Cluster diagrams with pvclust ggraph and ggplot2
Roeland Kindt
10/12/2020
1 Packages needed
library(BiodiversityR) # also loads vegan
library(pvclust)
library(ggplot2)
library(ggsci)
library(ggraph)
library(tidygraph)2 Introduction
The main objective of this document is to show a pathway of generating dendrograms via the ggraph package for results that were obtained via pvclust. I expect that similar pathways can be used to generate dendrograms with ggraph.
3 Step 1: Conduct the pvclust analysis
3.1 Load the data
The dune meadow data sets have been used as case study in Data Analysis in Community and Landscape Ecology, the Tree Diversity Analysis manual on community ecology and biodiversity analysis (this manual can be downloaded for free from this website ) and some recent examples of generating ordination graphs, species accumulation curves and Renyi diversity profiles.
data(dune)
str(dune)## 'data.frame': 20 obs. of 30 variables:
## $ Achimill: num 1 3 0 0 2 2 2 0 0 4 ...
## $ Agrostol: num 0 0 4 8 0 0 0 4 3 0 ...
## $ Airaprae: num 0 0 0 0 0 0 0 0 0 0 ...
## $ Alopgeni: num 0 2 7 2 0 0 0 5 3 0 ...
## $ Anthodor: num 0 0 0 0 4 3 2 0 0 4 ...
## $ Bellpere: num 0 3 2 2 2 0 0 0 0 2 ...
## $ Bromhord: num 0 4 0 3 2 0 2 0 0 4 ...
## $ Chenalbu: num 0 0 0 0 0 0 0 0 0 0 ...
## $ Cirsarve: num 0 0 0 2 0 0 0 0 0 0 ...
## $ Comapalu: num 0 0 0 0 0 0 0 0 0 0 ...
## $ Eleopalu: num 0 0 0 0 0 0 0 4 0 0 ...
## $ Elymrepe: num 4 4 4 4 4 0 0 0 6 0 ...
## $ Empenigr: num 0 0 0 0 0 0 0 0 0 0 ...
## $ Hyporadi: num 0 0 0 0 0 0 0 0 0 0 ...
## $ Juncarti: num 0 0 0 0 0 0 0 4 4 0 ...
## $ Juncbufo: num 0 0 0 0 0 0 2 0 4 0 ...
## $ Lolipere: num 7 5 6 5 2 6 6 4 2 6 ...
## $ Planlanc: num 0 0 0 0 5 5 5 0 0 3 ...
## $ Poaprat : num 4 4 5 4 2 3 4 4 4 4 ...
## $ Poatriv : num 2 7 6 5 6 4 5 4 5 4 ...
## $ Ranuflam: num 0 0 0 0 0 0 0 2 0 0 ...
## $ Rumeacet: num 0 0 0 0 5 6 3 0 2 0 ...
## $ Sagiproc: num 0 0 0 5 0 0 0 2 2 0 ...
## $ Salirepe: num 0 0 0 0 0 0 0 0 0 0 ...
## $ Scorautu: num 0 5 2 2 3 3 3 3 2 3 ...
## $ Trifprat: num 0 0 0 0 2 5 2 0 0 0 ...
## $ Trifrepe: num 0 5 2 1 2 5 2 2 3 6 ...
## $ Vicilath: num 0 0 0 0 0 0 0 0 0 1 ...
## $ Bracruta: num 0 0 2 2 2 6 2 2 2 2 ...
## $ Callcusp: num 0 0 0 0 0 0 0 0 0 0 ...data(dune.env)
summary(dune.env)## A1 Moisture Management Use Manure
## Min. : 2.800 1:7 BF:3 Hayfield:7 0:6
## 1st Qu.: 3.500 2:4 HF:5 Haypastu:8 1:3
## Median : 4.200 4:2 NM:6 Pasture :5 2:4
## Mean : 4.850 5:7 SF:6 3:4
## 3rd Qu.: 5.725 4:3
## Max. :11.5003.2 Conduct a Hellinger transformation
As the pvclust function can not operate with vegan::vegdist, we will use the Euclidean distance. Prior to calculating the clustering results, the community data set is transformed by the Hellinger method as recommended in this article and also shown in a pathway of generating ordination graphs with vegan, BiodiversityR and ggplot2.
