Barry Grant < http://thegrantlab.org/bimm143/ >
2020-09-15 (14:11:53 on Tue, Sep 15)

1. Overview

The ability to make clear and compelling data visualizations is a vital skill for scientists. The difference between good and bad figures can be the difference between a highly influential or an obscure paper, a grant or contract won or lost, a job interview gone well or poorly. In short, If you want to be successful in any technical field you will need to master these skills!

The goals for this hands-on session are for you to:

  • Understand major scientific plot types and when you might want to use them.
  • Learn the fundamentals of the ggplot2 package and what it can do for you.

2. Background

One of the biggest attractions of the R programming language is the ability to have complete programmatic control over the plotting of complex graphs and figures. R offers a ridiculously large set of tools and packages for data visualization. The core R language already provides a rich set of plotting functions and plot types. These plotting functions require users to specify how to plot each element on the canvas step by step. These “base R plots” offer complete control over virtually every pixel. However, they can be fiddly and time consuming to get just the way you want. By contrast, the ggplot2 package allows the specification of all plots through set of common plotting layers that minimally includes:

  • Specifying the input data (in the form of a data.frame),
  • How the data maps to aesthetic features of the plot (the x and y axis, color, plotting character, line type, etc.), and
  • The geometric layers used in the plot (such as points, bars, lines etc.)

Data visualization with ggplot always involves these steps. Once you have mastered this sequence of steps we will layer on additional customizations and you will see that beautiful and sophisticated plots come within your reach very quickly. Let’s get cracking!


Side note:

If you have not already, please watch this weeks intro videos, which cover why we want to visualize data graphically, what makes an effective figure, and an introduction to ggplot. You may also wish to (re) visit our previous introduction to R and RStudio as well as our video on major R data structures, data types, and using functions.



  • Q. For which phases is data visualization important in our scientific workflows?

  • Q. True or False? The ggplot2 package comes already installed with R?

Data visualization is important for all phases of our scientific workflows from exploratory data analysis (EDA), quality control and the detection of outliers, through formal analysis and the communication of results.

The ggplot2 package does not come pre-installed with R. Before you use it for the first time you will need to install it with the install.packages("ggplot2) command in your R console panel of RStudio.

We see in next week’s class (and throughout the rest of the course) that lot’s of useful functionality comes in the form of add-on R packages.
You can think of “base R” like your smartphone’s OS when you first take it out of the box and “R packages” like apps that you can optionally install to allow you to do more cool things.

3. Getting Organized

Creating a Project

We will begin by getting organized. This entails you opening up RStudio and creating a new RStudio Project, then creating a new R script for storing your work and notes for this session.

Side-note: If you are alrady fimilar with RMarkdown format documents feel free to use one of these rather than an RScript. If you have not yet heard of these, don’t worry we will be building towards these in our next class.

  • Begin by opening RStudio and creating a new Project File > New Project > New Directory > New Project make sure you are working in the directory (a.k.a. folder!) where you want to keep all your work for this class organized. For example, for me this is a directory on my Desktop with the class name (see animated figure below). We will create our project as a subdirectory called class05 in this location.



Side-note: The key step here is to name your project after this class session (i.e. “class05”) and make sure it is a sub-directory of where ever you are organizing all your work for this course.

Create an R script

Finally, open a new R script: File > New File > R Script and save as class05.R. This is nothing more than a text file where we can write and save our R code. The big advantage is that we will have a record of our work and thus be able to reproduce and automate our analysis later. We will also turn this into a fancy HTML and PDF report for sharing with others (more on this later…)

4. Common Plot Types

There are many plot types available that can help you understand different features and relationships in your data.

During the exploratory data analysis phase we typically want to detect the most obvious patterns by looking at each variable in isolation or by detecting relationships of variables against others. The used plot type is also determined by the data type of the input variables like continuous numeric or discrete categorical.

Scatter Plots

Scatter plots are used to visualize the relationship between two numeric variables. The position of each point represents the value of the variables on the x and y-axis.


Line Graphs

Line graphs are used to visualize the trajectory of one numeric variable against another which are connected through lines. They are well suited if values change continuously - like temperature over time.


Bar Charts and Histograms

Bar charts visualize numeric values grouped by categories. Each category is represented by one bar with a height defined by each numeric value. Histograms are specific bar charts that aim to summarize the number of occurrences of numeric values over a set of value ranges (a.k.a. bins). They are typically used to examine the distribution of numeric values.


