Last updated: 2018-02-23

Code version: b299bc0

Introduction/summary

In notations,

\[ y_{ij} = \mu + \tau_i + \beta_j + \gamma_k + \epsilon_{ij} \] where \(i = 1,2,..., I\) and \(j = 1,2,..., J\). The parameters are estimated under sum-to-zero constraints \(\sum \tau_i = 0\) and \(\sum \beta_j = 0\).

Note that in this model 1) not all \(y_{ij.}\) exists due to the incompleteness of the design, 2) the effects of individual and block are nonorthogonal, 3) the effects are additive due to the block design.

TO DO: Apply batch correction prior to background correction??

Data and packages

\(~\)

library(data.table)
library(dplyr)
library(ggplot2)
library(cowplot)
library(RColorBrewer)
library(Biobase)
library(scales)
library(car)
library(ashr)
library(lsmeans)

Read in filtered data.

df <- readRDS(file="../data/eset-filtered.rds")
pdata <- pData(df)
fdata <- fData(df)

Source of variation

Statistical tests show that for GFP, there’s significant individual effect, plate effect and position effect, and that for RFP and DAPI, there’s no signficant individual effect or position effect but there’s significant plate effect (all at P<.01).

lm.rfp <- lm(rfp.median.log10sum~factor(chip_id)+factor(experiment) + factor(image_label), 
             data = pdata)
lm.gfp <- lm(gfp.median.log10sum~factor(chip_id)+factor(experiment) + factor(image_label), 
             data = pdata)
lm.dapi <- lm(dapi.median.log10sum~factor(chip_id)+factor(experiment) + factor(image_label), 
             data = pdata)

aov.lm.rfp <- Anova(lm.rfp, type = "III")
aov.lm.gfp <- Anova(lm.gfp, type = "III")
aov.lm.dapi <- Anova(lm.dapi, type = "III")
aov.lm.rfp
Anova Table (Type III tests)

Response: rfp.median.log10sum
                     Sum Sq  Df  F value    Pr(>F)    
(Intercept)          44.799   1 193.7809 < 2.2e-16 ***
factor(chip_id)       2.061   5   1.7830  0.113731    
factor(experiment)    8.836  15   2.5480  0.001004 ** 
factor(image_label)  28.830  95   1.3127  0.029625 *  
Residuals           202.056 874                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
aov.lm.gfp
Anova Table (Type III tests)

Response: gfp.median.log10sum
                    Sum Sq  Df  F value    Pr(>F)    
(Intercept)         60.082   1 569.5986 < 2.2e-16 ***
factor(chip_id)      1.608   5   3.0492  0.009779 ** 
factor(experiment)  12.174  15   7.6944 2.293e-16 ***
factor(image_label) 14.688  95   1.4658  0.003756 ** 
Residuals           92.191 874                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
aov.lm.dapi
Anova Table (Type III tests)

Response: dapi.median.log10sum
                    Sum Sq  Df   F value  Pr(>F)    
(Intercept)         57.257   1 1474.3536 < 2e-16 ***
factor(chip_id)      0.568   5    2.9233 0.01262 *  
factor(experiment)  12.118  15   20.8019 < 2e-16 ***
factor(image_label)  3.333  95    0.9035 0.73028    
Residuals           33.942 874                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Indivdual and plate variation

Position variation

well.gfp.median <- pdata %>% group_by(image_label) %>% summarize(., median(gfp.median.log10sum))
well.rfp.median <- pdata %>% group_by(image_label) %>% summarize(., median(rfp.median.log10sum))
well.dapi.median <- pdata %>% group_by(image_label) %>% summarize(., median(dapi.median.log10sum))

well.pp <- data.frame(well=pdata$well, image_label=pdata$image_label)
well.pp <- well.pp[!duplicated(well.pp),]
colbrew <- brewer.pal(9, "Set1")
well.pp$cols <- rep(colbrew[9], 96)
well.pp$cols[which(well.pp$well %in% c("A03", "A02", "A01", "A09", "A08", "A07"))] <- colbrew[1]
well.pp$cols[which(well.pp$well %in% c("H03", "H02", "H01", "H09", "H08", "H07"))] <- colbrew[2]
well.pp$cols[which(well.pp$well %in% c("A06", "A05", "A04", "A12", "A11", "A10"))] <- colbrew[3]
well.pp$cols[which(well.pp$well %in% c("H06", "H05", "H04", "H12", "H11", "H10"))] <- colbrew[4]
well.pp <- well.pp[order(well.pp$image_label),]

