-02 CRAN (R 3.4.1)
## digest 0.6.12 2017-01-27 CRAN (R 3.4.1)
## evaluate 0.10.1 2017-06-24 cran (@0.10.1)
## gaston * 1.5 2017-05-25 CRAN (R 3.4.1)
## graphics * 3.4.1 2017-07-08 local
## grDevices * 3.4.1 2017-07-08 local
## grid 3.4.1 2017-07-08 local
## htmltools 0.3.6 2017-04-28 cran (@0.3.6)
## knitr 1.17 2017-08-10 cran (@1.17)
## lattice 0.20-35 2017-03-25 CRAN (R 3.3.3)
## magrittr 1.5 2014-11-22 CRAN (R 3.4.1)
## Matrix 1.2-11 2017-08-16 CRAN (R 3.4.1)
## memoise 1.1.0 2017-04-21 CRAN (R 3.4.1)
## methods * 3.4.1 2017-07-08 local
## nlme 3.1-131 2017-02-06 CRAN (R 3.4.0)
## Rcpp * 0.12.12 2017-07-15 CRAN (R 3.4.1)
## RcppParallel * 4.3.20 2016-08-16 CRAN (R 3.4.1)
## rmarkdown 1.6 2017-06-15 CRAN (R 3.4.1)
## rprojroot 1.2 2017-01-16 CRAN (R 3.4.1)
## splines 3.4.1 2017-07-08 local
## stats * 3.4.1 2017-07-08 local
## stringi 1.1.5 2017-04-07 CRAN (R 3.4.1)
## stringr 1.2.0 2017-02-18 CRAN (R 3.4.1)
## survival * 2.41-3 2017-04-04 CRAN (R 3.4.0)
## tools 3.4.1 2017-07-08 local
## utils * 3.4.1 2017-07-08 local
## withr 2.0.0 2017-09-18 Github (jimhester/withr@d1f0957)
## yaml 2.1.14 2016-11-12 cran (@2.1.14)
sahirbhatnagar/ggmix documentation built on April 18, 2021, 3:46 p.m.
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Fitting Linear Mixed Model
Load Packages
# install.packages("gaston")
library(gaston)
Set Simulation Parameters
# number of subjects
n <- 1000
# number of genotypes
p <- 1e4
# number of causal genotypes
p_causal <- 50
# Signal to noise ratio
signal_to_noise_ratio <- 2
# vector of allele frequencies from which to sample
probs <- c(0.05, 0.1, 0.3, 0.4)
Generate Sample Data with Missing Genotypes
set.seed(345321)
geno <- replicate(p, rbinom(n, 2, sample(probs, 1)))
dim(geno)
## [1] 1000 10000
geno[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 1 1 0
## [2,] 1 0 0 1 0
## [3,] 1 1 0 0 0
## [4,] 2 1 1 0 0
## [5,] 1 2 1 0 0
geno[sample(1:p, 100)] <- NA
geno[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 1 1 0
## [2,] 1 0 0 1 0
## [3,] 1 1 0 0 0
## [4,] 2 1 1 0 0
## [5,] 1 2 1 0 0
Convert to BED Matrix
This is so that we can use the functions in the gaston package for fitting mixed models
DT <- gaston::as.bed.matrix(geno)
# can access the data by
as.matrix(DT)[1:5, 1:5]
## M_1 M_2 M_3 M_4 M_5
## 1 0 0 1 1 0
## 2 1 0 0 1 0
## 3 1 1 0 0 0
## 4 2 1 1 0 0
## 5 1 2 1 0 0
# see the contents of DT
slotNames(DT)
## [1] "ped" "snps" "bed"
## [4] "p" "mu" "sigma"
## [7] "standardize_p" "standardize_mu_sigma"
# to access the different contents of DT use @
DT@snps[1:5,]