Others

Other frequently used plot types in the biosciences include:

  • Box plots: Show summary distributional information of numeric values grouped in categories as boxes. Great to quickly compare multiple distributions.
  • Violin plots: Same as box plots but show distributions as violins.
  • Heat Maps: Show interactions of variables - typically correlations - as rastered image highlighting areas of high interaction.
  • Network Graphs: Show connections (as lines - a.k.a. edges or vertices) between nodes (typically shown as spheres).
  • Density plots: A popular alternative to histograms.
  • Dendrograms: Also known as cluster trees display the results of a clustering analysis (we will have a separate class on these).


  • Q. Which plot types are typically NOT used to compare distributions of numeric variables?

  • Q. Which statement about data visualization with ggplot2 is incorrect?

There are multiple plot types that can be used to examine and compare numeric data distributions. We will examine only a handful of the most common types in this session.

For a detailed discussion and compassion of these approaches see Chapters 7-9 of Claus Wilke’s excellent book Fundamentalks of Data Visualization.

Due to the importance of data visualization R offers a very large set of tools and packages in this area. Indeed, the core R language itself provides a rich set of plotting functions and plot types. Personally, I love “base R plots” but I have been using them for many, many years and acknowledge that they are not the easiest for newcomers to pick up quickly. The ggplot2 package is a popular alternative but is only one of many. Particularly noteworthy alternatives include interactive packages like plotly, rgl, and various javascript based R graphics packages.

5. Creating Scatter Plots

In this section we will focus on:

  • Defining a dataset for your plot using the main ggplot() function.
  • Specifying how your data maps to plot aesthetics with the aes() function.
  • Adding geometric layers using the geom_point() function.
  • Combining the above function calls with + operator to make your plot.

Introduction to scatter plots

Scatter plots use points to visualize the relationship between two numeric variables. The position of each point represents the value of the variables on the x- and y-axis. Let’s see an example of a scatter plot to understand the relationship between the speed and the stopping distance of cars:


Each point in this plot represents a car. Each car starts to break at a speed given on the bottom x-axis and travels the distance shown on the side y-axis until full stop. If we take a look at all points in the plot, we can clearly see that it takes faster cars a longer distance until they are completely stopped.

Scatter plots like this one allow us to visualize the relationship between two numeric variables (in this case speed and stopping distance). The core of this plot is built with only three short lines of code that involve calling three ggplot2 functions as we will see in a moment (ggplot(), aes() and geom_point()).

We will use these same three functions to produce all sorts of scatter plots like the following plot that shows gene expression changes upon treating a particular cell line with a new anit-viral drug:


In this plot points represent individual genes. The red points are for genes that are up-regulated when treated with the drug (i.e. have more expression) and blue points are for down-regulated genes. Gray points are for genes with a non-significant difference in expression values when we look across replicate experiments. We will learn much more about these types of datasets in an upcoming class.

Our focus for this session is to learn how to produce a plot like this. The steps (and R functions) used are the same as those for the previous plot (namely ggplot(), aes(), and geom_point()). Are next sections will focus on each of these in turn. Once we understand these core functions we will be able to make small changes and additions to make all sorts of cool plots.

Specifing a dataset with ggplot()

To create plots with ggplot2 you first need to load the package using library(ggplot2).

After the package has been loaded specify the dataset to be used as an argument of the main ggplot() function. For example, we specify a plot using the in-built cars dataset. You should type (not paste) this code into your R console to see the effect:

library(ggplot2)

ggplot(cars)


N.B. Note that this command does not plot anything but a blank gray canvas yet. The ggplot() function alone just defines the dataset for the plot and creates an empty base on top of which we will add additional layers to build up our plot.

Side-note: If you get an error message at this stage then you likely need to install the ggplot2 package first with install.packages("ggplot2").

Specifing aesthetic mappings with aes()

The ggplot2 package uses the concept of aesthetics, which map variables (i.e. columns) from your dataset to the visual features of the plot. The most common aesthetics include x and y that determine the x- and y-axis coordinates of the points to plot (we will see others further below). The aesthetics are mapped with a call to the aes() function.