ord.gfp <- as.character(well.gfp.median$image_label[order(well.gfp.median$`median(gfp.median.log10sum)`)])
ord.rfp <- as.character(well.rfp.median$image_label[order(well.rfp.median$`median(rfp.median.log10sum)`)])
ord.dapi <- as.character(well.dapi.median$image_label[order(well.dapi.median$`median(dapi.median.log10sum)`)])

These are four corners previously found more likely to have high gene expression values in sequencing data.

par(mfrow=c(1,1))
plot(1:7, 1:7, pch="", axes=F, ann=F)
legend("center", legend = c("A_a", "H_a", "A_b", "H_b"), col=colbrew[c(1,2,3,4)],
       pch=16)

par(mfrow=c(3,1))
boxplot(rfp.median.log10sum ~ factor(image_label, levels=ord.rfp), 
        data=pdata, ylab = "RFP",
        col=well.pp$cols[as.numeric(ord.rfp)])
abline(h=0, lwd=2, col="royalblue")
boxplot(gfp.median.log10sum ~ factor(image_label, levels=ord.gfp), 
        data=pdata, ylab = "GFP",
        col=well.pp$cols[as.numeric(ord.gfp)])
abline(h=0, lwd=2, col="royalblue")
boxplot(dapi.median.log10sum ~ factor(image_label, levels=ord.dapi), 
        data=pdata, ylab = "GFP",
        col=well.pp$cols[as.numeric(ord.dapi)])
abline(h=0, lwd=2, col="royalblue")
title("Position variation", outer=TRUE, line = -1)

Estimate effects

Contrast test to estimate effects for for plate and position ID.

# make contrast matrix for plates
# each plate is compared to the average
n_plates <- uniqueN(pdata$experiment)
contrast_plates <- matrix(-1, nrow=n_plates, ncol=n_plates)
diag(contrast_plates) <- n_plates-1

# make contrast matrix for individuals
# each individual is compared to the average
n_pos <- uniqueN(pdata$image_label)
contrast_pos <- matrix(-1, nrow=n_pos, ncol=n_pos)
diag(contrast_pos) <- n_pos-1
gfp.plates <- summary(lsmeans(lm.gfp, specs="experiment", contrast=contrast_plates))
gfp.pos <- summary(lsmeans(lm.gfp, specs="image_label", contrast=contrast_pos))

rfp.plates <- summary(lsmeans(lm.rfp, specs="experiment", contrast=contrast_plates))
rfp.pos <- summary(lsmeans(lm.rfp, specs="image_label", contrast=contrast_pos))

dapi.plates <- summary(lsmeans(lm.dapi, specs="experiment", contrast=contrast_plates))
dapi.pos <- summary(lsmeans(lm.dapi, specs="image_label", contrast=contrast_pos))

Substract plate effect from the raw estimates.

## RFP
pdata$rfp.median.log10sum.adjust <- pdata$rfp.median.log10sum
rfp.plates$experiment <- as.character(rfp.plates$experiment)
rfp.pos$experiment <- as.character(rfp.pos$image_label)
pdata$experiment <- as.character(pdata$experiment)

exps <- unique(pdata$experiment)
for (i in 1:uniqueN(exps)) {
  exp <- exps[i]
  ii_exp <- which(pdata$experiment == exp)
  est_exp <- rfp.plates$lsmean[which(rfp.plates$experiment==exp)]
  pdata$rfp.median.log10sum.adjust[ii_exp] <- (pdata$rfp.median.log10sum[ii_exp] - est_exp)
}

pos <- unique(pdata$image_label)
for (i in 1:uniqueN(pos)) {
  p <- pos[i]
  ii_pos <- which(pdata$image_label == p)
  est_pos <- rfp.pos$lsmean[which(rfp.pos$image_label==p)]
  pdata$rfp.median.log10sum.adjust[ii_pos] <- (pdata$rfp.median.log10sum[ii_pos] - est_pos)
}