## chr id dist pos A1 A2 N0 N1 N2 NAs N0.f N1.f N2.f NAs.f callrate
## 1 NA M_1 NA NA NA NA 480 411 101 8 NA NA NA NA 0.992
## 2 NA M_2 NA NA NA NA 349 477 167 7 NA NA NA NA 0.993
## 3 NA M_3 NA NA NA NA 461 442 85 12 NA NA NA NA 0.988
## 4 NA M_4 NA NA NA NA 781 200 9 10 NA NA NA NA 0.990
## 5 NA M_5 NA NA NA NA 886 95 3 16 NA NA NA NA 0.984
## maf hz
## 1 0.30897177 0.41431452
## 2 0.40835851 0.48036254
## 3 0.30971660 0.44736842
## 4 0.11010101 0.20202020
## 5 0.05132114 0.09654472
DT@ped[1:5,]
## famid id father mother sex pheno N0 N1 N2 NAs N0.x N1.x N2.x NAs.x
## 1 1 1 0 0 0 0 6431 2916 652 1 0 0 0 0
## 2 2 2 0 0 0 0 6450 2873 677 0 0 0 0 0
## 3 3 3 0 0 0 0 6441 2899 660 0 0 0 0 0
## 4 4 4 0 0 0 0 6374 2986 640 0 0 0 0 0
## 5 5 5 0 0 0 0 6339 2978 683 0 0 0 0 0
## N0.y N1.y N2.y NAs.y N0.mt N1.mt N2.mt NAs.mt callrate hz callrate.x
## 1 0 0 0 0 0 0 0 0 -Inf -2916 NaN
## 2 0 0 0 0 0 0 0 0 NaN Inf NaN
## 3 0 0 0 0 0 0 0 0 NaN Inf NaN
## 4 0 0 0 0 0 0 0 0 NaN Inf NaN
## 5 0 0 0 0 0 0 0 0 NaN Inf NaN
## hz.x callrate.y hz.y callrate.mt hz.mt
## 1 NaN NaN NaN NaN NaN
## 2 NaN NaN NaN NaN NaN
## 3 NaN NaN NaN NaN NaN
## 4 NaN NaN NaN NaN NaN
## 5 NaN NaN NaN NaN NaN
# p contains the alternate allele frequency
# mu is equal to 2*p and is the expected value of the genotype (coded in 0, 1, 2)
# sigma is the genotype standard error
DT@p[1:10]
## [1] 0.30897177 0.40835851 0.30971660 0.11010101 0.05132114 0.05393145
## [7] 0.28556911 0.04984894 0.30502513 0.39635996
DT@mu[1:10]
## [1] 0.61794355 0.81671702 0.61943320 0.22020202 0.10264228 0.10786290
## [7] 0.57113821 0.09969789 0.61005025 0.79271992
DT@sigma[1:10]
## [1] 0.6607853 0.6950724 0.6350671 0.4338020 0.3110142 0.3155285 0.6256291
## [8] 0.3053246 0.6458457 0.6949752
plot(2*DT@p, DT@mu)
abline(a=0,b=1, col = "red")
If the Hardy-Weinberg equilibrium holds, sigma should be close to $\sqrt{2*p(1-p)}$. This is illustrated on the figure below
plot(DT@p, DT@sigma, xlim=c(0,1))
t <- seq(0,1,length=101);
lines(t, sqrt(2*t*(1-t)), col="red")
Standardized SNP matrix
# this will center the columns of DT to mean 0, and standard deviation sqrt(2p(1-p))
gaston::standardize(DT) <- "p"
X <- as.matrix(DT)
X[1:5, 1:5]
## M_1 M_2 M_3 M_4 M_5
## 1 -0.9456415 -1.1749151 0.5819959 1.7615748 -0.32893
## 2 0.5846625 -1.1749151 -0.9472912 1.7615748 -0.32893
## 3 0.5846625 0.2636678 -0.9472912 -0.4974395 -0.32893
## 4 2.1149665 0.2636678 0.5819959 -0.4974395 -0.32893
## 5 0.5846625 1.7022506 0.5819959 -0.4974395 -0.32893
Dealing With Missing Values
In standardized matrices, the NA values are replaced by zeroes, which amount to impute the missing genotypes by the mean genotype.