We will use the columns labeled speed and distance from the cars dataset to set the x and y aesthetics of our plot. Critically, we combine our call to the aes() function with our previous specification of the input dataset with the ggplot(cars) function call from above.

ggplot(cars) +
  aes(x=speed, y=dist)

N.B. Note how we combine the ggplot(cars) function call and the aes(x=speed, y=dist) function calls with the + plus sign. We will keep adding more layers to our plot in this same way as we will see in a moment.

As we can see above we don’t have any points in our plot yet. Adding them is the job of the geom_point() function discussed next.

Specifing a geom layer with geom_point()

Next we need to add one of ggplot’s geometric layers (or geoms) to define how we want to visualize our dataset. At this early stage we can thing of geoms as defining the plot type we actually want to produce. For example, geom_line() produces a line plot, geom_bar() produces a bar plot, geom_boxplot() a box plot (more on this later).


Here, we use geom_point() to add - you guessed it - points! Let’s take our code snippet from above and add a geom_point() function call:

ggplot(cars) +
  aes(x=speed, y=dist) +
  geom_point()

N.B. Note again that we use the + operator to connect the ggplot(cars) with aes(x=speed, y=dist) and now the geom_point() lines. Through the linking ggplot knows that the mapped speed and dist variables are taken from the cars dataset. finally the geom_point() line instructs ggplot to plot the mapped variables as points.

Side-note: The required steps to create a scatter plot with ggplot can be summarized as follows:

  • Load the package ggplot2 using library(ggplot2).
  • Specify the dataset to be plotted using ggplot().
  • Use the + operator to add layers to the plot.
  • Map variables from the dataset to aesthetic properties through the aes() function.
  • Add a geometric layer with to define the shapes to be plotted. In the case of scatter plots, use geom_point().

  • Q. Which geometric layer should be used to create scatter plots in ggplot2?

  • Q. In your own RStudio can you add a trend line layer to help show the relationship between the plot variables with the geom_smooth() function?

ggplot(cars) +
  aes(x=speed, y=dist) +
  geom_point() +
  geom_smooth()

  • Q. Can you finish this plot by adding various label annotations with the labs() function and changing the plot look to a more conservative “black & white” theme by adding the theme_bw() function:
ggplot(cars) + 
  aes(x=speed, y=dist) +
  geom_point() +
  labs(title="Speed and Stopping Distances of Cars",
       x="Speed (MPH)", 
       y="Stopping Distance (ft)",
       subtitle = "Your informative subtitle text here",
       caption="Dataset: 'cars'") +
  geom_smooth(method="lm", se=FALSE) +
  theme_bw()

Adding more plot aesthetics through aes()

In their most basic form scatter plots can only visualize datasets in two dimensions through the x and y aesthetics passed to the geom_point() layer. However, most data sets have more than two variables and thus might require additional plotting dimensions. Ggplot makes it very easy to map additional variables to different plotting aesthetics like size, transparency alpha and color.

Here we will cover how to:

  • Adjust the point size of a scatter plot using the size parameter.
  • Change the point color of a scatter plot using the color parameter.
  • Set a parameter alpha to change the transparency of all points.

We will also try to stress an important point that is often confusing for newcomers to ggplot: How to differentiate between aesthetic mappings (plot features you want mapped to variables in your data) and constant parameters (specifications of plot features you want to remain the same or otherwise come from elsewhere - i.e. not from your data).

Let’s turn for a moment to more relevant example data set. The code below reads the results of a differential expression analysis where a new anti-viral drug is being tested.

url <- "https://bioboot.github.io/bimm143_S20/class-material/up_down_expression.txt"
genes <- read.delim(url)
head(genes)

  • Q. Use the nrow() function to find out how many genes are in this dataset. What is your answer?

  • Q. Use the colnames() function and the ncol() function on the genes data frame to find out what the column names are (we will need these later) and how many columns there are. How many columns did you find?

  • Q. Use the table() function on the State column of this data.frame to find out how many ‘up’ regulated genes there are. What is your answer?

  • Q. Using your values above and 2 significant figures. What fraction of total genes is up-regulated in this dataset?

