The final article was, of course, edited down by their team to meet the 300 word guide. That there is a reproducibility crisis in psychological science—and arguably across all sciences—is, to me, beyond doubt.Murmurings of low reproducibility began in 2011— the so-called “year of horrors” for psychological science (Wagenmakers, 2012), with the infamous fraud case of Diedrik Stapel being its low-light. In 2015, the Open Science Collaboration published the findings of our large-scale effort to closely-replicate 100 studies in psychology (Open Science Collaboration, 2015).
There are two extensions in the current model over that of Lee & Wagenmakers: The model assumes—as does the one by Lee & Wagenmakers—that the data are modelled as draws from a multivariate normal distribution.
The parameters of this distribution are the means of each of the three variables (denoted ) and their standard deviations (), as well as the correlation coefficients that link them ().
But what if you want to explore the relationship between two variables whilst controlling for the effects of a third variable?
For example, you might be interested in the relationship between shoe size and height whilst controlling for age.
And the news was not good: Only 36% of studies were replicated.
Whilst low reproducibility is not unique to psychological science—indeed, cancer biology is currently reviewing its own reproducibility rate, and things are not looking great (see Baker & Dolgin, 2017)—psychology is leading the way in getting its house in order.
The Figure below shows the relationship with Regular (i.e., Pearson’s) and Partial correlation.
This is all well and good, but what if you want a Bayesian estimate of the partial correlation parameter?
I had a quick check of the internet, and couldn’t see anything.
(I should say that I didn’t spend long looking; I like to do some things from scratch just so I can learn something, and this was one of those times.) Inspired by the SUPERB book by Lee & Wagenmakers on Bayesian modelling, I devised a Bayesian graphical model of partial correlation: This model extended the one presented in the Lee & Wagenmakers book which was used to calculate the correlation coefficient between just two variables.
The distribution of interest for inference is now this new distribution pertaining to the partial correlation parameter.