X[is.na(X)] <- 0
X[1:5,1:5]
## M_1 M_2 M_3 M_4 M_5
## 1 -0.9456415 -1.1749151 0.5819959 1.7615748 -0.32893
## 2 0.5846625 -1.1749151 -0.9472912 1.7615748 -0.32893
## 3 0.5846625 0.2636678 -0.9472912 -0.4974395 -0.32893
## 4 2.1149665 0.2636678 0.5819959 -0.4974395 -0.32893
## 5 0.5846625 1.7022506 0.5819959 -0.4974395 -0.32893
The object X is what will be used as the data matrix in the LMM analysis. We also need to create the kinship matrix, which we do next.
Calculate Kinship Matrix
If $X_s$ is a standardized $n \times p$ matrix of genotypes, a Genetic Relationship Matrix of individuals can be computed as [GRM = \frac{1}{p-1} X_s X_s^\top] where $p$ is the number of SNPs and $n$ is the number of individuals. This computation is done by the gaston::GRM function. Note that we could also use
(1 / (p - 1)) * tcrossprod(X)
to calculate the kinship (covariance) matrix, but the gaston::GRM function is faster.
Note that the gaston::GRM function internally standardizes the genotype data, which is why we provide it the object DT. The object X will be used for fitting the model. We specify autosome.only = FALSE because we don't have that information.
kin <- gaston::GRM(DT, autosome.only = FALSE)
kin[1:5,1:5]
## 1 2 3 4 5
## 1 0.994037390 -0.020637164 -2.610047e-03 1.937922e-02 -0.0050506615
## 2 -0.020637164 0.970255971 1.689422e-03 -1.211568e-02 0.0014872842
## 3 -0.002610047 0.001689422 9.864578e-01 -9.975582e-05 0.0006858376
## 4 0.019379225 -0.012115683 -9.975582e-05 9.979130e-01 0.0061614369
## 5 -0.005050662 0.001487284 6.858376e-04 6.161437e-03 1.0134803057
Principal Components
From the GRM, we can compute the Principal components. The eigenvectors are normalized. The Principal Components (PC) can be computed by multiplying them by the square root of the associated eigenvalues
eiK <- eigen(kin)
# deal with a small negative eigen value
eiK$values[ eiK$values < 0 ] <- 0
PC <- sweep(eiK$vectors, 2, sqrt(eiK$values), "*")
dim(PC)
## [1] 1000 1000
plot(PC[,1], PC[,2])
Simulate Phenotype
p_causal SNPs are randomly assigned to a Uniform(0.9,1.1) distribution.
beta <- rep(0, p)
beta[sample(1:p, p_causal)] <- runif(p_causal, min = 0.9, max = 1.1)
y.star <- X %*% beta
error <- stats::rnorm(n)
k <- as.numeric(sqrt(stats::var(y.star)/(signal_to_noise_ratio*stats::var(error))))
Y <- y.star + k*error
Run Univariate LMM
There are two packages we can use to fit uni/multivariate LMMs.