Once you have read in the dataset with the read.delim() function and stored the results as a new object called genes (i.e. run the code above), you can issue the following command to see how many rows there are:

nrow(genes)

We can use the ncol() and colnames() like so:

colnames(genes)
## [1] "Gene"       "Condition1" "Condition2" "State"
ncol(genes)
## [1] 4

and the following command to get a summary table of the State column:

table(genes$State)

Finally we can combine these to find the answer to the last question. You can optionally use the round() function to round the numbers to however many figures you want:

round( table(genes$State)/nrow(genes) * 100, 2 )
## 
##       down unchanging         up 
##       1.39      96.17       2.44

We can make a first basic scatter plot of this dataset by following the same recipe we have already seen, namely:

  • Pass the genes data.frame as input to the ggplot() function.
  • Then use the aes() function to set the x and y aesthetic mappings to the Condition1 and Condition2 columns.
  • Finally add a geom_point() layer to add points to the plot.
  • Don’t forget to add layers step-wise with the + operator at the end of each line.
  • Q. Complete the code below to produce the following plot
ggplot(___) + 
    aes(x=Condition1, y=___) _
    _____

ggplot(genes) + 
    aes(x=Condition1, y=Condition2) +
    geom_point()

There is extra information in this dataset, namely the State column, which tells us whether the difference in expression values between conditions is statistically significant. Let’s map this column to point color:

p <- ggplot(genes) + 
    aes(x=Condition1, y=Condition2, col=State) +
    geom_point()
p

I am not a big fan of these default colors so let’s change them up by adding another layer to explicitly specifcy our color scale. Note how we saved our previous plot as the object p and can use it now to add more layers:

p + scale_colour_manual( values=c("blue","gray","red") )

  • Q. Nice, now add some plot annotations to the p object with the labs() function so your plot looks like the following:

p + scale_colour_manual(values=c("blue","gray","red")) +
    labs(title="Gene Expresion Changes Upon Drug Treatment",
         x="Control (no drug) ",
         y="Drug Treatment")

6. OPTIONAL: Going Further

The following sections are considered optional extensions for motivated students.

The gapminder dataset contains economic and demographic data about various countries since 1952. This dataset features in your DataCamp course for this week and you can find out more about the portion we are using here.

The data itself is available as either a tab-delimited file online, or via the gapmider package. You can use whichever method you are more comfortable with to obtain the dataset. I show both below:

install.packages("gapminder")
library(gapminder)

Or read the TSV file from online:

# File location online
url <- "https://raw.githubusercontent.com/jennybc/gapminder/master/inst/extdata/gapminder.tsv"

gapminder <- read.delim(url)

This dataset covers many years and many countries. Before we make some plots we will use some dplyr code to focus in on a single year. You can install the dplyr package with the command install.packages("dplyr"). We will learn more about dplyr in next weeks class. For now, feel free to just copy and paste the following line that takes gapmider data frame and filters to contain only the rows with a year value of 2007.

# install.packages("dplyr")  ## uncoment to install if needed
library(dplyr)

gapminder_2007 <- gapminder %>% filter(year==2007)

Let’s consider the gapminder_2007 dataset which contains the variables GDP per capita gdpPercap and life expectancy lifeExp for 142 countries in the year 2007

  • Q. Complete the code below to produce a first basic scater plot of this gapminder_2007 dataset:
ggplot(gapminder_2007) +
  aes(x=___, y=___) +
  ___
ggplot(gapminder_2007) +
  aes(x=gdpPercap, y=lifeExp) +
  geom_point()

There are quite a few points that are nearly on top of each other in the above plot. One useful approach here is to add an alpha=0.4 argument to your geom_point() call to make the points slightly transparent. This will help us see things a little more clearly later on.

ggplot(gapminder_2007) +
  aes(x=gdpPercap, y=lifeExp) +
  geom_point(alpha=0.5)

Adding more varables to aes()

By mapping the continent variable to the point color aesthetic and the population pop (in millions) through the point size argument to aes() we can obtain a much richer plot that now includes 4 different variables from the data set:

ggplot(gapminder_2007) +
  aes(x=gdpPercap, y=lifeExp, color=continent, size=pop) +
  geom_point(alpha=0.5)

N.B. Here each new aesthetic we add results in a new dimension to our scatter plot. We see that in the resulting plot each point is colored differently based on the continent of each country. ggplot uses the coloring scheme based on the categorical data type of the variable continent.

By contrast, let’s see how the plot looks like if we color the points by the numeric variable population pop:

ggplot(gapminder_2007) + 
  aes(x = gdpPercap, y = lifeExp, color = pop) +
  geom_point(alpha=0.8)

The scale immediately changes to continuous as can be seen in the legend and the light-blue points are now the countries with the highest population number (China and India).