# script generated by the BiodiversityR GUI from the constrained ordination menu
dune.Hellinger <- disttransform(dune, method='hellinger')3.3 pvclust analysis
To obtain a dendrogram that represents clustering of sites, the community data set needs to be transposed. Among the results of pvclust are a hclust object and a edges object.
set.seed(2)
dune.pv <- pvclust(t(dune.Hellinger),
method.hclust="mcquitty",
method.dist="euclidean",
nboot=10000)## Bootstrap (r = 0.5)... Done.
## Bootstrap (r = 0.6)... Done.
## Bootstrap (r = 0.7)... Done.
## Bootstrap (r = 0.8)... Done.
## Bootstrap (r = 0.9)... Done.
## Bootstrap (r = 1.0)... Done.
## Bootstrap (r = 1.1)... Done.
## Bootstrap (r = 1.2)... Done.
## Bootstrap (r = 1.3)... Done.
## Bootstrap (r = 1.4)... Done.attributes(dune.pv)## $names
## [1] "hclust" "edges" "count" "msfit" "nboot" "r" "store"
## [8] "version"
##
## $class
## [1] "pvclust"It is possible now to plot the dendrogram for pvlust. The rectangles correspond to thresholds applied to the AU (Approximately Unbiased) p-value (this is a better approximation to unbiased p-values than the BP value).
plot(dune.pv)
pvrect(dune.pv, alpha=0.89, pv="au")
In the following step, the data on edges from pvclust will be added to data on nodes for the ggraph (this may seem strange here, but makes sense in terms of the pathway).
dune.pv$edges## si au bp se.si se.au se.bp
## 1 0.67367308 0.8900162 0.6500593 0.009770065 0.004678329 0.001583062
## 2 0.62470794 0.8646679 0.6509359 0.010472806 0.005409584 0.001580079
## 3 0.52107100 0.8635731 0.4838776 0.012291046 0.005388623 0.001645255
## 4 0.52113062 0.8525799 0.5174110 0.012057944 0.005657406 0.001645565
## 5 0.95626554 0.9895152 0.8144038 0.002496067 0.000769902 0.001345803
## 6 0.58608873 0.8704015 0.5612605 0.011148662 0.005181026 0.001638187
## 7 0.05836824 0.6989306 0.3388617 0.015113634 0.008862102 0.001550838
## 8 0.00000000 0.6973572 0.2073154 0.000000000 0.009848418 0.001326916
## 9 0.49243363 0.8666564 0.4367195 0.013007827 0.005366611 0.001630304
## 10 0.00000000 0.8071869 0.1465370 0.000000000 0.008704205 0.001155411
## 11 0.29408270 0.8664202 0.2572468 0.018146901 0.005953144 0.001429867
## 12 0.42408147 0.8980210 0.2800115 0.016452438 0.004870102 0.001469841
## 13 0.00000000 0.7942302 0.1805106 0.000000000 0.008490440 0.001256977
## 14 0.71627120 0.9296202 0.5444732 0.009239222 0.003338046 0.001648334
## 15 0.67282027 0.8904846 0.6462837 0.009792951 0.004663206 0.001587233
## 16 0.00000000 0.7942395 0.1463469 0.000000000 0.009031488 0.001154938
## 17 0.00000000 0.6579862 0.1959493 0.000000000 0.010479640 0.001300267
## 18 0.00000000 0.5398721 0.1435282 0.000000000 0.012104246 0.001152906
## 19 1.00000000 1.0000000 1.0000000 0.000000000 0.000000000 0.000000000
## v c pchi
## 1 -0.80604734 0.4205668 0.3088314904
## 2 -0.74469145 0.3568429 0.4865154078
## 3 -0.52804525 0.5684691 0.4163645695
## 4 -0.54560966 0.5019529 0.0003789444
## 5 -1.60138616 0.7071441 0.1291560223
## 6 -0.64122844 0.4870626 0.0199835695
## 7 -0.05287773 0.4684495 0.2447419111
## 8 0.14947842 0.6662930 0.2247349650
## 9 -0.47571606 0.6350079 0.0258432352
## 10 0.09191271 0.9594893 0.6549884988
## 11 -0.22888526 0.8807420 0.0312624364
## 12 -0.34377404 0.9265815 0.4122681792
## 13 0.04611690 0.8673041 0.5504418008
## 14 -0.79233900 0.6806293 0.5457562329
## 15 -0.80220781 0.4269015 0.7693126115
## 16 0.11550532 0.9367252 0.9145713157
## 17 0.22460298 0.6315762 0.9794099838
## 18 0.48224497 0.5823565 0.7408072082
## 19 0.00000000 0.0000000 0.00000000004 Step 2: Add information for nodes
4.1 Make sure that all heights are unique
