Category: statistics

Bayesian Statistics using R, Python, and Stan

Bayesian Statistics using R, Python, and Stan

For a year now, this course on Bayesian statistics has been on my to-do list. So without further ado, I decided to share it with you already.

Richard McElreath is an evolutionary ecologist who is famous in the stats community for his work on Bayesian statistics.

At the Max Planck Institute for Evolutionary Anthropology, Richard teaches Bayesian statistics, and he was kind enough to put his whole course on Statistical Rethinking: Bayesian statistics using R & Stan open access online.

You can find the video lectures here on Youtube, and the slides are linked to here:

Richard also wrote a book that accompanies this course:

For more information abou the book, click here.

For the Python version of the code examples, click here.

10 Guidelines to Better Table Design

10 Guidelines to Better Table Design

Jon Schwabisch recently proposed ten guidelines for better table design.

Next to the academic paper, Jon shared his recommendations in a Twitter thread.

Let me summarize them for you:

  • Right-align your numbers
  • Left-align your texts
  • Use decimals appropriately (one or two is often enough)
  • Display units (e.g., $, %) sparsely (e.g., only on first row)
  • Highlight outliers
  • Highlight column headers
  • Use subtle highlights and dividers
  • Use white space between rows and columns
  • Use white space (or dividers) to highlight groups
  • Use visualizations for large tables
Highlights in a table. Via
Visualizations in a table. Via
Example of a well-organized table. Via
How most statistical tests are linear models

How most statistical tests are linear models

Jonas Kristoffer Lindeløv wrote a great visual explanation of how the most common statistical tests (t-test, ANOVA, ANCOVA, etc) are all linear models in the back-end.

Jonas’ original blog uses R programming to visually show how the tests work, what the linear models look like, and how different approaches result in the same statistics.

George Ho later remade a Python programming version of the same visual explanation.

If I was thought statistics and methodology this way, I sure would have struggled less! Have a look yourself:

Data Science vs. Data Alchemy – by Lucas Vermeer

Data Science vs. Data Alchemy – by Lucas Vermeer

How do scurvy, astronomy, alchemy and data science relate to each other?

In this goto conference presentation, Lucas Vermeer — Director of Experimentation at — uses some amazing storytelling to demonstrate how the value of data (science) is largely by organizations capability to gather the right data — the data they actually need.

It’s a definite recommendation to watch for data scientists and data science leaders out there.

Here are the slides, and they contain some great oneliners:

Create a publication-ready correlation matrix, with significance levels, in R

Create a publication-ready correlation matrix, with significance levels, in R

In most (observational) research papers you read, you will probably run into a correlation matrix. Often it looks something like this:


In Social Sciences, like Psychology, researchers like to denote the statistical significance levels of the correlation coefficients, often using asterisks (i.e., *). Then the table will look more like this:

Table 4 from Family moderators of relation between community ...

Regardless of my personal preferences and opinions, I had to make many of these tables for the scientific (non-)publications of my Ph.D..

I remember that, when I first started using R, I found it quite difficult to generate these correlation matrices automatically.

Yes, there is the cor function, but it does not include significance levels.

Then there the (in)famous Hmisc package, with its rcorr function. But this tool provides a whole new range of issues.

What’s this storage.mode, and what are we trying to coerce again?

Soon you figure out that Hmisc::rcorr only takes in matrices (thus with only numeric values). Hurray, now you can run a correlation analysis on your dataframe, you think…

Yet, the output is all but publication-ready!

You wanted one correlation matrix, but now you have two… Double the trouble?

To spare future scholars the struggle of the early day R programming, I would like to share my custom function correlation_matrix.

My correlation_matrix takes in a dataframe, selects only the numeric (and boolean/logical) columns, calculates the correlation coefficients and p-values, and outputs a fully formatted publication-ready correlation matrix!

You can specify many formatting options in correlation_matrix.

For instance, you can use only 2 decimals. You can focus on the lower triangle (as the lower and upper triangle values are identical). And you can drop the diagonal values:

Or maybe you are interested in a different type of correlation coefficients, and not so much in significance levels:

For other formatting options, do have a look at the source code below.

Now, to make matters even more easy, I wrote a second function (save_correlation_matrix) to directly save any created correlation matrices:

Once you open your new correlation matrix file in Excel, it is immediately ready to be copy-pasted into Word!

If you are looking for ways to visualize your correlations do have a look at the packages corrr and corrplot.

I hope my functions are of help to you!

Do reach out if you get to use them in any of your research papers!

I would be super interested and feel honored.