Adjusting point size

For the gapminder_2007 dataset we can plot the GDP per capita (x=gdpPercap) vs. the life expectancy (y=lifeExp) and set the point size based on the population (size=pop) of each country we can use:

ggplot(gapminder_2007) + 
  aes(x = gdpPercap, y = lifeExp, size = pop) +
  geom_point(alpha=0.5)

However, if you look closely we see that the point sizes in the plot above do not clearly reflect the population differences in each country. If we compare the point size representing a population of 250 million people with the one displaying 750 million, we can see, that their sizes are not proportional. Instead, the point sizes are binned by default. To reflect the actual population differences by the point size we can use the scale_size_area() function instead. The scaling information can be added like any other ggplot object with the + operator:

ggplot(gapminder_2007) + 
  geom_point(aes(x = gdpPercap, y = lifeExp,
                 size = pop), alpha=0.5) + 
  scale_size_area(max_size = 10)

Note that we have adjusted the point’s max_size which results in bigger point sizes.

-Q. Can you addapt the code you have leaqrned thus far to reproduce our gapminder scatter plot for the year 1957? What do you notice abouyt this plot is it easy to compare with the one for 2007?

Steps to produce your 1957 plot should include:

  • Use dplyr to filter the gapmider dataset to include only the year 1957 (check above for how we did this for 2007).
  • Save your result as gapminder_1957.
  • Use the ggplot() function and specify the gapminder_1957 dataset as input
  • Add a geom_point() layer to the plot and create a scatter plot showing the GDP per capita gdpPercap on the x-axis and the life expectancy lifeExp on the y-axis
  • Use the color aesthetic to indicate each continent by a different color
  • Use the size aesthetic to adjust the point size by the population pop
  • Use scale_size_area() so that the point sizes reflect the actual population differences and set the max_size of each point to 15 -Set the opacity/transparency of each point to 70% using the alpha=0.7 parameter
gapminder_1957 <- gapminder %>% filter(year==1957)

ggplot(gapminder_1957) + 
  aes(x = gdpPercap, y = lifeExp, color=continent,
                 size = pop) +
  geom_point(alpha=0.7) + 
  scale_size_area(max_size = 10)

Q. Do the same steps above but include 1957 and 2007 in your input dataset for ggplot(). You should now include the layer facet_wrap(~year) to produce the following plot:

gapminder_1957 <- gapminder %>% filter(year==1957 | year==2007)

ggplot(gapminder_1957) + 
  geom_point(aes(x = gdpPercap, y = lifeExp, color=continent,
                 size = pop), alpha=0.7) + 
  scale_size_area(max_size = 10) +
  facet_wrap(~year)

7. OPTIONAL: Bar Charts

In this section we will cover:

  • How to create bar charts using geom_col().
  • How to fill bars with color using the fill aesthetic.
  • Order your bars by the trend you are most interested in.
  • Flip (or rotate) your plots to help with presentation clarity.
  • Use alternative plot types when bar charts become too crowded.

Introduction to bar charts

Bar charts visualize numeric values grouped by categories. Each category is represented by one bar with a height defined by each numeric value.

Bar charts are well suited to compare values among different groups e.g. number of votes by parties, number of people in different countries or GDP per capita in different countries. Bar charts are a bit spacious and work best if the number of groups to compare is rather small.

Below you can find an example showing the number of people (in millions) in the five biggest countries by population in 2007:

gapminder_top5 <- gapminder %>% 
  filter(year==2007) %>% 
  arrange(desc(pop)) %>% 
  top_n(5, pop)

gapminder_top5

Creating a simple bar chart

In ggplot2, bar charts are created using the geom_col() geometric layer. The geom_col() layer requires the x aesthetic mapping which defines the different bars to be plotted. The height of each bar is defined by the variable specified in the y aesthetic mapping. Both mappings, x and y are required for geom_col(). Let’s create our first bar chart with the gapminder_top5 dataset. It contains population (in millions) and life expectancy data for the biggest countries by population in 2007.

ggplot(gapminder_top5) + 
  geom_col(aes(x = country, y = pop))

We see that the resulting bars are sorted by the country names in alphabetical order by default.

Q. Plot life expectancy by country

Create a bar chart showing the life expectancy of the five biggest countries by population in 2007.