When adding data on nodes (section 4.3), the nodes will be linked via the height variable. It is therefore necessary that all the heights are unique. One method of doing this is by adding a very small but also unique number…
anyDuplicated(dune.pv$hclust$height)## [1] 0if (anyDuplicated(dune.pv$hclust$height) != 0) {
.Machine$double.eps
small.add <- seq_along(dune.pv$hclust$height) * 1e-15
dune.pv$hclust$height <- dune.pv$hclust$height + small.add
}
anyDuplicated(dune.pv$hclust$height)## [1] 04.2 Obtain data sets for nodes and edges
When plotting a hclust object, ggraph internally converts the object to a tbl_graph object. The pathway shown here consists of adding variables to the nodes and edges of the tbl_graph data sets.
dune.graph2 <- as.list(as_tbl_graph(dune.pv$hclust))
dune.nodes <- data.frame(dune.graph2$nodes)
dune.edges <- data.frame(dune.graph2$edges)4.3 Add information on nodes about the merging process
Although the main fields that are needed are the ID and the clustering height, information is also added in the merging process. In the notations used by hclust, negative numbers refer to agglomerations of singletons, whereas positive numbers refer to non-singletons. The information that is added thus shows that row 3 is a branch node at height 0.6159… that has ID 6 (as also in the pvclust dendrogram above) and that corresponds to the merging of sites 12 and 13.
merge.data <- data.frame(cbind(dune.pv$hclust$merge, dune.pv$hclust$height))
names(merge.data) <- c("m1", "m2", "height")
merge.data$ID <- c(1:nrow(merge.data))
dune.nodes2 <- left_join(dune.nodes, merge.data, by="height")
head(dune.nodes2)## height leaf label members m1 m2 ID
## 1 0.0000000 TRUE 12 1 NA NA NA
## 2 0.0000000 TRUE 13 1 NA NA NA
## 3 0.6159341 FALSE 2 -12 -13 6
## 4 0.0000000 TRUE 8 1 NA NA NA
## 5 0.0000000 TRUE 9 1 NA NA NA
## 6 0.0000000 TRUE 3 1 NA NA NA4.4 Add probability values
Via the ID field (the sequence of hierarchical clustering), probability values are now added to the nodes.
edges.pv <- data.frame(dune.pv$edges)
edges.pv$ID <- c(1:nrow(edges.pv))
dune.nodes3 <- left_join(dune.nodes2, edges.pv, by="ID")
head(dune.nodes3)## height leaf label members m1 m2 ID si au bp
## 1 0.0000000 TRUE 12 1 NA NA NA NA NA NA
## 2 0.0000000 TRUE 13 1 NA NA NA NA NA NA
## 3 0.6159341 FALSE 2 -12 -13 6 0.5860887 0.8704015 0.5612605
## 4 0.0000000 TRUE 8 1 NA NA NA NA NA NA
## 5 0.0000000 TRUE 9 1 NA NA NA NA NA NA
## 6 0.0000000 TRUE 3 1 NA NA NA NA NA NA
## se.si se.au se.bp v c pchi
## 1 NA NA NA NA NA NA
## 2 NA NA NA NA NA NA
## 3 0.01114866 0.005181026 0.001638187 -0.6412284 0.4870626 0.01998357
## 4 NA NA NA NA NA NA
## 5 NA NA NA NA NA NA
## 6 NA NA NA NA NA NA4.5 Add environmental data to the leaves
Via the label of the leaves, information on environmental descriptors are added for the leaves.