## GFP
pdata$gfp.median.log10sum.adjust <- pdata$gfp.median.log10sum
gfp.plates$experiment <- as.character(gfp.plates$experiment)
gfp.pos$experiment <- as.character(gfp.pos$image_label)
pdata$experiment <- as.character(pdata$experiment)

exps <- unique(pdata$experiment)
for (i in 1:uniqueN(exps)) {
  exp <- exps[i]
  ii_exp <- which(pdata$experiment == exp)
  est_exp <- gfp.plates$lsmean[which(gfp.plates$experiment==exp)]
  pdata$gfp.median.log10sum.adjust[ii_exp] <- (pdata$gfp.median.log10sum[ii_exp] - est_exp)
}

pos <- unique(pdata$image_label)
for (i in 1:uniqueN(pos)) {
  p <- pos[i]
  ii_pos <- which(pdata$image_label == p)
  est_pos <- gfp.pos$lsmean[which(gfp.pos$image_label==p)]
  pdata$gfp.median.log10sum.adjust[ii_pos] <- (pdata$gfp.median.log10sum[ii_pos] - est_pos)
}



## DAPI
pdata$dapi.median.log10sum.adjust <- pdata$dapi.median.log10sum
dapi.plates$experiment <- as.character(dapi.plates$experiment)
dapi.pos$experiment <- as.character(dapi.pos$image_label)
pdata$experiment <- as.character(pdata$experiment)

exps <- unique(pdata$experiment)
for (i in 1:uniqueN(exps)) {
  exp <- exps[i]
  ii_exp <- which(pdata$experiment == exp)
  est_exp <- dapi.plates$lsmean[which(dapi.plates$experiment==exp)]
  pdata$dapi.median.log10sum.adjust[ii_exp] <- (pdata$dapi.median.log10sum[ii_exp] - est_exp)
}

pos <- unique(pdata$image_label)
for (i in 1:uniqueN(pos)) {
  p <- pos[i]
  ii_pos <- which(pdata$image_label == p)
  est_pos <- dapi.pos$lsmean[which(dapi.pos$image_label==p)]
  pdata$dapi.median.log10sum.adjust[ii_pos] <- (pdata$dapi.median.log10sum[ii_pos] - est_pos)
}

After adjustment

## These are four corners previously found more likely to have high gene expression values in sequencing data.
par(mfrow=c(1,1))
plot(1:7, 1:7, pch="", axes=F, ann=F)
legend("center", legend = c("A_a", "H_a", "A_b", "H_b"), col=colbrew[c(1,2,3,4)],
       pch=16)

par(mfrow=c(3,1))
boxplot(rfp.median.log10sum.adjust ~ factor(image_label, levels=ord.rfp), 
        data=pdata, ylab = "RFP",
        col=well.pp$cols[as.numeric(ord.rfp)])
abline(h=0, lwd=2, col="royalblue")
boxplot(gfp.median.log10sum.adjust ~ factor(image_label, levels=ord.gfp), 
        data=pdata, ylab = "GFP",
        col=well.pp$cols[as.numeric(ord.gfp)])
abline(h=0, lwd=2, col="royalblue")
boxplot(dapi.median.log10sum.adjust ~ factor(image_label, levels=ord.dapi), 
        data=pdata, ylab = "GFP",
        col=well.pp$cols[as.numeric(ord.dapi)])
abline(h=0, lwd=2, col="royalblue")
title("Position variation", outer=TRUE, line = -1)

Output results

Save corrected data to a temporary output folder.

saveRDS(pdata, file = "../output/images-normalize-anova.Rmd/pdata.adj.rds")

ash

apply shrinkage to position estimates

# # apply limma ebayes to shrink variances
# library(limma)
# gfp.pos.var <- squeezeVar(gfp.pos$SE^2, df = gfp.pos$df)$var.post
# gfp.pos.df <- squeezeVar(gfp.pos$SE^2, df = gfp.pos$df)$df.prior + gfp.pos$df
gfp.pos.ash <- ash(gfp.pos$lsmean, gfp.pos$SE, mixcompdist = "uniform",
                   lik = lik_t(df=gfp.pos$df[1]), mode = "estimate" )
# gfp.pos.ash.varpost <- ash(gfp.pos$lsmean, gfp.pos.var, mixcompdist = "uniform",
#                    lik = lik_t(df=gfp.pos.df), mode = "estimate" )