  • Use the ggplot() function and specify the gapminder_top5 dataset as input
  • Add a geom_col() layer to the plot
  • Plot one bar for each country (x aesthetic)
  • Use life expectancy lifeExp as bar height (y aesthetic)

Filling bars with color

Like other geoms geom_col() allows users to map additional dataset variables to the color attribute of the bar. The fill aesthetic can be used to fill the entire bars with color. A usual confusion is the color aesthetic which specifies the line color of each bar’s border instead of the fill color.

Based on the gapminder_top5 dataset we plot the population (in millions) of the biggest countries and use the continent variable to color each bar:

ggplot(gapminder_top5) + 
  geom_col(aes(x = country, y = pop, fill = continent))

Since the continent variable is a categorical variable the bars have a clear color scheme for each continent. Let’s see what happens if we use a numeric variable like life expectancy lifeExp instead:

ggplot(gapminder_top5) + 
  geom_col(aes(x = country, y = pop, fill = lifeExp))

The bar colors have now changed according the continuous legend on the right. We see that also numeric variables can be used to fill bars.

Q. Plot population size by country

Create a bar chart showing the population (in millions) of the five biggest countries by population in 2007.

  • Use the ggplot() function and specify the gapminder_top5 dataset as input
  • Add a geom_col() layer to the plot
  • Plot one bar for each country (x aesthetic)
  • Use population pop as bar height (y aesthetic)
  • Use the GDP per capita gdpPercap as fill aesthetic
ggplot(gapminder_top5) +
  aes(x=country, y=pop, fill=gdpPercap) +
  geom_col()

And change the order of the bars

ggplot(gapminder_top5) +
  aes(x=reorder(country, -pop), y=pop, fill=gdpPercap) +
  geom_col()

and just fill by country

ggplot(gapminder_top5) +
  aes(x=reorder(country, -pop), y=pop, fill=country) +
  geom_col(col="gray30") +
  guides(fill=FALSE)

Flipping bar charts

In some circumstances it might be useful to rotate (or “flip”) your plots to enable a more clear visualization. For this we can use the coord_flip() function. Lets look at an example considering arrest data in US states. This is another inbult dataset called USArrests.

head(USArrests)
USArrests$State <- rownames(USArrests)
ggplot(USArrests) +
  aes(x=reorder(State,Murder), y=Murder) +
  geom_col() +
  coord_flip()

Hmm… this is too crowded for an effective display in small format. Let’s try an alternative custom visualization by combining geom_point() and geom_segment():

ggplot(USArrests) +
  aes(x=reorder(State,Murder), y=Murder) +
  geom_point() +
  geom_segment(aes(x=State, 
                   xend=State, 
                   y=0, 
                   yend=Murder), color="blue") +
  coord_flip()

We will be exploring and producing different plot types in this way throughout the rest of our course going forward. Happy ggplotting!

About this document

Here we use the sessionInfo() function to report on our R systems setup at the time of document execution.

sessionInfo()
## R version 4.0.2 (2020-06-22)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Catalina 10.15.6
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] dplyr_1.0.2.9000 ggplot2_3.3.2    labsheet_0.1.2  
## 
## loaded via a namespace (and not attached):
##  [1] compiler_4.0.2   pillar_1.4.6     tools_4.0.2      digest_0.6.25   
##  [5] lattice_0.20-41  nlme_3.1-149     jsonlite_1.7.1   evaluate_0.14   
##  [9] lifecycle_0.2.0  tibble_3.0.3     gtable_0.3.0     mgcv_1.8-33     
## [13] pkgconfig_2.0.3  png_0.1-7        rlang_0.4.7      Matrix_1.2-18   
## [17] yaml_2.2.1       xfun_0.17        withr_2.2.0      stringr_1.4.0   
## [21] knitr_1.29       generics_0.0.2   vctrs_0.3.4      grid_4.0.2      
## [25] tidyselect_1.1.0 glue_1.4.2       R6_2.4.1         rmarkdown_2.3   
## [29] purrr_0.3.4      farver_2.0.3     magrittr_1.5     splines_4.0.2   
## [33] scales_1.1.1     ellipsis_0.3.1   htmltools_0.5.0  colorspace_1.4-1
## [37] labeling_0.3     stringi_1.5.3    munsell_0.5.0    crayon_1.3.4
 

Powered by labsheet