dune.env$label <- rownames(dune.env)
dune.nodes4 <- left_join(dune.nodes3, dune.env, by="label")
head(dune.nodes4)## height leaf label members m1 m2 ID si au bp
## 1 0.0000000 TRUE 12 1 NA NA NA NA NA NA
## 2 0.0000000 TRUE 13 1 NA NA NA NA NA NA
## 3 0.6159341 FALSE 2 -12 -13 6 0.5860887 0.8704015 0.5612605
## 4 0.0000000 TRUE 8 1 NA NA NA NA NA NA
## 5 0.0000000 TRUE 9 1 NA NA NA NA NA NA
## 6 0.0000000 TRUE 3 1 NA NA NA NA NA NA
## se.si se.au se.bp v c pchi A1
## 1 NA NA NA NA NA NA 5.8
## 2 NA NA NA NA NA NA 6.0
## 3 0.01114866 0.005181026 0.001638187 -0.6412284 0.4870626 0.01998357 NA
## 4 NA NA NA NA NA NA 4.2
## 5 NA NA NA NA NA NA 3.7
## 6 NA NA NA NA NA NA 4.3
## Moisture Management Use Manure
## 1 4 SF Haypastu 2
## 2 5 SF Haypastu 3
## 3 <NA> <NA> <NA> <NA>
## 4 5 HF Pasture 3
## 5 4 HF Hayfield 1
## 6 2 SF Haypastu 45 Step 3: Add information to edges
Now via the from field, information on probability values can be added to edges.
dune.nodes4$SEQ <- c(1:nrow(dune.nodes4))
dune.edges2 <- left_join(dune.edges,
dune.nodes4[, c("SEQ", "height", "au")],
by=c("from" = "SEQ"))
head(dune.edges2)## from to height au
## 1 3 1 0.6159341 0.8704015
## 2 3 2 0.6159341 0.8704015
## 3 8 6 0.5400578 0.8635731
## 4 8 7 0.5400578 0.8635731
## 5 9 5 0.6578777 0.6973572
## 6 9 8 0.6578777 0.69735726 Step 4: Generate the dendrogram with ggraph
With the various variables added to the nodes and the edges, a new tbl_graph object is created.
dune.graph2 <- tbl_graph(nodes=dune.nodes4, edges=dune.edges2, directed=TRUE)
dune.graph2## # A tbl_graph: 39 nodes and 38 edges
## #
## # A rooted tree
## #
## # Node Data: 39 x 22 (active)
## height leaf label members m1 m2 ID si au bp se.si
## <dbl> <lgl> <chr> <int> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl>
## 1 0 TRUE "12" 1 NA NA NA NA NA NA NA
## 2 0 TRUE "13" 1 NA NA NA NA NA NA NA
## 3 0.616 FALSE "" 2 -12 -13 6 0.586 0.870 0.561 0.0111
## 4 0 TRUE "8" 1 NA NA NA NA NA NA NA
## 5 0 TRUE "9" 1 NA NA NA NA NA NA NA
## 6 0 TRUE "3" 1 NA NA NA NA NA NA NA
## # ... with 33 more rows, and 11 more variables: se.au <dbl>, se.bp <dbl>,
## # v <dbl>, c <dbl>, pchi <dbl>, A1 <dbl>, Moisture <ord>, Management <fct>,
## # Use <ord>, Manure <ord>, SEQ <int>
## #
## # Edge Data: 38 x 4
## from to height au
## <int> <int> <dbl> <dbl>
## 1 3 1 0.616 0.870
## 2 3 2 0.616 0.870
## 3 8 6 0.540 0.864
## # ... with 35 more rowsIt now has become relatively straightforward to generate the ggraph by accessing the variables in various aesthetics calls.