# rfp.pos.var <- squeezeVar(rfp.pos$SE^2, df = rfp.pos$df)$var.post
# rfp.pos.df <- squeezeVar(rfp.pos$SE^2, df = rfp.pos$df)$df.prior + gfp.pos$df
rfp.pos.ash <- ash(rfp.pos$lsmean, rfp.pos$SE, mixcompdist = "uniform",
                   lik = lik_t(df=rfp.pos$df[1]), mode = "estimate" )
# rfp.pos.ash.varpost <- ash(rfp.pos$lsmean, rfp.pos.var, mixcompdist = "uniform",
#                    lik = lik_t(df=rfp.pos.df), mode = "estimate" )

# dapi.pos.var <- squeezeVar(dapi.pos$SE^2, df = dapi.pos$df)$var.post
# dapi.pos.df <- squeezeVar(dapi.pos$SE^2, df = dapi.pos$df)$df.prior + gfp.pos$df
dapi.pos.ash <- ash(dapi.pos$lsmean, dapi.pos$SE, mixcompdist = "uniform",
                   lik = lik_t(df=dapi.pos$df[1]), mode = "estimate" )
# dapi.pos.ash.varpost <- ash(dapi.pos$lsmean, dapi.pos.var, mixcompdist = "uniform",
#                    lik = lik_t(df=dapi.pos.df), mode = "estimate" )
# 

par(mfrow=c(2,2))
plot(gfp.pos.ash$result$betahat, gfp.pos.ash$result$PosteriorMean,
     xlab = "beta hat", ylab = "Shrunken estimate", main = "GFP")
abline(0,1, col = "royalblue")
plot(rfp.pos.ash$result$betahat, rfp.pos.ash$result$PosteriorMean,
     xlab = "beta hat", ylab = "Shrunken estimate", main = "RFP")
abline(0,1, col = "royalblue")
plot(dapi.pos.ash$result$betahat, dapi.pos.ash$result$PosteriorMean,
     xlab = "beta hat", ylab = "Shrunken estimate", main = "DAPI")
abline(0,1, col = "royalblue")
     
par(mfrow=c(2,2))

plot(gfp.pos.ash$result$sebetahat, gfp.pos.ash$result$PosteriorSD,
     xlab = "Standard Error", ylab = "Shrunken estimate", main = "GFP")
abline(0,1, col = "royalblue")
plot(rfp.pos.ash$result$sebetahat, rfp.pos.ash$result$PosteriorSD,
     xlab = "Standard Error", ylab = "Shrunken estimate", main = "RFP")
abline(0,1, col = "royalblue")
plot(dapi.pos.ash$result$sebetahat, dapi.pos.ash$result$PosteriorSD,
     xlab = "Standard Error", ylab = "Shrunken estimate", main = "DAPI")
abline(0,1, col = "royalblue")

Plate effect.

library(ashr)
gfp.plates.ash <- ash(gfp.plates$lsmean, gfp.plates$SE, mixcompdist = "uniform",
                   lik = lik_t(df=gfp.plates$df[1]), mode = "estimate")
rfp.plates.ash <- ash(rfp.plates$lsmean, rfp.plates$SE, mixcompdist = "uniform",
                   lik = lik_t(df=rfp.plates$df[1]), mode = "estimate")
dapi.plates.ash <- ash(dapi.plates$lsmean, dapi.plates$SE, mixcompdist = "uniform",
                    lik = lik_t(df=dapi.plates$df[1]), mode = "estimate")
  
par(mfrow=c(2,2))
plot(gfp.plates.ash$result$betahat, gfp.plates.ash$result$PosteriorMean,
     xlab = "beta hat", ylab = "Shrunken estimate", main = "GFP")
abline(0,1, col = "royalblue")
plot(rfp.plates.ash$result$betahat, rfp.plates.ash$result$PosteriorMean,
     xlab = "beta hat", ylab = "Shrunken estimate", main = "RFP")
abline(0,1, col = "royalblue")
plot(dapi.plates.ash$result$betahat, dapi.plates.ash$result$PosteriorMean,
     xlab = "beta hat", ylab = "Shrunken estimate", main = "DAPI")
abline(0,1, col = "royalblue")
     