#' correlation_matrix
#' Creates a publication-ready / formatted correlation matrix, using `Hmisc::rcorr` in the backend.
#' @param df dataframe; containing numeric and/or logical columns to calculate correlations for
#' @param type character; specifies the type of correlations to compute; gets passed to `Hmisc::rcorr`; options are `"pearson"` or `"spearman"`; defaults to `"pearson"`
#' @param digits integer/double; number of decimals to show in the correlation matrix; gets passed to `formatC`; defaults to `3`
#' @param decimal.mark character; which decimal.mark to use; gets passed to `formatC`; defaults to `.`
#' @param use character; which part of the correlation matrix to display; options are `"all"`, `"upper"`, `"lower"`; defaults to `"all"`
#' @param show_significance boolean; whether to add `*` to represent the significance levels for the correlations; defaults to `TRUE`
#' @param replace_diagonal boolean; whether to replace the correlations on the diagonal; defaults to `FALSE`
#' @param replacement character; what to replace the diagonal and/or upper/lower triangles with; defaults to `""` (empty string)
#' @return a correlation matrix
#' @export
#' @examples
#' `correlation_matrix(iris)`
#' `correlation_matrix(mtcars)`
correlation_matrix <- function(df, 
                               type = "pearson",
                               digits = 3, 
                               decimal.mark = ".",
                               use = "all", 
                               show_significance = TRUE, 
                               replace_diagonal = FALSE, 
                               replacement = ""){
  # check arguments
    digits >= 0
    use %in% c("all", "upper", "lower")
  # we need the Hmisc package for this
  # retain only numeric and boolean columns
  isNumericOrBoolean = vapply(df, function(x) is.numeric(x) | is.logical(x), logical(1))
  if (sum(!isNumericOrBoolean) > 0) {
    cat('Dropping non-numeric/-boolean column(s):', paste(names(isNumericOrBoolean)[!isNumericOrBoolean], collapse = ', '), '\n\n')
  df = df[isNumericOrBoolean]
  # transform input data frame to matrix
  x <- as.matrix(df)
  # run correlation analysis using Hmisc package
  correlation_matrix <- Hmisc::rcorr(x, type = type)
  R <- correlation_matrix$r # Matrix of correlation coeficients
  p <- correlation_matrix$P # Matrix of p-value 
  # transform correlations to specific character format
  Rformatted = formatC(R, format = 'f', digits = digits, decimal.mark = decimal.mark)
  # if there are any negative numbers, we want to put a space before the positives to align all
  if (sum(! & R < 0) > 0) {
    Rformatted = ifelse(! & R > 0, paste0(" ", Rformatted), Rformatted)

  # add significance levels if desired
  if (show_significance) {
    # define notions for significance levels; spacing is important.
    stars <- ifelse(, "", ifelse(p < .001, "***", ifelse(p < .01, "**", ifelse(p < .05, "*", ""))))
    Rformatted = paste0(Rformatted, stars)
  # make all character strings equally long
  max_length = max(nchar(Rformatted))
  Rformatted = vapply(Rformatted, function(x) {
    current_length = nchar(x)
    difference = max_length - current_length
    return(paste0(x, paste(rep(" ", difference), collapse = ''), sep = ''))
  }, FUN.VALUE = character(1))
  # build a new matrix that includes the formatted correlations and their significance stars
  Rnew <- matrix(Rformatted, ncol = ncol(x))
  rownames(Rnew) <- colnames(Rnew) <- colnames(x)
  # replace undesired values
  if (use == 'upper') {
    Rnew[lower.tri(Rnew, diag = replace_diagonal)] <- replacement
  } else if (use == 'lower') {
    Rnew[upper.tri(Rnew, diag = replace_diagonal)] <- replacement
  } else if (replace_diagonal) {
    diag(Rnew) <- replacement


#' save_correlation_matrix
#' Creates and save to file a fully formatted correlation matrix, using `correlation_matrix` and `Hmisc::rcorr` in the backend
#' @param df dataframe; passed to `correlation_matrix`
#' @param filename either a character string naming a file or a connection open for writing. "" indicates output to the console; passed to `write.csv`
#' @param ... any other arguments passed to `correlation_matrix`
#' @return NULL
#' @examples
#' `save_correlation_matrix(df = iris, filename = 'iris-correlation-matrix.csv')`
#' `save_correlation_matrix(df = mtcars, filename = 'mtcars-correlation-matrix.csv', digits = 3, use = 'lower')`
save_correlation_matrix = function(df, filename, ...) {
  return(write.csv2(correlation_matrix(df, ...), file = filename))

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Visualizing and interpreting Cohen’s d effect sizes

Visualizing and interpreting Cohen’s d effect sizes

Cohen’s d (wiki) is a statistic used to indicate the standardised difference between two means. Resarchers often use it to compare the averages between groups, for instance to determine that there are higher outcomes values in a experimental group than in a control group.

Researchers often use general guidelines to determine the size of an effect. Looking at Cohen’s d, psychologists often consider effects to be small when Cohen’s d is between 0.2 or 0.3, medium effects (whatever that may mean) are assumed for values around 0.5, and values of Cohen’s d larger than 0.8 would depict large effects (e.g., University of Bath).

The two groups’ distributions belonging to small, medium, and large effects visualized

Kristoffer Magnusson hosts this Cohen’s d effect size comparison tool on his website the R Psychologist, but recently updated the visualization and its interactivity. And the tool looks better than ever:

Moreover, Kristoffer adds some nice explanatons of the numbers and their interpretation in real life situations:

If you find the tool useful, please consider buying Kristoffer a coffee or buying one of his beautiful posters, like the one above, or below:

Frequentisme betekenis testen poster horizontaal image 0

By the way, Kristoffer hosts many other interesting visualization tools (most made with JavaScript’s D3 library) on statistics and statistical phenomena on his website, have a look!