First the theme is set to an empty canvas.
BioR.theme <- theme(
panel.background = element_blank(),
panel.border = element_blank(),
panel.grid = element_blank(),
axis.line.x = element_blank(),
axis.line.y = element_line("gray25"),
axis.ticks.y = element_line("gray25"),
text = element_text(size = 12),
axis.text.y = element_text(size = 10, colour = "gray25"),
axis.text.x = element_blank(),
axis.title = element_blank(),
legend.title = element_text(size = 14),
legend.text = element_text(size = 14),
legend.key = element_blank())And now the dendrogram is created.
Within the script, the width of edges and nodes reflects AU. The colour reflects a threshold of 0.89 for the AU.
ggraph1 <- ggraph(dune.graph2, layout="dendrogram", circular=FALSE, height=height) +
scale_y_continuous(sec.axis = dup_axis(name=NULL)) +
geom_edge_bend(aes(edge_width=au, colour=au>=0.89),
alpha=0.7, show.legend=FALSE) +
geom_node_point(aes(filter=leaf, shape=Management),
size=5, colour="black") +
geom_node_point(aes(size=au, fill=au>=0.89),
shape=21, show.legend=FALSE) +
geom_node_text(aes(filter=leaf, label=label),
hjust=1, nudge_y=-0.05, angle=90) +
geom_node_text(aes(filter=!leaf, label=as.character(round(au, 2))),
nudge_y=-0.05) +
BioR.theme +
scale_fill_npg() +
scale_edge_colour_manual(values=pal_npg()(2))
ggraph1## Warning: Removed 20 rows containing missing values (geom_point).
An alternative dendrogram is circular.
ggraph2 <- ggraph(dune.graph2, layout="dendrogram", circular=TRUE, height=NA) +
geom_edge_bend(aes(edge_width=au, colour=au>=0.89),
alpha=0.7, show.legend=FALSE) +
geom_node_point(aes(filter=leaf, shape=Management),
alpha=0.7, size=8, colour="darkolivegreen4") +
geom_node_text(aes(filter=leaf, label=label),
hjust=0.5, angle=0, colour="black") +
geom_node_text(aes(filter=!leaf, label=as.character(round(au, 2))),
colour="black", hjust=0.5) +
BioR.theme +
theme(axis.line.y = element_blank()) +
theme(axis.text.y = element_blank()) +
theme(axis.ticks.y = element_blank()) +
scale_fill_npg() +
scale_edge_colour_manual(values=pal_npg()(2)) +
coord_fixed()
ggraph2
7 Making a hanging tree
7.1 Get data from heights of ‘from’ edge
dune.nodes5 <- dune.nodes4
dune.nodes5$SEQ <- 1:nrow(dune.nodes5)
dune.nodes5$height.old <- dune.nodes5$height
from.heights <- left_join(dune.nodes4[dune.nodes4$leaf == TRUE, c("label", "SEQ")],
dune.edges2[, c("to", "height")],
by=c("SEQ" = "to"))
head(from.heights)## label SEQ height
## 1 12 1 0.6159341
## 2 13 2 0.6159341
## 3 8 4 0.6957780
## 4 9 5 0.6578777
## 5 3 6 0.5400578
## 6 4 7 0.54005787.2 Reduce the heights by desired amount of ‘hanging’
from.heights$hang.height <- from.heights$height - 0.2
dune.nodes6 <- left_join(dune.nodes5,
from.heights[, c("label", "hang.height")],
by="label")