par(mfrow=c(2,2))

plot(gfp.plates.ash$result$sebetahat, gfp.plates.ash$result$PosteriorSD,
     xlab = "Standard Error", ylab = "Shrunken estimate", main = "GFP")
abline(0,1, col = "royalblue")
plot(rfp.plates.ash$result$sebetahat, rfp.plates.ash$result$PosteriorSD,
     xlab = "Standard Error", ylab = "Shrunken estimate", main = "RFP")
abline(0,1, col = "royalblue")
plot(dapi.plates.ash$result$sebetahat, dapi.plates.ash$result$PosteriorSD,
     xlab = "Standard Error", ylab = "Shrunken estimate", main = "DAPI")
abline(0,1, col = "royalblue")

Substract plate effect from the raw estimates.

## RFP
pdata$rfp.median.log10sum.adjust.ash <- pdata$rfp.median.log10sum
rfp.plates$experiment <- as.character(rfp.plates$experiment)
rfp.pos$experiment <- as.character(rfp.pos$image_label)
pdata$experiment <- as.character(pdata$experiment)

exps <- unique(pdata$experiment)
for (i in 1:uniqueN(exps)) {
  exp <- exps[i]
  ii_exp <- which(pdata$experiment == exp)
  est_exp <- rfp.plates.ash$result$PosteriorMean[which(rfp.plates$experiment==exp)]
  pdata$rfp.median.log10sum.adjust.ash[ii_exp] <- (pdata$rfp.median.log10sum[ii_exp] - est_exp)
}

pos <- unique(pdata$image_label)
for (i in 1:uniqueN(pos)) {
  p <- pos[i]
  ii_pos <- which(pdata$image_label == p)
  est_pos <- rfp.pos.ash$result$PosteriorMean[which(rfp.pos$image_label==p)]
  pdata$rfp.median.log10sum.adjust.ash[ii_pos] <- (pdata$rfp.median.log10sum[ii_pos] - est_pos)
}


## GFP
pdata$gfp.median.log10sum.adjust.ash <- pdata$gfp.median.log10sum
gfp.plates$experiment <- as.character(gfp.plates$experiment)
gfp.pos$experiment <- as.character(gfp.pos$image_label)
pdata$experiment <- as.character(pdata$experiment)

exps <- unique(pdata$experiment)
for (i in 1:uniqueN(exps)) {
  exp <- exps[i]
  ii_exp <- which(pdata$experiment == exp)
  est_exp <- gfp.plates.ash$result$PosteriorMean[which(gfp.plates$experiment==exp)]
  pdata$gfp.median.log10sum.adjust.ash[ii_exp] <- (pdata$gfp.median.log10sum[ii_exp] - est_exp)
}

pos <- unique(pdata$image_label)
for (i in 1:uniqueN(pos)) {
  p <- pos[i]
  ii_pos <- which(pdata$image_label == p)
  est_pos <- gfp.pos.ash$result$PosteriorMean[which(gfp.pos$image_label==p)]
  pdata$gfp.median.log10sum.adjust.ash[ii_pos] <- (pdata$gfp.median.log10sum[ii_pos] - est_pos)
}



## DAPI
pdata$dapi.median.log10sum.adjust.ash <- pdata$dapi.median.log10sum
dapi.plates$experiment <- as.character(dapi.plates$experiment)
dapi.pos$experiment <- as.character(dapi.pos$image_label)
pdata$experiment <- as.character(pdata$experiment)

exps <- unique(pdata$experiment)
for (i in 1:uniqueN(exps)) {
  exp <- exps[i]
  ii_exp <- which(pdata$experiment == exp)
  est_exp <- dapi.plates.ash$result$PosteriorMean[which(dapi.plates$experiment==exp)]
  pdata$dapi.median.log10sum.adjust.ash[ii_exp] <- (pdata$dapi.median.log10sum[ii_exp] - est_exp)
}

pos <- unique(pdata$image_label)
for (i in 1:uniqueN(pos)) {
  p <- pos[i]
  ii_pos <- which(pdata$image_label == p)
  est_pos <- dapi.pos.ash$result$PosteriorMean[which(dapi.pos$image_label==p)]
  pdata$dapi.median.log10sum.adjust.ash[ii_pos] <- (pdata$dapi.median.log10sum[ii_pos] - est_pos)
}