gaston package
# make design matrix with intercept
x1 <- cbind(1, X[,1,drop=FALSE])
# with 1 random effect this function is faster than lmm.aireml
# in gaston::lmm.diago you provide the eigen decomposition
# in gaston::lmm.aireml you provide the kinship matrix
fit <- gaston::lmm.diago(Y, x1, eigenK = eiK)
## Optimization in interval [0, 1]
## Optimizing with p = 0
## [Iteration 1] Current point = 0 df = 28.579
## [Iteration 2] Current point = 0.54494 df = 2.00363
## [Iteration 3] Current point = 0.585132 df = -0.018836
## [Iteration 4] Current point = 0.584762 df = -1.81023e-06
# equivalently you can also fit using
# fit <- gaston::lmm.aireml(Y, x1, K = kin, verbose = FALSE)
# the second coefficient is x1, the first is the intercept
(z_score <- fit$BLUP_beta[2]/sqrt(fit$varbeta[2,2]))
## [1] 0.4642883
# pvalue
2*pnorm(z_score, lower.tail = F)
## [1] 0.6424412
# random effect variance
fit$tau
## [1] 44.29653
# error variance
fit$sigma2
## [1] 31.45491
# error sd
sqrt(fit$sigma2)
## [1] 5.608468
coxme package
# install.packages("coxme")
library(coxme)
# need an ID variable
dat <- data.frame(Y, x=X[,1], id = 1:n)
# provide the kinship matrix
gfit1 <- lmekin(Y ~ x + (1|id), data=dat, varlist=kin)
gfit1
## Linear mixed-effects kinship model fit by maximum likelihood
## Data: dat
## Log-likelihood = -3572.846
## n= 1000
##
##
## Model: Y ~ x + (1 | id)
## Fixed coefficients
## Value Std Error z p
## (Intercept) 0.3185476 0.1720343 1.85 0.064
## x 0.1320349 0.2748453 0.48 0.630
##
## Random effects
## Group Variable Std Dev Variance
## id Vmat.1 6.790932 46.116756
## Residual error= 5.440202
Session Info
## setting value
## version R version 3.4.1 (2017-06-30)
## system x86_64, linux-gnu
## ui X11
## language en_US
## collate en_US.UTF-8
## tz Canada/Eastern
## date 2017-10-12
##
## package * version date source
## backports 1.1.0 2017-05-22 cran (@1.1.0)
## base * 3.4.1 2017-07-08 local
## bdsmatrix * 1.3-2 2014-08-22 CRAN (R 3.4.1)
## compiler 3.4.1 2017-07-08 local
## coxme * 2.2-5 2015-06-15 CRAN (R 3.4.1)
## datasets * 3.4.1 2017-07-08 local
## devtools 1.13.3 2017-08-02 CRAN (R 3.4.1)
## digest 0.6.12 2017-01-27 CRAN (R 3.4.1)
## evaluate 0.10.1 2017-06-24 cran (@0.10.1)
## gaston * 1.5 2017-05-25 CRAN (R 3.4.1)
## graphics * 3.4.1 2017-07-08 local
## grDevices * 3.4.1 2017-07-08 local
## grid 3.4.1 2017-07-08 local
## htmltools 0.3.6 2017-04-28 cran (@0.3.6)
## knitr 1.17 2017-08-10 cran (@1.17)
## lattice 0.20-35 2017-03-25 CRAN (R 3.3.3)
## magrittr 1.5 2014-11-22 CRAN (R 3.4.1)
## Matrix 1.2-11 2017-08-16 CRAN (R 3.4.1)
## memoise 1.1.0 2017-04-21 CRAN (R 3.4.1)
## methods * 3.4.1 2017-07-08 local
## nlme 3.1-131 2017-02-06 CRAN (R 3.4.0)
## Rcpp * 0.12.12 2017-07-15 CRAN (R 3.4.1)
## RcppParallel * 4.3.20 2016-08-16 CRAN (R 3.4.1)
## rmarkdown 1.6 2017-06-15 CRAN (R 3.4.1)
## rprojroot 1.2 2017-01-16 CRAN (R 3.4.1)
## splines 3.4.1 2017-07-08 local
## stats * 3.4.1 2017-07-08 local
## stringi 1.1.5 2017-04-07 CRAN (R 3.4.1)
## stringr 1.2.0 2017-02-18 CRAN (R 3.4.1)
## survival * 2.41-3 2017-04-04 CRAN (R 3.4.0)
## tools 3.4.1 2017-07-08 local
## utils * 3.4.1 2017-07-08 local
## withr 2.0.0 2017-09-18 Github (jimhester/withr@d1f0957)
## yaml 2.1.14 2016-11-12 cran (@2.1.14)
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