dune.nodes6[dune.nodes6$leaf == TRUE, "height"] <- dune.nodes6[dune.nodes6$leaf == TRUE, "hang.height"]
head(dune.nodes6)## height leaf label members m1 m2 ID si au bp
## 1 0.4159341 TRUE 12 1 NA NA NA NA NA NA
## 2 0.4159341 TRUE 13 1 NA NA NA NA NA NA
## 3 0.6159341 FALSE 2 -12 -13 6 0.5860887 0.8704015 0.5612605
## 4 0.4957780 TRUE 8 1 NA NA NA NA NA NA
## 5 0.4578777 TRUE 9 1 NA NA NA NA NA NA
## 6 0.3400578 TRUE 3 1 NA NA NA NA NA NA
## se.si se.au se.bp v c pchi A1
## 1 NA NA NA NA NA NA 5.8
## 2 NA NA NA NA NA NA 6.0
## 3 0.01114866 0.005181026 0.001638187 -0.6412284 0.4870626 0.01998357 NA
## 4 NA NA NA NA NA NA 4.2
## 5 NA NA NA NA NA NA 3.7
## 6 NA NA NA NA NA NA 4.3
## Moisture Management Use Manure SEQ height.old hang.height
## 1 4 SF Haypastu 2 1 0.0000000 0.4159341
## 2 5 SF Haypastu 3 2 0.0000000 0.4159341
## 3 <NA> <NA> <NA> <NA> 3 0.6159341 NA
## 4 5 HF Pasture 3 4 0.0000000 0.4957780
## 5 4 HF Hayfield 1 5 0.0000000 0.4578777
## 6 2 SF Haypastu 4 6 0.0000000 0.34005787.3 Create the new tbl_graph object
dune.graph3 <- tbl_graph(nodes=dune.nodes6, edges=dune.edges2, directed=TRUE)
dune.graph3## # A tbl_graph: 39 nodes and 38 edges
## #
## # A rooted tree
## #
## # Node Data: 39 x 24 (active)
## height leaf label members m1 m2 ID si au bp se.si
## <dbl> <lgl> <chr> <int> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl>
## 1 0.416 TRUE "12" 1 NA NA NA NA NA NA NA
## 2 0.416 TRUE "13" 1 NA NA NA NA NA NA NA
## 3 0.616 FALSE "" 2 -12 -13 6 0.586 0.870 0.561 0.0111
## 4 0.496 TRUE "8" 1 NA NA NA NA NA NA NA
## 5 0.458 TRUE "9" 1 NA NA NA NA NA NA NA
## 6 0.340 TRUE "3" 1 NA NA NA NA NA NA NA
## # ... with 33 more rows, and 13 more variables: se.au <dbl>, se.bp <dbl>,
## # v <dbl>, c <dbl>, pchi <dbl>, A1 <dbl>, Moisture <ord>, Management <fct>,
## # Use <ord>, Manure <ord>, SEQ <int>, height.old <dbl>, hang.height <dbl>
## #
## # Edge Data: 38 x 4
## from to height au
## <int> <int> <dbl> <dbl>
## 1 3 1 0.616 0.870
## 2 3 2 0.616 0.870
## 3 8 6 0.540 0.864
## # ... with 35 more rows7.4 Generate the dendrogram
ggraph3 <- ggraph(dune.graph3, layout="dendrogram", circular=FALSE, height=height) +
scale_y_continuous(sec.axis = dup_axis(name=NULL)) +
geom_edge_bend(aes(edge_width=au, colour=au>=0.89),
alpha=0.7, show.legend=FALSE) +
geom_node_point(aes(filter=leaf, shape=Management),
size=5, colour="black") +
geom_node_point(aes(size=au, fill=au>=0.89),
shape=21, show.legend=FALSE) +
geom_node_text(aes(filter=leaf, label=label),
hjust=1, nudge_y=-0.05, angle=90) +
geom_node_text(aes(filter=!leaf, label=as.character(round(au, 2))),
nudge_y=-0.05) +
BioR.theme +
scale_fill_npg() +
scale_edge_colour_manual(values=pal_npg()(2))
ggraph3## Warning: Removed 20 rows containing missing values (geom_point).