After adjustment

## These are four corners previously found more likely to have high gene expression values in sequencing data.
par(mfrow=c(1,1))
plot(1:7, 1:7, pch="", axes=F, ann=F)
legend("center", legend = c("A_a", "H_a", "A_b", "H_b"), col=colbrew[c(1,2,3,4)],
       pch=16)

par(mfrow=c(3,1))
boxplot(rfp.median.log10sum.adjust.ash ~ factor(image_label, levels=ord.rfp), 
        data=pdata, ylab = "RFP",
        col=well.pp$cols[as.numeric(ord.rfp)])
abline(h=0, lwd=2, col="royalblue")
boxplot(gfp.median.log10sum.adjust.ash ~ factor(image_label, levels=ord.gfp), 
        data=pdata, ylab = "GFP",
        col=well.pp$cols[as.numeric(ord.gfp)])
abline(h=0, lwd=2, col="royalblue")
boxplot(dapi.median.log10sum.adjust.ash ~ factor(image_label, levels=ord.dapi), 
        data=pdata, ylab = "GFP",
        col=well.pp$cols[as.numeric(ord.dapi)])
abline(h=0, lwd=2, col="royalblue")
title("Position variation", outer=TRUE, line = -1)

Output results

Save corrected data to a temporary output folder.

saveRDS(pdata, file = "../output/images-normalize-anova.Rmd/pdata.adj.rds")

Session information

R version 3.4.1 (2017-06-30)
Platform: x86_64-redhat-linux-gnu (64-bit)
Running under: Scientific Linux 7.2 (Nitrogen)

Matrix products: default
BLAS/LAPACK: /usr/lib64/R/lib/libRblas.so

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] parallel  stats     graphics  grDevices utils     datasets  methods  
[8] base     

other attached packages:
 [1] bindrcpp_0.2        lsmeans_2.27-61     ashr_2.2-4         
 [4] car_2.1-6           scales_0.5.0        Biobase_2.38.0     
 [7] BiocGenerics_0.24.0 RColorBrewer_1.1-2  cowplot_0.9.2      
[10] ggplot2_2.2.1       dplyr_0.7.4         data.table_1.10.4-3

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.15       mvtnorm_1.0-7      lattice_0.20-35   
 [4] Rmosek_7.1.3       zoo_1.8-1          assertthat_0.2.0  
 [7] rprojroot_1.3-2    digest_0.6.15      foreach_1.4.4     
[10] truncnorm_1.0-7    R6_2.2.2           plyr_1.8.4        
[13] backports_1.1.2    MatrixModels_0.4-1 evaluate_0.10.1   
[16] coda_0.19-1        pillar_1.1.0       rlang_0.2.0       
[19] lazyeval_0.2.1     pscl_1.5.2         multcomp_1.4-8    
[22] minqa_1.2.4        SparseM_1.77       nloptr_1.0.4      
[25] Matrix_1.2-10      rmarkdown_1.8      labeling_0.3      
[28] splines_3.4.1      lme4_1.1-15        stringr_1.3.0     
[31] REBayes_1.3        munsell_0.4.3      compiler_3.4.1    
[34] pkgconfig_2.0.1    etrunct_0.1        SQUAREM_2017.10-1 
[37] mgcv_1.8-17        htmltools_0.3.6    nnet_7.3-12       
[40] tibble_1.4.2       codetools_0.2-15   MASS_7.3-47       
[43] grid_3.4.1         nlme_3.1-131       xtable_1.8-2      
[46] gtable_0.2.0       git2r_0.21.0       magrittr_1.5      
[49] estimability_1.3   stringi_1.1.6      doParallel_1.0.11 
[52] sandwich_2.4-0     TH.data_1.0-8      iterators_1.0.9   
[55] tools_3.4.1        glue_1.2.0         pbkrtest_0.4-7    
[58] survival_2.41-3    yaml_2.1.16        colorspace_1.3-2  
[61] knitr_1.20         bindr_0.1          quantreg_5.35

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