8 Session Information
sessionInfo()## R version 4.0.2 (2020-06-22)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 18363)
##
## Matrix products: default
##
## locale:
## [1] LC_COLLATE=English_United Kingdom.1252
## [2] LC_CTYPE=English_United Kingdom.1252
## [3] LC_MONETARY=English_United Kingdom.1252
## [4] LC_NUMERIC=C
## [5] LC_TIME=English_United Kingdom.1252
##
## attached base packages:
## [1] tcltk stats graphics grDevices utils datasets methods
## [8] base
##
## other attached packages:
## [1] tidygraph_1.2.0 ggraph_2.0.4 ggsci_2.9
## [4] ggplot2_3.3.2 pvclust_2.2-0 BiodiversityR_2.12-2
## [7] rgl_0.100.54 vegan3d_1.1-2 vegan_2.5-6
## [10] lattice_0.20-41 permute_0.9-5
##
## loaded via a namespace (and not attached):
## [1] minqa_1.2.4 colorspace_1.4-1 ellipsis_0.3.1
## [4] class_7.3-17 rio_0.5.16 htmlTable_2.0.0
## [7] base64enc_0.1-3 rstudioapi_0.11 farver_2.0.3
## [10] graphlayouts_0.7.1 ggrepel_0.8.2 fansi_0.4.1
## [13] splines_4.0.2 knitr_1.28 effects_4.1-4
## [16] polyclip_1.10-0 Formula_1.2-3 jsonlite_1.6.1
## [19] nloptr_1.2.2.1 cluster_2.1.0 Rcmdr_2.6-2
## [22] png_0.1-7 ggforce_0.3.2 shiny_1.4.0.2
## [25] compiler_4.0.2 backports_1.1.7 assertthat_0.2.1
## [28] Matrix_1.2-18 fastmap_1.0.1 cli_2.0.2
## [31] survey_4.0 tweenr_1.0.1 later_1.1.0.1
## [34] tcltk2_1.2-11 acepack_1.4.1 htmltools_0.5.0
## [37] tools_4.0.2 igraph_1.2.6 gtable_0.3.0
## [40] glue_1.4.1 dplyr_1.0.2 Rcpp_1.0.4.6
## [43] carData_3.0-4 cellranger_1.1.0 vctrs_0.3.4
## [46] nlme_3.1-148 crosstalk_1.1.0.1 xfun_0.15
## [49] stringr_1.4.0 openxlsx_4.1.5 lme4_1.1-23
## [52] mime_0.9 miniUI_0.1.1.1 lifecycle_0.2.0
## [55] RcmdrMisc_2.7-0 statmod_1.4.34 MASS_7.3-51.6
## [58] zoo_1.8-8 scales_1.1.1 hms_0.5.3
## [61] promises_1.1.1 parallel_4.0.2 sandwich_2.5-1
## [64] RColorBrewer_1.1-2 yaml_2.2.1 curl_4.3
## [67] gridExtra_2.3 rpart_4.1-15 latticeExtra_0.6-29
## [70] stringi_1.4.6 nortest_1.0-4 e1071_1.7-3
## [73] checkmate_2.0.0 boot_1.3-25 zip_2.0.4
## [76] manipulateWidget_0.10.1 rlang_0.4.8 pkgconfig_2.0.3
## [79] evaluate_0.14 purrr_0.3.4 labeling_0.3
## [82] htmlwidgets_1.5.1 tidyselect_1.1.0 magrittr_1.5
## [85] R6_2.4.1 generics_0.1.0 Hmisc_4.4-0
## [88] DBI_1.1.0 pillar_1.4.4 haven_2.3.1
## [91] foreign_0.8-80 withr_2.2.0 mgcv_1.8-31
## [94] survival_3.1-12 scatterplot3d_0.3-41 abind_1.4-5
## [97] nnet_7.3-14 tibble_3.0.1 crayon_1.3.4
## [100] car_3.0-8 utf8_1.1.4 relimp_1.0-5
## [103] rmarkdown_2.3 viridis_0.5.1 jpeg_0.1-8.1
## [106] grid_4.0.2 readxl_1.3.1 data.table_1.12.8
## [109] forcats_0.5.0 digest_0.6.25 webshot_0.5.2
## [112] xtable_1.8-4 tidyr_1.1.2 httpuv_1.5.4
## [115] munsell_0.5.0 viridisLite_0.3.0 mitools_2.4